Mixed StrategiesOverviewTennis AnyoneServingSlide 5The Game of TennisGame TableSlide 8Nash EquilibriumExtended GameCalculating Best ResponsesWhen to Defend LeftReceiver’s Best ResponseServer’s Best ResponseSlide 15Putting Things TogetherEquilibriumMixed Strategy EquilibriumGeneral Properties of Mixed Strategy EquilibriaGeneralized TennisSlide 21Sender’s Best ResponseSlide 23Minmax EquilibriumDoes Game Theory Work?Battle of the SexesHawk-DoveWars of AttritionPure Strategy EquilibriaMore Pure Strategy EquilibriaSymmetric Pure Strategy EquilibriaSymmetric EquilibriumWhen to concedeHazard RatesSlide 35ObservationsEconomic Costs of Wars of AttritionBig LessonWars of Attrition in PracticeAll-Pay AuctionsPure StrategiesSlide 42Best RespondingEquilibrium Mixed StrategySlide 45Properties of the All-Pay AuctionSlide 47Mixed StrategiesOverviewPrinciples of mixed strategy equilibriaWars of attritionAll-pay auctionsTennis AnyoneSRServingSRServingSRThe Game of TennisServer chooses to serve either left or rightReceiver defends either left or rightBetter chance to get a good return if you defend in the area the server is serving toGame TableReceiverServerLeft RightLeft ¼ ¾Right ¾ ¼Game TableReceiverServerLeft RightLeft ¼ ¾Right ¾ ¼For server: Best response to defend left is to serve rightBest response to defend right is to serve leftFor receiver: Just the oppositeNash EquilibriumNotice that there are no mutual best responses in this game.This means there are no Nash equilibria in pure strategiesBut games like this always have at least one Nash equilibriumWhat are we missing?Extended GameSuppose we allow each player to choose randomizing strategiesFor example, the server might serve left half the time and right half the time. In general, suppose the server serves left a fraction p of the timeWhat is the receiver’s best response?Calculating Best ResponsesClearly if p = 1, then the receiver should defend to the leftIf p = 0, the receiver should defend to the right.The expected payoff to the receiver is:p x ¾ + (1 – p) x ¼ if defending leftp x ¼ + (1 – p) x ¾ if defending rightTherefore, she should defend left ifp x ¾ + (1 – p) x ¼ > p x ¼ + (1 – p) x ¾When to Defend LeftWe said to defend left whenever:p x ¾ + (1 – p) x ¼ > p x ¼ + (1 – p) x ¾Rewritingp > 1 – p Orp > ½Receiver’s Best ResponsepLeftRight½Server’s Best ResponseSuppose that the receiver goes left with probability q.Clearly, if q = 1, the server should serve rightIf q = 0, the server should serve left.More generally, serve left if¼ x q + ¾ x (1 – q) > ¾ x q + ¼ x (1 – q)Simplifying, he should serve left ifq < ½Server’s Best ResponseLeftRight½ qPutting Things Together½ qpS’s best response1/2R’s bestresponseEquilibrium½ qpS’s best response1/2R’s bestresponseMutual best responsesMixed Strategy EquilibriumA mixed strategy equilibrium is a pair of mixed strategies that are mutual best responsesIn the tennis example, this occurred when each player chose a 50-50 mixture of left and right.General Properties of Mixed Strategy EquilibriaA player chooses his strategy so as to make his rival indifferentA player earns the same expected payoff for each pure strategy chosen with positive probabilityFunny property: When a player’s own payoff from a pure strategy goes up (or down), his mixture does not changeGeneralized TennisReceiverServerLeft RightLeft a, 1-a b, 1-bRight c, 1-c d, 1-dSuppose c > a, b > dSuppose 1 – a > 1 – b, 1 - d > 1 – c(equivalently: b > a, c > d)Receiver’s Best ResponseSuppose the sender plays left with probability p, then receiver should play left provided:(1-a)p + (1-c)(1-p) > (1-b)p + (1-d)(1-p)Or:p >= (c – d)/(c – d + b – a)Sender’s Best ResponseSame exercise only where the receiver plays left with probability q.The sender should serve left ifaq + b(1 – q) > cq + d(1 – q)Or:q <= (b – d)/(b – d + a – b)EquilibriumIn equilibrium, both sides are indifferent therefore:p = (c – d)/(c – d + b – a)q = (b – d)/(b – d + a – b)Minmax EquilibriumTennis is a constant sum gameIn such games, the mixed strategy equilibrium is also a minmax strategyThat is, each player plays assuming his opponent is out to mimimize his payoff (which he is) and therefore, the best response is to maximize this minimum.Does Game Theory Work?Walker and Wooders (2002)Ten grand slam tennis finalsCoded serves as left or rightDetermined who won each pointTests:Equal probability of winningPassSerial independence of choicesFailBattle of the SexesChrisPatOpera FightsOpera 3,1 0,0Fights 0,0 1,3Hawk-DoveKrushchevKennedyHawk DoveHawk 0, 0 4, 1Dove 1, 4 2, 2Wars of AttritionTwo sides are engaged in a costly conflictAs long as neither side concedes, it costs each side 1 per periodOnce one side concedes, the other wins a prize worth V.V is a common value and is commonly known by both partiesWhat advice can you give for this game?Pure Strategy EquilibriaSuppose that player 1 will concede after t1 periods and player 2 after t2 periodsWhere 0 < t1 < t2Is this an equilibrium?No: 1 should concede immediately in that caseThis is true of any equilibrium of this typeMore Pure Strategy EquilibriaSuppose 1 concedes immediatelySuppose 2 never concedesThis is an equilibrium though 2’s strategy is not credibleSymmetric Pure Strategy EquilibriaSuppose 1 and 2 will concede at time t.Is this an equilibrium?No – either can make more by waiting a split second longer to concedeOr, if t is a really long time, better to concede immediatelySymmetric EquilibriumThere is a symmetric equilibrium in this game, but it is in mixed strategiesSuppose each party concedes with probability p in each periodFor this to be an equilibrium, it must leave the other side indifferent between conceding and notWhen to concedeSuppose up to time t, no one has conceded:If I concede now, I earn –tIf I wait a split second to concede, I earn:V – t –  if my rival concedes– t – if notNotice the –t term is irrelevantIndifference:(V – x (f/(1 – F)) = - x (1 – f/(1-F))f/(1 – F) = 1/VHazard RatesThe term f/(1 – F) is called the hazard rate of a distributionIn words, this is the probability