Static Games of Incomplete Information Page 3 marginal distribution of player i's type is piA(i),where p0)=an8,8 (1) This number pi)is the probability that Nature draws a type profile whose i-th component is .5 For later technical convenience we remove from a player's type space any impossible types;i.e.we require, for all iel and all 0ii,that pi)>0. We assume that the probability distribution p,which generates the type profile,is common knowledge and that each player derives from p her subjective beliefs about the types of her opponents. We represent these beliefs by a conditional probability pi(il)that the opponents'types are a particular deleted type profile i-igiven that player i's known type is By Bayes'Rule we have6 P(0:&6-2-p) 8l0)=P8)=p0p° (2) Player i's knowledge of her own type may or may not affect her beliefs about the types of her opponents.When players'types are independent,the probability of a particular type profile 6 is just the product of the players'marginal distributions,each evaluated at the type 0;specified by 6,i.e.Vo, p)=IpKep. (3) Therefore,when players'types are independent,player i's subjective beliefs about others'types are independent of her own type: i0l0=p0= p() ?-jat (4) which does not involve 0i. Example:Micro type spaces Consider two players,Lucky and Dopey.They are in separate rooms and will compete against each other in a computer-intensive task.Clearly the brand of computer each employs will significantly affect her performance.Each player observes the brand of her own computer but not that of her opponent,so we describe each player's type by the brand of the computer with which she is endowed.The two available brands are Macintosh and IBM.Therefore both players have identical type spaces 5 The i subscript on the pidoes not mean that we are referring to player i's beliefs.Indeed,player i is the only player guaranteed to know player i's type.The subscript actually serves only a formal requirement:identifying the domain of this function,viz.i and the deleted type profile over which the summation should extend. 6 Here the subscript i on the pi happens to identify the holder of these beliefs;however,it actually refers to the component of on which this probability is conditioned.Note also that the denominator is positive,and therefore this quantity is well defined,because we have excluded impossible types. jim@virtualperfection.com Jim Ratliff virtualperfection.com/gametheoryStatic Games of Incomplete Information Page 3 jim@virtualperfection.com Jim Ratliff virtualperfection.com/gametheory marginal distribution of player i’s type is pi˙ǪØiº, where pi ªøi º= pªøi ,ø¥i S º ø¥i˙Ø¥i . (1) This number piªøiº is the probability that Nature draws a type profile ø whose i-th component is øi.5 For later technical convenience we remove from a player’s type space any impossible types; i.e. we require, for all i˙I and all øi˙Øi, that piªøiº>0. We assume that the probability distribution p, which generates the type profile, is common knowledge and that each player derives from p her subjective beliefs about the types of her opponents. We represent these beliefs by a conditional probability piªø¥i|øiº that the opponents’ types are a particular deleted type profile ø¥i˙Ø¥i given that player i’s known type is øi. By Bayes’ Rule we have6 pi ªø¥i|Ùøi º= pªøi Ù&Ùø¥iº pªøi º = pªøº pi ªøi º . (2) Player i’s knowledge of her own type may or may not affect her beliefs about the types of her opponents. When players’ types are independent, the probability of a particular type profile ø is just the product of the players’ marginal distributions, each evaluated at the type øj specified by ø, i.e. Åø˙Ø, pªøº= pj ªøj P º j˙I . (3) Therefore, when players’ types are independent, player i’s subjective beliefs about others’ types are independent of her own type: pi ªø¥i|Ùøi º= pªøº pi ªøi º = pj ªøj P º j˙I pi ªøi º = pj ªøj P º j˙I\{i} , (4) which does not involve øi. Example: Micro type spaces Consider two players, Lucky and Dopey. They are in separate rooms and will compete against each other in a computer-intensive task. Clearly the brand of computer each employs will significantly affect her performance. Each player observes the brand of her own computer but not that of her opponent, so we describe each player’s type by the brand of the computer with which she is endowed. The two available brands are Macintosh and IBM. Therefore both players have identical type spaces 5 The i subscript on the pi does not mean that we are referring to player i’s beliefs. Indeed, player i is the only player guaranteed to know player i’s type. The subscript actually serves only a formal requirement: identifying the domain of this function, viz. Øi , and the deleted type profile over which the summation should extend. 6 Here the subscript i on the pi happens to identify the holder of these beliefs; however, it actually refers to the component of ø on which this probability is conditioned. Note also that the denominator is positive, and therefore this quantity is well defined, because we have excluded impossible types