Presentation on theme: "Multi-player, non-zero-sum games"— Presentation transcript:
1 Multi-player, non-zero-sum games 4,3,24,3,21,5,24,3,27,4,11,5,27,7,1Utilities are tuplesEach player maximizes their own utility at each nodeUtilities get propagated (backed up) from children to parents
2 Game theoryGame theory deals with systems of interacting agents where the outcome for an agent depends on the actions of all the other agentsApplied in sociology, politics, economics, biology, and, of course, AIAgent design: determining the best strategy for a rational agent in a given gameMechanism design: how to set the rules of the game to ensure a desirable outcome
3 Simultaneous single-move games Players must choose their actions at the same time, without knowing what the others will doForm of partial observabilityNormal form representation:Player 10,01,-1-1,1Player 2Payoff matrix(row player’s utility is listed first)Note: this is a zero-sum game
4 Prisoner’s dilemmaTwo criminals have been arrested and the police visit them separatelyIf one player testifies against the other and the other refuses, the one who testified goes free and the one who refused gets a 10-year sentenceIf both players testify against each other, they each get a 5-year sentenceIf both refuse to testify, they each get a 1-year sentenceAlice:TestifyAlice: RefuseBob: Testify-5,-5-10,0Bob: Refuse0,-10-1,-1
5 Prisoner’s dilemma Alice’s reasoning: Suppose Bob testifies. Then I get 5 years if I testify and 10 years if I refuse. So I should testify.Suppose Bob refuses. Then I go free if I testify, and get 1 year if I refuse. So I should testify.Dominant strategy: A strategy whose outcome is better for the player regardless of the strategy chosen by the other playerAlice:TestifyAlice: RefuseBob: Testify-5,-5-10,0Bob: Refuse0,-10-1,-1
6 Prisoner’s dilemmaNash equilibrium: A pair of strategies such that no player can get a bigger payoff by switching strategies, provided the other player sticks with the same strategy(Testify, testify) is a dominant strategy equilibriumPareto optimal outcome: It is impossible to make one of the players better off without making another one worse offIn a non-zero-sum game, a Nash equilibrium is not necessarily Pareto optimal!Alice:TestifyAlice: RefuseBob: Testify-5,-5-10,0Bob: Refuse0,-10-1,-1
7 Prisoner’s dilemma in real life CooperateDefectWin – winWin big – lose bigLose big – win bigLose – losePrice warArms raceSteroid use
8 Is there any reasonable way to get a better answer? Superrationality (Douglas Hofstadter)Assume that the answer to a symmetric problem will be the same for both playersMaximize the payoff to each player while considering only identical strategiesNot a conventional model in game theory
9 Stag hunt Is there a dominant strategy for either player? Hunter 1: StagHunter 1: HareHunter 2: Stag2,21,0Hunter 2: Hare0,11,1Is there a dominant strategy for either player?Is there a Nash equilibrium?(Stag, stag) and (hare, hare)Model for cooperative activity
10 Prisoner’s dilemma vs. stag hunt Prisoner’ dilemmaCooperateDefectWin – winWin big – lose bigLose big – win bigLose – loseCooperateDefectWin big – win bigWin – loseLose – winWin – winPlayers can gain by defecting unilaterallyPlayers lose by defecting unilaterally
11 Coordination game (Battle of the sexes) Wife:BalletFootballHusband:3, 20, 0Husband:Football2, 3Wife:BalletFootballHusband:3, 20, 0Husband:Football1, 12, 3orIs there a dominant strategy?Is there a Nash equilibrium?(Ballet, ballet) or (football, football)How do we figure out which equilibrium to choose?
12 Game of Chicken Is there a dominant strategy for either player? -10, -10-1, 11, -10, 0StraightChickenIs there a dominant strategy for either player?Is there a Nash equilibrium?(Straight, chicken) or (chicken, straight)Anti-coordination game: it is mutually beneficial for the two players to choose different strategiesModel of escalated conflict in humans and animals (hawk-dove game)How are the players to decide what to do?Pre-commitment or threatsDifferent roles: the “hawk” is the territory owner and the “dove” is the intruder, or vice versa
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