The bottom line is that (i) assumptions about higher-order beliefs do influence equilib- rium outcomes, and (ii) it is very easy to analyze deviations from the"textbook"assumptions about higher-order beliefs in the framework of standard Bayesian Nash equilibrium analysis Priors and common priors The other buzzword that is often heard in connection with games with incomplete informa- tion is“ common prior Simply stated, this is the assumption that pi= p for all i E N. Note that, strictly peaking, the common prior assumption(CPA for short)is part of the"textbook"definition of a "game with incomplete information our slightly nonstandard terminology emphasizes that(1) we do not wish to treat priors(common or private)as part of the description of the model, but rather as part of the solution concept; and that(2) we certainly do not wish to impose the CPa in all circumstances. But is the CPa at all reasonable? The answer is, well, it depends. One often hears the following argument Prior beliefs reflect prior information. We assume that players approach the game with a common heritage, i. e. with the same prior information. Therefore, their prior beliefs should be the same On priors as summary of previously acquired information Let us first take this argument at face value. From a Bayesian standpoint, this makes sense only if we assume that players approach the game after having observed the same events for a very long time; only in this case, in fact, will their beliefs converge But perhaps we do not want to be really Bayesians; perhaps "prior information"is"no information"and we wish to invoke some variant of the principle of insufficient reason. I personally do not find this argument all that convincing, but you may differ On Priors and interactive beliefs However, the real problem with this sort of justification of the CPA is that, as we have illustrated above, the set Q actually conveys information about both payoff uncertainty and the players' infinite hierarchies of interactive beliefs. Therefore, it is not clear how players beliefs about infinite hierarchies of beliefs can "converge due to a long period of commor observations. " How do I"learn"your beliefs? The bottom line is that the only way to assess the validity of the CPa is via its imp ions for the infinite hierarchies of beliefs it generates IMore precisely: perhaps I can make inferences about your beliefs, with the aid of some auxiliary as ptions, but I can never observe your beliefsThe bottom line is that (i) assumptions about higher-order beliefs do influence equilib￾rium outcomes, and (ii) it is very easy to analyze deviations from the “textbook” assumptions about higher-order beliefs in the framework of standard Bayesian Nash equilibrium analysis. Priors and Common Priors The other buzzword that is often heard in connection with games with incomplete informa￾tion is “common prior.” Simply stated, this is the assumption that pi = p for all i ∈ N. Note that, strictly speaking, the common prior assumption (CPA for short) is part of the “textbook” definition of a “game with incomplete information”; our slightly nonstandard terminology emphasizes that (1) we do not wish to treat priors (common or private) as part of the description of the model, but rather as part of the solution concept; and that (2) we certainly do not wish to impose the CPA in all circumstances. But is the CPA at all reasonable? The answer is, well, it depends. One often hears the following argument: Prior beliefs reflect prior information. We assume that players approach the game with a common heritage, i.e. with the same prior information. Therefore, their prior beliefs should be the same. On priors as summary of previously acquired information Let us first take this argument at face value. From a Bayesian standpoint, this makes sense only if we assume that players approach the game after having observed the same events for a very long time; only in this case, in fact, will their beliefs converge. But perhaps we do not want to be really Bayesians; perhaps “prior information” is “no information” and we wish to invoke some variant of the principle of insufficient reason. I personally do not find this argument all that convincing, but you may differ. On Priors and interactive beliefs However, the real problem with this sort of justification of the CPA is that, as we have illustrated above, the set Ω actually conveys information about both payoff uncertainty and the players’ infinite hierarchies of interactive beliefs. Therefore, it is not clear how players’ beliefs about infinite hierarchies of beliefs can “converge due to a long period of common observations.” How do I “learn” your beliefs?1 The bottom line is that the only way to assess the validity of the CPA is via its implica￾tions for the infinite hierarchies of beliefs it generates. 1More precisely: perhaps I can make inferences about your beliefs, with the aid of some auxiliary as￾sumptions, but I can never observe your beliefs! 5