However,this is an informal statement. Recent contributions have shown how to con- truct formal models(based on the classical decision-theoretic setting) which allow one to formalize ideas such as "rationality"and"belief". In the framework of those models, the "implication"we have used to define rationalizability can actually be proved-i e. it is a theorem. This places rationalizability on very solid methodological ground Second, it might seem rather natural to assume that beliefs should be independent(and that moreover this ought to be "common belief"). After all, the basic model of simultaneous games postulates that players choose their actions independently, without any possibility of coordination. Thus, correlated beliefs might even appear to be unwarranted The debate on"causal vs. epistemic independence, "as the issue is often referred to, is lengthy and beyond the scope of these notes. Let me just invite you to reflect on the subject based on the following example Consider the following three-player "coordination"game: Players 1 and 2 choose L or R, and get 1 if they coordinate and 0 if they do not. Player 3 chooses C or N; C yields 1 if Players 1 and 2 coordinate, and 0 otherwise, while n yields 0 if they coordinate and 1 if they do not. That is, Player 3s choice represents a bet on whether or not Players 1 and 2 manage to coordinate Now suppose that Player 3 is fully aware that Players 1 and 2 choose independently however, she expects them to coordinate, but is unsure as to which action they will choose e. L or R). Thus, she assigns probability 2 to each of the profiles(L, L)and(R, R).This is a correlated belief, but it could be explained with the following simple story: Players 1 and 2 have been playing this game for a long time, and have "learned to coordinate. "Player 3 knows this, but has never observed their actual play--perhaps because she has also been playing the game for a long time, but she has only observed her own payoff in each play of and hence ascertained whether or not there was coordination) I welcome your comments on this issue! Rationalizability and dominance Consider the following definition Definition 6 Fix a finite game G=(N, (Ai, uiieN) and a player i E N. An action a; E Ai is strictly dominated for Player i iff there exists a; E A(Ai) such that a!∈AHowever, this is an informal statement. Recent contributions have shown how to con￾struct formal models (based on the classical decision-theoretic setting) which allow one to formalize ideas such as “rationality” and “belief”. In the framework of those models, the “implication” we have used to define rationalizability can actually be proved—i.e. it is a theorem. This places rationalizability on very solid methodological ground. Second, it might seem rather natural to assume that beliefs should be independent (and that moreover this ought to be “common belief”). After all, the basic model of simultaneous games postulates that players choose their actions independently, without any possibility of coordination. Thus, correlated beliefs might even appear to be unwarranted. The debate on “causal vs. epistemic independence,” as the issue is often referred to, is lengthy and beyond the scope of these notes. Let me just invite you to reflect on the subject, based on the following example. Consider the following three-player “coordination” game: Players 1 and 2 choose L or R, and get 1 if they coordinate and 0 if they do not. Player 3 chooses C or N; C yields 1 if Players 1 and 2 coordinate, and 0 otherwise, while N yields 0 if they coordinate and 1 if they do not. That is, Player 3’s choice represents a bet on whether or not Players 1 and 2 manage to coordinate. Now suppose that Player 3 is fully aware that Players 1 and 2 choose independently; however, she expects them to coordinate, but is unsure as to which action they will choose (i.e. L or R). Thus, she assigns probability 1 2 to each of the profiles (L,L) and (R,R). This is a correlated belief, but it could be explained with the following simple story: Players 1 and 2 have been playing this game for a long time, and have “learned to coordinate.” Player 3 knows this, but has never observed their actual play—perhaps because she has also been playing the game for a long time, but she has only observed her own payoff in each play of the game (and hence ascertained whether or not there was coordination). I welcome your comments on this issue! Rationalizability and Dominance Consider the following definition: Definition 6 Fix a finite game G = (N,(Ai , ui)i∈N ) and a player i ∈ N. An action ai ∈ Ai is strictly dominated for Player i iff there exists αi ∈ ∆(Ai) such that ∀a−i ∈ A−i , X a 0 i∈Ai ui(a 0 i , a−i)αi(ai) > ui(ai , a−i); 7