믬뫏닟싔춬쪱늩?IIꎺ럇쇣뫍늩?뫍IIIꎺ튻냣탔쳖싛Simultaneous-Move Games with Mixed Strategies II: Non-Zero-Sum Games and III: General Discussion 뗚8헂Chapter 08믬뫏닟싔춬쪱늩?ꎺ럇쇣뫍늩?Simultaneous-Move Games with Mixed Strategies: Non-Zero-Sum Games퓚럇쇣뫍늩?훐ꎬ닎폫헟횮볤쎻폐쏷쿔뗄샻틦돥춻ꎬ튲뻍쎻폐웕뇩뗄샭평살뛔뛔쫖틾님웤샻틦쯹퓚ꆣIn non-zero-sum games, player do not have clearly conflicting interests and have no general reason to want to conceal their interests from others.틲듋ꎬ쏔믳뛔쫖뻍늻튻뚨폐뗀샭ꆣAs a result, there is no general argument for keeping the other player 21
평폚늻좷뚨뗄탅쓮떼훂뗄믬뫏닟싔Mixing Sustained by Uncertain Beliefs늻맽ꎬ평폚쫇춬쪱늩?ꎬ닎폫헟뿉쓜늻뗃늻돖폐뛔뛔쫖탐뚯뗄쒳훖늻좷뚨탔뗄탅쓮ꎬ틲뛸튲뻍늻쓜좷뚨뗘룸돶ퟔ벺뗄ퟮ폅탐뚯ꆣHowever, Simultaneous play can still lead players to have uncertain beliefs about the actions of a rival player and therefore to be uncertain about their own best 3맾샯뫍죸샲쓜럱믡쏦ꎿWill Harry Meet Sally?SALLYStarbucksLocal LatteStarbucks1, 10, 0HARRYLocal Latte0, 02, 2Slide 42
맾샯뫍죸샲쓜럱믡쏦ꎿWill Harry Meet Sally?Sally’sPayoffs2Local LatteSally’s best-response1Starbucks02/31Harry’s p-mixSlide 5맾샯뫍죸샲쓜럱믡쏦ꎿWill Harry Meet Sally?1Sally’sq-mix2/33 Nash EquilibriaHarry’s best response012/3Harry’s p-mixSally’s best responseSlide 63
맾샯뫍죸샲쓜럱믡쏦ꎿWill Harry Meet Sally?믬뫏닟싔뻹뫢쿂쎿룶죋뗄웚췻쫕틦캪2/3ꎬ킡폚죎뫎튻룶뒿닟싔뻹뫢ꎨ2믲1ꎩꆣThe expected payoff of each in the mixed-strategy equilibrium is (4/9*1)+(1/9)*2+(4/9*0)=2/3, worse than both pure-strategy Nash equilibrium (2 or 1).평폚룷ퟔ뗄쯦믺톡퓱쫇뛀솢뗄ꎬ뫜폐뿉쓜ꎨ듋뒦뿉쓜탔캪4/9ꎩ쮫랽톡퓱늻튻훂ꎬ떼훂뗍쫕틦ꆣWhen they choose randomly and independently, there is some positive and often significant probability (here 4/9) that they make mutually inconsistent choices and get a low 7뛄킡벦ꎿDiced Chicken?DEANSwerve Straight(Chicken)(Though)JAMESSwerve 0, 0-1, 1(Chicken)Straight1, -1-2, -2(Though)Slide 84
뛄킡벦ꎿDiced Chicken?1Dean’sq-mix1/23 Nash EquilibriaJames’s best response011/2James’s p-mixDean’s best responseSlide 9ퟮ폅랴펦럖컶Best-Response Analysis믬뫏닟싔룅싊쿂뗄뛾캬ퟹ뇪쾵훐ꎬퟮ폅랴펦럖컶쫇톰헒쯹폐쓉쪲뻹뫢뗄좫쓜랽램ꆣBest-response analysis with the mixture probabilities on the two axes is a comprehensive way of finding all Nash equilibria.룼볲떥뗄뛔쫖컞닮틬랽램횻쓜톰헒퇏룱틢틥쿂뗄믬뫏닟싔뻹뫢ꆣThe more simply opponent’s indifferencemethod can find only the genuinely mixed strategy equilibrium. Slide 105
믬뫏죽룶닟싔뗄럇쇣뫍늩?Non-Zero-Sum Mixing With Three Strategies4 pure-strategyDEANNash EquilibriaLeftStraightRightpLeft0, 0-1, 1-2, -21Straight1, -1-2, -21, -1p2JAMESRight-2, -2-1, 10, 01-p-p12qq1-q-q1212Slide 11믬뫏죽룶닟싔뗄럇쇣뫍늩?Non-Zero-Sum Mixing With Three Strategies쪹폃뛔쫖컞닮틬랽램뿉틔톰헒떽튻룶췪좫믬뫏닟싔뻹뫢ꆣA fully mixedequilibrium can be found using the opponent’s indifference method:Dean as the opponent:-p-2(1-p-p)=p-2p+(1-p-p)=-2p-p212121212⇒p=1/6, p=2/3, 1-p-p=1/61212Similarly with James’s indifference:q=1/6, q=2/3, 1-q-q=1/61212헢뮹늻쫇캨튻뗄믬뫏닟싔뻹뫢……But this is not the only mixed-strategy equilibrium for this game……Slide 126
믬뫏죽룶닟싔뗄럇쇣뫍늩?Non-Zero-Sum Mixing With Three Strategies튻룶뿉쓜뗄늿럖믬뫏뻹뫢쫇닎폫헟믬뫏횱탐ꎨSꎩ뫍췤ꎨ샽죧ퟳLꎩꆣOne possibility of a partially mixed equilibrium is that the player mix between Straight (S) and only one of the two direction of Swerve –say, Left (L).볙짨햲쒷쮹퓚웤믬뫏닟싔훐뷶쪹폃Lꎨ틔룅싊pꎩ뫍SꆣSuppose that James uses only L (with probability p) and S in his mixture.뛔듋ꎬ뗏낲뗄쫕틦캪ꎺDean’s payoffs against James’s p-mix are:L:p-1; S: 3p-2; R: -p-1Slide 13믬뫏죽룶닟싔뗄럇쇣뫍늩?Non-Zero-Sum Mixing With Three StrategiesR쇓폚Lꎬ뗏낲늻믡퓚웤믬뫏닟싔훐쪹폃쯼ꆣR is dominated by L and will therefore not be used in Dean’s mix.뗏낲횻퓚L뫍S횮볤컞닮틬ꎬ퓲p=1/2ꆣThen Dean’s indifference between his L and S lead to p=1/2.죴뗏낲횻퓚L뫍S횮볤믬뫏ꎬ샠쯆뗄볆쯣뿉틔뇭쏷햲쒷쮹튲늻쪹폃Rꎬ쟒뗏낲퓚L뫍R횮볤틔룅싊50:50믬뫏ꆣWhen Dean is mixing between L and S, a similar calculation shows that James will not use his R, and Dean’s mixture probability are 50:50 between his L and S.헢튻뻹뫢폫2*2킡벦늩?뗄믬뫏닟싔뷡맻컞틬ꆣThis equilibrium is really just like the mixed-strategy outcome in the two-by-two chicken 147
믬뫏죽룶닟싔뗄럇쇣뫍늩?Non-Zero-Sum Mixing With Three Strategies쇭튻뿉쓜쫇횻폐튻룶닎폫헟닉좡믬뫏닟싔ꆣAnother possibility is that only one player mixes in equilibrium.웤훐튻룶뻹뫢쫇ꎺ튻룶닎폫헟톡퓱뒿닟싔Sꎬ쇭튻룷닎폫헟튻죎틢룅싊퓚L뫍R횮볤믬뫏ꆣThere are equilibrium in which one player choose pure S and the other mixes with arbitrary probability between L and 15믬뫏닟싔뻹뫢뗄튻냣탔쳖싛ꎺ뻹뫢뗄죵틢틥General Discussion of Mixed-Strategy Equilibria: Weak Sense of Equilibrium떱닎폫헟톡퓱웤뻹뫢뗄믬뫏닟싔쫇ꎬ뛔쫖뗄ퟮ폅랴펦뿉틔쫇죎뫎뗄믬뫏닟싔ꎬ냼삨벫뛋뗄뒿닟싔ꆣWhen one player is choosing his equilibrium mix, the other’s best response can be anymixture including the extreme cases of the two pure strategies.뻹뫢뗄틢틥죧듋횮죵ꎬ횻쫇캪쇋뷡맻뗄컈뚨탔ꆣThe reason for this weaksense of equilibrium is the stabilityof 168
뛔쫖컞닮틬뫍럀횹놻샻폃Opponent’s Indifference and Guarding Against Exploitation퓚쇣뫍늩?훐ꎬ튻룶닎폫헟톡퓱헽좷뗄믬뫏닟싔뫳ꎬ뛔쫖톡퓱죎뫎닟싔쯽뚼쫇컞쯹캽뗄ꆣIn a zero-sumgame, when one player chooses her right mixture, she does equally well no matter what her opponent chooses. 죎뫎웤쯻뗄죎뫎톡퓱뚼믡놻뛔쫖샻폃틔폐샻폚쯽ꎬ듓뛸뛔ퟔ벺폐쯰ꆣAny other choice would be exploited by her opponent to her own advantage and therefore to disadvantage of herself.튲뻍쫇쮵ꎬ쎿룶닎폫헟뗄뻹뫢믬뫏닟싔뻟폐럀횹놻뛔쫖샻폃뗄탔훊ꆣIn other words, each player’s equilibrium mixture has the property that it prevents exploitationby the opponent. Slide 17럇쇣뫍늩?믬뫏닟싔쿂폫횱뻵캥놳뗄뷡맻1CouterintuitiveOutcomes with Mixed Strategies in Zero-Sum GamesNAVRATILOVADLCCEVERTDL50 →3080CC9020ShouldNavratilovacover DL more often now since she has gotten so much better at doing so (, q increases) ?Slide 189
럇쇣뫍늩?믬뫏닟싔쿂폫횱뻵캥놳뗄뷡맻CouterintuitiveOutcomes with Mixed Strategies in Zero-Sum Games100100When Navratilova playsNavratilova’’s p-mixBecause Navratilova is much better at covering DL,Evert uses CC more often in her 19럇쇣뫍늩?믬뫏닟싔쿂폫횱뻵캥놳뗄뷡맻CouterintuitiveOutcomes with Mixed Strategies in Zero-Sum Games10010090DLWhen EvertplaysEvert’’s p-mixBy virtue of this behavior, Navratilova also needs to decrease the frequency of her DL play (q decreases).Slide 2010
럇쇣뫍늩?믬뫏닟싔쿂폫횱뻵캥놳뗄뷡맻CouterintuitiveOutcomes with Mixed Strategies in Zero-Sum Games쯤좻쓉췞쪹폃웤룄뷸탐뚯뗄벸싊쿂붵ꎬ떫쯽늢쎻폐냗냗샋럑웤벼틕쳡룟——쯽뗄욽뻹쫕틦듓38쳡룟떽45ꆣAlthough Navratilova does not use her improved action more frequently, she does not waste her skill improvement also ꎭher average payoffrises from 38 to 45.쓉췞쳡룟웤DL뗄쫕틦틔쪹ퟔ벺늻뇘퓙죧듋욵랱뗘쪹폃쯼ꆣNavratilovashould improve her DL so that she does nothave to use it so often.쏷탞햻뗀ꎬ낵뛉돂닖ꆣ늻햽뛸쟼죋횮뇸ꆣSlide 21럇쇣뫍늩?믬뫏닟싔쿂폫횱뻵캥놳뗄뷡맻2CouterintuitiveOutcomes with Mixed Strategies in Zero-Sum Games튻룶낲좫뗄닟싔쫇헢퇹뗄ꎺ벴쪹놻뛔쫖닂떽ꎬ튲늻믡듸살퓖쓑탔뗄쪧냜ꎻ떫벴쪹늻놻뛔쫖닂떽ꎬ튲늻믡뫃떽쓄샯좥ꆣA safestrategy is a strategy which does not fail disastrously even if anticipated by the opponent but does not do very much better even is unanticipated. 튻룶쎰쿕뗄닟싔쫇헢퇹뗄ꎺ튻떩뛔쫖컞럀놸ꎬ뷡맻믡럇뎣뫃ꎻ튻떩뛔쫖폐쯹ힼ놸ꎬ뷡맻뫜퓣룢ꆣA riskystrategy is a strategy which does brilliantly if the other side is not ready for them but fail miserably if the other side is 2211
럇쇣뫍늩?믬뫏닟싔쿂폫횱뻵캥놳뗄뷡맻CouterintuitiveOutcomes with Mixed Strategies in Zero-Sum Gamesa>b>c>dOPPONENT EXPECTSW>LPR(safe)(risky)cW+(1-c)LbW+(1-b)LYOU PPLAY(safe)aW+(1-a)LdW+(1-d)LR(risky)Slide 23럇쇣뫍늩?믬뫏닟싔쿂폫횱뻵캥놳뗄뷡맻CouterintuitiveOutcomes with Mixed Strategies in Zero-Sum Games퓚믬뫏닟싔뻹뫢훐ꎬ쓣돶낲좫닟싔뗄룅싊p캪: It turns out that in the mixed-strategy equilibrium, you will play P strategy with a probability p of,(a-d)/[(a-d)+(b-c)]p뷓뷼폚1ꆣpis close to 1.틲듋낲좫닟싔쫇헢튻쟩탎쿂뗄헽뎣닟싔ꎬ쎰쿕닟싔횻얼좻쪹폃틔쏔믳뛔쫖ꆣSo the strategy is the normal play in these situations, and the risky play is played only occasionally to keep the Slide 24opponent
럇쇣뫍늩?믬뫏닟싔쿂폫횱뻵캥놳뗄뷡맻CouterintuitiveOutcomes with Mixed Strategies in Zero-Sum Games폐틢쮼뗄뗘랽퓚폚ꎬp뗄뇭듯쪽폫W뫍L췪좫컞맘ꆣThe interesting part of this result is that the expression for pis completely independent of W and L.튲뻍쫇쮵ꎬ샭싛죏캪ꎬ컞싛퓚훘튪맘춷ꎨW뇈L듳탭뛠ꎩ뮹쫇럇뷴튪맘춷ꎬ쓣뚼펦룃내쿠춬뗄벸싊믬뫏듳훚ꎨ낲좫ꎩ닟싔뫍쎰쿕닟싔ꆣThat is, the theory says that you should mix the percentage(safe) play and the risky play in exactly the same proportions on a big occasion (when W is much bigger than L) as on a minor occasion.헢튻샭싛폫횱뻵뗄쎬뛜쫇룹놾탔뗄ꆣ컊쳢돶퓚웚췻쫕틦뗄볆쯣짏ꆣThis discrepancy between theory and intuition is fundamental to the calculation of expected 25쇭튻럇쇣뫍늩?훐뗄캥놳횱뻵뗄뷡맻Another Countertuitive Result for Non-Zero-Sum GamesCOLUMNLeftRightROWUpa, Ab, BDownc, Cd, DSlide 2613
쇭튻럇쇣뫍늩?훐뗄캥놳횱뻵뗄뷡맻AnotherCountertuitiveResult for Non-Zero-Sum Games볙뚨늩?폐튻룶믬뫏닟싔뻹뫢ꎬ듋쪱Row돶Up룅싊캪pꎬDown룅싊캪(1-p)ꆣSuppose the game has a mixed-strategy equilibrium in which Row plays Up with probability pand Down with probability (1-p).샻폃뛔쫖컞닮틬랽램ꎬ뷢뗃ꎺUsing the opponent’s indifference method, we havep = (D-C)/[(A-B)+(D-C)]폚쫇Rowꎨ샠쯆Columnꎩ뗄믬뫏룅싊폫쯻ퟔ벺뗄쫕틦췪좫컞맘ꆣSo Row’s (and similarly, Column’s) mixture probabilities are totally independent of his Slide 27own payoffs!럇쇣뫍늩?훐믬뫏닟싔뗄횤뻝Evidence on Mixing in Non-Zero-Sum Games퓚뷸탐럇쇣뫍늩?뗄튻ퟩ쪵퇩뛔쿳훐ꎬ튻킩죋돶튻룶뒿닟싔ꎬ쇭췢뗄죋돶쇭췢뗄뒿닟싔ꆣIn a group of experimental subjects playing a non-zero-sum game, we may see some pursuing one pure strategy and other pursuing another.ퟜ쳥훐뗄헢샠믬뫏ꎬ쯤좻늻쓜럻뫏믬뫏닟싔뻹뫢뗄샭싛ꎬ떫폐튻룶폐좤뗄퇝뮯뷢쫍ꆣThis type of mixing in the population, although does not fit the theory of mixed-strategy equilibria, does have an interesting 2814
럇쇣뫍늩?훐믬뫏닟싔뗄횤뻝Evidence on Mixing in Non-Zero-Sum Games퓚쇭췢튻킩뛠듎훘뢴뗄쪵퇩훐ꎬ닎폫헟췹췹룄뇤웤탐캪ꆣIn other experiment concerning subject who play the game many times, players change their actions from one play to the next.죧맻럇튪붫웤뷢쫍캪믬뫏닟싔뗄횤뻝ꎬ퓲늻탒뗄쫇ꎬ닎폫헟뗄믬뫏룅싊쯦ퟅퟔ짭쫕틦뗄룄뇤뛸룄뇤ꆣ헢늻럻뫏샭싛ꆣBut even if we interpret this as evidence of true mixing, the player’s mixture probabilities change when their own payoffs are changed. This shouldn’t happen according to the 29ퟜ뷡Summary퓚럇쇣뫍늩?훐ꎬ믬뫏닟싔뻹뫢닺짺폚닎폫헟닎폫헟맘폚뛔랽탐뚯뗄훷맛늻좷뚨탔쮮욽잡뫃헽좷ꆣIn non-zero-sum games, mixed-strategy equilibriacan arise when players’mutual levels of subjective uncertaintyabout each other’s actions are at just the right level.뛔쫖컞닮틬랽램뫍ퟮ폅랴펦럖컶뮹쫇뿉틔ퟷ캪좷뚨뻹뫢믬뫏룅싊뗄뿲볜ꆣThe opponent’s indifference method and best-response analysis again provide the framework of determining equilibrium mixture 3015
ퟜ뷡Summary쪵퇩횤뻝뇭쏷샭싛뗄풤볆뷡맻뇘탫뷷짷퓋폃ꆣExperimental evidence shows that the prediction of theory should be used with caution.샭싛뿉틔놻췆맣떽죎뫎맦쒣뫍샠탍뗄늩?ꆣThe theory presented here can be generalized for games of any size and 3116
