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What is Game Theory? • Created by Von Neumann and Morgenstern in 1944 (book: The Theory of Games and Economic Behaviour). • Has found application in economics, politics, biology, computer science, psychology, sociology, ...... Game Theory • Whenever people or other agents interact with each other they are playing a game. • Game Theory is the study of rational interactions in games i.e. it is the study of the logic of interaction in games. • It is the study of how agents can do as well as possible in games. • This sounds like classical AI!! • Will first consider examples of games, and then look at relevance to “Nouvelle AI”. Some Examples of Games Smugglers (adapted from Morton Davis). • Traditional games such as chess, draughts, • A drug smuggling ring operates by using two noughts-and-crosses. “carriers” to smuggle drugs via either the • These are not usually considered interesting (sole) airport or the (sole) seaport of a country. from a game theory point of view - this is • The police have 2 officers to try to stop them. because these games are games of perfect information, are finite, and have no chance • If one officer guards the exit used by both (random) moves. smugglers 70 kilos will get through, and for • In these circumstances there are theoretical each smuggler at an exit with no officers results showing that the results of such games present 50 kilos will get through. are essentially predetermined, assuming both • If there are at least as many officers as players play perfectly. smugglers no drugs will get through. 1 Enforcing speed limits (adapted from Police at Airport Morton Davis) 0 1 2 • A town council is trying to decide how strictly 0 0 70 100 to enforce speed limits. Smugglers at Airport 1 50 0 50 • By quantifying various costs and benefits to 2 100 70 0 the community and to drivers - the time saved by speeding, the danger to the driver, the danger to the general public, the penalties to What strategies should the police and smugglers adopt the speeding driver, the cost of enforcement, a to achieve their goals? game theorist arrives at the payoff matrix shown. (Bear in mind this game will be played repeatedly!) Game, Theory, Biology, and Nouvelle AI Council Enforces Law Yes No Bluegill Sunfish (adapted from Ken Binmore). • Bluegill sunfish males come in 2 varieties: Yes (-190, -25) (10, -5) – regular which take 7 years to reach maturity, and which Driver Speeds build a nest which attracts egg laying females. When eggs No (0, -20) (0, 0) are laid, he fertilizes them, and then defends resulting family, while female gets on with her life elsewhere. – rogue which reach maturity in 2 years, and is not capable • Is it “cheaper” for the council to enforce, or ignore the of rearing a family. Instead he lurks in hiding until a female has laid her eggs in response to a regular male, and then law? rushes out to fertilize the eggs before the regular male has • Is there any other alternative? a chance to do so. If successful, the regular male defends the resulting family, which are not related to him. 2 • If we regard the behaviour of bluegill sunfish males • Selecting the fittest to reproduce most often as a game in which each male is competing with the will therefore lead to a situation in which the others to get the maximum number of offspring, then this can be treated as an N-player game (where N is relative proportions of rogues and regulars is number of males in population). such that there is no particular advantage in • If there are enough regular males, then the payoff is being either. high for being a rogue, since there are plenty of • Applying game theory to biological modelling opportunities to apply the rogue strategy. • If there are too many rogues in the population, then it can therefore help us understand why is better to be a regular male since there is at least evolution leads to populations with particular some chance that the male will get to propagate its proportions of alternative strategies. genes, versus almost no chance for any individual rogue. • It can also help us understand similar • Therefore the fitness of an individual is determined phenomena in evolutionary computational by the relative proportions of the 2 types of male in processes. the population Variants Further Reading There are many variants of games and game theory. Games can be studied: • Davis, Morton (1970) Game Theory: A • With, or without, communication allowed between the players Nontechnical Introduction – promises Many of the examples in this lecture are adapted – threats – lies from this book. • With simultaneous play, or sequential (in which one player goes first, and then the other(s) get to choose their response). • With deterministic, or stochastic, payoffs. • Binmore, Ken (1992) Fun and games: A Text on • With 2 or more players. Game Theory • With, or without, “side-deals” allowed. • Repeated, or “one-shot”. • etc. • Smith, Maynard (1982) Evolution and the Theory of Games 3 Zero-Sum 2 Player Games A Political Example • These are 2 player games where the interests of the players are completely and utterly opposed to each other, and one player's gain is the other player's loss • Two political parties (the Acranial, and the (and vice-versa). Brainless) are in the process of deciding their • Can therefore be represented by a payoff matrix manifestos before an election. which just gives the payoffs to one of the players, and the payoffs for the other player can be computed from • Abortion is a major issue in this election, and these. each party must decide whether they will adopt • Sometimes, the payoffs for the two players will add a “pro” legal abortion position, an anti- to a non-zero constant value. In this case the games abortion position, or if they will simply avoid are still considered zero-sum as they can easily be the issue. converted to an equivalent game in which the payoffs do add to zero. Party B • Party researchers carry out opinion polls and Pro Anti Avoid discover that the following matrix represents the Pro 45% 50% 40% In this case this is percentage of the vote that the Acranials will receive very simple. if each party adopts the positions shown: Party A Anti 60% 55% 50% Party B Avoid 45% 55% 40% Pro Anti Avoid • Party B should avoid the issue, since it always does Pro 45% 50% 40% best with this option no matter what A does (remember: smaller numbers are better for B!) Party A Anti 60% 55% 50% • Party A should adopt an “anti” position since it always does at least as well with this as with any Avoid 45% 55% 40% other strategy no matter what B does. • Therefore the predicted best outcome for each party is What should the parties do? with these strategies and there will be a 50-50 split in the vote 4 What if the payoff matrix had been different? Points to Note Party B • In the last example the joint position A chooses “Anti” and B Pro Anti Avoid It is less clear now chooses “Pro” is known as a (Nash) equilibrium of the game, and these two choices are known as equilibrium strategies. Pro 35% 10% 60% what each party should do. • A (Nash) equilibrium of a game represents a choice of Party A Anti 45% 55% 50% strategy by each of the players such that no player is Avoid 40% 10% 65% unilaterally tempted to change strategy since they are guaranteed to do worse if they do. They can arrive at a solution by reasoning as follows: • Some games have more than one equilibrium. In zero-sum • B should NOT avoid the issue, since it will always do at least as well by games all such equilibria are equivalent in the sense that the choosing “Pro”. payoffs for all players are the same in each of the equilibria • Once A realises that “Avoid” is not an option for B, it becomes clear that A should adopt an “Anti” stance, since it does better with this than any other • Some games have no equilibria. Is it possible to have a option no matter what B does. sensible strategy in such games? • Once B realises that A will adopt an “Anti” position, B should adopt a “Pro” position since this maximizes B's vote (and minimizes A's) An Example (Smugglers) Pure and Mixed Strategies Police at Airport • In zero-sum games with Nash equilibria one can do 0 1 2 no better than to always play (one of) the equilibrium strategy. Since one always plays the same strategy, 0 0 70 100 this is known as a pure strategy. Smugglers at Airport 1 50 0 50 2 100 70 0 • In games with no Nash equilibria, one can only guarantee the value of the game by playing a so- called mixed strategy, which involves randomly • If this game is played repeatedly it turns out choosing (with appropriate probabilities) which pure that there are equilibrium strategies, but they strategy to play. are mixed. 5 • Police: Send both police to the airport 1/2 of the time, The Minimax Theorem and both to the seaport the other 1/2 of the time. • Von Neumann (yes, him again!) proved a theorem about finite • Smugglers: 2 player zero-sum games, which roughly says (again taken from Morton Davis): – 4/14 of the time send 1 smuggler to airport and 1 to seaport “Every finite 2 player zero-sum game has a value V which is – 5/14 of the time send both to the airport the amount player 1 will win on average if both players play – 5/14 of the time send both to the seaport sensibly” • The following should be noted about this theorem: • These strategies guarantee that on average the • There is a (mixed) strategy for player 1 that will guarantee smugglers will get 50 kilos through, and the police this. Nothing player 2 can do will prevent player 1 from will stop 50 kilos from getting through. If the getting V average. smugglers (always) used a different strategy then • There is a (mixed) strategy for player 2 that will guarantee that there is a strategy for the police which will mean the player 1 gets no more than V. smugglers get less through, and if the police use a • Since the game is zero-sum, what player 1 wins, player 2 loses. Since player 2 wants to minimise her losses, player 2 is different strategy then the smugglers can choose a motivated to limit player 1's average win to V. strategy which will get more through. • Note this last point does not hold for non-zero-sum games. • The mixed equilibrium strategies for the Smuggling game are an example of the theorem, where V = 50. Non-Zero-Sum 2 Player Games • Zero-sum games are games of “pure opposition” i.e. The Prisoner's Dilemma anything one player gains is balanced by equivalent losses by the other players, so the players simply Two people are arrested and charged with a serious compete with each other. crime. They are kept in separate cells, with no • In contrast, non-zero-sum games involve elements of opportunity to confer. If both people remain silent, both competition and cooperation, and it is possible and tell the police nothing, they will both be set free, for both players to gain, or both to lose, or for there to as there is insufficient evidence to convict them. If be a net gain or loss by the players. one “turns state's evidence” (and says the other • The payoff matrix for such games must specify the prisoner did it), that prisoner will be rewarded with payoffs for both players, since those for player 2 £20000, and the other prisoner will get a 6 year cannot be computed from those for player 1. sentence. If they both “turn state's evidence” (and • Note that in such games each player is trying to inform on each other), they both get a 1 year maximize their own payoff. This is NOT necessarily the same as minimizing the payoff to one's opponent. sentence. 6 P2 Iterated Prisoner's Dilemma C D Suppose instead that the players are given the P1 C (0, 0) (-60000, 20000) opportunity to repeatedly play the game against each D (20000, -60000) (-10000, -10000) other. Now the possibility exists to learn from past play, and to give the other player the opportunity to C = cooperate (= remain silent) cooperate, and if they don't, to “punish” them by D = defect (= inform) future play. The equilibrium of this game is where both players defect, and the inexorable logic of the fact that both players are trying to maximize their payoff drives them to Possible strategies include: this point. However, if they could agree to both cooperate, and stick to the agreement, they • random play would both do much better. However, even if they had agreed there is nothing to • always defect stop a player reneging on their promise (talk is cheap!), and for the player reneging, it is to their advantage to do so. • always cooperate For a one-shot game the Nash equilibrium is really the inevitable outcome of this • something “intelligent” depending on opponent's play game if both players play rationally. Evolution of Cooperation Something slightly paradoxical! • Note that there is something paradoxical about playing • Axelrod used this to study the evolution of cooperatively if the players know how many iterations there cooperative behaviour. Set up a computer are going to be (e.g. 10). • Even if (or maybe because) one has been playing tournament where programs (strategies) all cooperatively with one's opponent, on the last iteration there is play each other, and get to reproduce (have a temptation to defect to try and “sneak in” a final extra gain, since there is no possibility of being “punished” for it. copies of themselves made) in proportion to • Therefore the last iteration will therefore be just like a “!one- how well they do. shot” game. • Therefore the game before the last is in some sense really the • Do cooperative strategies win out? last. • Therefore one should play this like a one-shot game as well • More on this in exercise classes!! • And so on, for the previous game, and ultimately, all 10 games!! • What might be wrong with this argument? 7 Evolutionarily Stable Strategies • Consider the bluegill sunfish example mentioned earlier. • Suppose that the proportions of rogue and regular males are p1 and p2 (where p1 + p2 = 1). • From a game theoretic point of view it makes no difference if we have a population divided in these proportions, or if each player (a male sunfish) plays the mixed strategy rogue with probability p1 and regular with probability p2. • This mixed strategy is called an evolutionarily stable strategy if a male playing a different “mutant” strategy could not take over the population. • Much evolutionary biological modelling is devoted to finding evolutionarily stable strategies, or to showing that the evolutionarily stable strategies of the model correspond to what happens in reality. • Think about: Are strategies like tit-for-tat evolutionarily stable? 8

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Microsoft PowerPoint - Game Theory

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