Manipulation in games by Sunny
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Manipulation in games by Sunny



This seminar attempts to demonstrate how the outcome of a game can be manipulated by a non-participating external entity

This seminar attempts to demonstrate how the outcome of a game can be manipulated by a non-participating external entity



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Manipulation in games by Sunny Manipulation in games by Sunny Document Transcript

  • 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 ENGINEERINGCOCHIN 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 ACKNOWLEDGEMENTI am greatly indebted to Dr. David Peter, Head of Department, Division of ComputerScience, CUSAT for permitting me to undertake this work.I express my heartfelt gratitude to my respected Seminar guide Mr. SudheepElayidom for his kind and inspiring advise which helped me to understand the subjectand its semantic significance. I am also very thankful to my colleagues who helped and co-operated with me inconducting the seminar by their active participation. SUNNYDivision of Computer Science,SOE 3
  • Manipulation in Games ABSTRACT Games are strategic situations which involves a number ofparticipating players termed agents. Each agent makes decisions, called moves during thecourse of the game. Rational logic dictates that players make moves aimed at maximisingtheir welfare, given the available information. The choices made by the individual agentsdepend on the choices of others agents as well. The branch of mathematics which studiessuch interactions among the agents in a game is termed Game Theory. Under normal conditions, the players in a game are expected to actas rational beings. The choices they make during the game are guided by a commoninterest 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 togive up rational play by offering additional benefits. This seminar attempts todemonstrate how the outcome of a game can be manipulated by a non-participatingexternal entity.Division of Computer Science,SOE 4
  • Manipulation in Games TABLE OF CONTENTS Title Page No. 1. Game Theory – An Overview 6 1.1 What is a Game? 6 2. Representation of Games 2.1 Normal Form or Strategic Form representation 7 2.2 Extensive Form representation 8 3. Types of Games 3.1 Cooperative or noncooperative Games 9 3.2 Symmetric and asymmetric Games 9 3.3 Zero sum and non-zero sum 10 3.4 Simultaneous and sequential 11 3.5 Perfect information and imperfect information 12 3.6 Infinitely long games 12 3.7 Discrete and continuous games 12 4. Dominance 13 5. Nash Equilibrium 14 5.1 Dominance and Nash Equilibria 15 5.2 Iterated Elimination of Dominated Strategies (IED) 15 5.3 Prisoner’s Dilemma 17 6. Manipulating Games - Mechanism Design 18 6.1 Influencing rational play 19 6.2 Leverage 20 6.3 Extended Prisoner’s Dilemma 21 7. Conclusion 28 8. References 29Division of Computer Science,SOE 5
  • Manipulation in Games1. Game Theory – An Overview Game Theory is a branch of applied mathematics which studies the rationalinteractions among the players of a game. Game theory attempts to mathematicallycapture behavior in strategic situations, in which an individuals success in makingchoices depends on the choices of others. While initially developed to analyzecompetitions in which one individual does better at anothers expense (zero sum games),it has been expanded to treat a wide class of interactions, which are classified accordingto several criteria. Game Theory finds applications in fields as diverse as Biology, SocialSciences, Computer Science etc. Although some developments occurred before it, the field of game theory cameinto being with the 1944 book Theory of Games and Economic Behaviour by John vonNeumann and Oskar Morgenstern. This theory was developed extensively in the 1950sby 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 beenwidely 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 ofstrategies.The participating agents are assumed to be rational. A rational player willchoose the action which he or she expects to give the best consequences, where “best” isaccording to the agent’s personal set of preferences. For example, people typically prefermore money to less money, or pleasure to pain. A decision maker is assumed to have afixed range of alternatives to choose from, and the choice adopted influences the outcomeof the situation. Each possible outcome is associated with a real number– its utility. Apayoff function associates the combination of This can be subjective (how much theoutcome is desired) or objective (how good the outcome actually is for the player).Division of Computer Science,SOE 6
  • Manipulation in Games2. Representation of Games2.1 Normal Form or Strategic Form representation A normal form representation of a game is a specification of players strategyspaces and payoff functions. A strategy space for a player is the set of all strategiesavailable to that player, where a strategy is a complete plan of action for every stage ofthe game, regardless of whether that stage actually arises in play. A payoff function for aplayer is a mapping from the cross-product of players strategy spaces to that players setof payoffs of a player, i.e. the payoff function of a player takes as its input a strategyprofile and yields a representation of payoff as its output. The normal representation ofa 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 iscalled a strategy profile. The normal (or strategic form) game is usually represented by a matrix whichshows the players, strategies, and payoffs (see the example to the right). More generally itcan be represented by any function that associates a payoff for each player with everypossible combination of actions. In the below figure, there are two players; one choosesthe row and the other chooses the column. Each player has two strategies, which arespecified by the number of rows and the number of columns. The payoffs are provided inthe interior. The first number is the payoff received by the row player (Player 1 in ourexample); 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 payoffof 4, and Player 2 gets 3.Division of Computer Science,SOE 7
  • Manipulation in Games Player 2 chooses Player 2 chooses Left RightPlayer 1 chooses Up 4, 3 -1, -1Player 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 actssimultaneously or, at least, without knowing the actions of the other. If players have someinformation about the choices of other players, the game is usually presented in extensiveform.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 ofchoice for a player. The player is specified by a number listed by the vertex. The lines outof the vertex represent a possible action for that player. The payoffs are specified at thebottom 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 chooseseither F or U. Player 2 sees Player 1s move and then chooses A or R. Suppose thatDivision of Computer Science,SOE 8
  • Manipulation in GamesPlayer 1 chooses U and then Player 2 chooses A, then Player 1 gets 8 and Player 2gets 2. Unlike the normal form, the extensive form allows explicit modeling ofinteractions in which a player makes more than one move during the game, and movescontingent upon varying states.3. Types of Games3.1 Cooperative or noncooperative Games A game is cooperative if the players are able to form binding commitments. Forinstance the legal system requires them to adhere to their promises. In noncooperativegames this is not possible. Often it is assumed that communication among players isallowed in cooperative games, but not in noncooperative ones. This classification on twobinary criteria has been rejected A non-cooperative game is a one in which players can cooperate, but anycooperation must be self-enforcing. Of the two types of games, noncooperative games areable to model situations to the finest details, producing accurate results. Cooperativegames focus on the game at large. Considerable efforts have been made to link the twoapproaches. The so-called Nash-programme has already established many of thecooperative 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 strategydepend only on the other strategies employed, not on who is playing them. If theidentities 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. TheDivision of Computer Science,SOE 9
  • Manipulation in Gamesstandard representations of chicken, the prisoners dilemma, and the stag hunt are allsymmetric games. Some scholars would consider certain asymmetric games as examplesof these games as well. However, the most common payoffs for each of these games aresymmetric. Most commonly studied asymmetric games are games where there are notidentical strategy sets for both players. For instance, the ultimatum game and similarlythe dictator game have different strategies for each player. It is possible, however, for agame to have identical strategies for both players, yet be asymmetric. For example, thegame pictured below is asymmetric despite having identical strategy sets for both players. E F E 1, 2 0, 0 F 0, 0 1, 2 Fig. An asymmetric game3.3 Zero sum and non-zero sum Zero sum games are a special case of constant sum games, in which choices byplayers can neither increase nor decrease the available resources. In zero-sum games thetotal benefit to all players in the game, for every combination of strategies, always addsto zero (more informally, a player benefits only at the equal expense of others). Pokerexemplifies a zero-sum game (ignoring the possibility of the houses cut), because onewins exactly the amount ones opponents lose. Other zero sum games include matchingpennies and most classical board games including Go and chess. Many games studied by game theorists (including the famous prisoners dilemma)are non-zero-sum games, because some outcomes have net results greater or less thanDivision of Computer Science,SOE 10
  • Manipulation in Gameszero. Informally, in non-zero-sum games, a gain by one player does not necessarilycorrespond with a loss by another. Constant sum games correspond to activities like theft and gambling, but not tothe fundamental economic situation in which there are potential gains from trade. It ispossible to transform any game into a (possibly asymmetric) zero-sum game by addingan additional dummy player (often called "the board"), whose losses compensate theplayers net winnings. A B A -1, -1 3, -3 B 0, 0 -2, -2 Fig. A zero-sum game3.4 Simultaneous and sequential Simultaneous games are games where both players move simultaneously, or ifthey do not move simultaneously, the later players are unaware of the earlier playersactions (making them effectively simultaneous). Sequential games (or dynamic games)are games where later players have some knowledge about earlier actions. This need notbe perfect information about every action of earlier players; it might be very littleknowledge. For instance, a player may know that an earlier player did not perform oneparticular action, while he does not know which of the other available actions the firstplayer actually performed. The difference between simultaneous and sequential games is captured in thedifferent representations discussed above. Often, normal form is used to representsimultaneous games, and extensive form is used to represent sequential ones; althoughthis isnt a strict rule in a technical sense.Division of Computer Science,SOE 11
  • Manipulation in Games3.5 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 byall other players. Thus, only sequential games can be games of perfect information, sincein simultaneous games not every player knows the actions of the others. Most gamesstudied in game theory are imperfect information games, although there are someinteresting examples of perfect information games, including the ultimatum game andcentipede game. Perfect information games include also chess, go, mancala, and arimaa. Perfect information is often confused with complete information, which is asimilar concept. Complete information requires that every player know the strategies andpayoffs 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 generallyfinished in a finite number of moves. Pure mathematicians are not so constrained, and settheorists in particular study games that last for an infinite number of moves, with thewinner (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 sucha game, but simply on whether one or the other player has a winning strategy. (It can beproven, 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 winningstrategy.) The existence of such strategies, for cleverly designed games, has importantconsequences in descriptive set theory.3.7 Discrete and continuous games Much of game theory is concerned with finite, discrete games, that have a finitenumber of players, moves, events, outcomes, etc. Many concepts can be extended,however. Continuous games allow players to choose a strategy from a continuousDivision of Computer Science,SOE 12
  • Manipulation in Gamesstrategy set. For instance, Cournot competition is typically modeled with playersstrategies being any non-negative quantities, including fractional quantities. Differential games such as the continuous pursuit and evasion game arecontinuous games.4. Dominance In game theory, dominance (also called strategic dominance) occurs when onestrategy is better than another strategy for one player, no matter how that playersopponents 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 anotherstrategy for one player, depending on how the players opponents may play. In the normal-form game {S1, S2, ..., Sn, u1, u2, ..., un}, let si, si’  Si be feasiblestrategies 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 siregardless of other players’ choices. In the normal-form game {S1, S2, ..., Sn, u1, u2, ..., un}, let si, si’  Si be feasiblestrategies 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 inequalitysi’ is at least as good as si. A rational player will never choose a strictly dominated strategy during the courseof a game. The player may, however, choose a weakly dominated strategy depending oncircumstances.Division of Computer Science,SOE 13
  • Manipulation in Games5. Nash Equilibrium In game theory, a solution concept is a formal rule for predicting how the gamewill be played. These predictions are called "solutions", and describe which strategieswill be adopted by players, therefore predicting the result of the game. The mostcommonly used solution concepts are equilibrium concepts, most famously Nashequilibrium. The Nash equilibrium (named after John Forbes Nash, who proposed it) is asolution concept of a game involving two or more players, in which each player isassumed to know the equilibrium strategies of the other players, and no player hasanything 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 herstrategy while the other players keep theirs unchanged, then the current set of strategychoices 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 wouldwant to do something different after being informed about the others strategies, then thatset 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 Nashequilibrium. 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 bestdecision that they can, taking into account the decisions of the others. However, Nashequilibrium does not necessarily mean the best cumulative payoff for all the playersinvolved; in many cases all the players might improve their payoffs if they couldsomehow agree on strategies different from the Nash equilibrium.Division of Computer Science,SOE 14
  • Manipulation in Games5.1 Dominance and Nash Equilibria If a strictly dominant strategy exists for one player in a game, that player will playthat strategy in each of the games Nash equilibria. If both players have a strictlydominant strategy, the game has only one unique Nash equilibrium. However, that Nashequilibrium is not necessarily Pareto optimal, meaning that there may be non-equilibriumoutcomes of the game that would be better for both players. The classic game used toillustrate this is the Prisoners 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 dominatedstrategies may be part of Nash equilibria.5.2 Iterated Elimination of Dominated Strategies (IED) The iterated elimination (or deletion) of dominated strategies is one commontechnique for solving games that involves iteratively removing dominated strategies. Inthe first step, all dominated strategies of the game are removed, since rational players willnot play them. This results in a new, smaller game. Some strategies—that were notdominated before—may be dominated in the smaller game. These are removed, creatinga new even smaller game, and so on. This process is valid since it is assumed thatrationality among players is common knowledge, that is, each player know that the rest ofthe players are rational, and each player know that the rest of the players know that heknows that the rest of the players are rational, and so on ad infinitum. There are two versions of this process. One version involves only eliminatingstrictly dominated strategies. If, after completing this process, there is only one strategyfor each player remaining, that strategy set is the unique Nash equilibrium. Another version involves eliminating both strictly and weakly dominatedstrategies. If, at the end of the process, there is a single strategy for each player, thisstrategy set is also a Nash equilibrium. However, unlike the first process, elimination ofweakly dominated strategies may eliminate some Nash equilibria. As a result, the NashDivision of Computer Science,SOE 15
  • Manipulation in Gamesequilibrium found by eliminating weakly dominated strategies may not be the only Nashequilibrium. For example, consider the normal form game as given by the payoff matrix 1shown below. Initially, the strategy Right strictly dominates the strategy Middle forplayer 2. Hence, it can be eliminated giving the modified payoff matrix 2. In this case, thestrategy Left is strictly dominated by strategy Middle for player 2. Also, the strategy Upstrictly 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, 1 Up 1, 0Player 1 0, 1 2, 0 0, 0 Down Fig. Payoff Matrix 1 Player 2 Left Middle 1, 2 Up 1, 0 Nash EquilibriumPlayer 1 0, 1 0, 0 Down Fig. Payoff Matrix 2Division of Computer Science,SOE 16
  • Manipulation in Games5.3 Prisoner’s Dilemma The Prisoners Dilemma constitutes a problem in game theory. It was originallyframed by Merrill Flood and Melvin Dresher working at RAND in 1950.In its "classical" form, the prisoners dilemma (PD) is presented as follows: Two suspects are arrested by the police. The police have insufficient evidence fora conviction, and, having separated both prisoners, visit each of them to offer the samedeal. If one testifies ("defects") for the prosecution against the other and the other remainssilent, 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 minorcharge. If each betrays the other, each receives a five-year sentence. Each prisoner mustchoose to betray the other or to remain silent. Each one is assured that the other wouldnot know about the betrayal before the end of the investigation. How should the prisonersact? If it is assumed that each player prefers shorter sentences to longer ones, and thateach gets no utility out of lowering the other players sentence, and that there are noreputation effects from a players decision, then the prisoners 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 individualplayer ("prisoner") is maximizing his/her own payoff, without any concern for the otherplayers 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 eachplayers individual reward would be greater if they both played cooperately. In the classic form of this game, cooperating is strictly dominated by defecting, sothat the only possible equilibrium for the game is for all players to defect. In simplerterms, no matter what the other player does, one player will always gain a greater payoffby playing defect. Since in any situation playing defect is more beneficial thancooperating, all rational players will play defect, all things being equal.Division of Computer Science,SOE 17
  • Manipulation in GamesThe payoff matrix for Prisoner’s Dilemma is as shown below: Prisoner 2 Silent Testify -10, 0 Silent -0.5, -0.5Prisoner 1 -5, -5 0, 10 Testify Nash Equilibrium Fig. Payoff Matrix for Prisoner’s Dilemma In this game, regardless of what the opponent chooses, each player alwaysreceives a higher payoff (lesser sentence) by betraying; that is to say that betraying is thestrictly dominant strategy. For instance, Prisoner A can accurately say, "No matter whatPrisoner B does, I personally am better off betraying than staying silent. Therefore, formy own sake, I should betray." However, if the other player acts similarly, then they bothbetray and both get a lower payoff than they would get by staying silent. Rational self-interested decisions result in each prisoners being worse off than if each chose to lessenthe sentence of the accomplice at the cost of staying a little longer in jail himself. Hence aseeming dilemma. In game theory, this demonstrates very elegantly that in a non-zerosum 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 rulesof 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 tobehave according to the rules. The resulting mechanism is then said to implement thedesired outcome. The strength of such a result depends on the solution concept used inDivision of Computer Science,SOE 18
  • Manipulation in Gamesthe rules. It is related to metagame analysis, which uses the techniques of game theory todevelop rules for a game.The rules implemented by the mechanism designers may often contradict with therational intents of the participating players in the game. However, the incentivespromised can influence the players to deter from rational play, leading to the outcomedesired by the mechanism designers. Mechanism designers belong to two classes:benevolent and malicious. Benevolent mechanism designers are interested in increasingthe welfare of the players in the game. On the other hand, malicious mechanism designersaim 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 isimpossible to guarantee optimal results for all four outcomes simultaneously in manysituations, particularly in markets where buyers can also be sellers, thus significantresearch in mechanism design involves making trade-offs between these qualities. Otherdesirable criteria that may be achieved include fairness (minimizing variance betweenparticipants utilities), maximizing the auction holders revenue, and Pareto efficiency.More advanced mechanisms sometimes attempt to resist harmful coalitions of players. Acommon exercise in mechanism design is to achieve the desired outcome according to aspecific solution concept. One branch of mechanism design is the creation of markets, auctions, andcombinatorial auctions. Another is the design of matching algorithms, such as the oneused to pair medical school graduates with internships. A third application is to theprovision 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 themechanism designers if they intend to manipulate the outcome of the game. Theseincentives are often referred to as payments. These payments are described by a tuple ofnon-negative payoff functionsDivision of Computer Science,SOE 19
  • Manipulation in Games + V = (V1, V2,… , Vn) where Vi: S->R Payments depend on the strategy that player i selects as well as the choices ofothers. 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 designersmain objective is to force the players to choose a certain strategy profile or a set ofstrategy 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. Thereexists two diametrically opposed kinds of interested parties, the first one beingbenevolent towards the participants of the game, and the other being malicious. While theformer is interested in increasing a games social gain, the latter seeks to minimize theplayers welfare. It is required to define a measure indicating whether the mechanism ofimplementation enables them to modify a game in a favorable way such that their gainexceeds the manipulations cost. These measures are called leverage and maliciousleverage, 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 theimplementation cost. Regarding the payments offered by the mechanism designer assome form of insurance, it seems natural that outcomes of a game can be improved at nocosts. A malicious mechanism designer can in some cases even reduce the social welfareat no costs. Several optimization problems related to the leverage are NP-hard.Division of Computer Science,SOE 20
  • Manipulation in Games As the concept of leverage depends on the implementation costs, there existsworst-case and uniform leverage. The worst-case leverage is a lower bound on themechanism designers influence: it is assumed that without the additional payments, theplayers 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 mechanismdesigners cost is minimized. The value of the leverage is given by the net social gainachieved by this implementation minus the amount of money the mechanism designerhad to spend. For malicious mechanism designers it is needed to invert signs and swapmax and min. Moreover, the payments made by the mechanism designer have to besubtracted twice, because for a malicious mechanism designer, the money received by theplayers are considered a loss. The concept of leverage is illustrated by the ExtendedPrisoner’s Dilemma as follows.6.3 Extended Prisoner’s Dilemma In the context of manipulating games by providing incentives, the classic exampleof Prisoner’s Dilemma is analyzed as follows. The rules of the game remain the same asearlier with slight modifications. The Extended Prisoner’s Dilemma is as follows: Two suspects are arrested by the police. The police have insufficient evidence fora conviction, but they have sufficient evidence to convict them for a minor crime. Havingseparated both prisoners, the police visit each of them to offer the same deal. If onetestifies for the prosecution against the other and the other remains silent, the betrayergoes free and the silent accomplice receives 3 years in prison and an additonal year forthe minor crime. If both remain silent, both prisoners are sentenced to only one year forthe 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. Howshould the prisoners act?Division of Computer Science,SOE 21
  • Manipulation in Games The payoff matrix for the above game in its unmodified form is as shown in thefigure below. The entries in each cell indicates the number of years saved by theindividual 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 dominatedby testify for both prisoners. Also the strategy confess is dominated by both silent andtestify for either of the prisoners. As such, they can be eliminated, leaving behind thestrategies (testify, testify) which is the Nash Equilibrium for the game. Hence, bothprisoners choose to betray each other. However, this combination of strategies is not themost optimum solution for the game, as the prisoners would be much better off if bothremain silent. Now let us consider two third parties who are interested in the choices made bythe prisoners and eventually, the outcome of the game. The first is the Crime Boss, whowants his gang members to spend as little time as possible in prison. The second is thePolice Chief, who wants the convicts to spend maximum time in prison. To realize theirpersonal objectives, both the individuals attempt to alter the mechanism of the game byDivision of Computer Science,SOE 22
  • Manipulation in Gamesoffering rewards to the prisoners to overlook rational play. The incentives prompt theprisoners to make choices that cause the outcome of the game to sway in favor of theinterested party. The manipulation implemented by each of the interested outsiders is asfollows:i. Mechanism Design by the Crime Boss: The crime boss wants the prisoners to spendminimum time in prison. As seen earlier, the choice made by the prisoners namely, tobetray each other, is not the best solution in terms of years spent in prison. The beststrategy for the prisoners is to cooperate and remain silent which buys them minimumtime in prison. The crime boss, therefore wants to shift the equilibrium of the game to thiscombination of choices. To this end, he offers the prisoners monetary rewards which areas 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 prisonThe 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, 0Division of Computer Science,SOE 23
  • Manipulation in Games + 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 confess 0, 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 theprisoners to remain silent and thus cooperate with each other. The equilibrium of thegame shifts to (silent, silent) which ensures that both the prisoners spend minimum timeDivision of Computer Science,SOE 24
  • Manipulation in Gamesin 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 forthe crime boss is thus, 2 years. The crime boss is a benevolent mechanism designer, whois interested in improving the welfare of the prisoners.ii. Mechanism Design by the Police: The police want the prisoners to spend themaximum time in prison. Under the original rules, each prisoner spends three years inprison, which is one year less than the maximum possible sentence of four years. This isrealized if either of the prisoner confesses the crime. Therefore, the police chief changesthe 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 prisonThe 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 +Division of Computer Science,SOE 25
  • Manipulation in Games Police Chief’s monetary promises silent testify confess silent 0, 5 testify 0, 2 confess 5, 0 2, 0 = Prisoner 2 silent testify confess silent 3, 3 0, 4 0, 5Prisoner 1 4, 0 1, 1 0, 2 testify 5, 0 2, 0 0, 0 confess 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 toconfess 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. Themodified 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. TheDivision of Computer Science,SOE 26
  • Manipulation in Gamesnet gain for the police, is therefore 2 years. The police chief is a malicious mechanismdesigner, who is interested in worsening the welfare of the prisoners. The above example shows that a game can be influenced by the mechanismdesigner to yield a particular outcome. The designer incurs some cost in bringing aboutthe desired outcome. In certain conditions, a particular result can be realized at no cost atall to the designer. The problem of finding a strategy profile’s exact uniformimplementation cost is NP-Hard.Division of Computer Science,SOE 27
  • Manipulation in Games CONCLUSION Games are common occurrence in every day world. Games involve a set of agentswith a common interest to maximize their benefits. Game theory studies the patterns thatarise when multiple agents compete with each other. It attempts to predict the behaviourof participating agents in conflict scenarios. Games being omnipresent in almost everyconceivable aspect of this world, Game theory is a comprehensive field transcending anumber of diverse disciplines. Games assume the participation of rational players. It ispossible for to influence the players of the game to give up rational play, by altering themechanism design. In my paper, I have sought to demonstrate how games can bemanipulated by interested non-participating outsiders.Division of Computer Science,SOE 28
  • Manipulation in Games REFERENCES1. Manipulation in Games - Raphael Eidenbenz, Yvonne Anne Oswald, Stefan Schmid,and Roger Wattenhofer, Computer Engineering and Networks Laboratory, ETH Zurich,Switzerland2. of Computer Science,SOE 29