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Game Theory Eduardo Costa

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Contents What is game theory? Representation of games Types of games Applications of game theory Interesting Examples

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What is game theory? Game theory is the study of strategies for decision making. Formally: “the study of mathematical models of conflict and cooperation between intelligent rational decision- makers” (Roger B. Myerson, 1991)

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Representation of games Extensive Form. Game is represented by a series of moves on a tree. Each node is a choice by a player. Lines out of a node is a possible action. Payoffs are specified at the bottom.

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Representation of games (2) Normal-form game Game represented by a matrix, with every possible combination of actions. For this representation, players act simultaneously or without knowing the other player actions.

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Representation of games (3) Characteristic function form. Used for cooperative games. A characteristic function is seen as (N,v) –With N being the number of agents; –With v: 2 N R It defines the payoff for a certain coalition of players.

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Type of games There are many type of games, depending on what real-life situation they are trying to mimmic. Games can be cooperative or non- cooperative. In asymmetric games, the payoff is not the same for all players.

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Type of games(2) In zero-sum games, the payoff in every combination of strategies always adds to 0. Poker, Go or Chess are real examples of zero-sum games. In non-zero-sum games, a gain by one player does not imply a loss by another.

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Types of games(3) In a perfect information game, all players know the moves previously made by others. Only applies to sequential games. A game can also have incomplete information. Complete information requires a player to know all strategies and payoffs available to the other players.

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Applications of game theory Game theory was initially developed in the studies of economics, to understand behaviors of firms, markets and consumers. Now used in many other fields as political science, sociological and psychological behaviors.

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Applications of game theory – description and modelling Game theory can be used to describe and model how human populations behave. But humans do not always play in a way to maximize their wins. Example: guess 2/3 of average game –Each person tries to guess what 2/3 of the average of their guesses will be (between 0 and 100). –Everyone guess 0 is the perfect strategy.

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Applications of game theory – Prescriptive analysis Game theory may not be used as a prediction of human behavior, but it can be used a suggestion of what actions should people take. Playing a strategy that is part of Nash Equilibrium – best response to actions of other players – should be the best strategy.

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Applications of game theory – Prescriptive analysis (2) But other players may not take the best possible actions, and in these cases the player should also play non-equilibrium strategy. Example: Prisioner’s dilemma

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Applications of game theory – economics and business Game theory is an important method in mathematical economics and business. Useful to model behavior of competing or cooperative agents. A wide list of economic situation can be model by game theory: –Auctions, bargaining, oligopolies, voting systems, industrial organization and political economy.

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Applications of game theory – economics and business (2) Example: oligopoly – market dominated by small number of sellers. Bertrand competition: –Firm A and B sell the same product; –Customers always buy the cheaper product; Nash equilibrium is when price of product for both firms equals the cost of producing it (no profit).

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Applications of game theory – economics and business (3) Second example: auction. For 2 bidders: –v – value of the item –other bidder uses strategy β(v2) = av2 –if you bid b: (v − b)prob(you win) = (v − b)prob(b > av2) = (v − b)prob(v2 < b/a) = (v − b) b/a Maximizing: b = v / 2

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Applications of game theory – biology In biology application, the focus is less on equilibrium strategies but on ones maintained by evolutionary forces. Usually Evolutionary Game Theory (EGT) is used instead of classic game theory.

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Applications of game theory – biology (2) Example: game of chicken or hawk-dove game. Used to analyze fighting behavior and territoriality. First presented by John Maynard Smith and George Price in a 1973 Nature paper.

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Applications of game theory – philosophy Game theory has several uses in philosophy, specially in questions of morality and self-interest. In the prisoner’s dilemma game, the rational decision would be to betray the other. In real life, there is a bias towards stay silent.

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Game Theory – Examples (Volunteer’s dilemma) N players have to decide to make a small sacrifice for all to benefit, or wait for another one to do it. Scenario example: electricity goes out in a neighborhood. At least one person has to contact the electricity company for the problem to get fixed.

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Game Theory – Examples (Pirate game) 5 pirates – A, B, C, D, E. 100 gold coins The first pirate – by alphabetical order – proposes a distribution. All pirates vote. If distribution is not accepted, the proposer is thrown into the sea and another round happens – with 1 less pirate.

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Game Theory – Examples (Pirate game) (2) What is your choice as pirate A? Give many coins to the others so you at least survive? Actually, pirate A may take 98 coins in his first proposal and it will be the best for pirate C and E.

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Game Theory – Examples (Pirate game) (3) Imagine when there is only D and E left. D will give himself 100 coins (tie in voting is acceptance). If C, D and E are left: –E knows that in next round he will get nothing, so he will accept only 1 coin. –C may offer 99 coins for himself.

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Game Theory – Examples (Pirate game) (4) If B, C, D and E are left: –D knows he will get nothing in the next round, so he may accept only 1 coin. –B may offer 99 coins for himself. So in the first round, if the pirates think of all this: –C and E know that they will get nothing in the next round, so they may accept only 1 coin. –A may offer himself 98 coins.

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