믬뫏닟싔춬쪱늩?Iꎺ쇣뫍늩?Simultaneous-Move Games with Mixed Strategies I: Zero-Sum Games뗚07헂Chapter 07믬뫏닟싔춬쪱늩?Simultaneous-Move Games with Mixed Strategies폐튻샠춬쪱늩?쫇쎻폐뒿닟싔쓉쪲뻹뫢뗄ꆣThere is a class of simultaneous-move games that has no Nash equilibria in purestrategies.캪쇋풤닢헢킩늩?뗄뷡맻ꎬ컒쏇탨튪삩햹맘폚닟싔뫍뻹뫢뗄룅쓮ꆣTo predict outcomes for such games, we need an extension of our concepts of strategies and equilibria.냬램쫇붫탐뚯쯦믺뮯ꆣThis is to be found in the randomization Slide 2of
믬뫏닟싔춬쪱늩?Simultaneous-Move Games with Mixed Strategies쯦믺뮯탐뚯뗄튪쟳튻냣돶쿖퓚튻룶늩?헟욫낮쒳훖탐뚯ퟩ뫏ꎬ뛸웤뛔쫖좴쫔춼뇜쏢쯻ꆣThe need for randomized moves in the play of a game usually arises when one player prefers a coincidence of actions, while her rival prefers to avoid it.쇣뫍늩?뎣뎣닺짺샠쯆뗄횱뷓돥춻ꆣSuch direct conflict between players arises in zero-sum games.샽죧ꎺ뻼쫂ꎬ쳥폽, military conflicts, sporting contestsSlide 3믬뫏닟싔춬쪱늩?ꎺ튻룶샽ퟓSimultaneous-Move Games with Mixed Strategies: An ExampleNAVRATILOVADLCCEVERTDL5080CC9020Slide 42
쪲쎴쫇믬뫏닟싔ꎿWhat Is A Mixed Strategy?튻룶믬뫏닟싔쫇뒿닟싔뗄튻훖쯦믺뗄믬뫏ꆣA mixed strategyis a random mixture of pure strategies.퓚췸쟲늩?훐ꎬ틔룅싊p톡퓱DLꎬ틔룅싊1-p톡퓱CCꎬp캪0떽1횮볤뗄쪵쫽ꎬ뻍쫇닎폫헟뗄튻룶믬뫏닟싔ꆣIn the tennis-point game above, choosing DL with probability pand CC with probability (1-p) , where p is some real numberranging from 0 to 1, or (p*DL, (1-p)*CC) for short, is a particular mixed strategy available for the player (Evertor Navratilova).헢퇹뗄믬뫏닟싔폐헻룶뗄솬탸랶캧ꎬ뒿닟싔쫇웤벫뛋쟩탎ꆣThere is a whole continuous range of such mixed strategies, including pure strategies as extreme special 5쪲쎴쫇믬뫏닟싔ꎿWhat Is A Mixed Strategy?살ퟔ믬뫏닟싔뗄쫕틦ꎬ뻍쫇릹돉룃믬뫏닟싔뗄뒿닟싔ퟩ뫏쯹뛔펦쫕틦뗄룅싊볓좨욽뻹ꎭ웚췻쫕틦ꆣThe payoffs from a mixed strategy are defined as the corresponding probability-weighted averages of the payoffs from its constituent pure strategies –expected payoffs.컊ꎺ쏦뛔쓉췞뗄DLꎬ낣?쳘뗄믬뫏닟싔(, )듸룸쯽쫕틦쫇뛠짙ꎿQ: Against Navratilova’s DL, what is Evert’s payoff of her mixture?Slide 63
쪲쎴쫇믬뫏닟싔ꎿWhat Is A Mixed Strategy?쓉쪲뻹뫢뻍쫇튻ퟩ믬뫏닟싔ꎬ웤훐쎿튻룶닟싔뛔펦튻룶닎폫헟ꎬ싺ퟣ틔쿂쳵볾ꎺ쎿룶죋뗄톡퓱뚼쫇쯽뗄ퟮ폅톡퓱ꎬ튲뻍쫇쮵ꎬ룸뚨웤쯻죋뗄믬뫏닟싔ꎬ룃톡퓱듸룸쯽ퟮ룟뗄쫕틦ꆣNash equilibriumis defined as a list of mixed strategies, one for each player, such that the choice of each is her best choice, in the sense of yielding the highest expected payoff for her, given the mixed strategies of the 7쪲쎴쫇믬뫏닟싔ꎿWhat Is A Mixed Strategy?ퟷ캪탅쓮뫍랴펳뗄쾵춳ꎬ쓉쪲뻹뫢튲뿉틔뚨틥캪ꎺAs a system of beliefs and response, Nash equilibrium can also be define as:쎿룶닎폫헟튪탎돉뛔웤쯻닎폫헟쯹돶믬뫏닟싔룅싊뗄튻룶탅쓮ꎬ늢헫뛔쯼ퟶ돶ퟮ폅랴펦ꆣEach player forms beliefs about the probability of the mixture that the other is choosing and chooses her own best response to this.탅쓮헽좷ꆣ뻍쫇쮵ꎬ닎폫헟횪뗀ꆢ쯣돶믲닂돶웤쯻닎폫헟듓쯻믹폚뗄뒿닟싔훐뷸탐톡퓱뗄헽좷룅싊ꆣThe beliefs are correct. That is, the player knows, or calculated or guesses the correct probabilities with which the other player chooses from among her underlying basic or pure 84
쪲쎴쫇믬뫏닟싔ꎿWhat Is A Mixed Strategy?퓊탭늩?훐돶쿖믬뫏닟싔ꎬퟔ뚯뫍벸뫵췪좫뗘뷢뻶쇋뿉쓜돶쿖뗄컞쓉쪲뻹뫢뗄컊쳢ꆣ헢튻컊쳢퓚뒿닟싔훐뎣믡엶떽ꆣAllowing for mixed strategies in a game solves the problem of possible nonexistence of Nash equilibrium, which we encountered for pure strategies, automatically and almost entirely.훸쏻뗄쓉쪲뚨샭뇭쏷ꎺ퓚튻룶쿠떱튻냣뗄쳵볾쿂ꎬ믬뫏닟싔뗄쓉쪲뻹뫢뚨믡듦퓚ꆣNash’s celebrated theorem shows that, under very general circumstances, a Nash equilibrium in mixed strategies 9늻좷뚨탔탐캪ꎺ캪쏔믳뛔쫖뛸믬뫏닟싔Uncertain Actions: Mixing Moves to Keep the Opponent Guessing떱쯹폐닎폫헟뚼횻폐솽룶믹뒡뗄뒿닟싔쪱ꎬ믬뫏닟싔웤쪵뻍쫇튻훖쳘쫢뗄솬탸뇤솿닟싔ꆣWhen all the players have only two underlying pure strategies, mixed strategies are a special kind of continuously variable strategy.틲뛸뿉틔쪹폃ퟮ폅랴펦럖컶ꆣThus we can use the same solution method –best response 105
ퟮ폅랴펦럖컶Best-Response AnalysisNAVRATILOVADLCCq-mix50, 5080, 2050q+80(1-q), DL50q+20(1-q)90, 1020, 8090q+20(1-q), CCEVERT10q+80(1-q)50p+90(1-p), 80p+20(1-p), p-mix50p+10(1-p)20p+80(1-p)Slide 11ퟮ폅랴펦럖컶Best-Response Analysis100100Navratilova’s best-Navratilova’s80CCExpectedresponse ’s p-mixSlide 126
ퟮ폅랴펦럖컶Best-Response AnalysisNavratilova’s qbest-response ’s best-response 13뛔쫖컞닮틬탔훊Opponent’s Indifference Property낣?쳘뗄뻹뫢p횵잡뫃쪹뗃쓉췞퓚웤쯹폐뗄뒿닟싔믲믬뫏닟싔횮볤컞닮틬ꆣEvert’s equilibrium pis where Navratilova is indifferentamong all her strategies, pure or mixed.쓉췞뗄뻹뫢q횵튲쪹뗃낣?쳘퓚웤쯹폐닟싔횮볤컞닮틬ꆣNavratilova’s equilibrium qis where Evert is indifferent among all her strategies.뛔쫖컞닮틬탔훊ꎺ쎿룶닎폫헟뗄뻹뫢믬뫏닟싔뚼쪹뗃웤쯻닎폫헟퓚쯽뗄쯹폐닟싔횮볤컞닮틬ꆣOpponent’s indifference property: eachplayer’s equilibrium mixture is such that the otherplayer is indifferent among all her 147
뛔쫖컞닮틬탔훊Opponent’s Indifference Property퓚쇣뫍늩?훐ꎬ떱뛔쫖컞닮틬쪱ꎬ쓣튲죧듋ꆣIn a zero-sum game, when the opponent is indifferent, it follows yourself is also.쎿룶닎폫헟뗄뻹뫢믬뫏닟싔룅싊쪹뗃쯽ퟔ벺뛔폚뛔쫖톡퓱뫎뗈뒿닟싔쫇컞닮틬뗄ꆣThus each player’s equilibrium mixture probability is such that she is indifferent about which of her opponent’s pure strategies the opponent chooses.늩?닎폫헟폐틢뗘톡퓱뇤뮻웤탐뚯틔럀횹뛔쫖샻폃웤뿉풤닢탔ꆣGame players deliberately mix their moves to prevent the opponent from taking advantage Slide 15of their predictability.ퟮ킡뮯ퟮ듳랽램The Minimax–Methodퟮ킡뮯ퟮ듳랽램퓚쓉쪲뻹뫢듦퓚쪱ꎬퟜ믡떼돶쯼ꆣThe minimaxmethod will lead to a Nash equilibrium if one exists.쓉쪲뚨샭쮵뗄뻍쫇믬뫏닟싔뗄쓉쪲뻹뫢뚨믡듦퓚ꆣNash’s theorem shows that a Nash equilibrium in mixed strategies exists.틲듋ퟮ킡뮯ퟮ듳랽램ퟜ뿉틔냯훺컒쏇톰헒떽믬뫏닟싔쿂뗄쓉쪲뻹뫢ꆣSo the minimax method can always help us to find the mixed-strategy Nash Slide
ퟮ킡뮯ퟮ듳랽램The Minimax–MethodNAVRATILOVADLCCDL5080Min=50EVERTCC9020Min=20Max=90Max=80Maximin≠Minimax, no purestrategy NESlide 17ퟮ킡뮯ퟮ듳랽램The Minimax–Method100100The ‘Max’part: Navratilova’s80Navratilova’s best-CCExpectedresponsepayoffs5038Evert’s minimax strategy of EvertEvert’s p-mixSo:Evert’s NEstrategy is (,), Slide 18her payoff is 100-38=
ퟮ킡뮯ퟮ듳랽램The Minimax –Method 낣?쳘뗄ퟮ폅믬뫏닟싔쪹쯽붫뒿닟싔쿂뗄ퟮ듳뮯ퟮ킡횵50ꎨ헢쫇쯽횴ퟅ폚죎뫎뒿닟싔쪱뿉틔뗃떽뗄ퟮ룟횧뢶ꎩ쳡룟떽쿖퓚뗄62ꆣEvert’s best mixing enables her to raise her maximin from 50 (in pure strategy) to 62, which is the highest payoff she can imagine when she sticks to any pure strategy.죧맻쇣뫍늩?쎻폐뒿닟싔쓉쪲뻹뫢ꎬ닎폫헟쾵춳뗘횴탐쒳룶뒿닟싔뻍쫇ퟔ헒떹쎹ꎬ룼뫃뗄냬램쫇쪼훕쏔믳힡뛔쫖ꆣIf a zero-sum game has no pure-strategy Nash equilibrium, then each player would be ill-advised to stick to any pure action systematically and would do better to keep the other player 19떱튻룶닎폫헟폐죽룶믲룼뛠뒿닟싔쪱뗄믬뫏닟싔Mixing When One player Has Three or More Pure Strategies퓚쇣뫍늩?훐ꎬ떱튻룶닎폫헟뷶폐솽룶뒿닟싔뛸쇭튻룶폐죽룶믲룼뛠뒿닟싔쪱ꎬ뫳헟춨뎣횻퓚뻹뫢쪱쪹폃웤훐뗄솽룶ꆣIn a zero-sum game where one of the players has only two pure strategies and the other has more, the player who has three (or more) pure strategies typically uses only two of them in 2010
떱튻룶닎폫헟폐죽룶믲룼뛠뒿닟싔쪱뗄믬뫏닟싔Mixing When One player Has Three or More Pure StrategiesNAVRATILOVADLCCq-mixDL508050q+80(1-q)EVERTCC902090q+20(1-q)Lob706070q+60(1-q)p-mix50p+90p+80p+20p+6121270(1-p-p)0(1-p-p)1212Slide 21떱튻룶닎폫헟폐죽룶믲룼뛠뒿닟싔쪱뗄믬뫏닟싔Mixing When One player Has Three or More Pure Strategies쟳뷢듋샠쇣뫍늩?뗄믬뫏닟싔뻹뫢뗄맘볼쫇듓뷶폐솽룶뒿닟싔뗄닎폫헟뗄뷇뛈ꎬ쪹폃ퟮ킡뮯ퟮ듳랽램ꆣThe key to solving for a mixed-strategy equilibrium in a zero(constant)-sum game of this type is to use the minimax method from the perspective of the player who has just two pure strategies.헢튻닎폫헟뗄믬뫏닟싔뿉틔폃뷶튻룶뇤솿ꎨ룅싊ꎩ살뿌뮭ꆣThis player’s mixture can be specified by using just one variable –the single probability used to define her mixed 2211
떱튻룶닎폫헟폐죽룶믲룼뛠뒿닟싔쪱뗄믬뫏닟싔Mixing When One player Has Three or More Pure StrategiesEvert’sEvert’s best-When Evert playsExpectedreponsePayoffsDL, Lob, and CC 9080706050Navratilova’s ’s q-mixSo the NE strategy for Navratilova is (, 0,5CC)Given Navratilova’s NE strategy, Evert will not use CC at 23샽췢쟩탎1Exceptional CasesEvert’sEvert’s best-When Evert playsExpectedreponsePayoffsDL, Lob, and CC 9080707050Navratilova’s 20Minimax01/ q-mixA NE strategies are:Navratilova plays (qDL, (1-q)CC), 1/3≤q≤5/7;Evert plays 2412
샽췢쟩탎2Exceptional CasesEvert’sEvert’s best-When Evert playsExpectedreponsePayoffsDL, Lob, and CC 9080665650Navratilova’s ’s q-mixIn the NE equilibrium, Navratilova must use the mixture with q=, Evert is indifferent between all three Slide 25strategies –equilibrium mixture is not fully determinate.쇓폚쒳튻믬뫏닟싔뗄쟩탎Case of Domination by a Mixed StrategyEvert’sEvert’s best-When Evert ’s q-mixLob is never the best response for must be dominated by a mixture of DL and 2613
쇓폚쒳튻믬뫏닟싔뗄쟩탎Case of Domination by a Mixed Strategy죧맻튻룶닟싔뛔폚죎뫎뗄뒿닟싔믲믬뫏닟싔뚼뻸럇ퟮ폅랴펦ꎬ쓇쎴쯼뇘뚨쇓폚쒳튻룶믬뫏닟싔ꆣIf a strategy is never a best response to any pure or mixed strategy, then there must be a mixed strategy that dominate it.랴횮튲돉솢ꎺ죧맻쒳튻룶닟싔쇓폚웤쯻닟싔ꎬ벴쪹쫇믬뫏닟싔ꎬ쯼뻍늻뿉쓜쫇뛔쫖죎뫎닟싔뗄ퟮ폅랴펦ꆣThe converse also is true: if a strategy is dominated by another strategy, albeit a mixed one, it can never be a best response to any of the other player’s 27떱솽룶닎폫헟뚼폐죽룶닟싔쪱Mixing When Both Player Have Three StrategiesGOALIELeftCenterRightLeft459090pLCenter85085KICKERpCRight959560pRqqqLCRSlide 2814
떱솽룶닎폫헟뚼폐죽룶닟싔쪱Mixing When Both Player Have Three Strategies뛔폚짤쫖뗄쒳믬뫏닟싔ꎬ쫘쏅풱죽룶뒿닟싔뗄쫕틦쫇ꎺAgainst a kicker’s mixture, the opponent’s (goalie’s) payoffs of her three pure strategies are:Left: -45p-85p-95p=-45p-85(1-p-p)-LCRLLR95p=40p-10p-85RLRCenter: -90p-95pLRRight: -5p+25p-85LR샻폃쓉쪲뻹뫢쿂뛔쫖컞닮틬풭샭ꎺUsing the principle of the opponent’s indifference, under NE, the goalie is indifferent among her three pure strategies:Left=Right: 40p-10p-85= -5p+25p-85 LRLRCenter=Right: -90p-95p= -5p+25p-85LRLRSlide 29떱솽룶닎폫헟뚼폐죽룶닟싔쪱Mixing When Both Player Have Three Strategies짤쫖뗄쓉쪲뻹뫢믬뫏닟싔캪ꎺThen the kicker’s Nash equilibrium mixture is: p=, p=, p=쫘쏅풱돶죎뫎튻룶뒿닟싔뗄쫕틦캪ꆣAnd the goalie’s payoff of playing any of the three pure strategy is: .춬샭쟳뗃쫘쏅풱뗄믬뫏닟싔ꆣSimilar to the goalie’s 3015
쒳킩닟싔캴놻쪹폃뗄뻹뫢믬뫏닟싔Equilibrium Mixtures with Some Strategies UnusedGOALIELeftCenterRightLeft459090Center70070KICKERRight959560Slide 31쒳킩닟싔캴놻쪹폃뗄뻹뫢믬뫏닟싔Equilibrium Mixtures with Some Strategies Unused듓쫘쏅풱뗄뷇뛈ꎬ컒쏇쫔춼헒떽웤믬뫏룅싊q,q뫍LRqꆣ쪹폃짤쫖뗄컞닮틬탔훊ꆣCFrom the goalie’s perspective, we try to find her mixture probability by using the condition that the kicker should be indifferent among all three of her pure strategies when played against this mixture.짤쫖퓚죽룶뒿닟싔쿂뗄쫕틦캪ꎺThe kicker’s payoffs from her pure strategies are,Left: -45q+90LCenter: 70q+70q= 70(1-q)LR CRight: -35q+95R뷢뗃ꎺThe solution: Slide 32q=, q=, q=<0!LRC16
쒳킩닟싔캴놻쪹폃뗄뻹뫢믬뫏닟싔Equilibrium Mixtures with Some Strategies Unused짤쫖풸틢놣쇴닮잿죋틢뗄훐슷닟싔ꎬ뷶뷶쫇퓚쫘쏅풱쪹폃웤ퟮ폅펦뛔닟싔ꎭ쫘훐슷ꎭ뗄욵싊ퟣ릻뗍쪱ꆣThe kicker can be kept willing to use the poor Center strategy, only if the goalie is using her best counter –the goalie’s own Center –sufficiently infrequently.듋샽짵훁튪쟳쫘쏅풱쪹폃쫘훐슷뗄룅싊뗍떽뢺쫽ꆣIn this example that logic has to be carried so far that the goalie’s probability of Center has to become negative.헢쪹뗃컒쏇뾼싇붫쫘쏅풱톡퓱쫘훐슷뗄룅싊붵뗍떽쇣ꎬ춬쪱죃짤쫖늻톡퓱짤훐슷ꆣThe best that can be done in reality is to push the goalie’s probability of choosing Center as low as possible –to zero. But that leaves the kicker Slide 33unwilling to use her own Center.쒳킩닟싔캴놻쪹폃뗄뻹뫢믬뫏닟싔Equilibrium Mixtures with Some Strategies Unused헢퇹컒쏇뗃떽뗄쟩탎쫇ꎬ쎿룶닎폫헟뚼퓚웤믬뫏닟싔훐늻쪹폃쒳룶뒿닟싔ꆣThus we get a situation in which each player is not using one of her pure strategies in her mixture.퓚헢튻볲뮯뗄2*2늩?훐ꎬ톰헒믬뫏닟싔쓉쪲뻹뫢뻍뇤뗃죝틗ꆣIn this reduced two-by-two game game, we can easily find its mixed-strategy equilibrium.ퟮ뫳뇘탫볬닩ꎬ퓚룸뚨뛔쫖뗄룃솽닟싔믬뫏뫳ꎬ쎿룶닎폫헟뚼늻믡랢쿖돶웤뗚죽룶닟싔ꎨ훐슷ꎩ쫇폐샻뗄ꆣAfter that, we must check that neither player finds it desirable to bring in her third strategy (Center), given the mixture of two strategies Slide 34chosen by the other
쒳킩닟싔캴놻쪹폃뗄뻹뫢믬뫏닟싔Equilibrium Mixtures with Some Strategies Unused캪쇋퓊탭퓚뻹뫢뗄믬뫏닟싔쿂쒳킩닟싔늻놻쪹폃뗄쟩탎ꎬ컒쏇뇘탫탞룄“뛔쫖컞닮틬”풭샭ꎬ돉캪뮥늹쯉돚탔풭샭ꆣTo allowing for the possibility that some strategies may go unused in an equilibrium mixture, we must modify or extend the “opponent’s indifference”principle to the principle of complementary slackness: 헫뛔뛔쫖뗄뻹뫢믬뫏닟싔ꎬ퓚쓣ퟔ벺뻹뫢뗄믬뫏닟싔훐쪹폃뗄쯤폐닟싔뇘탫듸룸쓣춬퇹뗄웚췻쫕틦ꎬ뛸쟒튪룟폚쓣캴쪹폃닟싔듸룸쓣뗄쫕틦ꆣAgainst the opponent’s equilibrium mix, all of the strategies used in your own equilibrium mix should give you the same expected payoffs, which in turn should be higher than what you would get from any of your unused strategies. Slide 35쇣뫍늩?믬뫏닟싔뗄횤뻝Evidence on Mixing in Zero-Sum Games떱풤볆뗄뻹뫢뷡맻뇭쿖캴솽룶믲뛠룶뒿닟싔뗄믬뫏쪱ꎬ쪵퇩뗄뷡맻좷쪵쿔쪾돶좺쳥훐뗄쒳킩놻쪵퇩헟닉폃쒳튻뒿닟싔ꎬ웤쯻뗄닉폃웤쯻닟싔ꆣ떫헢늢늻쓜쫓ퟷ춬튻닎폫헟뗄믬뫏닟싔ꆣWhen the predicted equilibrium entails mixing two or more pure strategies, experimental results do show some subjects in the group pursuing one of the pure strategies and others pursuing another, but this does not constitute true mixing by an individual player.떱훘뢴뷸탐쇣뫍늩?쪱ꎬ떥룶닎폫헟뺭뎣톡퓱퓚늻춬쪱볤톡퓱늻춬뗄뒿닟싔ꆣ떫쯻쏇쯆뫵컳붫닟싔쯦믺뮯쫓ퟷ닟싔붻쳦ꆣWhen subjects play zero-sum games repeatedly, individual players often choose different pure strategies over time. But they seems to mistake alternation for randomization. Slide 3618
쇣뫍늩?믬뫏닟싔뗄횤뻝Evidence on Mixing in Zero-Sum Games쿖쪵샽ퟓ쯆뫵룼쓜횤쏷샭싛ꆣReal life seems ahead of the laboratory.샽ퟓꎺExamples:40쓪듺십살냫떺뗄펢뻼British army in Malaya, 1940s컂췸훐룟쫖쏇뗄뷓랢쟲Serve-and-return play of top-level players at Wimbledonퟣ쟲훐뗄뗣쟲Penalty kicks in soccerSlide 37ퟜ뷡Summary쇣뫍늩?춨뎣쎻폐뒿닟싔쓉쪲뻹뫢ꆣZero-sum games often have no Nash equilibrium in pure strategies.퓚헢킩늩?훐ꎬ쎿룶닎폫헟뚼늻튪죃죋닂춸ꎬ틲듋쪹폃튻룶믬뫏닟싔ꎬ벴퓚웤뒿닟싔벯뫏짏뚨틥튻룶룅싊럖늼ꆣIn these games, each player wants to be unpredictable and so uses a mixed strategythat specifies a probability distribution over his set of pure 3819
ퟜ뷡Summaryퟮ폅랴펦럖컶쓜릻놻폃살뷢돶믬뫏닟싔뻹뫢ꆣBest-response analysis can be used to solve for mixed-strategy equilibria.믬뫏닟싔뗄뛔쫖컞닮틬탔훊뇭쏷쎿룶닎폫헟뗄뻹뫢믬뫏닟싔믡쪹뗃뛔쫖퓚웤쯹폐믬뫏닟싔횮볤컞닮틬ꆣThe opponent’s indifferenceproperty of mixed-strategy equilibria indicates that each player’s equilibrium mixture is such that the other player is indifferent among all her mixes.ퟮ킡뮯ퟮ듳럖컶튲쓜릻폃살헒떽닎폫헟뗄뻹뫢믬뫏닟싔ꆣMinimaxanalysis also can be used to find player’s equilibrium mixtures. Slide 39ퟜ뷡Summary떱튻룶늩?헟뫍쮫랽늩?헟폐죽룶믲룼뛠닟싔쫇ꎬ뻹뫢뗄믬뫏닟싔뿉쓜믡틔헽뗄룅싊돶쯹폐뒿닟싔ꎬ튲뿉쓜횻냼삨늿럖뒿닟싔ꆣWhen one or both players have three (or more) strategies, equilibrium mixed strategies may put positive probability on all pure strategies or may include only a subset of the pure strategies.믬뫏닟싔뗄쪵퇩쫒횤뻝뫜쒣뫽ꎬ떫퓚룼캪쿖쪵뗄뮷뺳쿂ꎬ폐횤뻝뇭쏷뻹뫢믬뫏닟싔뗄쪹폃ꆣEvidence on mixed-strategy play in the laboratory is mixed, but there is evidences of the use of equilibrium mixes in other more realistic 4020
