review article hypergame theory: a model for conflict...

Review ArticleHypergame Theory A Model for ConflictMisperception and Deception

Nicholas S Kovach Alan S Gibson and Gary B Lamont

Department of Electrical and Computer Engineering Graduate School of Engineering and ManagementAir Force Institute of Technology Wright-Patterson AFB OH 45433-7765 USA

Correspondence should be addressed to Nicholas S Kovach nicholaskovachafitedu

Received 24 May 2015 Accepted 9 July 2015

Academic Editor Tonu Puu

Copyright copy 2015 Nicholas S Kovach et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

When dealing with conflicts game theory and decision theory can be used to model the interactions of the decision-makers Todate game theory and decision theory have received considerable modeling focus while hypergame theory has not A metagameknown as a hypergame occurs when one player does not know or fully understand all the strategies of a game Hypergame theoryextends the advantages of game theory by allowing a player to outmaneuver an opponent and obtaining a more preferred outcomewith a higher utility The ability to outmaneuver an opponent occurs in the hypergame because the different views (perceptionor deception) of opponents are captured in the model through the incorporation of information unknown to other players(misperception or intentional deception)Thehypergamemodelmore accurately provides solutions for complex theoreticmodelingof conflicts than those modeled by game theory and excels where perception or information differences exist between players Thispaper explores the current research in hypergame theory and presents a broad overview of the historical literature on hypergametheory

1 Introduction

ldquoA conflict is a situation in which there is a lsquocondition ofoppositionrsquo [1] and parties with opposing goals affect oneanother [2]rdquo The study of how decision-makers interactduring a conflict is known as game theory [3] while the studyof how decision-makers make rational decisions is known asdecision theory [4]

Game theory has been used to model diverse areassuch as economics natural selection battles in past warsand many other types of conflict [5] The main influencebehind the creation of game theory is the resolution of suchcompetitions Game theory models have many propertiesassociated with them that influence the outcome and howgame analysis proceeds but often fail to model the situationwhen one player has an advantage over the other in a conflictWhen one or more players lack a full understanding or havea misunderstanding or incorrect view of the nature of theconflict hypergame theory can be used to model the conflict

Decision theory on the other hand is concerned withgoal-directed behavior when options exist with differentpossible outcomes The main influence behind the creationof decision theory is the rational behavior of the decision-maker [6] Problems decision theory tries to answer includethe following ldquoshall I bring an umbrella todayrdquo or ldquoI amlooking for a house to buy shall I buy this onerdquo Decisiontheoretic models often fail to model the notion of fear whereanother playermay be able to outmaneuver during game playInstead models rely heavily on probability distributions todetermine the preferred outcome

Hypergame theory is an extension of game theory thataddresses the kind of conflict games where misperceptionexists The term hypergame was coined by Bennett in 1977It seeks to explain how players in a game can have differingviews of the conflict [7] This advance in game theory showshow one player can believe that decisions of the other playerare irrational but the opponent is actually making a rationaldecision based upon the perceived game model

Hindawi Publishing CorporationGame eoryVolume 2015 Article ID 570639 20 pageshttpdxdoiorg1011552015570639

2 GameTheory

Hypergame analysis extends game theory by providingthe larger game that is really being played whether or notboth players are aware of it A different game model canrepresent each playerrsquos view of the conflict but often theplayerrsquos views will overlap where common knowledge existsFiguring outwhat strategy a playerwill use is dependent uponnot only his or her observation of the game but also howthat player believes their opponent is viewing the game Thiscreatesmany different gamemodels that are examined for thesolution to be obtained The goal of hypergame analysis is toprovide insight into real-world situations that are often morecomplex than a game where the choices of strategy presentthemselves as obvious

After its introduction hypergame theory was used tomodel past military conflicts which are prone to having mis-perceptions and missing information in the process of theirunfolding to showhow the outcomeswere achievedAnalysisof past conflicts also lends itself to ease of understandingsince the fog of war has cleared and the outcome has beendetermined Options selected by each side in the conflictare shown to be the rational choice by way of defining thegame that each side perceived In thismanner the hypergameanalysis shows why unexpected results were obtained whenone or both sides misconstrued the conflict Hypergameanalysis offers advantageous reasoning of strategy selectionthrough situational awareness

Throughout this paper the players are often referred toas human entities engaged in a conflict The perceptionor misperception of the players in all cases is the result ofsensors whether mechanical or human senses are combinedwith brain power Human players mechanical players orartificial intelligence (AI) players are interchangeable in thehypergame models

Examples in this paper are presented using two playerswith a limited number of actionsstrategies This allows thecomplexity of the example to be reduced and the visualizationincluded in the figures to be clear As more players andactionsstrategies increase the complexity will also increase

2 Foundations of Hypergame Theory

Game theory decision theory and hypergame theory can beused to model conflicts as games When very little is knownabout the opponents game theory is used for adversarial rea-soning Decision theory is a better choice if the opponents arewell known which is often the case in complete informationgames If one or more of the opponents are playing differentgames because they are not fully aware of the nature of thegame hypergames can be used to reason about subgames thatare shared between opponentsThe following provides a briefoverview of decision theory and game theory

Hypergames extend game theory by allowing for anunbalanced game model that contains different view of thegame representing the differences in each playerrsquos informa-tion or beliefs The unbalanced game model allows for adifferent game model for each playerrsquos view while havingoverlap where there is common knowledge Decision theoryhas been used in hypergames to model the fear of being

outguessed The fear of being outguessed is common in agame model where the different playerrsquos perceived games areunbalanced

21 GameTheory GameTheory is a set of analytical tools de-signed to help understand the phenomena that are observedwhen decision-makers interact [3] The assumption is madein game theory that human beings are rational and alwaysseek the best alternative when presented with a set of possiblechoices Game theory is highly mathematical and assumesthat all human interactions can be understood and navigatedby presumptions Game theoretic models seek to answer twoquestions about the interaction of the decision-makers [12]

(i) How do individuals behave in strategic situations(ii) How should these individuals behave

Answers to the two questions do not always coincide [13]Often the answers to the two questions may be in conflict

Game theory models have many properties associatedwith those that influence the outcome and how game anal-ysis proceeds Cooperative games can exist where there iscommunication between players but more often games areseen as noncooperative where the players do not attempt togive up any information to their rivals Simultaneous gamesare when players make decisions at essentially the same timeand do not know of the opponentrsquos move in advance (ierock paper and scissors) Conversely sequential games arewhen the opponentrsquosmove is known before a decision ismade(ie betting in poker or chess) Perfect information versusimperfect information is when all previous moves are knownto all players instead of some being hidden These previoustwo concepts are often confused with complete informationand incomplete information which are actually intended tobe the knowledge of all playerrsquos strategies and payoffs Thepayoffs also known as utilities have a common property ofzero sum or nonzero sum

The majority of game theory models are identified aseither strategic games (normal form) used to represent asimultaneous game or extensive games more often used torepresent a sequential game even though it can representsimultaneous one as well A strategic (normal form) gameis shown in Figure 1 Strategic games have utilities that aredetermined by which strategy is selected by each playerWhen a playerrsquos strategy is selected all strategies are availableto choose from and therefore the game is represented in agrid or matrix format Extensive gamesrsquo outcomes are insteadrepresented as a tree structure where the initial player is atthe top with branches leading off for each strategy availableas shown in Figure 2 After the first player chooses an actionthe next player has strategies available creating branches foreach of its strategies available All strategies that a player canuse may or may not be available at a particular node of thetree dependent upon the previous playerrsquos selected strategy

The normal form is used to represent games where playermoves or turns are simultaneous This is often the case ingames where information is imperfect and allows for betteridentifying strictly dominated strategies and Nash equilibriaThe extensive form is better suited for games where player

GameTheory 3

Arm

Disa

rm

Arm Disarm

Nation B

Nat

ion

A

(2 2) (4 1)

(1 4) (3 3)

Figure 1 Cyber Arms Race represented as a classical prisonerrsquosdilemma shown as a normal form game

Arm Disarm

Nation A

Arm ArmDisarm Disarm

(2 2) (4 1) (1 4) (3 3)

Nation B Nation B

Figure 2 Prisonerrsquos dilemma shown in extensive form

moves are sequential (ie Player 1 moves first observed byPlayer 2 and then Player 2 moves)

In the early 1950rsquos Nash contributed to noncooperativeand cooperative game theory [40ndash42] Nash [43] built on vonNeumann and Morgensternrsquos work by assuming the absenceof coalitions where each player acts independently His workproves for each finite noncooperative game that there is atleast one equilibrium point assuming the players are rationalANash equilibrium is a strategywhere none of the players canimprove their payoff by unilaterally changing their strategyIn a game of mixed strategies every game will have at leastone Nash equilibrium

The Nash equilibrium means that none of the playerscan improve its own outcome (payoffutility) by unilaterallychanging strategies The following definition of a Nashequilibrium is stated for two players but can be applied toany number of players [3] The goal is to determine a uniqueoutcome [120590 120591] for the game given a strategy pair 120590 and 120591 fortwo players denoted as P1 and P2 The unique outcome ofthe game is obtained by each player playing their respectivestrategy (P1 plays 120590 and P2 plays 120591) against each other

Definition (Nash equilibrium [44]) A strategy pair 120590 120591 is aNash equilibrium if for no 1205901015840 = 120590 [1205901015840 120591]gtP1[120590 120591] and for1205911015840= 120591 [120590 1205911015840]gtP2[120590 120591]

Considering the Cyber Arms Race shown in Figures 1and 2 and applying the definition of a Nash equilibriumthe resulting solution to the Cyber Arms Race is (2 2) orboth players choose ldquoArmrdquo leading to an arms race Neithernation can improve its own outcome or utility by unilaterallychanging strategies

Deception in game theory has been mostly studied inturn-based or dynamic games where a player chooses an

action and then reports the action or outcome to the otherplayer This type of game is called signaling games afterthe ldquosignalrdquo is sent between players The signal is subject todeception since the player can be truthful or deceptive orchoose not to send a signal

Carroll and Grosu [45] study network defense usingdeceptive signaling games In their research the defender candisguise a normal computer as a honeypot or a honeypotas a normal computer or use no disguising techniques Theattacker has the ability to test the system type and thedefender sends the appropriate signal deceptive or truthfulThe authors showed that deception is an equilibrium strategyfor the defender either by disguising all honeypots as normalcomputers or all normal computers as honeypots providingan increase in utility for the defender over using only truthfulsignals

Multiturn attacker-defender games are used by Zhuanget al to study deception [46] In the game a defender type israndomly selected from a set of possible defender types andat each turn of the game the defender selects a strategy andldquosignalsrdquo the attacker of the selected strategy The defendermay be either truthful or deceptive The attacker then usesthe signal to update his belief of the defenderrsquos true type andselects an attack strategy After each turn the payoffs are usedto update the belief state until the game ends The authorsstate that given their game deception can be a beneficialstrategy for the defender

Hespanha et al [47] modeled an attacker-defender gamewhere the defender has three units available to defend twolocations In the game the defender signals the locationsof the units by either sending a truthful or deceptive signalor not camouflaging the units revealed to the attacker Theauthors also discuss the possibility of a malfunction of eitherthe attackerrsquos sensors or the defender camouflage whichmay mean the signal seen may not be correct The authorsconclude that the use of deception can render the informationcollected from sensors and other methods to be useless to theattacker

Deception has also been studied in repeated games Inthis type of game the players both choose an action andmake their moves simultaneously Depending on the gamethe players may receive information about how the environ-ment state changed between selectingmoves Pursuer-evadergames are commonly modeled with this type of repeatedgame Yavin [48] studies pursuer-evader deception whereboth players choose a strategy based on the bearing of theother player and the distance between them by corruptingthe evaderrsquos bearing signal to the pursuerThe authorrsquos goal isto determine the optimal (or near-optimal) pursuit strategiesfor a pursuer when faced with deceptive or incompleteinformation

22 DecisionTheory In any given situation there are actionswhich a player can choose between making a choice ina nonrandom way The choice between actions are goal-directed activities [49] Given a set of actions decision theoryis concerned with goal-directed behavior to reach a desiredoutcome

4 GameTheory

Decision theory is a formal mathematical theory abouthow decision-makers make rational decisions It is alsoknown as normative decision theory [4 50] Bayesian deci-sion theory [49] rational choice theory [51] and statisticaldecision theory [52] Decision theory predates the develop-ment of game theory and can be divided into three partsnormative descriptive and prescriptive [53 54]

(i) Normative decision theory [4 50] studies the idealagent and the decisions that this perfectly rationalagent would make often referred to as the study ofhow decisions should be made

(ii) Descriptive decision theory [55] studies the nonidealagent such as humans and how they make decisionsoften referred to the study of how decisions are madein reality

(iii) Prescriptive decision theory [56] studies how non-ideal agents given their imperfections can improvethe decisions

A person uses his own preferences to determine hisaction according to rational choice theory [57] A rationalperson selects his action according to the one that maximizeshis preferences [58] The Nash equilibrium builds on thisconcept adding consideration for the other player(s) andwhat can be done to maximize the outcome unilaterally

The problem is that preferences do not just represent thedecision maker but a rational person can consider moralethical social (peer-pressure social expectance etc) andorother norms when establishing their preferences Intentionalor unintentional deceptions can also affect a playerrsquos prefer-encesThis problem highlights the complexity of preferenceswhich may lead to odd choices in real-world situations

Luce and Raiffa presented a classic example of thisproblem [59] In the example they compare two alternativescenarios of visiting a restaurant In the first visit only salmonand steak are offered on the menu In this case the customerdecides to order salmon even though the customer normallyprefers steak The customer refrains from the steak becausethe small menu indicates the cook may not know how toprefer a steak In the second visit the menu includes lobsterand clams in addition to salmon and steak Here the customerchooses the steak If the customer does not like lobster orclams the fact they are offered on the menu indicates therestaurant is good and should know how to prepare a steak

In this scenario the customer may seem irrational sincehe did not choose the steak which he prefersThe reason whyhe is not irrational is because the menu does not only list thechoices but also conveys information of value to the customer(which does not have to be true) The addition of lobster andclams on the second menu indicates the cook has the abilityto prepare these delicate foods while the first menu just hasthe basics In this case not only do the items offered on themenu get considered in the preferences but also the type ofrestaurant (or the perceived type of restaurant)

Based on todayrsquos Internet connected pollution the pre-vious scenario can include deception For example insteadof relying on the menu options he may instead consult acommunity rankingwebsite Given that anyone can post their

opinion the restaurant may have paid for favorable rankingstherefore adding deception in the customerrsquos preferences

23 Bounded Rationality Bounded rationality is where aplayerrsquos rationality is limited in the decision-making processby the information the player has cognitive limitations oftheir minds and time available to make the decision [60]Simon originally proposed the concept of bounded ratio-nality as an improvement to the model of human decision-making [61] Bounded rationality helps to explain why themost rational decision is not always the decision chosen bythe player in game theory or decision theory

Bounded rationality does not mean irrationality sinceplayers want to make rational decisions but cannot alwaysdo so [62] Players are often very complex but in order to befully rational they need unlimited cognitive capabilities [63]The cognitive capabilities of players are limited and thereforecannot conform to full rationality Players will use thecognitive resources they have with the information availableand often within time constraints to reach a decision that isas rational as possible Bounded rationality allows the playerto make a decision based on their perceived state of the gameor environment leading to multiple players having differentperceptions of the game or interaction

3 Hypergame Theory

Hypergame theory extends game theory by allowing for anunbalanced game model that contains a different view rep-resenting the differences in each playerrsquos information beliefsand understating of the game The unbalanced game modelallows for a different game model for each playerrsquos viewwhile having overlap where there is common knowledgeTheoutcome or solution to the hypergame model is dependenton the playerrsquos perception of the game model including howthe player views the game and how the player believes theopponent is viewing the game Because of multiple gamemodels each model has to be analyzed in order to determinethe outcome to the hypergame This allows hypergames tomore accurately provide solutions for complex real-worldconflicts than thosemodeled by game theory and excel whereperception or information differences exist between players

Two papers explain the transition from game theory tohypergames early in the history of hypergames The firstpaper [64] discusses the development apart from classicalgame theory towards hypergame theory It explains thechanges and focuses on descriptive modeling and does notcover the issue surrounding attempts to influence decision-makersThe second paper [65] provides illustrative case stud-ies and presents a methodological framework for applyinghypergames to complex decision problems

31 Theory Foundations Hypergames first discussed byBennett [7] are used to model the games where one ormore players are playing different games [66] Hypergametheory decomposes a single situation into multiple games Byreasoning about multiple games the outcome to the singleproblem can be improved Each player in a game has their

GameTheory 5

Column

Row0 4 1 3

4 0 3 1

Column

0 4 1 3 1 3Row4 0 3 1 3 1

3 1 5 0 1 3

C1 C2

C1 C2 C3

R1

R2

R1

R2

R3

Figure 3 An example of hypergame where each player has a dif-ferent view of the game Rowrsquos view is on the left while columnrsquosview is on the right

own perspective of how the other players view the game withregard to the possible actions and player preferences Bryant[67 68] discussed the difference in the set of players pointingout that the set may vary in real life as players perceivedifferently In a hypergame each player may [2]

(i) have a false or misled understanding of the prefer-ences of the other players

(ii) have incorrect or incomplete comprehension of theactions available to the other players

(iii) not have awareness of all the players in a game(iv) have any combination of the above faulty incorrect

incomplete or misled interpretations

A playerrsquos choice of actions (decisions) reflects the playerrsquosunderstanding of the game outcomes the player choosesactions based on the way they perceive reality whichmay notbe the true state of reality Figure 3 shows a basic two-playerhypergame between ldquorowrdquo and ldquocolumnrdquo where119862

119894and119877

119894are

different actions each player could takeHypergame analysis is conducted by first examining

Rowrsquos belief about Columnrsquos reasoning and then by examin-ing Rowrsquos available actions [69 70] In Figure 3 the game onthe left shows how Row believes Column will reason aboutthe game Based on this Column will play 119862

2while Row

plays 1198772 the Nash equilibrium concept from game theory

This allows the experience and intuition of the decisionmakerto be incorporated into hypergames For example this couldapply to planning variables such as a novel course of actionfor Row or Columnrsquos lack of time to plan or to situationalvariables such as the hidden location of Rowrsquos resource [2]

Hypergames allow for domain knowledge incorporationtherefore it does not require the game theory equilibriumcondition [2] Furthermore the standard rationality argu-ments from game theory are replaced by knowledge of howthe opponentwill reason [2] It is also valid to assumeunequalavailability of information in hypergames whenmany playersin games have imperfect information In Figure 4 Kopp givesa graphical comparison of the general differences betweena standard game model and a hypergame model This rep-resentation depicts a general overview of how a hypergameincorporates different aspects of the conflict being modeled

32 Hypergame Levels Wang et al [71] proposed differentlevels for developing mathematical hypergame models based

on perceptions of the players The lowest level (level 0) is abasic game with no misperceptions among the players In afirst level hypergame players have different views of the gamebut are not aware of the other playersrsquo games In a secondlevel hypergame at least one player is aware there are differentgames being played and that misperceptions exist A thirdlevel hypergame is possible and occurs when at least oneplayer is aware that at least one other player is aware thatdifferent games are being played An 119899th level hypergamecan be described but the authors state this does not addto the hypergame model instead it adds complication andexcess information for the hypergame analysis This allowsthe perceptions of the players to be incorporated into thehypergame model but with varying degrees of perceptions inorder to reach a more complete game model

321 First Level Hypergame The levels of hypergames wereoriginally presented by Fraser and Hipel [2] A game 119866 isdefined by a set of preference vectors119881

119899 for all game players

where 119899 is the number of players and 119881119894is the preferences

vector for player 119894

119866 = 1198811 1198812 119881119899 (1)

In game of complete information all players know theother playerrsquos preference vectors therefore all players areplaying the exact same game In hypergames one or moreplayers may have incomplete information which leads play-ers to form slightly different versions of the same game orcompletely different games altogether A game formed byplayer 119902 includes any and all lack of information about theconflict which is denoted by

119866119902= 1198811119902 1198812119902 119881

119899119902 (2)

where 119881119894119902

represents the preference vector of player 119894 asunderstood (perceived) by player 119902

A first level hypergame119867 is a set of games as understoodfrom each player

119867 = 1198661 1198662 119866

119899 (3)

An example of a hypergame in this form is shown inTable 1 in matrix form Since players may have differentmisperceptions each player may make a different decisionwhich will result in a different outcome to the conflict Amapping function can be used to relate the outcomes betweenthe playerrsquos individual games Bennett [72] gives an algebraicdescription of this problem while an application is presentedin Bennett et al [20]

Game analysis is performed by treating each playerrsquos gameseparately This means player 119902rsquos game is analyzed from 119902rsquosunderstanding about the conflict The decisions made andthe strategies chosen by 119902 depend on 119902rsquos interpretation ofthe conflict therefore a given player may not perceive alloutcomes of a game The player cannot unilaterally changefrom a perceived outcome so for the purpose of stabilityanalysis the outcome is stable for that player [2] Thereforean unknown outcome to a player can be stable in the

6 GameTheory

Decisionfunction Actions Decision

functionActions

Outcomemodel A

Outcomemodel B

Player A game

Mod

el fo

r a g

ame

Mod

el fo

r a h

yper

gam

e

Player B game


functionActions

Outcomemodel A

Outcomemodel B

Player A game Player B game

Information Information

Perceivedmodel A

Perceivedmodel B

Playerrsquos account for opponentrsquos strategies

Accuracyof the perceivedgames depends

on available information

Figure 4 Comparison of a game theory and a hypergame model [8]

Table 1 An example of a hypergame in matrix form

Player perceived Game perceived by player1 119881

1111988112

sdot sdot sdot 1198811119899

2 11988121

11988122


119899 1198811198991

1198811198992

sdot sdot sdot 119881119899119899

1198661

1198662


hypergame analysis A strategic surprise occurs when a gamecontains an unknown outcome

For player 119902rsquos game an outcome is stable if the outcomeis stable in each of 119902rsquos preference vectors This means theequilibriums of 119902rsquos game are only the outcomes 119902 believeswould resolve the conflict even if other equilibriums existin the full game Hypergame equilibriums depend on eachplayerrsquos perception of the stability of the outcomes Whendetermining equilibriums of hypergames the equilibriumsof each playerrsquos game are not needed but these individualequilibriums can be useful to demonstrate what each playerbelieves will happen

322 Second Level Hypergame A second level hypergame isa hypergame where at least one player is aware that a hyper-game is being playedThis situation can happen if at least oneplayer perceives another playerrsquos misperception [2] Player 119902rsquoshypergames is defined as the (hyper)gameperceived by player119902 This hypergame is denoted as

119867119902= 1198661119902 1198662119902 119866

119899119902 (4)

where 119866119894119902is the game of the 119894th player as it is perceived by

player 119902 It is not necessary for player 119902 to be one of theplayers who are aware that a hypergame is being played Ifset 119867119902is missing a playerrsquos game it is because player 119902 does

not perceive the gameA second level hypergame is a set of hypergames per-

ceived by each player denoted as

1198672= 1198671 1198672 119867

119899 (5)

Table 2 shows a second level hypergame in matrix formwhere the hypergame for player 119901 is the 119901th column Eachelement of thematrix is a gamemade upof a preference vectorfor each player

Similar to a first level hypergame analysis game analysisof second level hypergames is performed by treating each

GameTheory 7

Column 1 Column 2 Column 3

Row 1

Row 2

RMSs

CMSs

Full game

middot middot middot







middot middot middot middot middot middot


middot middot middot middot middot middot middot middot middot middot middot middotmiddot middot middot middot middot middot middot middot middot middot middot middot










middot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

PKminus1

RKminus1

rk1

rk2

rkm

P1

R1

r11

r12

r1m

R0 = full game

r01

r02

r0m

CΣ

CKminus1

C1

C0

Row m

S1 S2 S3 Sn

Ck1 Ck2 Ck3

C11 C12 C13

C01 C02 C03

u11 u12 u13

u21 u22 u23

um1 um2 um3

Column n

Ckn

C1n

C0n

u1n

u2n

umn

Belief-contexts

P0Kminus1i=1 Pk1= minus sum

Figure 5 Hypergame Normal Form as proposed by Dr Vane III

Table 2 An example of a second level hypergame in matrix form


1111986612


2 11986621

11986622


119899 1198661198991

1198661198992

sdot sdot sdot 119866119899119899

1198671

1198672


playerrsquos game separately This allows stability informationto be determined for every preference vector in a conflictThis information can further be used to determine eachgamersquos equilibrium The preference vectors of each playerrsquosgame provides the stability information that determines theequilibriums of the second level hypergame ldquoJust as theequilibriums of a game within a hypergame are not neededto determine the equilibriums of that hypergame the equi-libriums of a hypergame within a higher level hypergame arenot needed to determine the equilibriums of that higher levelhypergamerdquo [2]

33 Hypergame Normal Form (HNF) In [34] Dr Vane IIIoffers a different approach to hypergame modeling by pro-viding the incorporation of a playerrsquos beliefs on an opponentrsquospossible actions He also provides a graphic representationof the hypergame that is reminiscent of the normal strategicform used in standard game theory analysis

The new model is referred to as Hypergame NormalForm (HNF) see Figure 5 The full game is the familiargrid form with Row and Column strategies labeled and theutility values (119906

11minus 119906119899119898) in the cells cross-referenced from

the strategies The additional sections are the hypergamesituational informationThe rowmixed strategies (RMSs) arehyperstrategies gleaned from what the row player believesabout the game being played by the Column player

They are called hyperstrategies because they do notencompass the view of the full game except for 119877

0which

is the full game Nash equilibrium (NE) The column mixedstrategies (CMSs) are rowrsquos belief about the mixed strategypercentages that Column will play when selecting a strategy1198620is columnrsquos NE view of the full game When a CMS cell

contains a 0 this is an indication that there is a subgame thatcolumn is believed to be playing where the correspondingstrategies are either unknown to Column or discounted asnot worthwhileThe final section is the belief-contexts whichcorrespond to the percentage of which Row believes thatthe adjacent CMS will be played by column Since they arepercentages the belief-contexts sum up to one with the sumof values 119875

1through 119875

119870minus1being at most 1 and the leftover

constitutes the NE belief-context Filling in the HNF with thevalues associated with the game provides the avenue for thehypergame analysis with HNF

Determining the utility values allows for a NE for thefull game to be calculated which provides the input into 119877

0

and 1198620 CMSs are then entered in the section above the full

game ACMS can be determinedmanually that is knowing aplayerrsquos preference for selecting rock in a game of rock paperand scissors or a NE for the Column player can be used fromthe analysis of the subgame that theColumnplayer is believedto be using Each CMS is assigned a belief-context valuewhich serves to weight the Row playerrsquos belief that Columnwill choose that CMS These values are used to calculate119862119875 the aggregate amount which directly affects the expected

utility that Row hopes to achieve Rowrsquos hyperstrategies arethen input into the RMS section Expected utility values for

8 GameTheory

0

Expe

cted

util

ity

1

EffectivePartially

NEMSIneffective

120575

EU(R0 CΣ)

EU(R0 C0)

Figure 6 Hyperstrategies effectiveness in Hypergame NormalForm

each strategy set listed in the RMS section are calculated forthe full game NE CMS119862

0 and the aggregate belief CMS119862

119875

These values determine the effectiveness for which an RMShyperstrategy is a practical selection for Row to apply

RMS effectiveness is categorized into three levels ofusefulness fully effective partially effective and ineffective(Figure 6) A fully effective strategy set will provide at worstcase the same expected utility that Rowrsquos 119877

0strategy set

achieves for 1198620but has a greater expected utility at 119862

119875 Thus

given that row is viewing the game correctly a fully effectiveRMS is always a good choice Partially effective strategy setsalso provide a greater expected utility at 119862

119875than 119877

0but have

a lower utility expectation at 1198620 Given Rowrsquos information a

partially effective RMS could provide a good outcome but itis not always assured The ineffective strategy set provides noincrease in utility and at best can only get to that expectedby the NE so there is no reason to select it It is reasonableto assume that fully effective strategies sets should always beused but that does not mean there is not some inherent riskinvolved because the utility values are only expected and arenot foolproof Worst case scenarios can also be included inthis determination to help mitigate risk

Risk assessment is built into the hypergame analysisthrough a method termed quantified outguessing Thismethod introduces the fear of the player that he or she willbe outmaneuvered and the worst case utility will be the endresult Three types of hyperstrategy sets are described forthis analysis modeling opponent (MO) pick subgame (PS)and weighted subgame (WS) MO is simply selecting thestrategy for row that will provide the highest utility given allof rowrsquos strategy selections and when considering the beliefof how column views the game In contrast the PS strategyset consists of the NE for the same game view that wasconsidered forMOWS uses the PS strategy valuesmultipliedby the belief-context percentage for that CMS and adds

0

Hyp

erga

me e

xpec

ted

utili

ty

1g

WS

PS

NEMS

MO

Crossover point

Figure 7 The value of HEU is dependent on the value of 119892

the 1198770multiplied by the belief-context for 119862

0 which results

in a hybrid strategy set between PS and the NEMS for thefull game Each hyperstrategy is then assessed against the fullgame to quantify the worst case utility (119866) or the utility valueexpected when Column selects the correct counter strategyThe expected utility (EU) and 119866 once determined allow thehypergame expected utility (HEU) to be calculated by alsoconsidering 119892 the percentage chance Row believes they willbe outguessed (Equation (6)) As can be seen the distancebetween EU and 119866 has a quantifiable effect on the value ofHEU for the hyperstrategy (hs)

HEU (hs) = EU (hs) minus (EU (hs) minus119866 (hs)) lowast 119892 (6)

As the fear of being outguessed increases the abilityof any hyperstrategy to provide better utility (when com-pared to the NEMS solution of the full game) decreases(Figure 7) When the fear of being outguessed is low the MOhyperstrategy is the best selection but as that fear increaseseventually PS dominates for a short period until the crossoverpoint where NEMS for the full game is dominant (note thatWS is always dominated and does not provide a suitablechoice)Therefore with good information on the intent of theadversary hyperstrategy selection that provides better utilitythan standard game analysis is achievable

Further research in the use of the HNF has been con-ducted after its creation The ideas about hypergame analysisare expanded upon in [73] by Russell Vane The presence ofluck and robustness of strategy plan are examined but forthe most part the research provides further evidence of theusefulness of hypergame analysis for the strategy selectionprocess A real-world example of how to use the HNF isprovided in [74] which examines a terrorist attack Theanalysis entails applying belief-context values to expectedtypes of attackers so that a strategic decision can be madeto best protect first responders This reiterates the idea thatuncertainty exists and needs to be assessed when planning

Perhaps the most interesting application of the HNFis in [75] where it is used to model the fall of France in

GameTheory 9

1940 The model is compared to the dual standard gamemodel presented in [16] and to a preference vector modellike that in [76] Specifically it is outlined that using the HNFapproach allows all information to continue to be presentedand not removed from the model Even when a strategy iscompletely discounted by applying a percentage chance of useof zero it remains in the total game NEMS analysis and is notcompletely removed from the model These research effortsshow insight into the usefulness of the HNF

4 Hypergame Modeling

Huxham and Bennett [77] introduce the idea of preliminaryproblem structuring In this phase the problem is exploredand the relevant participants are identified along with thepossible interactionsThe authors try to build up a structuredpicture in hypergame terms of the situation instead of ahypergame model The idea is to explore how the variouspieces fit together The structured picture will often be toocomplex to form into a formal hypergame model It is there-fore necessary to abstract farther making simplifications byasking specific questions [77]

(i) How do two different problem aspects relate

(ii) Where are the complexities of the system

(iii) Can simplifications be made while retaining theessential structure

(iv) Which participants aremost important or influential

Hipel and Dagnino present an algorithm for modelingbargaining situations with two or more decision-makerswhere one or more of players have misperceptions [9] Thealgorithm is called the hypergame cooperative conflict anal-ysis system (HCCAS) HCCAS unifies work in hypergametheory [71 72 78ndash80] conflict analysis [2] and cooperativeconflict analysis in bargaining [81 82]TheHCCAS algorithmis shown in Figure 8

The real-world situation is represented at the top of thealgorithm and provides critical information for the algo-rithm The first step is to use the real-world information todefine the structure of the bargaining situation This stageinvolves selecting a point in time at which the analysis will beconducted as well as identifying the participants and poten-tial interactions The second step in HCCAS is modelingwhere the actions and outcomes of the players are identifiedThe third step of HCCAS is the hypergame frameworkwhere the bargaining situation structure and the levels ofmisperception for each player are identified Following thisstep the preference vectors for each player are formed usinginformation from the previous steps this is referring tothe preference assessment in Figure 8 Stability analysis ofthe hypergame is performed in the fifth step After this astrategy is selected and can be used to explain the real-worldevents The authors then apply the HCCAS algorithm to theSeymour landfill case between Eau Claire city and the townof Seymour in Wisconsin

The real world

Problem structuring

Modeling

Hypergame framework

Preference assessment

Stability analysis

Strategy selection

Figure 8 The HCCAS algorithm [9]

All possible outcomes

SMR

FHQ

R

GMR

Figure 9 Venn diagram of stability analysis outcomes for 119899-players[10]

5 Other Related Works

In this section we summarize additional research related tohypergame theory This work adds to the theory of hyper-game and there are many contributions from previous re-searchers

51 Stability Analysis Wang et al explores stability analysisfor 119899-players in [10] The authors present a relationshipof possible outcomes as shown in the Venn Diagram inFigure 9 Nash stability is when players make a rationaldecision based on the best outcome for the player this typeof outcome is considered rational (R) Nash stability is harderto achieve when misperceptions exist between players Ageneral metarational (GMR) outcome is where other players

10 GameTheory

All outcomes

HN(H)

N(BG)

SHN(H)

Figure 10 Venn diagram of stability relationships among outcomes[11]

have joint action for player 119894 and player 119894 cannot achieve abetter outcome than the original A symmetric metarational(SMR) outcome is when there is one jointly sequentialstrategy selection that results in player 119894 achieving the sameoutcome If a response to a playerrsquos strategy results in thatplayer not achieving a better outcome and the respondingplayer not being able to possibly achieve aworse outcome it isknown as a sequential stable (FHQ) The contribution of thisresearch is an FHQ outcome exists in all hypergame levelswhich implies a GMR outcome also existing in all hypergamelevels

Another view of hypergame stability is given in [11]When there exist hyper Nash equilibria in a hypergame if allof them are not Nash equilibrium in the base game theredoes not exist stable hyper-Nash equilibrium An intuitiveinterpretation of the paperrsquos theorem is that when we antici-pate all outcomeswhich seem to happen actually (hyper-Nashequilibrium) eachwould not happen if all themisperceptionsare eliminated and those outcomes are necessarily unsta-ble Hence the stability relationships among the solutionconcepts in a hypergame can be depicted by Figure 10 Therelationships are defined as the hypergame (H) hyper-Nashequilibrium of H (HN(H)) base game (BG) and Nashequilibrium of H (SHN(H)) A hyper-Nash equilibrium isdefined as a profile of such strategies that each agent playsaccording to their Nash strategy in their own subjectivegame This allows for generalization of Nashrsquos theorem aboutnoncooperative games [40] to hypergames in every finitehypergame with mixed strategies there is at least one hyperNash equilibrium [83] A hyper-Nash equilibrium providesan equilibrium solution for a simple hypergame This alsoallows for hypergames with cardinal utilities while previousresearch only dealt with ordinal utilities

52 Player Beliefs Vane and Lehner [84] deal with beliefsover games The hypergame framework allows a player tohedge its risk about what the other opponents are doingThisis done by selecting a set of possible games that representthe action the opponents may take and then a probabilitydistribution is built over this set of games and evaluated

using the maximum expected utility This allows the playerto hedge its risk by using the probably that an opponent willselect an action increasing payoffs by lowering the effect ofmisperceptions on the hypergame model

53 PerceptionsDeception Hypergames have been used tomodel interactive decisions through matrices trees andtableaux [85 86] The authors expand this repertoire byshowing preliminary problem structuring where there aregameswithin games and build the perception in hypergamesThey also expand the repertoire by combining hypergameswith different methods to solve complex decisions

Mateski et al explore perception misperception anddeception in conflict using hypergames [87] They introducea diagrammatic representation for hypergames called thehypergame perception model (HPM) The HPM was usedto model misperception and deception during the CubanMissile Crisis where perception played a critical role in theconflict The HPM diagram is shown in Figure 11

Gharesifard and Cortes [88] show that for a game withrational players where the past outcomes are perfectly ob-servable repeated play converges to equilibriaThis results inthe hypergame having an acyclic structure They also presentthe notion of inconsistent equilibrium in the repeated playof first-level hypergames with two players [89] Inconsistentequilibrium refers to the equilibria of the hypergame whereat least one player expects the other to move away fromthe equilibria Just the existence of inconsistent equilibriummeans there is some misperception about the game amongone of the players A class of actions called exploratory arealso identified by the authors to allow players to move awayfrom inconsistent equilibria and decrease the misperceptionIf only one player in the game uses exploratory actionsthen the hypergame will arrive at an outcome rational forthe player If both players use exploratory actions then therepeated play may finish in a cycle

They [90] also study the situations where the perceptionsof players in the game are inconsistent and evolving Theauthors present a new method called swap learning whichallows the incorporation of information gained by observingtheir opponents actions into the playerrsquos beliefs This methodallows a player to decrease misperceptions but at a costof incorporating inconsistencies into their beliefs Sincethe swap of preferences does not take into account theother outcomes inconsistencies can form in the beliefs ofplayer A To eliminate the inconsistencies the modified swaplearning method is presented This method assumes thatthe opponent has perfect information and plays their beststrategy but yields consistent beliefs and decreases playermisperceptionThe swap learning method place the origin ofthemisperception on the player performing the belief update

Again Gharesifard and Cortes [91 92] focus on conflictswith incomplete information where players may have differ-ent perceptions about the conflict Specifically they focus ona 2-player hypergame where one player the deceiver has fullinformation about his opponentrsquos game and wants to intro-duce a certain belief in itThey use their previously developedH-digraph [93] a special class of digraph used to encode

GameTheory 11

Player 1 Player 2

Player 1

Player 1 Player 2

Player 1Awareness notation

p11 p12 p13 p14 p21 p22 p23 p24p99840021

p99840022

p99840023

p99840024

p99840011

p99840012

p99840013

p99840014

times

times

times

Figure 11 HPM diagrammatic representation

Not attack Attack

Attacker

Def

ende

r Def

end

Not

def

end

(0 0)

Prov

ide

ruse

Shut

dow

n

Zero-dayexploit

(minusWi Wi minus CaWi) (minusWi Wi minus CzWi)

(minusWi Wi minus CzWi)

(minusWi minus CrWi 0) (Va minus CrWi Wi minus CaWi) (minusWi Wi minus CzWi)

(minusWi minus CtWi 0) (Va minus Wi minus CtWi Wi) (Va minus CtWi minusWi)

(minusbCfWi minus CmWi 0) (minus(1 minus 2a)Wi minus CmWi(1 minus 2a)Wi minus CaWi)

Figure 12 Gibsonrsquos normal form game model

the belief structure of the hypergame players Using the H-digraph they are able to characterize deceptionwhen stealthyactions are possible in the game Their papers [90 93 94]also present two algorithms for updating perception in thehypergame These methods can decrease the misperceptionbetween the playerrsquos perceived game and true payoffs

54 Dynamic Payoff Functions Gibson presents a modelbased on the intrusion model presented by Chen and Leneu-tre [95] and the Hypergame Normal Form model presentedby Vane [33 34] Table 3 shows the symbols used in the payofffunctions while Figure 12 shows the game in normal formThe author achieves a model that has a changeable nonzero-sum utility values with a process for delineation of strategyselection [39] In order to achieve this model the Chen andLeneutre intrusionmodel is extended by adding strategies forboth the attacker and defender while the HNFmodel is usedto hide or discount strategies from the other player

Table 3 Gibsonrsquos experimental model variables

Variable symbol Meaning119886 Detection rate119887 False alarm rate119862119886

Cost of attack119862119891

Cost of false alarm119862119898

Cost of monitoring119862119903

Cost of providing ruse119862119905

Cost of time down119862119911

Cost of zero-day Exploit119881119886

Value of attacker119882119894

Value of target

The attacker and defender are given additional strategiesover the original model presented by Chen and Lenectre

12 GameTheory

Themost important contribution ofGibsonrsquosmodel is that bycombining the Chen and Leneutremodel withHNF dynamicvariables are added to the payoff functions in HNF as shownin Figure 12 This allows for dynamic play and updating ofvariables as the game is played

55 Mutual Interaction Inohara et al discuss the ability ofplayers to engage in multiple games simultaneously [96]Each game a player engages in may have interactions withother games which can affect outcomes They integratedifferent games in order to capture the interactions whichis realistic of real-life situations An example is given usingthe hypergame methodology in order to model hypergamesthat are mutually interactive and increase perception abilityof players

56 Fuzzy Logic Song et al [97 98] present a novel methodthat uses fuzzy logic to obtain the outcome preference infirst-level hypergame models A fuzzy aggregate algorithm isapplied to get the group fuzzy perception of the opponentsrsquooutcome preference The preference sets are then obtainedby solving linear programming models The authors obtainthe crisp perception for the opponentsrsquo outcome preferenceby using a defuzzification function and the Newton-Cotesnumerical integration formulaThe authors then use the con-cept of consensus winner to determine the preference vectorsin the hypergame models In [99] artificial neural networks(ANNs) are trained to learn the criteria for comparing fuzzyoutcome preference numbers

Qu et al [100] use fuzzy pattern recognition to establisha nonlinear programming model This model is used to inte-grate different outcome preferences for opponents perceivedby different experts Each expert perceives the outcome of thegame and this information is processed using fuzzy patternrecognition to obtain a standard outcome

Zeng et al [101] develop an integration model for hyper-games with fuzzy preference perceptions In conflicts playerscannot perceive information about the opponentrsquos gameclearly so an integration model of multiple perceived fuzzygames using hypergames is developed Each player has fuzzypreference perceptions The authors use linguistic values forthe outcome preferences over the outcome space whichare represented as triangular fuzzy numbers Hypergameswith fuzzy preference perceptions are demonstrated with amilitary example about two countryrsquos navies

57 Comparison to Bayesian Games Sasaki and Kijima [102]propose a Bayesian representation of hypergames by usingHarsanyirsquos theory that any game of incomplete informationcan be transformed into a game of complete informationTheauthors make the claim that ldquoany hypergame can naturally bereformulated in terms of Bayesian games in a unified wayrdquoThis claim is much stronger than the method they actuallypropose There are limitations that result in hypergames thatcannot be reformulated in terms of a Bayesian game Theauthors discuss the limitations of their method which limitsthe ability to reformulate a hypergame in terms of a Bayesiangame Sasaki and Kijima only apply Harsanyirsquos claims to

the original hypergamemodel developed by Bennett [7] theydo not discuss ormention the extension to hypergame theorybyRussell Vane in his doctoral dissertation published in 2000

58 Multiagent Environments Chaib-Dara [103] uses hyper-games to analyze differences in perceptions in multiagentenvironments The author shows how multiagents can inter-act using a third party while having different views andperceptions of the game The third party is used to observethe exact perceptions of the players from an external contextThe players can then choose to trust the external observationand update their perceptions of the game

59 Combining Approaches Huxham and Bennett [104]explore combining hypergames with cognitive mappingsince they both deal with the subjective world of decision-makers They start with the idea that maps could be builtup and then the players preferences and outcomes couldbe extracted The authors determined this process wasnot straightforward They then structure the problem inhypergame form and then used piecemeal maps to explorecertain outcomes The relationship between hypergames andcognitive mapping is explored theoretically by Bryant [67]

Bennett and Cropper [105] examine combining hyper-games with Strategic Choice to provide an effective methodfor modeling decision problems Strategic Choice deals withuncertainty [106] where a participant moves between theactivities of problem shaping generating alternatives com-paring solutions and finally choosing how to act Whilehypergames and Strategic Choice often deal with uncertaintythey both offer different perspectives In Strategic Choice theemphasis is on the need to coordinate between parties wherein hypergames the emphasis is on communication as ameansto makes threats bluffs or deception [105]

Putro et al [33 107ndash109] combine hypergames withgenetic algorithms to produce adaptive learning proceduresThe genetic algorithm is used to choose naturersquos strategies inorder to improve perceptions They present three learningmethods where each method varies a part of the geneticalgorithm (such as fitness evaluationmodified crossover andaction choice) The authors present two experiments thatanalyze the effect of uncertainty and crossover rates on theoutcome of the learning procedures

Kanazawa et al [110ndash112] study hypergames and evolu-tionary game theoryTheyuse hypergames to addperceptionsto evolutionary game theory which result in evolutionaryhypergames Interpretation functions which specify therelationship between the playerrsquos strategies and those oftheir opponent(s) from hypergames are introduced intoevolutionary games These interpretation functions are thenused to create the replicator dynamics for the evolutionarygame which describe the selection process for the distribu-tion of the strategies in a given population This process isdemonstrated using the original application by Bennett tosoccer hooliganism [111]

510 LG Hypergames While not directly related to hyper-game theory as envisioned by Bennett LG hypergames have

GameTheory 13

Cyber

Militaryconflicts

Business

Hypergameapplications

Resourceallocation

Sports

Conflict analysis and modeling

Figure 13 Hypergame application characterization

a similar goal to ldquoaccount for drastic mutual influence ofmultiple subgamesrdquo and are applied to abstract board games(ASBs) [113] Linguistic geometry (LG) hypergame was firstdemonstrated in [113] where it was used to infer the directand indirect effects Each ASB is dynamically linked togetherby interlinking maps a concept similar to hyperlinks inan HTML document [114] A detailed application of LGhypergames is given in [115]

6 Examples and Applications

Hypergame theory has been used to examine past militaryconflicts which by their nature are conducted with missinginformation and misperceptions Past conflicts lend to anal-ysis because the excitement and fog of war have cleared aswell as the outcome already being determined Hypergametheory has also been applied to sports resource allocationand business where competitive nature and proprietaryinformation often lead to missing information and a desireto introducemisperceptions Recently hypergame theory hasbeen applied to cyber in the form of attackdefender models

We have separated applications of hypergames into thesefive topic categories military conflict sports resource allo-cation business and cyber holding the majority of thehypergame application work as shown in Figure 13 An over-view of the numerous applications in hypergame theory issummarized in Table 4 Each is listed chronologically anddenoted with the corresponding year and topic category

61 Military Conflicts Bennett and Dando [15 16] firstapplied hypergames to the first real-world application duringtheir analysis of the Fall of France during WWII They usedhypergame theory to show how misperceptions between thetwo countries can lead to unexpected outcomes

Wright et al [18 19] presented a more complex hyper-game example in their analysis of the nationalization of

Table 4 Listing of hypergame applications chronological

1st Author and citation Year CategoryGiesen [14] 1978 BusinessBennett [15 16] 1979 Military conflictsGiesen [17] 1979 BusinessWright [18] Shupe [19] 1980 Military conflictsBennett [20] 1980 SportsFraser [21 22] 1980 BusinessBennett [23] 1980 BusinessBennett [24] 1981 BusinessFraser [25] 1981 BusinessSaid [26] 1982 Military conflictsBennett [27] 1982 Military conflictsFraser [28] 1983 Military conflictsStokes [29] 1983 BusinessOkada [30] 1985 Resource allocationHipel [31] 1988 Military conflictsGraham [32] 1992 BusinessVane [33] 1999 CyberVane [34] 2000 CyberMaxime [35] 2002 BusinessKopp [8] 2002 CyberHamandawana [36] 2007 Resource allocationNovani [37] 2010 BusinessHouse [38] 2010 CyberGibson [39] 2013 Cyber

the Suez Canal in the 1950s This hypergame shows howone player waiting to participate in the conflict can leadto strategies changing over time While this is a temporalconcept the analysis is onlymade for one point in timeduringthe conflict

Said and Hartley use hypergame theory to analyze the1973 Middle East War [26] Their analysis shows that eachplayer behaves in a rational manner within their own percep-tual beliefs They also propose a methodology for applyinghypergame theory to the crisis

Rott [57] examines the FalklandMalvinas conflictbetween Britain and Argentina in 1982The author approach-es the conflict from a different angle in the analysis of theconflict between Britain and Argentina The hypergameanalysis of the conflict is used to show how misperceptionsdictated an outcome that was unexpected by all sides Thisanalysis uses three specific points in the conflict to conductthree different hypergame analysis While multiple timepoints are used each is picked and really does not containany temporal aspects

Bennett and Dando also model an arms race as a hyper-game in [27]Their analysis forces themodeler to consider theperceptions beliefs and actions of all parties involved whichthey claim to lead to a more competent analysis

Fraser et al [28] apply five conflict analysis models toa possible nuclear confrontation between USA and USSRThe five conflict analysis models are normal form analysisfrom game theory the extensive form of the game metagame

14 GameTheory

analysis [116] hypergame analysis [2 117] and the statetransition model [25 118] Their analysis determines that thehypergame analysis of conflicts is the best for modeling real-world conflicts

62 Sports Bennett et al model soccer hooliganism [20]which appears in UK soccer around the late 1970s Theyuse the hooligan fans and the authorities as the playersEmpirical studies were used to build up possible games thatmay be played between the players The hypergame analysisshowed that there were three critical variables (1) the fansinterpretation of how the authorities prepared for possibleconflict (2) how the authorities interpret the ldquoplay hooliganrdquostrategy by the fans (3) the effect previous incidents have onperception for future conflictsThe result of the analysis is thattolerance should be used by the authorities This reduces theoverpreparation and expectation that everyone is a hooliganand in time reduces the effect of previous incidents

When the hypergame goes through a number of itera-tions additional forces put pressure on players in the gameFor example previous incidents will place pressure on theauthorities to be seen taking firmmeasures andmay cause theauthorities to expect trouble If this is the case then authori-ties will start using toughermeasures If the authorities expectmalevolent fans then there is the possibility that the fanswill become malevolent and start playing the role after beingcategorized Over several rounds if each player is unhappyabout the previous interaction then they will start to see theother player as increasingly malevolent

63 Resource Allocation Okada et al first applied hypergameanalysis to water resource allocation in Japanrsquos Lake Biwaconflict in the early 1970rsquos [30] The conflict is a waterresource management problem where the downstream usersdesire more water from the upstream water source but thecontrollers of the water source are unresponsive While eachplayer in the Lake Biwa conflict had misperceptions aboutthe other playerrsquos preferences the hypergame analysis wasable to correctly identify the compromise that resolved theconflict historically This hypergame has three players theShiga Prefecture downstream prefectures and the nationalgovernmentThe authors use the notation fromHoward [116]and the metagame analysis in [28] to solve the hypergameWhile this game is unique in that it models three players thedetails of the analysis are similar to [28]

Hamandawana et al again applied a game theoreticanalysis to a water management conflict [36] They use amethod similar to hypergame analysis to model the interstateconflict between Angola Botswana and Namibia over theshared water resource of the Okavango River The authorsuse a hypothetical game to build a framework for developingsharing arrangements that minimize conflict where playersmake compensatory sacrifices to offset the losses of otherplayers

Their model introduces the idea of perceived comprisedstrategic relationships There are three types fate controlreflexive control and behavior control In fate control theplayerrsquos outcome may be influenced by the actions of other

players With reflexive control the player has some degreeof control over the outcome regardless of the actions ofother players Behavior control is the case where the playerrsquosoutcome is only feasible through interdependent actions ofcopartners This idea follows that of Bennett with perceivedgames and Fraser with enforceablecredible equilibriums

64 Business

641 Applications to Shipping Hypergame theory wasapplied to a conflict in the oil shipping business in [1417] The incident in 1954 almost led to the bankruptcy ofAristotle Onassis an oil tanker fleet owner The hypergameanalysis showed that decisions made by a player whichappear to be irrational under a conventional game theorymodel are actually rational when the perceptual limitationsand differences in information are considered in hypergametheory

Hypergame analysis was applied to an ongoing shipbuilding conflict in [24] The authors were invited by staffof a UK shipping company Ship building had taken off inthe 1970s in UK but due to developing countries buildingcompleting fleets and the oil crisis in 1973 The hypergameanalysis helped to show how different countries supportedthe crisis in different ways For example Japanrsquos profitableindustries support the less profitable ones which allow Japanto keep producing ships when the ship market went into adepression Other developing countries had labor rates thatwere below those in UK and support the ship buildingindustry which was lacking in UK

642 Negotiation and Contracting Fraser and Hipel explorecontract bargaining using hypergame theory [21] They builda model using the information available to the bargainerand look at the effects of providing opponents with mis-information They use the model to predict the expectedcourse of events during a negotiation session The authorsprovide the first implementation of hypergame analysis ona microprocessor called Conflict Analysis Program (CAP)discussed later

Fraser and Hipel [25] explore labor-management nego-tiations where they apply hypergame analysis to a hypo-thetical labor-management conflictThe hypothetical conflictis developed in detail in [22] The authors again use theConflict Analysis Program (CAP) to show that the bestmodeldoes not always conform to the way things should be butsometimes will conform to how things actually are Forexample they build their model without considering uniondemands fairness of salaries benefits or working conditionsInstead they model the power of the individual players

Bennett used a hypergame analysis to explore a conflictwheremultiple bidders negotiate with a dispenser who is ableto accept the most generous offer [23] This is a case of twonations bidding to get a multinational corporation to relocateto their jurisdiction The model focuses on the ability of thedispenser to play bidders against each other

Graham et al [32] apply hypergame theory to studysupply relationships and modify control systems They use

GameTheory 15

hypergames to identify misperceptions in the process that arecausing inefficiencyThesemisperceptions are then identifiedand targeted for correction to improve efficiency in the supplyrelationships of the players While the authors are studyingtwelve pairs of companies they discuss the types of gamescreated to study the relationship between a vendor of forgingsand an engineering company

643 Trade and E-Commerce Stokes and Hipel use hyper-game theory to study an international trade dispute overgovernment subsidized export credits [29] They model theawarding of large contracts to supply subway cars inNewYorkCity which involves the US and Canada as well as the NewYork transit authority Their analysis of the hypergame high-lights the role of strategic deception in awarding contractsand presents logically reasonable resolutions

Hypergame theory is applied to e-commerce by Leclercand Chaib-draa in [35] They use hypergame theory as ananalysis tool for a multiagent environment They show howmultiple agents interact through communication and amedi-ator when each has differing views of the conflict A discus-sion is also provided on how agents can take advantage ofmisperceptions

Novani and Kijima [37] use a symbiotic hypergamemodel to examine themutual understanding process betweencustomer expectation and provider capability They try toformalize the playersrsquo internal model dealing with the wayeach player identifies the situation subjectively and the inter-pretation function concerning how each player interprets theset of strategies This model is then applied to different typesof customers and providers that the authors develop

65 Cyber Unfortunately very little in the way of hypergametheory application has been done in the arena of cyber war-fare Although much recent work has used game theoreticalapplications in an attempt to model network security the useof hypergame theory is not considered on the same levelHints to its effectiveness have been suggested Certainly acomputer-based tool can provide easy access to information[74] It is easily conceivable that a cyber-defense system canbe infused with a hypergame model in order to influencedecisions on network defense Even Vane although not ina sense directed at defense agents suggests that hypergametheory should be used for decision theory and game theoryagents [119] Providing a computer-based tool the ability toanalyze a situation with a hypergame model creates a quickand efficient technique for strategy selection

651 Information Warfare Kopp uses hypergame theory tomodel Information Warfare [8] He uses a hypergame todescribe how the manipulation of an information channel isreflected in the behavior of the adversaries Figure 2 providesan overview of how hypergames improve upon the gametheoretic model

The focus is on the Information Warfare techniquesof denial of information or degradation deception andcorruption disruption and destruction and subversion

The hypergame provides a tool for understanding thenature of InformationWarfare and allows for quantifying theeffects of the action during warfare The author determinesthat the hypergame theory can be used to model InformationWarfare because the strategies map directly into hypergamemodels

652 Model with Obfuscation There has been at least asmall amount of work performed in the use of hypergametheory applied to cyber warfare House and Cybenko in theirpaper [38] use hypergame theory to model a generic cyber-attack where the defender can choose a specific subgame orthe full game dependent upon the experience level of thecurrent defending administrator Their model is designedusing Dr Vanersquos HNF and static utility values and with theRow player as the attacker Using learningmodels it is shownthat Row can eventually determine the percentages whichin the HNF are the belief-contexts that each subgame isbeing used In simulations the results indicate that within3000 to 5000 iterations these percentages are known withinplusmn5 of the actual usage percentages These results show thatthere is at least some measure of confidence that using alearning strategy one player can begin to understand thestrategy selection of the other This represents the abilityto understand onersquos adversary in order to select onersquos ownstrategy to provide the best possibility at maximum utility

The research does not conclude with these findingsInstead the possibility of the defender (or Column player inthis case) obfuscating the learning ability of the attacker isexamined It is shown that there does exist at least a limitedability for the Column player using an obfuscation NEMSto disrupt the ability of Row to learn Columnrsquos true usagepercentagesThis obfuscationNEMS is created by remodelingthe utility values that were present in the initial Row learningexperiments The result is that Row will have a more difficultprocess to determine the strategy that has been played againstit in each iterationThis approach is somewhat manufacturedin order to provideColumnwith the ability to select strategiesthatmay bemisinterpretedHowever the fact that payoffs andsubgame definitions are the heart of any hypergame scenariois presented as the reason for this approach

The results of this research effort are of most interest dueto the usage of hypergame theory in the attempt to modela cyber-warfare situation more than the actual content ofthe findings House and Cybenko do not use the full HNFmodel choosing to concentrate on learning the nature ofthe game through playing instead of attempting to outguessthe opponent as discussed in [33 34] Despite this factthe usage of the HNF to model the experiments shows theviability of themodel to be incorporated into network defenseinitiatives More avenues of the use of hypergame theory andspecifically of models using the HNF are required to provethe effectiveness of the approach The exploration of howhypergame theory can affect cyber defense and offense is onlyin its infancy

653 Attacker-Defender Model As discussed previouslyGibson [39] presents a hypergame model (based on HNF) of

16 GameTheory

Figure 14 HNF analysis tool (HAT) software

the work of Chen and Lenectre [95] At the heart this modelis an attacker-defender game It keeps the functional and non-zero-sum utilities from the Chen and Lenectre model

With Gibsonrsquos model the attacker is given a new strategyzero-day exploit which is an attack where there is no defensesince the vulnerability is undiscovered The defender is giventwo new strategies providing ruse or shutdown A defendermay provide a ruse by fooling the attacker into attacking ahoneypot while collecting information about the type andstyle of the attack The shutdown option allows the defenderto remove the system from the network and stop the attackin its tracks but also removes the system from operation evenfor mission critical activities

7 Hypergame Analysis Software

Hypergame analysis is possible by hand but it is not rec-ommended it is tedious belief-contexts between each gameiteration The ability of software to execute a hypergameanalysis is discussed as different tools are explored forhypergame analysis Microsoft Excel Minitab and otherstatistical software packages are able to model hypergamesand can easily calculate utility functions from mathematicalequations as shown at the IEEWINFORMS Joint Programfor Capital Science [120]Themain disadvantage is these pro-grams have the inability to run multiple game iterations andupdate variables between iterations Mathematical softwaresuch as MATLAB or Mathematica can calculate the utilityfunctions from mathematical equations and run multiplegame iterations and update variables as well as player beliefsbetween iterations but this software is not specialized forhypergame analysis This means that for each game modelthe entire model has to be built from scratch there is nostandardization of hypergames between researchers

The first implementation of hypergame analysis on amicroprocessor was done by Fraser and Hipel and calledConflict Analysis Program (CAP) Detailed information onCAP can be found in [21 121 122] The authors use CAPto solve complex conflicts in [21 25] during hypergameanalysis While this software was the first for hypergames it

may be outdated and there is no indication that it has beenmaintained

Gambit is software designed for analyzing finite nonco-operative games using the strategic form [123] Players andstrategies can be added using the Gambit interface to quicklycreate a game for analysis It has the ability to exchange gamemodel to external tools creating a standard for game theorymodel data The main disadvantage of this software is lack ofsupport for the complex hypergame model there is no wayin the Gambit interface to enter different games based oneach playerrsquos perceptions or to use mathematical equations tocalculate utility values during game analysis

A software tool specifically designed for hypergame anal-ysis called HYPANT was written by Brumley [124] It usesa standard notation referred to as a language to representhypergame models called Hypergame Markup Language(HML) The HML allows the hypergame model data to besaved restored and transported as well as supported sub-games based on the playerrsquos perceptionsThe disadvantages ofHYPANT are the lack of support for functional utility valuesit only supports the stability and unilateral improvementvalues used by Frasier andHipel in their analysis of theCubanMissile Crisis [2]

Another hypergame analysis program based on VanersquosHNF theory is called security policy assistant (SPA) SPAwas created to assist in deciding if classified documents arereleased or withheld from foreign disclosure [125] While thesoftware manual was available the software is not given itssensitive nature in decision-making with classified informa-tion This software supports the application of hypergametheory beyond the previous applications of military sportsand business conflicts given this softwarersquos ability to assist indecision-making about classified documents

The lack of suitable softwaremeeting all the requirementsfor hypergame analysis caused Gibson to create the HNFanalysis tool (HAT) software [39] as shown in Figure 14HATis written in Java and supports using the Extensible MarkupLanguage (XML) to input and save game design XML is animprovement over the HML used by HYPANT because XMLis widely supported and has many tools to create read and

GameTheory 17

verify and it is not proprietary like HML as well Once agame in XML is loaded HAT allows multiple game iterationsto be run supporting static or random strategy selectionIt also allows variables and belief-contexts to be updatedbetween hypergame iterations Given the availability of theHAT software and its ability to handle hypergames usingHNF concepts developed by Vane [34] this software is usedand updated throughout this research effort

8 Hypergame Theory Fitness and Benefits

Evidence tends to point out that given credible informationhypergame theory can do no worse than game theory modelsbut has the potential to outdo them With this in mind verylittle research has been completed using hypergame theoryto model cyber operations (attack defense etc) Althoughthere has been a limited research it has been recommendedthat more attempts using hypergame theory applied to cyberwarfare should be made House and Cybenko have providedsome limited insight about how hypergame theory can givecredence to the approach They were able to show that theHNF is a feasible modeling technique for experimentation inthis arena Further research into the use of hypergame theoryfor cyber defense and offense needs to be accomplished

Little has also been donewith hypergames in the temporaldomain Most conflicts develop over time and can havemany rounds by combining hypergames with the temporaldomain a more representative conflict model may be able tobe created leading to a better understanding about outcomesWork with temporal game theoretic models indicates that atemporalmodel for hypergame theory is not only possible butwould further be the foundation of the theory

Hypergame analysis can become complex so softwaretools are necessary to help analysis complex models ina timely manner Software tools are still in the maturingprocess with the first tool released byAlanGibson specificallyfor hypergames based on Vanersquos HNF model Continuedsoftware development and refinement will be necessary tosupport hypergame theory in the future which will lead toimprovements in conflict models under hypergame theory

Disclaimer

Theviews expressed in this paper are those of the authors anddo not reflect the official policy of the United States Air ForceDepartment of Defense or the US Government

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was performed at the Air Force Institute of Tech-nologyrsquos (AFIT) Center for Cyberspace Research (CCR) TheCCR at AFIT is a designated Air Force Cyberspace TechnicalCenter of Excellence (CyTCoE)

References

[1] Funk and Wagnells Standard Dictionary Conflict Fitzhenry ampWhiteside Toronto Canada 1974

[2] N M Fraser and K W Hipel Conflict Analysis Models andResolutions vol 2 ofNorth-Holland Series in System Science andEngineering edited by A P Sage North-Holland 1984

[3] M J Osborne and A Rubinstein A Course in Game TheoryMIT Press Cambridge Mass USA 1994

[4] M Peterson An Introduction to Decision Theory CambridgeIntroductions to Philosophy Cambridge University Press Cam-bridge UK 2009

[5] M J OsborneAn Introduction to GameTheory OxfordUniver-sity Press 2004

[6] W R Reitman Cognition and Thought Wiley New York NYUSA 1965

[7] P Bennett ldquoToward a theory of hypergamesrdquoOmega vol 5 no6 pp 749ndash751 1977

[8] C Kopp ldquoShannon hypergames and information warfarerdquoin Proceedings of the 3rd Australian Information Warfare ampSecurity Conference (IWAR rsquo02) W Hutchinson Ed PerthAustralia November 2002

[9] KW Hipel and A Dagnino ldquoA hypergame algorithm for mod-eling misperceptions in bargainingrdquo Journal of EnvironmentalManagement vol 27 pp 131ndash152 1988

[10] M Wang K W Hipel and N M Fraser ldquoSolution conceptsin hypergamesrdquoAppliedMathematics and Computation vol 34no 3 pp 147ndash171 1989

[11] Y Sasaki N Kobayashi and K Kijima ldquoMixed extensionof hypergames and its application to inspection gamesrdquo inProceedings of the 51st Annual Meeting of the ISSS vol 777 pp1ndash9 Tokyo Japan 2007

[12] R J Weber ldquoNoncooperative gamesrdquo in Game Theory andIts Applications vol 24 of Proceedings of Symposia in AppliedMathematics pp 83ndash125 American Mathematical SocietyProvidence RI USA 1981

[13] US Department of Defense Joint Doctrine for InformationOperations Joint Publication 1998

[14] M O Giesen The hypergame modelmdasha critical examinationusing case-studies [BSc Dissertation] Department of Opera-tional Research University of Sussex Brighton UK 1978

[15] P G Bennett and M R Dando ldquoFall Gelb and other games ahypergame perspective of the fall of France 1940rdquo Journal of theConflict Research Society vol 1 no 2 pp 1ndash32 1979

[16] P G Bennett and M R Dando ldquoComplex strategic analysis ahypergame study of the fall of Francerdquo Journal of theOperationalResearch Society vol 30 no 1 pp 23ndash32 1979

[17] M O Giesen and P G Bennett ldquoAristotlersquos fallacy a hypergamein the oil shipping business rdquoOmega vol 7 no 4 pp 309ndash3201979

[18] W M Wright M C Shupe N M Fraser and K W Hipel ldquoAconflict analysis of the Suez Canal invasion of 1956rdquo ConflictManagement and Peace Science vol 5 no 1 pp 27ndash40 1980

[19] M C Shupe W M Wright K W Hipel and N M FraserldquoNationalization of the Suez Canal a hypergame analysisrdquoJournal of Conflict Resolution vol 24 no 3 pp 477ndash493 1980

[20] PG BennettM RDando andRG Sharp ldquoUsing hypergamestomodel difficult social issues an approach to the case of soccerhooliganismrdquo Journal of the Operational Research Society vol31 no 7 pp 621ndash635 1980

18 GameTheory

[21] N M Fraser and K W Hipel ldquoConflict analysis and bar-gainingrdquo in Proceedings of the International Conference onCybernetics and Society pp 225ndash229 IEEE Systems Man andCybernetics Society Cambridge Mass USA October 1980

[22] N M Fraser and K W Hipel ldquoComputer analysis of a labor-management conflictrdquo Tech Rep 71-SM270880 Department ofSystems Design Engineering University of Waterloo 1980

[23] P G Bennett ldquoBidders and dispenser manipulative hyper-games in a multinational contextrdquo European Journal of Oper-ational Research vol 4 no 5 pp 293ndash306 1980

[24] P G Bennett C S Huxham and M R Dando ldquoShippingin crisismdasha trial run for lsquoLiversquo application of the hypergameapproachrdquo Omega vol 9 no 6 pp 579ndash594 1981

[25] N M Fraser and K W Hipel ldquoComputer assistance in labor-management negotiationrdquo Interfaces vol 11 no 2 pp 22ndash301981

[26] A K Said and D A Hartley ldquoA hypergame approach to crisisdecision-making the 1973 middle east warrdquo The Journal of theOperational Research Society vol 33 no 10 pp 937ndash948 1982

[27] P G Bennett andM R Dando ldquoThe arms race as a hypergameA study of routes towards a safer worldrdquo Futures vol 14 no 4pp 293ndash306 1982

[28] N M Fraser K W Hipel and J R del Monte ldquoApproachesto conflict modeling a study of a possible USA-USSR nuclearconfrontationrdquo Journal of Policy Modeling vol 5 no 3 pp 397ndash417 1983

[29] N W Stokes and K W Hipel ldquoConflict analysis of an exportcredit trade disputerdquo Omega vol 11 no 4 pp 365ndash376 1983

[30] N Okada K W Hipel and Y Oka ldquoHypergame analysis of theLake Biwa conflictrdquoWater Resources Research vol 21 no 7 pp917ndash926 1985

[31] KWHipelMWang andNM Fraser ldquoHypergame analysis ofthe Falkland Island crisisrdquo International Studies Quarterly vol32 pp 335ndash358 1988

[32] I Graham F OrsquoDoherty A McKinnon and L Baxter ldquoHyper-game analysis of the stability of relationships between comput-erbased logistics systemsrdquo International Journal of ProductionEconomics vol 26 no 1ndash3 pp 303ndash310 1992

[33] R R Vane and P E Lehner ldquoUsing hypergames to select plansin adversarial environmentsrdquo in Proceedings of the 1st WorkshoponGameTheoretic andDecisionTheoretic Agents S Parsons andM J Wooldridge Eds pp 103ndash111 1999

[34] R R Vane Using hypergames to select plans in competitiveenvironments [PhD thesis] George Mason University 2000

[35] L Maxime and B Chaib-draa Hypergame Analysis in E-Commerce A Preliminary Report Scientific Series CIRANO2002

[36] H Hamandawana R Chanda and F Eckardt ldquoHypergameanalysis and hydroconflicts in the Okavango drainage basinrdquoWater International vol 32 no 4 pp 538ndash557 2007

[37] S Novani andK Kijima Symbiotic HypergameAnalysis of ValueCo-Creation Process in Service System IEEE 2010

[38] J T House and G Cybenko ldquoHypergame theory applied tocyber attack and defenserdquo in Sensors and Command ControlCommunications and Intelligence (C3I) Technologies for Home-land Security and Homeland Defense IX E M Carapezza Edvol 7666 of Proceedings of SPIE Orlando Fla USA May 2010

[39] A S Gibson Applied hypergame theory for network defense[MS thesis] Air Force Institute of Technology Wright-Patterson Air Force Base Ohio USA 2013

[40] J Nash ldquoEquilibrium points in 119899-person gamesrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 36 no 1 pp 48ndash49 1950

[41] J F Nash ldquoThe bargaining problemrdquo Econometrica vol 18 no2 pp 155ndash162 1950

[42] J F Nash ldquoTwo-person cooperative gamesrdquo Econometrica vol21 no 1 pp 128ndash140 1953

[43] J Nash ldquoNon-cooperative gamesrdquo Annals of Mathematics vol54 no 2 pp 286ndash295 1951

[44] J van Benthem Logic in Games The MIT Press CambridgeMass USA 2014

[45] T E Carroll and D Grosu ldquoA game theoretic investigationof deception in network securityrdquo in Proceedings of the 18thInternatonal Conference on Computer Communications andNetworks (ICCCN rsquo09) pp 1ndash6 IEEE San Francisco Calif USAAugust 2009

[46] J Zhuang V M Bier and O Alagoz ldquoModeling secrecyand deception in a multiple-period attacker-defender signalinggamerdquo European Journal of Operational Research vol 203 no2 pp 409ndash418 2010

[47] J P Hespanha Y S Ateskan and H H Kizilocak ldquoDeceptionin non-cooperative games with partial informationrdquo in Pro-ceedings of the 2nd DARPA-JFACC Symposium on Advances inEnterprise Control July 2000

[48] Y Yavin ldquoPursuit-evasion differential games with deceptionor interrupted observationrdquo Computers and Mathematics withApplications vol 13 no 1ndash3 pp 191ndash203 1987

[49] S O Hansson ldquoDecision theory a brief introductionrdquo TechRep Royal Institute of Technology Stockholm Sweden 2005

[50] M Resnik ChoicesmdashAn Introduction to Decision Theory Uni-versity of Minnesota Press 1987

[51] J Scott ldquoRational choice theoryrdquo in Understanding Contempo-rary Society Theories of the Present Sage 2000

[52] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer Series in Statistics Springer 2nd edition 1993

[53] J Baron Thinking and Deciding Cambridge University Press4th edition 2007

[54] S Grant and T Van Zandt ldquoExpected utility theoryrdquo inHandbook of Rational and Social Choice Oxford UniversityPress Oxford UK 2008

[55] K Stanovich Rationality and the Reflective Mind OxfordUniversity Press 2010

[56] R Hastie and R M Dawes Rational Choice in an UncertainWorldmdashThe Psychology of Judgment and Decision MakingSAGE Publications 2nd edition 2009

[57] H Rott ldquoOdd choices on the rationality of some allegedanomalies of decision and inferencerdquo Topoi vol 30 no 1 pp59ndash69 2011

[58] M Friedman Essays in Positive Economics University ofChicago Press 1953

[59] RD Luce andHRaiffaGames andDecisionsWileyNewYorkNY USA 1957

[60] H A Simon ldquoTheories of bounded rationalityrdquo inDecision andOrganization C B McGuire and R Radner Eds pp 161ndash176North-Holland 1972

[61] HA SimonModels ofManWileyamp SonsNewYorkNYUSA1957

[62] B D Jones ldquoBounded rationalityrdquo Annual Review of PoliticalScience vol 2 pp 297ndash321 1999

GameTheory 19

[63] R Selten ldquoWhat is bounded rationalityrdquo in Proceedings of theDahlem Conference SFBDiscussion Paper B-454 pp 1ndash25May1999

[64] P G Bennett ldquoBeyond game theory-whererdquo in AnalyzingConflict and Its Resolution P G Bennett Ed pp 43ndash69 OxfordUniversity Press London UK 1987

[65] P G Bennett and R R Bussel ldquoHypergame theory andmethodology the current lsquostate of the artrsquordquo inTheManagementof Uncertainty Approaches Methods and Applications vol 32of NATO ASI Series pp 158ndash181 Springer Amsterdam TheNetherlands 1986

[66] P G Bennett and C S Huxham ldquoHypergames and what theydo a lsquosoft ORrsquo approachrdquo Journal of the Operational ResearchSociety vol 33 no 1 pp 41ndash50 1982

[67] J W Bryant ldquoHypermaps a representation of perceptions inconflictsrdquo Omega vol 11 no 6 pp 575ndash586 1983

[68] J W Bryant ldquoModelling alternative realities in conflict andnegotiationrdquo Journal of the Operational Research Society vol 35no 11 pp 985ndash993 1984

[69] R R Vane and P E Lehner ldquoApplications of hypergame theoryto adversarial planningrdquo in Proceedings of the JDL SymposiumNaval Postgraduate School Monterey Calif USA 1990

[70] R R Vane and P E Lehner ldquoHypergames and AI in planningrdquoin Proceedings of the DARPA Research Symposium San DiegoCalif USA September 1990

[71] MWang KW Hipel and NM Fraser ldquoModeling mispercep-tions in gamesrdquo Behavioral Science vol 33 no 3 pp 207ndash2231988

[72] P G Bennett ldquoHypergames the development of an approach tomodeling conflictsrdquo Futures vol 12 no 6 pp 489ndash507 1980

[73] R RVane III ldquoAdvances in hypergame theoryrdquo inProceedings ofthe Workshop on GameTheoretic and DecisionTheoretic AgentsHakodate Japan 2006

[74] R R Vane III ldquoPlanning for terrorist-caused emergenciesrdquo inProceedings of the Winter Simulation Conference (WSC rsquo05) FB Armstrong M E Kuhl N M Steiger and J A Joines EdsOrlando Fla USA December 2005

[75] M Bennett andR R Vane III ldquoUsing hypergames for deceptionplanning and counter deception analysisrdquo Defense IntelligenceJournal vol 15 pp 117ndash138 2006

[76] KWHipelMWang andNM Fraser ldquoHypergame analysis ofthe FalklandMalvinas conflictrdquo International Studies Quarterlyvol 32 no 3 pp 335ndash358 1988

[77] C S Huxham and P G Bennett ldquoHypergames and designdecisionsrdquo Design Studies vol 4 no 4 pp 227ndash232 1983

[78] M A Takahashi N M Fraser and K W Hipel ldquoA procedurefor analyzing hypergamesrdquo European Journal of OperationalResearch vol 18 no 1 pp 111ndash122 1984

[79] MWang KWHipel andNM Fraser ldquoHypergamesmdashflexiblemathematical tools to describe misperceptions in conflictsrdquo inProceedings of the Internaional Federation of Automatic Control(UFAC) Workshop on Modeling Decisions and Games withApplication to Social Phenomena Beijing China 1986

[80] M Wang K W Hipel and N M Fraser ldquoResolving environ-mental conflicts having misperceptionsrdquo Journal of Environ-mental Management vol 27 no 2 pp 163ndash178 1988

[81] A Dagnino N M Fraser and KW Hipel ldquoConflict analysis ofan environmental disputerdquo in Systems Analysis Applied toWaterand Related Land Resources Proceedings of the InternationalFederation of Automatic Control (IFAC) Conference Held inLisbon Portugal 2ndash5 October 1985 L V Tavares and J E SilvaEds pp 253ndash258 1985

[82] A Dagnino K W Hipel and N M Fraser ldquoGame theoryanalysis of a groundwater contamination disputerdquo Journal ofthe Geological Society of India vol 29 no 1 pp 6ndash22 1987

[83] Y Sasaki and K Kijima ldquoPreservation of misperceptionsmdashstability analysis of hypergamesrdquo in Proceedings of the 52ndAnnual Conference of the International Society for the SystemsSciences (ISSS rsquo08) pp 1ndash5 July 2008

[84] R Vane and P Lehner ldquoUsing hypergames to increase plannedpayoff and reduce riskrdquo Autonomous Agents and Multi-AgentSystems vol 5 no 3 pp 365ndash380 2002

[85] P G Bennett S Cropper and C Huxham ldquoModelling interac-tive decisions the hypergame focusrdquo in Rational Analysis for aProblematic World John Wiley amp Sons Chichester UK 1989

[86] P G Bennett S Cropper and C Huxham ldquoUsing the hyper-game perspective a case studyrdquo in Rational Analysis for aProblematic World J Rosenhead Ed pp 315ndash340 John Wileyamp Sons Chichester UK 1989

[87] M E Mateski T A Mazzuchi and S Sarkani ldquoThe hyper-game perception model a diagrammatic approach to modelingperception misperception and deceptionrdquoMilitary OperationsResearch vol 15 no 2 pp 21ndash37 2010

[88] B Gharesifard and J Cortes ldquoAcyclic structure of hypergamesand convergence to equilibriardquo Games and Economic BehaviorIn press

[89] B Gharesifard and J Cortes ldquoExploration of misperceptions inhypergamesrdquo in Proceedings of the 49th Annual Allerton Confer-ence on Communication Control and Computing (Allerton rsquo11)pp 1565ndash1570 September 2011

[90] B Gharesifard and J Cortes ldquoEvolution of playersrsquo misper-ceptions in hypergames under perfect observationsrdquo IEEETransactions on Automatic Control vol 57 no 7 pp 1627ndash16402012

[91] B Gharesifard and J Cortes ldquoStealthy strategies for deceptionin hypergames with asymmetric informationrdquo in Proceedings ofthe 50th IEEE Conference on Decision and Control and EuropeanControl Conference (CDC-ECC rsquo11) pp 5762ndash5767 December2011

[92] B Gharesifard and J Cortes ldquoStealthy deception in hypergamesunder informational asymmetryrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 44 no 6 pp 785ndash7952014

[93] B Gharesifard and J Cortes ldquoEvolution of the perception aboutthe opponent in hypergamesrdquo in Proceedings of the 49th IEEEConference on Decision and Control (CDC rsquo10) pp 1076ndash1081Atlanta Ga USA December 2010

[94] B Gharesifard and J Cortes ldquoLearning of equilibria andmisperceptions in hypergames with perfect observationsrdquo inProceedings of the American Control Conference pp 4045ndash4050San Francisco Calif USA 2010

[95] L Chen and J Leneutre ldquoA game theoretical framework onintrusion detection in heterogeneous networksrdquo IEEE Transac-tions on Information Forensics and Security vol 4 no 2 pp 165ndash178 2009

[96] T Inohara S Takahashi and B Nakano ldquoIntegration of gamesand hypergames generated from a class of gamesrdquo Journal of theOperational Research Society vol 48 no 4 pp 423ndash432 1997

[97] Y X Song Z J Li and Y Q Chen ldquoFuzzy information fusionfor hypergame outcome preference perceptionrdquo in IntelligentControl andAutomation vol 344 ofLectureNotes in Control andInformation Sciences pp 882ndash887 Springer Berlin Germany2006

20 GameTheory

[98] Y X Song M Q Dai and Y Cui ldquoMilitary conflict decisionmodeling based on fuzzy hypergamerdquo in Proceedings of the 6thInternational Conference on Management pp 410ndash415 WuhanChina 2007

[99] Y-X Song Y Qu Z-R Liu and Y-Q Chen ldquoFusion andautomatic ranking of fuzzy outcome preferences informationin hypergame modelsrdquo in Proceedings of the InternationalConference on Machine Learning and Cybernetics (ICMLC rsquo05)vol 5 pp 2711ndash2715 August 2005

[100] Y Qu Y Song and J Zhang ldquoAn outcome preference infor-mation aggregation model and its algorithm in hypergamesituationsrdquo in Proceedings of the 5th International Conference onFuzzy Systems and Knowledge Discovery (FSKD rsquo08) vol 4 pp494ndash497 IEEE Jinan China October 2008

[101] X Zeng Y Song and B Luan ldquoInteractive integrationmodel ofhypergames with fuzzy preference perceptions inmulti-conflictsituationsrdquo in Proceedings of the IEEE International Conferenceon Intelligent Computing and Intelligent Systems (ICIS rsquo09) vol2 pp 646ndash650 IEEE Shanghai China November 2009

[102] Y Sasaki and K Kijima ldquoHypergames and Bayesian games atheoretical comparison of the models of games with incompleteinformationrdquo Journal of Systems Science amp Complexity vol 25no 4 pp 720ndash735 2012

[103] B Chaib-Dara ldquoHypergame analysis in multiagent environ-mentsrdquo in Proceedings of the AAAI Spring Symposium pp 147ndash149 Stanford Calif USA March 2001

[104] C Huxham and P Bennett ldquoFloating ideasmdashan experiment inenhancing hypergames with mapsrdquo Omega vol 13 no 4 pp331ndash347 1985

[105] P Bennett and S Cropper ldquoUncertainty and conflict combiningconflict analysis and strategic choicerdquo Journal of BehavioralDecision Making vol 3 no 1 pp 29ndash45 1990

[106] J K Friend andAHicklingPlanning under Pressure PergamonPress Oxford UK 1987

[107] U S Putro K Kijima and S Takahashi ldquoSimulation approachto learning problem in hypergame situation by genetic algo-rithmrdquo in Proceedings of the IEEE International Conference onSystems Man and Cybernetics vol 4 pp 260ndash265 October1999

[108] U S Putro K Kijima and S Takahashi ldquoSimulation of adap-tation process in hypergame situation by genetic algorithmrdquoSystems Analysis Modelling Simulation vol 40 no 1 pp 15ndash372001

[109] U S Putro K Kijima and S Takahashi ldquoAdaptive learningof hypergame situations using a genetic algorithmrdquo IEEETransactions on Systems Man and Cybernetics Part ASystemsand Humans vol 30 no 5 pp 562ndash572 2000

[110] T Kanazawa T Ushio and T Yamasaki ldquoReplicator dynamicsof evolutionary hypergamesrdquo inProceedings of the IEEE Interna-tional Conference on Systems Man and Cybernetics pp 3828ndash3833 October 2003

[111] T Kanazawa andTUshio ldquoMulti-population replicator dynam-ics with changes of interpretations of strategiesrdquo in Proceedingsof the International Symposium on Nonlinear Theory and itsApplications (NOLTA rsquo05) Bruges Belgium October 2005

[112] T Kanazawa T Ushio and T Yamasaki ldquoReplicator dynamicsof evolutionary hypergamesrdquo IEEE Transactions on SystemsMan and CyberneticsmdashPart ASystems and Humans vol 37 no1 pp 132ndash138 2007

[113] B Stilman V Yakhnis and O Umanskiy ldquoIntroduction to LGhypergames for practical wargamingrdquo in Enabling Technologies

for Simulation ScienceVI vol 4716 ofProceedings of SPIE p 386Orlando Fla USA July 2002

[114] B Stilman V Yakhnis O Umanskiy and M McCrabb ldquoLGhypergames for effects-based operationsrdquo in Enabling Technolo-gies for Simulation Science VII vol 5091 of Proceedings of SPIESeptember 2003

[115] R Weber B Stilman and V Yakhnis ldquoExtension of theLG hypergame to inner games played over the topology ofcompeting mind netsrdquo in Enabling Technologies for SimulationScience IX vol 5805 of Proceedings of SPIE p 224 May 2005

[116] N Howard Paradoxes of Rationality Theory of Metagames andPolitical Behavior MIT Press Cambridge Mass USA 1971

[117] N M Fraser and KW Hipel ldquoSolving complex conflictsrdquo IEEETransactions on Systems Man and Cybernetics vol 9 no 12 pp805ndash816 1979

[118] NM Fraser andKWHipel ldquoDynamicmodelling of the cubanmissile crisisrdquo Conflict Management and Peace Science vol 6no 2 pp 1ndash18 1982

[119] R R Vane Hypergame Theory for DTGT Agents AmericanAssociation for Artificial Intelligence 2000

[120] The Washington Institute and the Management Sciences forOperations Research ldquoFeedbackrdquo Newsletter of WINFORMSvol 27 no 2 pp 1ndash6 2006 httpwwwinformsorgcontentdownload2668042524773filefeedbackvol27no2pdf

[121] N M Fraser and K W Hipel ldquoComputer assistance in conflictanalysisrdquo in Proceedings of the International Conference onCybernetics and Society pp 205ndash209 1979

[122] N M Fraser and K W Hipel ldquoComputational techniques inconflict analysisrdquo Advances in Engineering Software vol 2 no4 pp 181ndash185 1980

[123] R D McKelvey A M McLennan and T L Turocy ldquoGam-bit Software Tools for Game Theoryrdquo 2010 httpwwwgam-bit-projectorg

[124] L BrumleyHYPANT a hypergame analysis tool [Honors thesis]Monash University 2003

[125] R Vane III ldquoVeridian security policy assistant users manualrdquoUser Manual 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 GameTheory

Hypergame analysis extends game theory by providingthe larger game that is really being played whether or notboth players are aware of it A different game model canrepresent each playerrsquos view of the conflict but often theplayerrsquos views will overlap where common knowledge existsFiguring outwhat strategy a playerwill use is dependent uponnot only his or her observation of the game but also howthat player believes their opponent is viewing the game Thiscreatesmany different gamemodels that are examined for thesolution to be obtained The goal of hypergame analysis is toprovide insight into real-world situations that are often morecomplex than a game where the choices of strategy presentthemselves as obvious

After its introduction hypergame theory was used tomodel past military conflicts which are prone to having mis-perceptions and missing information in the process of theirunfolding to showhow the outcomeswere achievedAnalysisof past conflicts also lends itself to ease of understandingsince the fog of war has cleared and the outcome has beendetermined Options selected by each side in the conflictare shown to be the rational choice by way of defining thegame that each side perceived In thismanner the hypergameanalysis shows why unexpected results were obtained whenone or both sides misconstrued the conflict Hypergameanalysis offers advantageous reasoning of strategy selectionthrough situational awareness

Throughout this paper the players are often referred toas human entities engaged in a conflict The perceptionor misperception of the players in all cases is the result ofsensors whether mechanical or human senses are combinedwith brain power Human players mechanical players orartificial intelligence (AI) players are interchangeable in thehypergame models

Examples in this paper are presented using two playerswith a limited number of actionsstrategies This allows thecomplexity of the example to be reduced and the visualizationincluded in the figures to be clear As more players andactionsstrategies increase the complexity will also increase

2 Foundations of Hypergame Theory

Game theory decision theory and hypergame theory can beused to model conflicts as games When very little is knownabout the opponents game theory is used for adversarial rea-soning Decision theory is a better choice if the opponents arewell known which is often the case in complete informationgames If one or more of the opponents are playing differentgames because they are not fully aware of the nature of thegame hypergames can be used to reason about subgames thatare shared between opponentsThe following provides a briefoverview of decision theory and game theory

Hypergames extend game theory by allowing for anunbalanced game model that contains different view of thegame representing the differences in each playerrsquos informa-tion or beliefs The unbalanced game model allows for adifferent game model for each playerrsquos view while havingoverlap where there is common knowledge Decision theoryhas been used in hypergames to model the fear of being

outguessed The fear of being outguessed is common in agame model where the different playerrsquos perceived games areunbalanced

21 GameTheory GameTheory is a set of analytical tools de-signed to help understand the phenomena that are observedwhen decision-makers interact [3] The assumption is madein game theory that human beings are rational and alwaysseek the best alternative when presented with a set of possiblechoices Game theory is highly mathematical and assumesthat all human interactions can be understood and navigatedby presumptions Game theoretic models seek to answer twoquestions about the interaction of the decision-makers [12]

(i) How do individuals behave in strategic situations(ii) How should these individuals behave

Answers to the two questions do not always coincide [13]Often the answers to the two questions may be in conflict

Game theory models have many properties associatedwith those that influence the outcome and how game anal-ysis proceeds Cooperative games can exist where there iscommunication between players but more often games areseen as noncooperative where the players do not attempt togive up any information to their rivals Simultaneous gamesare when players make decisions at essentially the same timeand do not know of the opponentrsquos move in advance (ierock paper and scissors) Conversely sequential games arewhen the opponentrsquosmove is known before a decision ismade(ie betting in poker or chess) Perfect information versusimperfect information is when all previous moves are knownto all players instead of some being hidden These previoustwo concepts are often confused with complete informationand incomplete information which are actually intended tobe the knowledge of all playerrsquos strategies and payoffs Thepayoffs also known as utilities have a common property ofzero sum or nonzero sum

The majority of game theory models are identified aseither strategic games (normal form) used to represent asimultaneous game or extensive games more often used torepresent a sequential game even though it can representsimultaneous one as well A strategic (normal form) gameis shown in Figure 1 Strategic games have utilities that aredetermined by which strategy is selected by each playerWhen a playerrsquos strategy is selected all strategies are availableto choose from and therefore the game is represented in agrid or matrix format Extensive gamesrsquo outcomes are insteadrepresented as a tree structure where the initial player is atthe top with branches leading off for each strategy availableas shown in Figure 2 After the first player chooses an actionthe next player has strategies available creating branches foreach of its strategies available All strategies that a player canuse may or may not be available at a particular node of thetree dependent upon the previous playerrsquos selected strategy

The normal form is used to represent games where playermoves or turns are simultaneous This is often the case ingames where information is imperfect and allows for betteridentifying strictly dominated strategies and Nash equilibriaThe extensive form is better suited for games where player

GameTheory 3

Arm

Disa

rm

Arm Disarm

Nation B

Nat

ion

A

(2 2) (4 1)

(1 4) (3 3)


Arm Disarm

Nation A


(2 2) (4 1) (1 4) (3 3)

Nation B Nation B














4 GameTheory













3 Hypergame Theory




GameTheory 5

Column

Row0 4 1 3

4 0 3 1

Column

0 4 1 3 1 3Row4 0 3 1 3 1

3 1 5 0 1 3

C1 C2

C1 C2 C3

R1

R2

R1

R2

R3








119894and119877

119894are



2while Row










119866 = 1198811 1198812 119881119899 (1)


119866119902= 1198811119902 1198812119902 119881

119899119902 (2)

where 119881119894119902



119867 = 1198661 1198662 119866

119899 (3)



6 GameTheory


functionActions

Outcomemodel A

Outcomemodel B

Player A game

Mod

el fo

r a g

ame

Mod

el fo

r a h

yper

gam

e

Player B game


functionActions

Outcomemodel A

Outcomemodel B



Perceivedmodel A

Perceivedmodel B







1111988112


2 11988121

11988122


119899 1198811198991

1198811198992

sdot sdot sdot 119881119899119899

1198661

1198662





119867119902= 1198661119902 1198662119902 119866

119899119902 (4)





1198672= 1198671 1198672 119867

119899 (5)



GameTheory 7


Row 1

Row 2

RMSs

CMSs

Full game





















PKminus1

RKminus1

rk1

rk2

rkm

P1

R1

r11

r12

r1m

R0 = full game

r01

r02

r0m

CΣ

CKminus1

C1

C0

Row m

S1 S2 S3 Sn

Ck1 Ck2 Ck3

C11 C12 C13

C01 C02 C03

u11 u12 u13

u21 u22 u23

um1 um2 um3

Column n

Ckn

C1n

C0n

u1n

u2n

umn

Belief-contexts





1111986612


2 11986621

11986622


119899 1198661198991

1198661198992

sdot sdot sdot 119866119899119899

1198671

1198672








0which



1through 119875




0




8 GameTheory

0

Expe

cted

util

ity

1

EffectivePartially

NEMSIneffective

120575

EU(R0 CΣ)

EU(R0 C0)




119875



0strategy set


119875 Thus


119875than 119877

0but have




0

Hyp

erga

me e

xpec

ted

utili

ty

1g

WS

PS

NEMS

MO

Crossover point



0 which results






GameTheory 9










The real world

Problem structuring

Modeling

Hypergame framework


Stability analysis

Strategy selection



SMR

FHQ

R

GMR





10 GameTheory

All outcomes

HN(H)

N(BG)

SHN(H)











GameTheory 11

Player 1 Player 2

Player 1

Player 1 Player 2


p11 p12 p13 p14 p21 p22 p23 p24p99840021

p99840022

p99840023

p99840024

p99840011

p99840012

p99840013

p99840014

times

times

times


Not attack Attack

Attacker

Def

ende

r Def

end

Not

def

end

(0 0)

Prov

ide

ruse

Shut

dow

n

Zero-dayexploit


















Value of target


12 GameTheory














GameTheory 13

Cyber

Militaryconflicts

Business


Resourceallocation

Sports
















14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 3

Arm

Disa

rm

Arm Disarm

Nation B

Nat

ion

A

(2 2) (4 1)

(1 4) (3 3)


Arm Disarm

Nation A


(2 2) (4 1) (1 4) (3 3)

Nation B Nation B














4 GameTheory













3 Hypergame Theory




GameTheory 5

Column

Row0 4 1 3

4 0 3 1

Column

0 4 1 3 1 3Row4 0 3 1 3 1

3 1 5 0 1 3

C1 C2

C1 C2 C3

R1

R2

R1

R2

R3








119894and119877

119894are



2while Row










119866 = 1198811 1198812 119881119899 (1)


119866119902= 1198811119902 1198812119902 119881

119899119902 (2)

where 119881119894119902



119867 = 1198661 1198662 119866

119899 (3)



6 GameTheory


functionActions

Outcomemodel A

Outcomemodel B

Player A game

Mod

el fo

r a g

ame

Mod

el fo

r a h

yper

gam

e

Player B game


functionActions

Outcomemodel A

Outcomemodel B



Perceivedmodel A

Perceivedmodel B







1111988112


2 11988121

11988122


119899 1198811198991

1198811198992

sdot sdot sdot 119881119899119899

1198661

1198662





119867119902= 1198661119902 1198662119902 119866

119899119902 (4)





1198672= 1198671 1198672 119867

119899 (5)



GameTheory 7


Row 1

Row 2

RMSs

CMSs

Full game





















PKminus1

RKminus1

rk1

rk2

rkm

P1

R1

r11

r12

r1m

R0 = full game

r01

r02

r0m

CΣ

CKminus1

C1

C0

Row m

S1 S2 S3 Sn

Ck1 Ck2 Ck3

C11 C12 C13

C01 C02 C03

u11 u12 u13

u21 u22 u23

um1 um2 um3

Column n

Ckn

C1n

C0n

u1n

u2n

umn

Belief-contexts





1111986612


2 11986621

11986622


119899 1198661198991

1198661198992

sdot sdot sdot 119866119899119899

1198671

1198672








0which



1through 119875




0




8 GameTheory

0

Expe

cted

util

ity

1

EffectivePartially

NEMSIneffective

120575

EU(R0 CΣ)

EU(R0 C0)




119875



0strategy set


119875 Thus


119875than 119877

0but have




0

Hyp

erga

me e

xpec

ted

utili

ty

1g

WS

PS

NEMS

MO

Crossover point



0 which results






GameTheory 9










The real world

Problem structuring

Modeling

Hypergame framework


Stability analysis

Strategy selection



SMR

FHQ

R

GMR





10 GameTheory

All outcomes

HN(H)

N(BG)

SHN(H)











GameTheory 11

Player 1 Player 2

Player 1

Player 1 Player 2


p11 p12 p13 p14 p21 p22 p23 p24p99840021

p99840022

p99840023

p99840024

p99840011

p99840012

p99840013

p99840014

times

times

times


Not attack Attack

Attacker

Def

ende

r Def

end

Not

def

end

(0 0)

Prov

ide

ruse

Shut

dow

n

Zero-dayexploit


















Value of target


12 GameTheory














GameTheory 13

Cyber

Militaryconflicts

Business


Resourceallocation

Sports
















14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 GameTheory













3 Hypergame Theory




GameTheory 5

Column

Row0 4 1 3

4 0 3 1

Column

0 4 1 3 1 3Row4 0 3 1 3 1

3 1 5 0 1 3

C1 C2

C1 C2 C3

R1

R2

R1

R2

R3








119894and119877

119894are



2while Row










119866 = 1198811 1198812 119881119899 (1)


119866119902= 1198811119902 1198812119902 119881

119899119902 (2)

where 119881119894119902



119867 = 1198661 1198662 119866

119899 (3)



6 GameTheory


functionActions

Outcomemodel A

Outcomemodel B

Player A game

Mod

el fo

r a g

ame

Mod

el fo

r a h

yper

gam

e

Player B game


functionActions

Outcomemodel A

Outcomemodel B



Perceivedmodel A

Perceivedmodel B







1111988112


2 11988121

11988122


119899 1198811198991

1198811198992

sdot sdot sdot 119881119899119899

1198661

1198662





119867119902= 1198661119902 1198662119902 119866

119899119902 (4)





1198672= 1198671 1198672 119867

119899 (5)



GameTheory 7


Row 1

Row 2

RMSs

CMSs

Full game





















PKminus1

RKminus1

rk1

rk2

rkm

P1

R1

r11

r12

r1m

R0 = full game

r01

r02

r0m

CΣ

CKminus1

C1

C0

Row m

S1 S2 S3 Sn

Ck1 Ck2 Ck3

C11 C12 C13

C01 C02 C03

u11 u12 u13

u21 u22 u23

um1 um2 um3

Column n

Ckn

C1n

C0n

u1n

u2n

umn

Belief-contexts





1111986612


2 11986621

11986622


119899 1198661198991

1198661198992

sdot sdot sdot 119866119899119899

1198671

1198672








0which



1through 119875




0




8 GameTheory

0

Expe

cted

util

ity

1

EffectivePartially

NEMSIneffective

120575

EU(R0 CΣ)

EU(R0 C0)




119875



0strategy set


119875 Thus


119875than 119877

0but have




0

Hyp

erga

me e

xpec

ted

utili

ty

1g

WS

PS

NEMS

MO

Crossover point



0 which results






GameTheory 9










The real world

Problem structuring

Modeling

Hypergame framework


Stability analysis

Strategy selection



SMR

FHQ

R

GMR





10 GameTheory

All outcomes

HN(H)

N(BG)

SHN(H)











GameTheory 11

Player 1 Player 2

Player 1

Player 1 Player 2


p11 p12 p13 p14 p21 p22 p23 p24p99840021

p99840022

p99840023

p99840024

p99840011

p99840012

p99840013

p99840014

times

times

times


Not attack Attack

Attacker

Def

ende

r Def

end

Not

def

end

(0 0)

Prov

ide

ruse

Shut

dow

n

Zero-dayexploit


















Value of target


12 GameTheory














GameTheory 13

Cyber

Militaryconflicts

Business


Resourceallocation

Sports
















14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 5

Column

Row0 4 1 3

4 0 3 1

Column

0 4 1 3 1 3Row4 0 3 1 3 1

3 1 5 0 1 3

C1 C2

C1 C2 C3

R1

R2

R1

R2

R3








119894and119877

119894are



2while Row










119866 = 1198811 1198812 119881119899 (1)


119866119902= 1198811119902 1198812119902 119881

119899119902 (2)

where 119881119894119902



119867 = 1198661 1198662 119866

119899 (3)



6 GameTheory


functionActions

Outcomemodel A

Outcomemodel B

Player A game

Mod

el fo

r a g

ame

Mod

el fo

r a h

yper

gam

e

Player B game


functionActions

Outcomemodel A

Outcomemodel B



Perceivedmodel A

Perceivedmodel B







1111988112


2 11988121

11988122


119899 1198811198991

1198811198992

sdot sdot sdot 119881119899119899

1198661

1198662





119867119902= 1198661119902 1198662119902 119866

119899119902 (4)





1198672= 1198671 1198672 119867

119899 (5)



GameTheory 7


Row 1

Row 2

RMSs

CMSs

Full game





















PKminus1

RKminus1

rk1

rk2

rkm

P1

R1

r11

r12

r1m

R0 = full game

r01

r02

r0m

CΣ

CKminus1

C1

C0

Row m

S1 S2 S3 Sn

Ck1 Ck2 Ck3

C11 C12 C13

C01 C02 C03

u11 u12 u13

u21 u22 u23

um1 um2 um3

Column n

Ckn

C1n

C0n

u1n

u2n

umn

Belief-contexts





1111986612


2 11986621

11986622


119899 1198661198991

1198661198992

sdot sdot sdot 119866119899119899

1198671

1198672








0which



1through 119875




0




8 GameTheory

0

Expe

cted

util

ity

1

EffectivePartially

NEMSIneffective

120575

EU(R0 CΣ)

EU(R0 C0)




119875



0strategy set


119875 Thus


119875than 119877

0but have




0

Hyp

erga

me e

xpec

ted

utili

ty

1g

WS

PS

NEMS

MO

Crossover point



0 which results






GameTheory 9










The real world

Problem structuring

Modeling

Hypergame framework


Stability analysis

Strategy selection



SMR

FHQ

R

GMR





10 GameTheory

All outcomes

HN(H)

N(BG)

SHN(H)











GameTheory 11

Player 1 Player 2

Player 1

Player 1 Player 2


p11 p12 p13 p14 p21 p22 p23 p24p99840021

p99840022

p99840023

p99840024

p99840011

p99840012

p99840013

p99840014

times

times

times


Not attack Attack

Attacker

Def

ende

r Def

end

Not

def

end

(0 0)

Prov

ide

ruse

Shut

dow

n

Zero-dayexploit


















Value of target


12 GameTheory














GameTheory 13

Cyber

Militaryconflicts

Business


Resourceallocation

Sports
















14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









6 GameTheory


functionActions

Outcomemodel A

Outcomemodel B

Player A game

Mod

el fo

r a g

ame

Mod

el fo

r a h

yper

gam

e

Player B game


functionActions

Outcomemodel A

Outcomemodel B



Perceivedmodel A

Perceivedmodel B







1111988112


2 11988121

11988122


119899 1198811198991

1198811198992

sdot sdot sdot 119881119899119899

1198661

1198662





119867119902= 1198661119902 1198662119902 119866

119899119902 (4)





1198672= 1198671 1198672 119867

119899 (5)



GameTheory 7


Row 1

Row 2

RMSs

CMSs

Full game





















PKminus1

RKminus1

rk1

rk2

rkm

P1

R1

r11

r12

r1m

R0 = full game

r01

r02

r0m

CΣ

CKminus1

C1

C0

Row m

S1 S2 S3 Sn

Ck1 Ck2 Ck3

C11 C12 C13

C01 C02 C03

u11 u12 u13

u21 u22 u23

um1 um2 um3

Column n

Ckn

C1n

C0n

u1n

u2n

umn

Belief-contexts





1111986612


2 11986621

11986622


119899 1198661198991

1198661198992

sdot sdot sdot 119866119899119899

1198671

1198672








0which



1through 119875




0




8 GameTheory

0

Expe

cted

util

ity

1

EffectivePartially

NEMSIneffective

120575

EU(R0 CΣ)

EU(R0 C0)




119875



0strategy set


119875 Thus


119875than 119877

0but have




0

Hyp

erga

me e

xpec

ted

utili

ty

1g

WS

PS

NEMS

MO

Crossover point



0 which results






GameTheory 9










The real world

Problem structuring

Modeling

Hypergame framework


Stability analysis

Strategy selection



SMR

FHQ

R

GMR





10 GameTheory

All outcomes

HN(H)

N(BG)

SHN(H)











GameTheory 11

Player 1 Player 2

Player 1

Player 1 Player 2


p11 p12 p13 p14 p21 p22 p23 p24p99840021

p99840022

p99840023

p99840024

p99840011

p99840012

p99840013

p99840014

times

times

times


Not attack Attack

Attacker

Def

ende

r Def

end

Not

def

end

(0 0)

Prov

ide

ruse

Shut

dow

n

Zero-dayexploit


















Value of target


12 GameTheory














GameTheory 13

Cyber

Militaryconflicts

Business


Resourceallocation

Sports
















14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 7


Row 1

Row 2

RMSs

CMSs

Full game





















PKminus1

RKminus1

rk1

rk2

rkm

P1

R1

r11

r12

r1m

R0 = full game

r01

r02

r0m

CΣ

CKminus1

C1

C0

Row m

S1 S2 S3 Sn

Ck1 Ck2 Ck3

C11 C12 C13

C01 C02 C03

u11 u12 u13

u21 u22 u23

um1 um2 um3

Column n

Ckn

C1n

C0n

u1n

u2n

umn

Belief-contexts





1111986612


2 11986621

11986622


119899 1198661198991

1198661198992

sdot sdot sdot 119866119899119899

1198671

1198672








0which



1through 119875




0




8 GameTheory

0

Expe

cted

util

ity

1

EffectivePartially

NEMSIneffective

120575

EU(R0 CΣ)

EU(R0 C0)




119875



0strategy set


119875 Thus


119875than 119877

0but have




0

Hyp

erga

me e

xpec

ted

utili

ty

1g

WS

PS

NEMS

MO

Crossover point



0 which results






GameTheory 9










The real world

Problem structuring

Modeling

Hypergame framework


Stability analysis

Strategy selection



SMR

FHQ

R

GMR





10 GameTheory

All outcomes

HN(H)

N(BG)

SHN(H)











GameTheory 11

Player 1 Player 2

Player 1

Player 1 Player 2


p11 p12 p13 p14 p21 p22 p23 p24p99840021

p99840022

p99840023

p99840024

p99840011

p99840012

p99840013

p99840014

times

times

times


Not attack Attack

Attacker

Def

ende

r Def

end

Not

def

end

(0 0)

Prov

ide

ruse

Shut

dow

n

Zero-dayexploit


















Value of target


12 GameTheory














GameTheory 13

Cyber

Militaryconflicts

Business


Resourceallocation

Sports
















14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









8 GameTheory

0

Expe

cted

util

ity

1

EffectivePartially

NEMSIneffective

120575

EU(R0 CΣ)

EU(R0 C0)




119875



0strategy set


119875 Thus


119875than 119877

0but have




0

Hyp

erga

me e

xpec

ted

utili

ty

1g

WS

PS

NEMS

MO

Crossover point



0 which results






GameTheory 9










The real world

Problem structuring

Modeling

Hypergame framework


Stability analysis

Strategy selection



SMR

FHQ

R

GMR





10 GameTheory

All outcomes

HN(H)

N(BG)

SHN(H)











GameTheory 11

Player 1 Player 2

Player 1

Player 1 Player 2


p11 p12 p13 p14 p21 p22 p23 p24p99840021

p99840022

p99840023

p99840024

p99840011

p99840012

p99840013

p99840014

times

times

times


Not attack Attack

Attacker

Def

ende

r Def

end

Not

def

end

(0 0)

Prov

ide

ruse

Shut

dow

n

Zero-dayexploit


















Value of target


12 GameTheory














GameTheory 13

Cyber

Militaryconflicts

Business


Resourceallocation

Sports
















14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 9










The real world

Problem structuring

Modeling

Hypergame framework


Stability analysis

Strategy selection



SMR

FHQ

R

GMR





10 GameTheory

All outcomes

HN(H)

N(BG)

SHN(H)











GameTheory 11

Player 1 Player 2

Player 1

Player 1 Player 2


p11 p12 p13 p14 p21 p22 p23 p24p99840021

p99840022

p99840023

p99840024

p99840011

p99840012

p99840013

p99840014

times

times

times


Not attack Attack

Attacker

Def

ende

r Def

end

Not

def

end

(0 0)

Prov

ide

ruse

Shut

dow

n

Zero-dayexploit


















Value of target


12 GameTheory














GameTheory 13

Cyber

Militaryconflicts

Business


Resourceallocation

Sports
















14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









10 GameTheory

All outcomes

HN(H)

N(BG)

SHN(H)











GameTheory 11

Player 1 Player 2

Player 1

Player 1 Player 2


p11 p12 p13 p14 p21 p22 p23 p24p99840021

p99840022

p99840023

p99840024

p99840011

p99840012

p99840013

p99840014

times

times

times


Not attack Attack

Attacker

Def

ende

r Def

end

Not

def

end

(0 0)

Prov

ide

ruse

Shut

dow

n

Zero-dayexploit


















Value of target


12 GameTheory














GameTheory 13

Cyber

Militaryconflicts

Business


Resourceallocation

Sports
















14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 11

Player 1 Player 2

Player 1

Player 1 Player 2


p11 p12 p13 p14 p21 p22 p23 p24p99840021

p99840022

p99840023

p99840024

p99840011

p99840012

p99840013

p99840014

times

times

times


Not attack Attack

Attacker

Def

ende

r Def

end

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Value of target


12 GameTheory














GameTheory 13

Cyber

Militaryconflicts

Business


Resourceallocation

Sports
















14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









12 GameTheory














GameTheory 13

Cyber

Militaryconflicts

Business


Resourceallocation

Sports
















14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 13

Cyber

Militaryconflicts

Business


Resourceallocation

Sports
















14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









14 GameTheory








64 Business







GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 15













16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









16 GameTheory












GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 17






Disclaimer




Acknowledgments


References





















18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









18 GameTheory











































GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 19




































20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









20 GameTheory





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









review article hypergame theory: a model for conflict...

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