The basic idea Recall that, in order to represent payoff uncertainty, we introduced a set Q2 of states of the world, and made the players payoff functions depend on the realization w E Q, as well as on the profile(alien E Lien A; of actions chosen by the players This allowed us to represent hierarchical beliefs about the state of the world; however we are no more capable of describing hierarchical beliefs about actions(at least not without introducing additional information, such as a specification of equilibrium actions for each type of each player) Thus, a natural extension suggests itself. For simplicity, I will consider games without payoff uncertainty, but the extension should be obvious Definition 1 Consider a simultaneous game G=(N, (Ai, wi)ien)(without payoff uncer tainty). A frame for G is a tuple F=(Q, (Ti, alieN) such that, for every player iE N, Tiis a partition of Q2, and ai is a map a;: Q2- Ai such that (a1)≠0→a1(a)∈T Continue to denote by ti(w) the cell of the partition Ti containing w. Finally, a model forG is a tuple M=(F, (pilieN), where F is a frame for G and each P; is a probability distribution I distinguish between frames and models to emphasize that probabilistic beliefs convey additional information-which we wish to relate to solution concepts. The distintion is also often made in the literature le, The main innovation is the introduction of the functions ai (.). This is not so far-fetched er all, uncertainty about opponents' actions is clearly a form of payoff uncertainty--one that arises in any strategic situation. However, by making players ' choices part of the state of the world, it is possible to discuss the players' hierarchical beliefs about them. Ultimately, we wish to relate solution concepts to precisely such assumptions There is one technical issue which deserves to be noted. We are assuming that"actions e measurable with respect to types, to use a conventional expression; that is, whenever w, w'e ti E Ti, the action chosen by Player i at w has to be the same as the action she chooses at w. This is natural: after all, in any given state, a player only knows her type, so it would be impossible for her to implement a contingent action plan which specifies different choices at different states consistent with her type. Our definition of a frame captures this Putting the model to work sider one concrete example to fix ideas. Fi exhibits a game and a model forThe basic idea Recall that, in order to represent payoff uncertainty, we introduced a set Ω of states of the world, and made the players’ payoff functions depend on the realization ω ∈ Ω, as well as on the profile (ai)i∈N ∈ Q i∈N Ai of actions chosen by the players. This allowed us to represent hierarchical beliefs about the state of the world; however, we are no more capable of describing hierarchical beliefs about actions (at least not without introducing additional information, such as a specification of equilibrium actions for each type of each player). Thus, a natural extension suggests itself. For simplicity, I will consider games without payoff uncertainty, but the extension should be obvious. Definition 1 Consider a simultaneous game G = (N,(Ai , ui)i∈N ) (without payoff uncer￾tainty). A frame for G is a tuple F = (Ω,(Ti , ai)i∈N ) such that, for every player i ∈ N, Ti is a partition of Ω, and ai is a map ai : Ω → Ai such that a −1 i (ai) 6= ∅ ⇒ a −1 i (ai) ∈ Ti . Continue to denote by ti(ω) the cell of the partition Ti containing ω. Finally, a model for G is a tuple M = (F,(pi)i∈N ), where F is a frame for G and each pi is a probability distribution on ∆(Ω). I distinguish between frames and models to emphasize that probabilistic beliefs convey additional information—which we wish to relate to solution concepts. The distintion is also often made in the literature. The main innovation is the introduction of the functions ai(·). This is not so far-fetched: after all, uncertainty about opponents’ actions is clearly a form of payoff uncertainty—one that arises in any strategic situation. However, by making players’ choices part of the state of the world, it is possible to discuss the players’ hierarchical beliefs about them. Ultimately, we wish to relate solution concepts to precisely such assumptions. There is one technical issue which deserves to be noted. We are assuming that “actions be measurable with respect to types,” to use a conventional expression; that is, whenever ω, ω0 ∈ ti ∈ Ti , the action chosen by Player i at ω has to be the same as the action she chooses at ω 0 . This is natural: after all, in any given state, a player only knows her type, so it would be impossible for her to implement a contingent action plan which specifies different choices at different states consistent with her type. Our definition of a frame captures this. Putting the model to work Let us consider one concrete example to fix ideas. Figure 1 exhibits a game and a model for it. 2