Download - 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
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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.
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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
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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).
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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.
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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
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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
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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
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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.
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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
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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.
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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.
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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
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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
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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.
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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
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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
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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.
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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?
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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
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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 testify confess
silent 3, 3 0, 4 0, 0
testify 4, 0 1, 1 0, 0
Nash
Equilibrium
confess 0, 0 0, 0 0, 0
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+
Crime Boss’s monetary promises
silent testify confess
silent 1, 1 2, 0
testify 0, 2
confess
Prisoner 2
silent testify confess
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
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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.
silent testify confess
3, 3 0, 4 0, 0
silent
4, 0 1, 1 0, 0
testify Nash
Equilibrium
confess 0, 0 0, 0 0, 0
+
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silent
testify
confess
=
silent
Prisoner 1
testify
confess
Manipulation in Games
Police Chief’s monetary promises
silent testify confess
0, 5
0, 2
5, 0 2, 0
Prisoner 2
silent testify confess
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
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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.
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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.
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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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