Engineering Decisions

Transcription

1 GSOE9210 Solving games 1 Solutions of zero-sum games est response eliefs; rationalisation Non stritly ompetitive games Cooperation in games Games against Nature

2 Outline 1 Solutions of zero-sum games est response eliefs; rationalisation Non stritly ompetitive games Cooperation in games Games against Nature Solutions of zero-sum games To player zero-sum games: dominane Consider the folloing zero-sum game (matrix entries are the payoffs for the ro player): b 4 a a a a Using dominane, the solution of this game is the play (a 2, b 2 ).

3 Solutions of zero-sum games Rational behaviour and strategi unertainty In games the unertainty for eah player inludes the behaviour of other players; i.e., hih strategy they ll hoose This unertainty an be redued if players have ommon knoledge about the preferenes and rationality of other players Dominane redues strategi unertainty about rational behaviour of other players (e.g., rational players ill never play dominated strategies) General priniple about rational behaviour: best response... est response est response Consider again the previous zero-sum game: b 4 a a a a Play (a 2, b 2 ) is maximal in its olumn and minimal in its ro i.e., if olumn player plays b 2, then a 2 gives best possible outome for ro player Conversely, if ro player plays a 2, then b 2 gives best possible outome for olumn player

4 est response: zero-sum games est response Definition (est response) player s strategy s is a best response to another player s strategy s if it gives a preferene maximal outome against s. In a zero-sum game: b 1 b 2 a a for any strategy of the olumn player, a best response of the ro player is a strategy hih maximises the olumn value ( ) for any strategy of the ro player, a best response of the olumn player is a strategy hih minimises the ro value ( ) est response est response a a a min max Column player s best responses are minimal in their ro Ro player s are maximal in their olumn gainst any strategy there is at least one best response; possibly more than one (e.g., ro 2) If there are multiple best responses, then they have the same payoff

5 est response: Maximin est response a a a min max Ro player s Maximin strategy is best strategy against perfet play by opponent bove, ro player s Maximin strategy is a 2 ; Column player s Maximin strategy (i.e., minimax strategy) is b 2 Maximin is rational play if, e.g., opponent an see your move Repeated play a a a Suppose initially ro player plays a 3, hoping for best outome; similarly olumn player plays b 1 ; play (a 3, b 1 ) Ro player happy (best response) Column player unhappy, so sithes to best response b 2 ; in response ro player plays a 2 ;... Play stabilises at (a 2, b 2 )

6 Equilibrium The stable play (a 2, b 2 ) has property that eah of its strategies is a best response to the other. a a a John F. Nash ( ) Definition (Nash equilibrium) play is in equilibrium if eah of its strategies is a best response to the others. Equilibrium: belief interpretation a a a If ro player believes olumn player ill play b 2, then ro player annot improve outome, and vie versa More generally, if eah player believes the other ill play aording to their equilibrium strategy, then neither an improve their outome by deviating from their equilibrium strategy

7 Equilibrium: existene and uniqueness Not all games have an equilibrium... in pure strategies b 1 b 2 a a Some games may have multiple equilibria: b 4 a a a a Zero-sum games: saddle points Definition (Saddle point) n entry in a zero-sum game is alled a saddle point iff it is minimal in its ro and maximal in its olumn. a a a Theorem (Minimax) In a zero sum game, saddle points represent equilibria.

8 Zero sum games: solutions Theorem If a zero sum game has an equilibrium, then it orresponds to the players playing Maximin strategies. a a a min max eause the matrix entries are the payoffs for the ro player, the olumn player s Maximin strategy translates to a minimax strategy. Zero-sum games: equilibrium a a a min max Theorem (Unique value) ll equilibria in a zero sum game yield the same payoffs. This payoff is said to be the value of the game. The value of the game above is 5 Equilibria in zero-sum games are Maximin strategies (minimax for olumn player)

9 Zero-sum games: finding saddle points a a a min max Saddle points are Maximin strategies To find them: Use Maximin to evaluate eah of the players strategies (i.e., minimax for olumn player) If the Maximin values agree for any play (e.g., 5 above), then that is a saddle point of the game ehaviour and beliefs eliefs; rationalisation game matrix represents possible outomes, but says nothing about the players behaviour; i.e., hih strategies the players should play Dominane and best response are priniples about rational behaviour n agent s behaviour should depend on its beliefs about the other players behaviour (inluding likelihoods) In order to better explain behaviour e must formulate an agent s beliefs

10 Rational behaviour: rationalisation eliefs; rationalisation Rational behaviour priniple: best response rational player should not play an strategy hih is not a best responses to any of its opponent s strategies. Definition (Rationalisable strategies) strategy is rationalisable for a player if it is a best response to some rational strategy of the other players. Only rationalisable strategies should be onsidered by players; i.e., non-rationalisable strategies an be eliminated dominated strategy is never rationalisable Theorem rationalisable play ill survive elimination by iterated dominane. eliefs and behaviour eliefs; rationalisation eliefs about the other players play an be represented by a mixture of the other players pure strategies Player assigns to player s strategy b j a proportion p j if s belief in the degree of likelihood that ill play b j is p j Reall that utilities enode preferenes in the presene of unertainty (risk)

11 eliefs; rationalisation est response to beliefs: zero-sum games Suppose player believes that player is tie as likely to play b 2 as b 1 ; i.e., ill play b 1 ith probability 1 3 and b 2 ith probability 2 3 Let β ( 1 3, 2 3 ) represent s belief about s behaviour b 1 b 2 a a For belief β alulate the ayes values of s strategies: V β (a 1) = 1 3 (2) (0) = 2 3 V β (a 2) = 1 3 (1) (3) = 7 3 Therefore, s best response given belief β about is a 2. eliefs; rationalisation Rationalisation of behaviour and belief ny strategy that is not a best response for any belief β about the other players ill not be played; i.e., it should reeive degree of belief (i.e., probability) 0 In general, a strategy is rationalisable iff it is ayes for some belief β (not just for some pure strategy) Compare rationalisability and admissibility In a zero-sum game, a player s rationalisable strategies must be on the player s admissibility frontier

12 Non zero-sum games: best response Non stritly ompetitive games C W 5, 3 4, 4 9, 1 0, 0 C 5, 533 4, 44 4 W 9, 911 0, 0 If lie ere to ait, then ob s best ounter-move ould be to limb Conversely, if ob ere to limb, then lie s best ounter-move ould be to ait belo Solving games Non stritly ompetitive games What if lie moves first? C W 5, 3 4, 4 9, 1 0, 0 Exerises What is ob s best response to lie aiting? To lie Climbing? re there any equilibrium pairs/points? If so, hih are they?

13 Equilibrium and solutions Non stritly ompetitive games 5, 3 C 5, 3 C W 4, 4 9, 1 0, 0 W C W 4, 4 9, 1 0, 0 Exerise For the problems above, find all the equilibrium plays. In games that aren t stritly ompetitive, determining hih equilibrium points are solutions is less lear, beause opportunities for o-operation should be onsidered Other onsiderations inlude: group benefit (Pareto optimality), initial tendenies (equilibrium), et. Non stritly ompetitive games Cooperation in games Example (The Prisoner s Dilemma) lie and ob are suspets in a joint rime. The polie doesn t yet have enough evidene to onvit both/either, so it is trying to get either to impliate the other. The polie inspetor offers eah separately a redued sentene if they defet (D) by impliating their aomplie. If both suspets defet they ill get a moderate sentene eah (2 years). suspet ho defets ill get immunity, and the other ill get the full sentene (3 years). If neither defets i.e., they both ooperate (C) ith eah other both ill be harged for only a minor offene (1 year). d D 1, 111 3, 0 C 0, 033 2, 2 The payoff is the redution in the player s sentene: 3 s, here s {0, 1, 2, 3} is the length of the sentene.

14 Cooperation in games Cooperation in games d D 1, 111 3, 0 C 0, 033 2, 2 Individual rationalisation (dominane) suggests that they should both defet (Dd); hoever mutual ooperation (C) is better for both In games that aren t stritly ompetitive ooperation may be possible What s best for individuals (individual rationalisation) may not be best for the group, and vie versa Here play C gives eah player a better payoff than the individually rationalisable play Dd The Prisoner s Dilemma Cooperation in games Definition (Pareto optimality) n outome is Pareto optimal iff there is no other outome hih is at least as good or better for all the agents. Pareto priniple Pareto optimal outomes are optimal for the group. Consider the to-player play diagram on the right, here: v 1 is the payoff to Prisoner 1 v 2 is the payoff to Prisoner 2 Pareto optimal outomes represented by points on solid line Cd 3 C 2 Dd 1 D

15 The Prisoner s Dilemma Cooperation in games d D 1, , 0 C 0, 033 2, 2 The equilibrium is Dd (irled) The Pareto optimal outomes are: C, Cd, D Cd 3 C 2 Dd 1 D Play C, hih is Pareto optimal, is better than Dd for both players Conlusion In to-player non stritly ompetitive games, hat s best for the individual may not be best for the group; i.e., ooperation preferable. Nature as a player Games against Nature C Tr E E D Single agent deisions an be regarded as games against a neutral player alled Nature, or Chane, ho has no preferenes Game in hih some of the players preferenes are unknon are said to have inomplete informtation as opposed to imperfet information, in hih information sets may have multiple nodes In extensive form, Nature s moves take plae at hane nodes, and its moves orrespond to hane events u u v b P b L D W Co C D

16 Summary Games against Nature est response strategies Equilibrium in games Rationalisation Group preferene and Pareto optimality; ooperation Single agent deisions are games against nature