hus, the key innovations are the sets e and the measures p;. The former enter as parameters in the payoff functions, and thus capture payoff uncertainty; note that Player j's payoffs may depend on Player i's payoff type (i.e. values may be interdependent, using our auction-theoretic terminology The probabilities Pi determine a common prior over e. This is not yet apparent, but it will be when we define a solution concept that applies to such games The implicit assumption one makes is that, before the game begins, payoff types are drawn according to the probabilities pi, .. PN; each player i E N learns her own payoff type Bi, and the product measure p_i E A(e-i) defined by (-)=I (63)≠i∈ j≠i serves as her prior belief about her opponents' payoff types Two important comments are in order. First of all, one might visualize the select of payoff types as a chance move. Clearly, to make things interesting, it must be the that this initial move by chance is not perfectly observed by all players. Hence, one needs a model of extensive games without observed actions; this is precisely the model we are going to develop in the next section Indeed, one might even say that games with incomplete information are the main reason why the theory has traditionally focused on extensive games without observable actions. My point here(shared by OR, as well as Fudenberg and Tirole) is that the general notion is perhaps more than is strictly required to model dynamic games of economic interest I should also add that reducing a situation with incomplete information/payoff uncer- tainty to one with imperfect information(in particular, unobserved actions)is not an entirely harmless assumption-chiefly because doing so requires imposing the common -prior assump- tion(can you see why?) Second. note that i am deviating somewhat from our basic framework for interactive decision making. First, I am incorporating probabilities in the description of the model, and I am making the players'priors over payoff type profiles common. This is why I am using the term "incomplete information"as opposed to "payoff uncertainty-in keeping with the literature Second, for simultaneous-move games, the setup does not immediately reduce to the one we adopt for static Bayesian games. However, to relate the two constructions, it is sufficient to note that 0 is the counterpart of @2 in our static Bayesian setting, and each e; defines a partition of e whose cells are of the form (0i x_i1 While the static Bayesian model is more general than this derived formulation(think about a setting in which all players'payoffs depend on a single random draw which noThus, the key innovations are the sets Θi and the measures pi . The former enter as parameters in the payoff functions, and thus capture payoff uncertainty; note that Player j’s payoffs may depend on Player i’s payoff type (i.e. values may be interdependent, using our auction-theoretic terminology). The probabilities pi determine a common prior over Θ. This is not yet apparent, but it will be when we define a solution concept that applies to such games. The implicit assumption one makes is that, before the game begins, payoff types are drawn according to the probabilities p1, . . . , pN ; each player i ∈ N learns her own payoff type θi , and the product measure p−i ∈ ∆(Θ−i) defined by p−i(θ−i) = Y j6=i pj (θj ) ∀θ−i = (θj )6=i ∈ Θ−i serves as her prior belief about her opponents’ payoff types. Two important comments are in order. First of all, one might visualize the selection of payoff types as a chance move. Clearly, to make things interesting, it must be the case that this initial move by chance is not perfectly observed by all players. Hence, one needs a model of extensive games without observed actions; this is precisely the model we are going to develop in the next section. Indeed, one might even say that games with incomplete information are the main reason why the theory has traditionally focused on extensive games without observable actions. My point here (shared by OR, as well as Fudenberg and Tirole) is that the general notion is perhaps more than is strictly required to model dynamic games of economic interest. I should also add that reducing a situation with incomplete information/payoff uncer￾tainty to one with imperfect information (in particular, unobserved actions) is not an entirely harmless assumption—chiefly because doing so requires imposing the common-prior assump￾tion (can you see why?) Second, note that I am deviating somewhat from our basic framework for interactive decision making. First, I am incorporating probabilities in the description of the model, and I am making the players’ priors over payoff type profiles common. This is why I am using the term “incomplete information” as opposed to “payoff uncertainty”—in keeping with the literature. Second, for simultaneous-move games, the setup does not immediately reduce to the one we adopt for static Bayesian games. However, to relate the two constructions, it is sufficient to note that Θ is the counterpart of Ω in our static Bayesian setting, and each Θi defines a partition of Θ whose cells are of the form {{θi} × Θ−i}. While the static Bayesian model is more general than this derived formulation (think about a setting in which all players’ payoffs depend on a single random draw which no 2