Static Games of Incomplete Information Page 4 OL=Θp={Mac,IBM}. The type-profile space is ⊙=⊙L×⊙D={Mac,Mac),Mac,IBM),(IBM,Mac),IBM,IBM)}. First assume computers are being handed out by independent fair-coin tosses such that heads dispenses a Macintosh and tails an IBM.In this scenario the types are independent;e.g p((Mac,IBM))=pL(Mac)pp(IBM)=.Lucky's prior subjective beliefs-prior,that is,to observing her own type-about Dopey's computer are given by pD(Mac)=p((Mac,Mac))+p((IBM,Mac))=1.7 After observing that she was awarded a Mac,her posterior beliefs about Dopey's computer are L(Mac|Mac)=p((Mac,Mac))/pL(Mac)=pL(Mac)pp(Mac)/pL(Mac)=pD(Mac)=;i.e.the prior and the posterior are the same. Alternatively,assume that there is only one Mac,and it is awarded to Lucky or Dopey on the basis of a coin toss.The unfortunate remaining player receives an IBM.In this case the probability distribution over the type-profile space is p((Mac,IBM))=p((IBM,Mac))=,p((Mac,Mac))=p((IBM,IBM))=0. Although Lucky's prior about Dopey's computer is still pp(Mac)=pp(IBM)=,after Lucky observes that she was awarded the Mac,she updates her prior to reflect her certainty that Dopey received the IBM: P(BMIMa)=D(Mac IBM)=1. PL(Mac) 2二12 This joint probability distribution and the associated marginal probabilities are shown in the table below. Dopey Mac IBM Mac 0 Lucky 12 12=PL(Mac) IBM 2 0 1五-n(IBM0 12 五 Pp(Mac)=Pp(IBM) Strategies We will now see that it is not sufficient for a strategy for player ie/in a Bayesian game to merely specify an action for that player;it must specify an action for every type 0iei of player i.Player i's payoff function depends upon her type.Therefore for given actions by,and types of,her opponent,each type of player i will be solving a different maximization problem yielding different best responses;each 7 Note,again,that the D subscript on the pp refers to the component of with respect to which the marginal distribution is computed,not to the player whose beliefs we are discussing. jim@virtualperfection.com Jim Ratliff virtualperfection.com/gametheoryStatic Games of Incomplete Information Page 4 jim@virtualperfection.com Jim Ratliff virtualperfection.com/gametheory ØL=ØD={Mac,IBM}. The type-profile space is Ø=ØL˜ØD={(Mac,Mac),(Mac,IBM),(IBM,Mac),(IBM,IBM)}. First assume computers are being handed out by independent fair-coin tosses such that heads dispenses a Macintosh and tails an IBM. In this scenario the types are independent; e.g. pª(Mac,IBM)º=pLªMacºpDªIBMº=¤. Lucky’s prior subjective beliefs—prior, that is, to observing her own type—about Dopey’s computer are given by pDªMacº=pª(Mac,Mac)º+pª(IBM,Mac)º=™. 7 After observing that she was awarded a Mac, her posterior beliefs about Dopey’s computer are pLªMac|Macº=pª(Mac,Mac)º/pLªMacº=pLªMacºpDªMacº/pLªMacº=pDªMacº=™; i.e. the prior and the posterior are the same. Alternatively, assume that there is only one Mac, and it is awarded to Lucky or Dopey on the basis of a coin toss. The unfortunate remaining player receives an IBM. In this case the probability distribution over the type-profile space is pª(Mac,IBM)º=pª(IBM,Mac)º=™,!!pª(Mac,Mac)º=pª(IBM,IBM)º=0. Although Lucky’s prior about Dopey’s computer is still pDªMacº=pDªIBMº=™, after Lucky observes that she was awarded the Mac, she updates her prior to reflect her certainty that Dopey received the IBM: pLªIBM|Macº= pª(Mac,IBM)º pLªMacº =1 2 1 2 =1. This joint probability distribution and the associated marginal probabilities are shown in the table below. Strategies We will now see that it is not sufficient for a strategy for player i˙I in a Bayesian game to merely specify an action for that player; it must specify an action for every type øi˙Øi of player i. Player i’s payoff function depends upon her type. Therefore for given actions by, and types of, her opponent, each type of player i will be solving a different maximization problem yielding different best responses; each 7 Note, again, that the D subscript on the pD refers to the component of ø with respect to which the marginal distribution is computed, not to the player whose beliefs we are discussing