manipulation in games by sunny

A Seminar Report

On

MANIPULATION IN GAMES

submitted by

SUNNY

In partial fulfillment of the requirements for the Degree of

Bachelor of Technology (B.Tech)In

Computer Science & Engineering

DEPARTMENT OF COMPUTER SCIENCE

SCHOOL OF ENGINEERING

COCHIN UNIVERSTY OF SCIENCE AND TECHNOLOGY

KOCHI-682022

JULY 2010

Division of Computer Engineering

School of EngineeringCochin University of Science & Technology

Kochi-682022_________________________________________________________

CERTIFICATE

Certified that this is a bonafied record of the seminar report titled

MANIPULATION IN GAMES Presented by

SUNNY

of VII semester Computer Science & Engineering in the year 2010 in partial fulfillment of

the requirements for the award of Degree of Bachelor of Technology in Computer Science &

Engineering of Cochin University of Science & Technology.

Dr.David Peter S Mr. Sudheep Elayidom

Head of the Division Seminar Guide

Manipulation in Games

ACKNOWLEDGEMENT

I am greatly indebted to Dr. David Peter, Head of Department, Division of Computer Science, CUSAT for permitting me to undertake this work.

I express my heartfelt gratitude to my respected Seminar guide Mr. Sudheep Elayidom for his kind and inspiring advise which helped me to understand the subject and its semantic significance.

I am also very thankful to my colleagues who helped and co-operated with me in

conducting the seminar by their active participation.

SUNNY

Division of Computer Science,SOE 3


ABSTRACT

Games are strategic situations which involves a number of

participating players termed agents. Each agent makes decisions, called moves during the

course of the game. Rational logic dictates that players make moves aimed at maximising

their welfare, given the available information. The choices made by the individual agents

depend on the choices of others agents as well. The branch of mathematics which studies

such interactions among the agents in a game is termed Game Theory.

Under normal conditions, the players in a game are expected to act

as rational beings. The choices they make during the game are guided by a common

interest to sway the outcome of the game in their favour. However, an external agent,

interested in altering the outcome of a game, can influence the participating agents to

give up rational play by offering additional benefits. This seminar attempts to

demonstrate how the outcome of a game can be manipulated by a non-participating

external entity.



TABLE OF CONTENTS

Title Page No.

1. Game Theory – An Overview 6

1.1 What is a Game? 6

2. Representation of Games2.1 Normal Form or Strategic Form representation 72.2 Extensive Form representation 8

3. Types of Games3.1 Cooperative or noncooperative Games 93.2 Symmetric and asymmetric Games 93.3 Zero sum and non-zero sum 103.4 Simultaneous and sequential 113.5 Perfect information and imperfect information 123.6 Infinitely long games 123.7 Discrete and continuous games 12

4. Dominance 13

5. Nash Equilibrium 145.1 Dominance and Nash Equilibria 155.2 Iterated Elimination of Dominated Strategies (IED) 155.3 Prisoner’s Dilemma 17

6. Manipulating Games - Mechanism Design 186.1 Influencing rational play 196.2 Leverage 206.3 Extended Prisoner’s Dilemma 21

7. Conclusion 28

8. References 29



1. Game Theory – An Overview

Game Theory is a branch of applied mathematics which studies the rational

interactions among the players of a game. Game theory attempts to mathematically

capture behavior in strategic situations, in which an individual's success in making

choices depends on the choices of others. While initially developed to analyze

competitions in which one individual does better at another's expense (zero sum games),

it has been expanded to treat a wide class of interactions, which are classified according

to several criteria. Game Theory finds applications in fields as diverse as Biology, Social

Sciences, Computer Science etc.

Although some developments occurred before it, the field of game theory came

into being with the 1944 book Theory of Games and Economic Behaviour by John von

Neumann and Oskar Morgenstern. This theory was developed extensively in the 1950s

by many scholars. Game theory was later explicitly applied to biology in the 1970s,

although similar developments go back at least as far as the 1930s. Game theory has been

widely recognized as an important tool in many fields.

1.1. What is a Game?

A game consists of a set of players or agents, a set of moves (or strategies)

available to those players, and a specification of payoffs for each combination of

strategies.The participating agents are assumed to be rational. A rational player will

choose the action which he or she expects to give the best consequences, where “best” is

according to the agent’s personal set of preferences. For example, people typically prefer

more money to less money, or pleasure to pain. A decision maker is assumed to have a

fixed range of alternatives to choose from, and the choice adopted influences the outcome

of the situation. Each possible outcome is associated with a real number– its utility. A

payoff function associates the combination of This can be subjective (how much the

outcome is desired) or objective (how good the outcome actually is for the player).



2. Representation of Games

2.1 Normal Form or Strategic Form representation

A normal form representation of a game is a specification of players' strategy

spaces and payoff functions. A strategy space for a player is the set of all strategies

available to that player, where a strategy is a complete plan of action for every stage of

the game, regardless of whether that stage actually arises in play. A payoff function for a

player is a mapping from the cross-product of players' strategy spaces to that player's set

of payoffs of a player, i.e. the payoff function of a player takes as its input a strategy

profile and yields a representation of payoff as its output. The normal representation of

a game G specifies:

- a finite set of players {1, 2, ..., n},

- players’ strategy spaces S1, S2 ... Sn and

- their payoff functions u1, u2 ... un where ui : S1 × S2 × ...× Sn→R

The set (s1, s2, … , sn) representing the strategies adopted by all the players is

called a strategy profile.

The normal (or strategic form) game is usually represented by a matrix which

shows the players, strategies, and payoffs (see the example to the right). More generally it

can be represented by any function that associates a payoff for each player with every

possible combination of actions. In the below figure, there are two players; one chooses

the row and the other chooses the column. Each player has two strategies, which are

specified by the number of rows and the number of columns. The payoffs are provided in

the interior. The first number is the payoff received by the row player (Player 1 in our

example); the second is the payoff for the column player (Player 2 in our example).

Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff

of 4, and Player 2 gets 3.



Player 2 chooses Player 2 choosesLeft Right

Player 1 chooses Up 4, 3 -1, -1

Player 1 chooses Down 0, 0 3, 4

Fig. Normal form or payoff matrix of a 2-player, 2-strategy game

When a game is presented in normal form, it is presumed that each player acts

simultaneously or, at least, without knowing the actions of the other. If players have some

information about the choices of other players, the game is usually presented in extensive

form.

2.2 Extensive form representation

The extensive form can be used to formalize games with some important order.

Games here are often presented as trees. Here each vertex (or node) represents a point of

choice for a player. The player is specified by a number listed by the vertex. The lines out

of the vertex represent a possible action for that player. The payoffs are specified at the

bottom of the tree.

1

F U

2 2

A R A R

5;5 0;0 8;2 0;0

Fig. Extensive form representation of a game

In the game pictured here, there are two players. Player 1 moves first and chooses

either F or U. Player 2 sees Player 1's move and then chooses A or R. Suppose that



Player 1 chooses U and then Player 2 chooses A, then Player 1 gets 8 and Player 2

gets 2.

Unlike the normal form, the extensive form allows explicit modeling of

interactions in which a player makes more than one move during the game, and moves

contingent upon varying states.

3. Types of Games

3.1 Cooperative or noncooperative Games

A game is cooperative if the players are able to form binding commitments. For

instance the legal system requires them to adhere to their promises. In noncooperative

games this is not possible. Often it is assumed that communication among players is

allowed in cooperative games, but not in noncooperative ones. This classification on two

binary criteria has been rejected

A non-cooperative game is a one in which players can cooperate, but any

cooperation must be self-enforcing. Of the two types of games, noncooperative games are

able to model situations to the finest details, producing accurate results. Cooperative

games focus on the game at large. Considerable efforts have been made to link the two

approaches. The so-called Nash-programme has already established many of the

cooperative solutions as noncooperative equilibria.

Hybrid games contain cooperative and non-cooperative elements. For instance,

coalitions of players are formed in a cooperative game, but these play in a non-

cooperative fashion.

3.2 Symmetric and asymmetric Games

A symmetric game is a game where the payoffs for playing a particular strategy

depend only on the other strategies employed, not on who is playing them. If the

identities of the players can be changed without changing the payoff to the strategies,

then a game is symmetric. Many of the commonly studied 2×2 games are symmetric. The



standard representations of chicken, the prisoner's dilemma, and the stag hunt are all

symmetric games. Some scholars would consider certain asymmetric games as examples

of these games as well. However, the most common payoffs for each of these games are

symmetric.

Most commonly studied asymmetric games are games where there are not

identical strategy sets for both players. For instance, the ultimatum game and similarly

the dictator game have different strategies for each player. It is possible, however, for a

game to have identical strategies for both players, yet be asymmetric. For example, the

game pictured below is asymmetric despite having identical strategy sets for both players.

E F

E 1, 2 0, 0

F0, 0 1, 2

Fig. An asymmetric game

3.3 Zero sum and non-zero sum

Zero sum games are a special case of constant sum games, in which choices by

players can neither increase nor decrease the available resources. In zero-sum games the

total benefit to all players in the game, for every combination of strategies, always adds

to zero (more informally, a player benefits only at the equal expense of others). Poker

exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one

wins exactly the amount one's opponents lose. Other zero sum games include matching

pennies and most classical board games including Go and chess.

Many games studied by game theorists (including the famous prisoner's dilemma)

are non-zero-sum games, because some outcomes have net results greater or less than



zero. Informally, in non-zero-sum games, a gain by one player does not necessarily

correspond with a loss by another.

Constant sum games correspond to activities like theft and gambling, but not to

the fundamental economic situation in which there are potential gains from trade. It is

possible to transform any game into a (possibly asymmetric) zero-sum game by adding

an additional dummy player (often called "the board"), whose losses compensate the

players' net winnings.

A B

A -1, -1 3, -3

B 0, 0 -2, -2

Fig. A zero-sum game

3.4 Simultaneous and sequential

Simultaneous games are games where both players move simultaneously, or if

they do not move simultaneously, the later players are unaware of the earlier players'

actions (making them effectively simultaneous). Sequential games (or dynamic games)

are games where later players have some knowledge about earlier actions. This need not

be perfect information about every action of earlier players; it might be very little

knowledge. For instance, a player may know that an earlier player did not perform one

particular action, while he does not know which of the other available actions the first

player actually performed.

The difference between simultaneous and sequential games is captured in the

different representations discussed above. Often, normal form is used to represent

simultaneous games, and extensive form is used to represent sequential ones; although

this isn't a strict rule in a technical sense.



3.5 Perfect information and imperfect information

An important subset of sequential games consists of games of perfect information.

A game is one of perfect information if all players know the moves previously made by

all other players. Thus, only sequential games can be games of perfect information, since

in simultaneous games not every player knows the actions of the others. Most games

studied in game theory are imperfect information games, although there are some

interesting examples of perfect information games, including the ultimatum game and

centipede game. Perfect information games include also chess, go, mancala, and arimaa.

Perfect information is often confused with complete information, which is a

similar concept. Complete information requires that every player know the strategies and

payoffs of the other players but not necessarily the actions.

3.6 Infinitely long games

Games, as studied by economists and real-world game players, are generally

finished in a finite number of moves. Pure mathematicians are not so constrained, and set

theorists in particular study games that last for an infinite number of moves, with the

winner (or other payoff) not known until after all those moves are completed.

The focus of attention is usually not so much on what is the best way to play such

a game, but simply on whether one or the other player has a winning strategy. (It can be

proven, using the axiom of choice, that there are games—even with perfect information,

and where the only outcomes are "win" or "lose"—for which neither player has a winning

strategy.) The existence of such strategies, for cleverly designed games, has important

consequences in descriptive set theory.

3.7 Discrete and continuous games

Much of game theory is concerned with finite, discrete games, that have a finite

number of players, moves, events, outcomes, etc. Many concepts can be extended,

however. Continuous games allow players to choose a strategy from a continuous



strategy set. For instance, Cournot competition is typically modeled with players'

strategies being any non-negative quantities, including fractional quantities.

Differential games such as the continuous pursuit and evasion game are

continuous games.

4. Dominance

In game theory, dominance (also called strategic dominance) occurs when one

strategy is better than another strategy for one player, no matter how that player's

opponents may play. Many simple games can be solved using dominance. The opposite,

intransitivity, occurs in games where one strategy may be better or worse than another

strategy for one player, depending on how the player's opponents may play.

In the normal-form game {S1, S2, ..., Sn, u1, u2, ..., un}, let si, si’ Si be feasible

strategies for player i. Strategy si is strictly dominated by strategy si’ if

ui(si, s-i) < ui(si’, s-i),

where s-i is the strategy profile of all players except i. si’ is strictly better than si

regardless of other players’ choices.

In the normal-form game {S1, S2, ..., Sn, u1, u2, ..., un}, let si, si’ Si be feasible

strategies for player i. Strategy si is weakly dominated by strategy si’ if

ui(si, s-i) ≤ ui(si’, s-i), with at least one strict inequality

si’ is at least as good as si.

A rational player will never choose a strictly dominated strategy during the course

of a game. The player may, however, choose a weakly dominated strategy depending on

circumstances.



5. Nash Equilibrium

In game theory, a solution concept is a formal rule for predicting how the game

will be played. These predictions are called "solutions", and describe which strategies

will be adopted by players, therefore predicting the result of the game. The most

commonly used solution concepts are equilibrium concepts, most famously Nash

equilibrium.

The Nash equilibrium (named after John Forbes Nash, who proposed it) is a

solution concept of a game involving two or more players, in which each player is

assumed to know the equilibrium strategies of the other players, and no player has

anything to gain by changing only his or her own strategy (i.e., by changing unilaterally).

If each player has chosen a strategy and no player can benefit by changing his or her

strategy while the other players keep theirs unchanged, then the current set of strategy

choices and the corresponding payoffs constitute a Nash equilibrium. . As a heuristic,

suppose that each player is told the strategies of the other players. If any player would

want to do something different after being informed about the others' strategies, then that

set of strategies is not a Nash equilibrium. If, however, the player does not want to switch

(or is indifferent between switching and not) then the set of strategies is a Nash

equilibrium.

A strategy profile s* = (s*1, ...., s*n) constitutes a Nash Equilibrium if for every i,

ui(s*i, s*-i) ≥ ui(si', s*-i) for all si' Si

The players in a game are in Nash equilibrium if each one is making the best

decision that they can, taking into account the decisions of the others. However, Nash

equilibrium does not necessarily mean the best cumulative payoff for all the players

involved; in many cases all the players might improve their payoffs if they could

somehow agree on strategies different from the Nash equilibrium.



5.1 Dominance and Nash Equilibria

If a strictly dominant strategy exists for one player in a game, that player will play

that strategy in each of the game's Nash equilibria. If both players have a strictly

dominant strategy, the game has only one unique Nash equilibrium. However, that Nash

equilibrium is not necessarily Pareto optimal, meaning that there may be non-equilibrium

outcomes of the game that would be better for both players. The classic game used to

illustrate this is the Prisoner's Dilemma.

Strictly dominated strategies cannot be a part of a Nash equilibrium, and as such,

it is irrational for any player to play them. On the other hand, weakly dominated

strategies may be part of Nash equilibria.

5.2 Iterated Elimination of Dominated Strategies (IED)

The iterated elimination (or deletion) of dominated strategies is one common

technique for solving games that involves iteratively removing dominated strategies. In

the first step, all dominated strategies of the game are removed, since rational players will

not play them. This results in a new, smaller game. Some strategies—that were not

dominated before—may be dominated in the smaller game. These are removed, creating

a new even smaller game, and so on. This process is valid since it is assumed that

rationality among players is common knowledge, that is, each player know that the rest of

the players are rational, and each player know that the rest of the players know that he

knows that the rest of the players are rational, and so on ad infinitum.

There are two versions of this process. One version involves only eliminating

strictly dominated strategies. If, after completing this process, there is only one strategy

for each player remaining, that strategy set is the unique Nash equilibrium.

Another version involves eliminating both strictly and weakly dominated

strategies. If, at the end of the process, there is a single strategy for each player, this

strategy set is also a Nash equilibrium. However, unlike the first process, elimination of

weakly dominated strategies may eliminate some Nash equilibria. As a result, the Nash



equilibrium found by eliminating weakly dominated strategies may not be the only Nash

equilibrium.

For example, consider the normal form game as given by the payoff matrix 1

shown below. Initially, the strategy Right strictly dominates the strategy Middle for

player 2. Hence, it can be eliminated giving the modified payoff matrix 2. In this case, the

strategy Left is strictly dominated by strategy Middle for player 2. Also, the strategy Up

strictly dominates strategy Down for player 1. Hence, both these rows can be eliminated,

giving the Nash Equilibrium for the game which is namely the strategy (Up, Middle).

Player 2

Left Middle Right

1, 2 0, 1Up 1, 0

Player 1 0, 1 2, 0

Down0, 0

Fig. Payoff Matrix 1

Player 2

Left Middle

1, 2Up 1, 0

Nash EquilibriumPlayer 1 0, 1

Down0, 0

Fig. Payoff Matrix 2



5.3 Prisoner’s Dilemma

The Prisoner's Dilemma constitutes a problem in game theory. It was originally

framed by Merrill Flood and Melvin Dresher working at RAND in 1950.

In its "classical" form, the prisoner's dilemma (PD) is presented as follows:

Two suspects are arrested by the police. The police have insufficient evidence for

a conviction, and, having separated both prisoners, visit each of them to offer the same

deal. If one testifies ("defects") for the prosecution against the other and the other remains

silent, the betrayer goes free and the silent accomplice receives the full 10-year sentence.

If both remain silent, both prisoners are sentenced to only six months in jail for a minor

charge. If each betrays the other, each receives a five-year sentence. Each prisoner must

choose to betray the other or to remain silent. Each one is assured that the other would

not know about the betrayal before the end of the investigation. How should the prisoners

act?

If it is assumed that each player prefers shorter sentences to longer ones, and that

each gets no utility out of lowering the other player's sentence, and that there are no

reputation effects from a player's decision, then the prisoner's dilemma forms a non-zero-

sum game in which two players may each "cooperate" with or "defect" from (i.e., betray)

the other player. In this game, as in all game theory, the only concern of each individual

player ("prisoner") is maximizing his/her own payoff, without any concern for the other

player's payoff. The unique equilibrium for this game is a Pareto-suboptimal solution—

that is, rational choice leads the two players to both play defectly even though each

player's individual reward would be greater if they both played cooperately.

In the classic form of this game, cooperating is strictly dominated by defecting, so

that the only possible equilibrium for the game is for all players to defect. In simpler

terms, no matter what the other player does, one player will always gain a greater payoff

by playing defect. Since in any situation playing defect is more beneficial than

cooperating, all rational players will play defect, all things being equal.



The payoff matrix for Prisoner’s Dilemma is as shown below:

Prisoner 2

Silent Testify

-10, 0Silent -0.5, -0.5

Prisoner 1 -5, -50, 10

Testify Nash Equilibrium

Fig. Payoff Matrix for Prisoner’s Dilemma

In this game, regardless of what the opponent chooses, each player always

receives a higher payoff (lesser sentence) by betraying; that is to say that betraying is the

strictly dominant strategy. For instance, Prisoner A can accurately say, "No matter what

Prisoner B does, I personally am better off betraying than staying silent. Therefore, for

my own sake, I should betray." However, if the other player acts similarly, then they both

betray and both get a lower payoff than they would get by staying silent. Rational self-

interested decisions result in each prisoner's being worse off than if each chose to lessen

the sentence of the accomplice at the cost of staying a little longer in jail himself. Hence a

seeming dilemma. In game theory, this demonstrates very elegantly that in a non-zero

sum game a Nash Equilibrium need not be a Pareto optimum.

6. Manipulating Games - Mechanism Design

In economics and game theory, mechanism design is the study of designing rules

of a game or system to achieve a specific outcome, even though each agent may be self-

interested. This is done by setting up a structure in which agents have an incentive to

behave according to the rules. The resulting mechanism is then said to implement the

desired outcome. The strength of such a result depends on the solution concept used in



the rules. It is related to metagame analysis, which uses the techniques of game theory to

develop rules for a game.

The rules implemented by the mechanism designers may often contradict with the

rational intents of the participating players in the game. However, the incentives

promised can influence the players to deter from rational play, leading to the outcome

desired by the mechanism designers. Mechanism designers belong to two classes:

benevolent and malicious. Benevolent mechanism designers are interested in increasing

the welfare of the players in the game. On the other hand, malicious mechanism designers

aim to worsen the welfare of the players in the game.

Mechanism designers commonly try to achieve the following basic outcomes:

truthfulness, individual rationality, budget balance, and social welfare. However, it is

impossible to guarantee optimal results for all four outcomes simultaneously in many

situations, particularly in markets where buyers can also be sellers, thus significant

research in mechanism design involves making trade-offs between these qualities. Other

desirable criteria that may be achieved include fairness (minimizing variance between

participants' utilities), maximizing the auction holder's revenue, and Pareto efficiency.

More advanced mechanisms sometimes attempt to resist harmful coalitions of players. A

common exercise in mechanism design is to achieve the desired outcome according to a

specific solution concept.

One branch of mechanism design is the creation of markets, auctions, and

combinatorial auctions. Another is the design of matching algorithms, such as the one

used to pair medical school graduates with internships. A third application is to the

provision of public goods and to the optimal design of taxation schemes by governments.

6.1 Influencing rational play

As described earlier, incentives are to be provided to the players in a game by the

mechanism designers if they intend to manipulate the outcome of the game. These

incentives are often referred to as payments. These payments are described by a tuple of

non-negative payoff functions



V = (V1, V2,… , Vn) where Vi: S->R+

Payments depend on the strategy that player i selects as well as the choices of

others. As a result of the payments, the original game

G = (N, S, U)

is modified to

G(V) = (N, S, [U + V]), where [U + V]i(s) = Ui(s) + Vi(s)

Each player i obtains a payoff of Vi in addition to Ui. The mechanism designer's

main objective is to force the players to choose a certain strategy profile or a set of

strategy profiles, without spending too much.

6.2 Leverage

Mechanism designers can implement desired outcomes in games at certain costs.

This raises the question for which games it makes sense to take influence at all. There

exists two diametrically opposed kinds of interested parties, the first one being

benevolent towards the participants of the game, and the other being malicious. While the

former is interested in increasing a game's social gain, the latter seeks to minimize the

players' welfare. It is required to define a measure indicating whether the mechanism of

implementation enables them to modify a game in a favorable way such that their gain

exceeds the manipulation's cost. These measures are called leverage and malicious

leverage, respectively. The concept of leverage is used to measure the change of players’

behaviour a mechanism design can inflict, taking into account the social gain and the

implementation cost. Regarding the payments offered by the mechanism designer as

some form of insurance, it seems natural that outcomes of a game can be improved at no

costs. A malicious mechanism designer can in some cases even reduce the social welfare

at no costs. Several optimization problems related to the leverage are NP-hard.



As the concept of leverage depends on the implementation costs, there exists

worst-case and uniform leverage. The worst-case leverage is a lower bound on the

mechanism designer's influence: it is assumed that without the additional payments, the

players choose a strategy profile in the original game where the social gain is maximal,

while in the modified game, they select a strategy profile among the newly non-

dominated profiles where the difference between the social gain and the mechanism

designer's cost is minimized. The value of the leverage is given by the net social gain

achieved by this implementation minus the amount of money the mechanism designer

had to spend. For malicious mechanism designers it is needed to invert signs and swap

max and min. Moreover, the payments made by the mechanism designer have to be

subtracted twice, because for a malicious mechanism designer, the money received by the

players are considered a loss. The concept of leverage is illustrated by the Extended

Prisoner’s Dilemma as follows.

6.3 Extended Prisoner’s Dilemma

In the context of manipulating games by providing incentives, the classic example

of Prisoner’s Dilemma is analyzed as follows. The rules of the game remain the same as

earlier with slight modifications. The Extended Prisoner’s Dilemma is as follows:

Two suspects are arrested by the police. The police have insufficient evidence for

a conviction, but they have sufficient evidence to convict them for a minor crime. Having

separated both prisoners, the police visit each of them to offer the same deal. If one

testifies for the prosecution against the other and the other remains silent, the betrayer

goes free and the silent accomplice receives 3 years in prison and an additonal year for

the minor crime. If both remain silent, both prisoners are sentenced to only one year for

the minor crime. If each betrays the other, each receives a three year sentence. However,

if either one of the prisoners confess their crime, both get four years in prison. How

should the prisoners act?



The payoff matrix for the above game in its unmodified form is as shown in the

figure below. The entries in each cell indicates the number of years saved by the

individual prisoners for that particular combination of strategies.

Prisoner 2

silent testify confess

silent 3, 3 0, 4 0, 0

Prisoner 1

4, 0 1, 1 0, 0

testify Nash

Equilibrium

confess 0, 0 0, 0 0, 0

Fig. Payoff Matrix for Extended Prisoner’s Dilemma

In the above payoff matrix, it can be observed that the strategy silent is dominated

by testify for both prisoners. Also the strategy confess is dominated by both silent and

testify for either of the prisoners. As such, they can be eliminated, leaving behind the

strategies (testify, testify) which is the Nash Equilibrium for the game. Hence, both

prisoners choose to betray each other. However, this combination of strategies is not the

most optimum solution for the game, as the prisoners would be much better off if both

remain silent.

Now let us consider two third parties who are interested in the choices made by

the prisoners and eventually, the outcome of the game. The first is the Crime Boss, who

wants his gang members to spend as little time as possible in prison. The second is the

Police Chief, who wants the convicts to spend maximum time in prison. To realize their

personal objectives, both the individuals attempt to alter the mechanism of the game by



offering rewards to the prisoners to overlook rational play. The incentives prompt the

prisoners to make choices that cause the outcome of the game to sway in favor of the

interested party. The manipulation implemented by each of the interested outsiders is as

follows:

i. Mechanism Design by the Crime Boss: The crime boss wants the prisoners to spend

minimum time in prison. As seen earlier, the choice made by the prisoners namely, to

betray each other, is not the best solution in terms of years spent in prison. The best

strategy for the prisoners is to cooperate and remain silent which buys them minimum

time in prison. The crime boss, therefore wants to shift the equilibrium of the game to this

combination of choices. To this end, he offers the prisoners monetary rewards which are

as below:

1. If both prisoners remain silent, they will each be rewarded with monetary

benefits amounting to one year in prison.

2. If one prisoner remains silent, and the other betrays him, the former will

be paid benefits worth two years in prison

The above incentives cause the mechanism of the game to be modified as below


silent 3, 3 0, 4 0, 0

testify 4, 0 1, 1 0, 0

Nash

Equilibrium

confess 0, 0 0, 0 0, 0



+

Crime Boss’s monetary promises


silent 1, 1 2, 0

testify 0, 2

confess

Prisoner 2


silent 4, 4 2, 4 0, 0

Nash

Prisoner 1 Equilibrium

= testify 4, 2 1, 1 0, 0

confess0, 0 0, 0 0, 0

Fig. Payoff Matrix for mechanism design by the crime boss

As depicted above, the incentives promised by the crime boss prompt the

prisoners to remain silent and thus cooperate with each other. The equilibrium of the

game shifts to (silent, silent) which ensures that both the prisoners spend minimum time



in prison. The crime boss needs to pay money worth two years in prison to the convicts.

Each prisoner saves two years in prison by adopting the new strategy. The net gain for

the crime boss is thus, 2 years. The crime boss is a benevolent mechanism designer, who

is interested in improving the welfare of the prisoners.

ii. Mechanism Design by the Police: The police want the prisoners to spend the

maximum time in prison. Under the original rules, each prisoner spends three years in

prison, which is one year less than the maximum possible sentence of four years. This is

realized if either of the prisoner confesses the crime. Therefore, the police chief changes

the mechanism design of the game by offering the prisoners incentives as follows:

1. If one prisoner remains silent, and the other betrays him, the former’s sentence

will be reduced by two years.

2. If one prisoner confesses and the other remains silent, the former will be

allowed to go free and rewarded with money worth one year in prison

The modification in the game brought about by the above incentives is illustrated below.


3, 3 0, 4 0, 0

silent

4, 0 1, 1 0, 0

testify Nash

Equilibrium

confess 0, 0 0, 0 0, 0

+


silent

testify

confess

=

silent

Prisoner 1

testify

confess


Police Chief’s monetary promises


0, 5

0, 2

5, 0 2, 0

Prisoner 2


3, 3 0, 4 0, 5

4, 0 1, 1 0, 2

5, 0 2, 0 0, 0

Nash

Equilibrium

Fig. Payoff Matrix for mechanism design by the Police chief

As depicted above the promises made by the police chief prompt the prisoners to

confess to the crime. The equilibrium of the game shifts to the strategy combination

(confess, confess) thereby ensuring that the prisoners spend maximum time in prison. The

modified design is implemented at no cost to the police. The prisoners, on the other hand,

has to spend one year in prison in addition to the maximum sentence of four years. The



net gain for the police, is therefore 2 years. The police chief is a malicious mechanism

designer, who is interested in worsening the welfare of the prisoners.

The above example shows that a game can be influenced by the mechanism

designer to yield a particular outcome. The designer incurs some cost in bringing about

the desired outcome. In certain conditions, a particular result can be realized at no cost at

all to the designer. The problem of finding a strategy profile’s exact uniform

implementation cost is NP-Hard.



CONCLUSION

Games are common occurrence in every day world. Games involve a set of agents

with a common interest to maximize their benefits. Game theory studies the patterns that

arise when multiple agents compete with each other. It attempts to predict the behaviour

of participating agents in conflict scenarios. Games being omnipresent in almost every

conceivable aspect of this world, Game theory is a comprehensive field transcending a

number of diverse disciplines. Games assume the participation of rational players. It is

possible for to influence the players of the game to give up rational play, by altering the

mechanism design. In my paper, I have sought to demonstrate how games can be

manipulated by interested non-participating outsiders.



REFERENCES

1. Manipulation in Games - Raphael Eidenbenz, Yvonne Anne Oswald, Stefan Schmid, and Roger Wattenhofer, Computer Engineering and Networks Laboratory, ETH Zurich, Switzerland

2. http://en.wikipedia.org/wiki/Game_theory

3. http://www.cse.iitd.ernet.in/~rahul/cs905/

4. http://www.gametheory.net/

.


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