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# Game Theory ECON 309

UMass

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This 34 page Class Notes was uploaded by Mr. Kay Bergstrom on Friday October 30, 2015. The Class Notes belongs to ECON 309 at University of Massachusetts taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/232324/econ-309-university-of-massachusetts in Economcs at University of Massachusetts.

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Date Created: 10/30/15

Mixed Strategy Nash equilibrium Example Matching Pennies Matching pennies is an example of a zerosum game Suppose two players are John and Wooj in Head Tail Head 1 1 1 1 Tail 1 l l l It has no pure strategy Nash equilibrium Nevertheless there is a mixed strategy NE To see this suppose the probability distribution 01p1 p is assigned to John s strategy space and the probability distribution of a q1 q is assigned to Woojin s strategy space John s expected payoff from playing Head is given by U151 2 H702 q 1 Q 1 2Q while his expected payoff from playing Tail is U181 T702 q l Q 261 1 Hence if John believes that qlt05 then playing Head is his best response to Woojin s mixed strategy of playing HeadTailq1 q If John believes that q gt 05 then his best response to Woojin s mixed strategy of playing HeadTailq1 q is Tail If he believes that q 05 then he is indifferent between the two strategies In a similar manner we can compute the expected payoff of Woojin Woojin s expected payoff from playing Head is U20182 Hp 1 p2p 1 while his expected payoff from playing Tail is U20182 T p1 p 1 2p Hence Woojin s best response to John s mixed strategy HeadaTail 191 19 is Head if he believes that p gt 05 Whereas his best response to the John s mixed strategy is Tail if he believes that p lt 05 Again Woojin is indifferent if he believes that p 05 The mixed strategy NE is at the intersection of two best response curves Hence the mixed strategy equilibrium is pqquot 0505 Note that the two players are indifferent between the two pure strategies at the equilibrium 9 Woojin s best response John s best response P Example HawkDove Two animals are fighting over some prey Each can behave like a dove or like a hawk The best outcome for each animal is that in which it acts like a hawk while the other acts like a dove the worst outcome is that in which both animals act like hawks Each animal prefers to be hawkish if its opponent is dovish and dovish if its opponent is hawkish A game that captures this situation is shown below The game has two NE Hawk Dove and D0veHawk Dove Hawk Dove 3 3 14 Hawk 41 00 One can easily show that there is a complete mixed strategy Suppose we assign probability distributions 01 191 p to player 1 and 02 q1 q to player 2 Player 1 s expected payoff is 3q 1 q M 1 from playing Dove and 4g from playing Hawk Hence his best response is Dove if q lt 05 Player 2 s expected payoff is 3 19 1 19 217 1 from playing Dove and 417 from playing Hawk Her best response is Dove if p lt 05 Hence a complete mixed strategy NE is pq 0505 But there are two more mixed strategy Nash equilibria because pure strategy NE are also considered mixed strategy NE For instance a pure strategy NE of Hawk Dove is a mixed strategy of Pi 01 and a pure strategy NE of DoveHawk is a mixed strategy of 1016f 10 Her best response His best response P Refinements of Nash equilibrium When there are more than one Nash equilibria it is in general difficult to predict what equilibrium will be played A refinement may be needed but there is no generally agreed refinement concept We briefly discuss a few of them here In this section we will discuss Nash equilibrium and its refinements in terms of mixed strategies 1 Schelling s focal point When a game has several NE the assumption that a NE is played relies upon there being some mechanism process or culture that leads all the players to eXpect the same equilibrium Schelling 1960 argues that in some situations players may be able to coordinate on a particular equilibrium by using some information that is abstracted away in the strategic form eg social conventions Consequently one of the many Nash equilibria might stand out from the others Sometimes names of the strategies may have some focal power Of course the focalness of various strategies depends on the culture of players society and their past experience 1 Name Heads or Tails If you and your partner name the same both of you win a prize otherwise you win nothing 86 of his sample chose Heads thus Head is the focal point 2 You are to meet your friend in New York City You have not been instructed where and when to meet and you cannot communicate with your friend You and your friend have to guess where and when to meet More than half of Schelling s sample proposed meeting at the information booth in Grand Central Station at 12 noon 3 There are five candidates in an election and on the first ballot candidates polled as follows Smith 19 Jones 26 Brown 15 Robinson 29 White 9 The second ballot is about to be taken You will be rewarded if someone gets a majority on the second ballot and you voted for him The same is true for every other voter For whom do people vote on the second ballot Over 80 of Schelling s sample voted for Robinson 12 Strict Nash equilibrium A B X 00 00 Y 00 11 The game has two Nash equilibria X A and Y B Intuitively however the first equilibrium is not very reasonable because it is weakly dominated by the second If player 2 keeps choosing A player 1 neither gains nor loses from deviating from X to Y If player 2 chose B by mistake on the other hand such a deviation surely would bring in payoff gains to player 1 Thus player 1 has an incentive to deviate to Y The same logic applies to player 2 In other words the first equilibrium is very fragile or unstable with respect to small perturbations of the equilibrium and there seem to be good reasons to reject it as a robust equilibrium configuration 14 MaynardSmith s Evolutionarin Stable Nash Eguilibrium According to the Darwinian theory of evolution a mode of behavior survives only if no other mode of behavior is more successful at producing offspring In an environment in which organisms interact the reproductive success of a mode of behavior may depend on the behavior of other organisms We focus here on a context where the individuals of a single large population are assumed to undergo a series of parallel and identical pairwise contests ie games In this model each player s set of actions consists of the modes of behavior an organism could acquire and its payoffs measure its biological fitness or reproductive success Each player is programmed to follow a certain mode of behavior which comes from two sources with high probability it is inherited from the player s parent parents and with low probability it is assigned to the player as the result of mutation Suppose there is a single large population of organisms the members of which interact with each other pairwise ie pairwise random matching Each organism can behave either like a dove or a hawk Thus the pure strategy space of the organism is BD0veHawlt The population is divided into normals with fraction 1 6 and mutants with fraction 6 Normals can be matched with normals or mutants and mutants can also be matched with normals or mutants Suppose ub1b2 measures each organism s ability to survive if it uses b1 as its strategy and matched with an organism which uses b2 as strategy Suppose normals take the action If while mutants take the strategy b Then the expected payoff of a normal is 1 ub b ubquotb and the expected payoff of a mutant is 1 may bquot may b The notion of evolutionarily stable equilibrium is designed to capture a steady state in which all organisms take an action so that no mutant can invade the population Therefore De nition 1 19 is an evolutionarily stable strategy if 1 5ubb aubquotb gt 1 5ubb 5ub b for all values of sufficiently small 6 and b c b Theorem 17 is an evolutionarily stable strategy if a ub7b 2 ubb for all bthat is biz is a symmetric Nash equilibrium and b Vb If ub7b ub7b ub7b gt ubbthat is for every best response I to bquot with b 7 If ubb gt ubb Example Suppose 0 s c lt 2 Hawk Dove Hawk 2 2 0 0 Dove 0 0 c c Note that there are two pure strategy symmetric Nash equilibria Hawk Hawk and Dove Dove Are both of them evolutionarily stable Suppose every organism normally takes action Hawk Then a normal s expected payoff is 21808 218 and a mutant s expected payoff is 01 8c8 c8 If lt 2 then a normal s expected payoff is higher than a mutant s expected payoff 8 Thus Hawk is evolutionarily stable Note also that Hawk Hawk is a strict Nash equilibrium Now suppose every organism normally takes action Dove Then a normal s expected payoff is cl 808cl 8 and a mutant s expected payoff is 0 20 7 01 828 28 If slt then a normal s expected payoff is higher than a 0 20 mutant s expected payoff But slt is possible only when cgt0 if c0 it is impossible Thus Dove is evolutionarily stable if cgt0 if c0 it is not evolutionarily stable Dynamic Games under Complete Information Games under complete information can be classi ed into two cases Perfect inf0 each player knows where she is on the game tree when it is her turn to move Imperfect inf0 some players have to move without knowing the previous moves of other players We can de ne perfect and imperfect information more precisely using the concept of an information set Definition An information set of player i is the set of nodes among which player i cannot distinguish If each information set is a singleton then the game has perfect information If some information sets are not singletons the game is one of imperfect information 0 1 0 1 2 1 2 1 1 3 1 3 Imperfect information Perfect information We have to impose some information set rules if we want to avoid a strange structure of games Rule 1 All the nodes in any information set must belong to the same player This violates Rule 1 Rule 2 If a node D1 is a predecessor of node D2 then D1 and D2 cannot be contained in the same information set This violates Rule 2 Rule 3 Exactly the same set of moves can be taken at each of the decision nodes in an information set This violates Rule 3 Actions versus strategies Actions are different from strategies in dynamic games under complete information An action or a move is a choice a player can make at each decision node A strategy on the other hand is a detailed set of plans for playing the game that specifies the player s move for every contingency Thus a strategy maps out a plan of actions under all eventualities Because a Nash equilibrium is de ned in terms of strategies not of actions we must translate an extensive form represented in terms of actions into its normal form representation in terms of strategies to find a Nash equilibrium Example 1 L 10 01 21 13 L R 11 11 21 13 10 L L L R KL RR 1 0 10 01 0 1 2 1 13 21 1 3 Backward Induction and Incredible threats Consider the following game of perfect information 1 Player 1 chooses an action 611 from hisher feasible set 2 After observing a1 player 2 chooses an action a from hisher feasible set 3 Payoffs are given by r1ala2 and 461161 One way to approach the problem of prediction in this game is to simply derive its normal form representation and then apply the Nash equilibrium concept to it In this type of games however the Nash equilibrium may include incredible threats Selten believed that any Nash equilibrium incorporating incredible threats is a poor predictor of human behavior and thus should be eliminated Nash equilibria after eliminating those incorporating incredible threats are called subgame perfect Nash equilibria In games of perfect information subgame perfect Nash equilibria can be found by using the method of backward induction Example Entrant versus Incumbent 3825 Enter 430 4010 Stay Out 800 Enter Enter Enter SO 80 Enter SO 80 Expand 38 25 38 25 43 0 43 0 Don t 40 10 80 0 40 10 80 0 Expand There are two pure strategy Nash equilibna Don t Expand Enter Enter and Expand Stay Out Enter Are both of them reasonable Consider Don t Expand Enter Enter The strategy of Enter Enter can be interpreted as a situation in which the Entrant tries to convince Incumbent to play Don t Expand by threatening that he would play Enter regardless of the choice made by Incumbent Bus is this threat really credible If Incumbent actually adopt Expand by ignoring this threat there is nothing to stop Entrant from backing away from its threat and not entering at all Thus it is an incredible threat for Incumbent if he is wise enough Example Ulysses and the Sirens You must bind me hard and fast so that I cannot stir from the spot where you will stand me and if I beg you to release me you must tighten and add to my bonds The Odyssey The problem of an incredible threat is a problem of dynamic inconsistency To avoid dynamic inconsistency involved in an incredible threat we need to impose the principle of sequential rationality equilibrium strategies should specify optimal behavior from any point in the game onward This principle is intimately related to the procedure of backward induction Now we will describe the method of backward induction in a general way When player 2 gets the move at the second stage of the game heshe will choose a Eargmaxr2ala2 Given the action a the solution to this problem is thus given by a function a aal Since player 1 can solve player 2 s problem as well as player 2 can player 1 should anticipate player 2 s reaction to each a so player l s problem at stage 1 amounts to a E argmaxrxapaxal The pair 64305461 is called a backward induction outcome of this game whereas the pair 4304611 is called a backward induction equilibrium of this game Note that 0524 is a numbers while oczal is a function of a1 Example Entrant versus Incumbent continued Stay Out if afEXpand In this example afExpand and oca1 Thus the backward induction Enter if afDon39t equilibrium of this game is Expand Stay Out Enter while the backward induction outcome of the game is Expand Stay Out Example Stackelberg duopoly There are two companies in the bottled industry Sparkling Water Co and Clear Water Inc They sell the exact same product in the same market and set how much water to put up for sale to maximize pro ts The price of water is set by the market then Sparkling Water is the industry leader so that it sets the quantity rst afterwards Clear Water chooses its quantity The total cost of each rm is T C 3Q where i S C and inverse demand function is where Q QC QS 10 Q istlO 0 ifQgt10 7 QQ istlO Thus the pro t ofeach rm is 7r 3Qi ifQ gt10 A strategy for Sparkling Water is a real number QS while a strategy for Clear Water is a function RQS which determines QC for each level of Q3 Let us solve Clear Water s problem rst The rst order condition for Clear Water is a QSZQC 0 thus R Q 35 s This is the sequentially rational strategy of Clear Water Now Sparkling Water solves maX7 RQS Qs QS which gives Q 35 Thus the backward induction equilibrium is 35 RQS35 QS while the backward induction outcome is 35 1725 In this game the rst mover has an advantage because 715 6125 gt 71C 3063 But in other games the second mover may have an advantage Note If this game were played simultaneously then the equilibrium would be the Coumot equilibrium which is Q Q g Definition A proper subgame in an extensive form game is a set of nodes that 1 begins at a noninitial decision node n that is a singleton 2 includes all the decision and terminal nodes following that decision node n and 3 does not cut any information sets The entire game is also a subgame Subgames that are not the entire game are called proper subgames 10 01 21 Three subgames and two proper subgames No proper subgame Definition A strategy pro le is a subgame perfect Nash equilibrium if it is a Nash equilibrium of every subgame Remark 1 A backward induction equilibrium is a Nash equilibrium which is consistent with the backward induction procedure Thus it is subgame game perfect Thus subgame perfection is a generalization of backward induction procedure 2 In simultaneous move games under complete information the entire game is the only subgame Thus the Nash equilibrium in that case is trivially subgame perfect 3 A subgame perfect Nash equilibrium is a Nash equilibrium but not vice versa A pure strategy SPNE may not exist even in nite games D L 3 2 0 3 Thus there are two pure strategy Nash equilibria U R d and ULd None of the pure strategy Nash equilibria are subgame perfect indeed player II will never play d when he is given a chance to move because u strictly dominates d for him whatever player I does afterwards The notion of subgame perfection is powerful enough to capture the principle of sequential rationality in most cases But sometimes especially in games under imperfect information it may still fail to eliminate the equilibria that incorporate incredible threats 1 1 There are two pure strategy NE InI Accommodate Out Fight Both of them are also SPNE since the game has no proper subgames Nonetheless Out Fight depends on an incumbent s incredible threat since Accommodate strictly dominates Fight for incumbent the entrant would not be induced to play Out by incumbent s threat to play Fight if given his move Thus the criterion of subgame perfection is of no use here to eliminate the equilibrium incorporating an incredible threat

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