Static Games of Incomplete Information Page 5 type of player i is playing a different game from her sisters.Therefore a pure strategy for player i in a static Bayesian game is type contingent;it is a function si:iAi.The space of all such functions and hence player i's pure-strategy space is Si=A For a particular type 0 of player i,her strategysi specifies some action ai=si()EAi.A mixed strategy oi:ii for player i specifies a mixed action ifor each type of player i;i.e.i).As usuali is the space of all player-i mixed strategies. At this point you might say:"Fine.I agree that different types would play different actions;but,since only one type of each player is participating in the game at any time,it is only necessary for an equilibrium to specify that type's action."In order to rebut this line of reasoning let's consider some player j,who wants to play a best response to i's strategy.The problem for player j is that player i's action will depend upon her type 0i,and player j doesn't know what type of player i he is facing.He considers perhaps several types of player i as possibilities for his opponent.He needs to consider the actions of all those types of player i,because he can't rule any of them out.Since every player must be able to compute a best response,every player must know the planned actions of all types of all other players.Therefore any well-defined strategy profile must define an action for every type of every player. Therefore a strategy profile s maps type profiles into action profiles;i.e.s:-A,sES=XieSi. Bayesian equilibrium Consider a particular player ie/and a particular one of her types 0iei.Assume that her n-1 opponents'types are described by some deleted type profile 6-i-i and that they play some deleted action profile a-iA-i.If player i then chooses an action aiAi,her utility will be ui((ai,a-i),(0i,0i)). More generally,if the players choose a mixed-action profile o4,player i's expected utility is u(,a-),(0,8-). Now assume player i knows the type-contingent mixed strategies o-iE-i her opponents are playing; i.e.she knows what mixed actions they would take for any given set of types.However,she doesn't know their realized types,so she doesn't know the actual deleted mixed-action profile a-i which will occur as a result of their type-contingent strategies.What action aiA;should player i choose?Although player i doesn't know i,she does know the probability distribution p by which Nature generates type profiles;and she also knows her own type 0i,upon which she conditions her subjective probability about the types 0-i of her opponents.For any particular combination 0-i of other players'types,player i assesses this combination the probability pi(-ili).Therefore she also adds this probability to the event that her opponents will choose the particular deleted mixed-action profile o-i(0-i)Es4-i.Player i's expected utility,then,given her knowledge of her own type 6;and of her opponents'type-contingent strategies o-i,if she chooses the action aiAi,is 高00ua,c-0八0,0》 (5) 8 AB is the set of all functions from BA. jim@virtualperfection.com Jim Ratliff virtualperfection.com/gametheoryStatic Games of Incomplete Information Page 5 jim@virtualperfection.com Jim Ratliff virtualperfection.com/gametheory type of player i is playing a different game from her sisters. Therefore a pure strategy for player i in a static Bayesian game is type contingent; it is a function si:Øi§Ai. The space of all such functions and hence player i’s pure-strategy space is Si=Ai Øi .8 For a particular type øi of player i, her strategy si specifies some action ai=siªøiº˙Ai. A mixed strategy ßi:Øi§Ai for player i specifies a mixed action åi˙Ai for each type of player i; i.e. Åøi˙Øi, ßiªøiº˙Ai. As usual Íi=Ai Øi is the space of all player-i mixed strategies. At this point you might say: “Fine. I agree that different types would play different actions; but, since only one type of each player is participating in the game at any time, it is only necessary for an equilibrium to specify that type’s action.” In order to rebut this line of reasoning let’s consider some player j, who wants to play a best response to i’s strategy. The problem for player j is that player i’s action will depend upon her type øi, and player j doesn’t know what type of player i he is facing. He considers perhaps several types of player i as possibilities for his opponent. He needs to consider the actions of all those types of player i, because he can’t rule any of them out. Since every player must be able to compute a best response, every player must know the planned actions of all types of all other players. Therefore any well-defined strategy profile must define an action for every type of every player. Therefore a strategy profile s maps type profiles into action profiles; i.e. s:اA, s˙S=Xi˙ISi. Bayesian equilibrium Consider a particular player i˙I and a particular one of her types øi˙Øi. Assume that her n_1 opponents’ types are described by some deleted type profile ø¥i˙Ø¥i and that they play some deleted action profile a¥i˙A¥i. If player i then chooses an action ai˙Ai, her utility will be uiª(ai,a¥i),(øi,ø¥i)º. More generally, if the players choose a mixed-action profile å˙A, player i’s expected utility is uiª(åi,å¥i),(øi,ø¥i)º. Now assume player i knows the type-contingent mixed strategies ߥi˙Í¥i her opponents are playing; i.e. she knows what mixed actions they would take for any given set of types. However, she doesn’t know their realized types, so she doesn’t know the actual deleted mixed-action profile å¥i which will occur as a result of their type-contingent strategies. What action ai˙Ai should player i choose? Although player i doesn’t know ø¥i, she does know the probability distribution p by which Nature generates type profiles; and she also knows her own type øi, upon which she conditions her subjective probability about the types ø¥i of her opponents. For any particular combination ø¥i of other players’ types, player i assesses this combination the probability piªø¥i|øiº. Therefore she also adds this probability to the event that her opponents will choose the particular deleted mixed-action profile ߥiªø¥iº˙A¥i. Player i’s expected utility, then, given her knowledge of her own type øi and of her opponents’ type-contingent strategies ߥi, if she chooses the action ai˙Ai, is pi ªø¥i|Ùøi ºui ª(ai ,ߥiªø¥iº),(øi ,ø¥i S )º ø¥i˙Ø¥i . (5) 8 AB is the set of all functions from B§A