Correlated Equilibrium As a first application of this formalism, I will provide a characterization of the notion of correlated equilibrium, due to R.Aumann I have already argued that the fact that players choose their actions independently of each other does not imply that beliefs should necessarily be stochastically independent(recall the " betting on coordination"game). Correlated equilibrium provides a way to allow for correlated beliefs that is consistent with the equilibrium approach Definition 5 Fix a game G=(N, (Ai, ui)ieN). A correlated equilibrium of G is a probability distribution a E A(A) such that, for every player iE N, and every function di: Ai ∑a2(a1-)a(a,a-)≥∑ (aa,a-a)∈A (aa,a-1)∈A The above is the standard definition of correlated equilibrium. However Proposition 0.2 Fix a game=(N, (Ai, ui)ieN) and a probability distribution a E A(A) Then a is a correlated equilibrium of G iff, for any player i E N and action a; E Ai such that a({a}×A-)>0, and for all a∈A, ∑(an,a-)a(a-la)≥∑ta,a-)a(a-lan) a-;∈A- a-;∈A- where a(a_ilai=a(ai,a-i] x A-i Proof: Fix a player i E N. Observe first that, for any function f: Ai-Ais ∑a2(f(a,a-)a(a12a-)=∑∑at(f(a),a-)a(a,a-)= (a1,a-)∈A a∈A1a-i∈A ∑a({a}×A-)∑at(f(a),a-)a(a-la aa({a}×A-i)>0 a-i∈A- Suppose first that there exists an action ai E Ai with adlai)x A-i)>0 such that ∑a∈A,1(a,a-)a(a-l|a)<∑a∈A,(,a-)a(a-la1). Then the function d,:A A: defined by di(ai)=a' and di(ai= ai for all ait a constitutes a profitable er-ante deviation(see the above observation), so a cannot be a correlated equilibrium Conversely, suppose that the above inequality holds for all ai and a; as in the claim Now consider any function d,:A→A; by assumption,∑a∈A,t(a2,a-)a(a-lan)≥ 2ai ea_ ui(di(ai),a-i)a(a-ilai)for any ai such that a(lai] x A-i)>0. The claim follows from our initial observation,Correlated Equilibrium As a first application of this formalism, I will provide a characterization of the notion of correlated equilibrium, due to R. Aumann. I have already argued that the fact that players choose their actions independently of each other does not imply that beliefs should necessarily be stochastically independent (recall the “betting on coordination” game). Correlated equilibrium provides a way to allow for correlated beliefs that is consistent with the equilibrium approach. Definition 5 Fix a game G = (N,(Ai , ui)i∈N ). A correlated equilibrium of G is a probability distribution α ∈ ∆(A) such that, for every player i ∈ N, and every function di : Ai → Ai , X (ai,a−i)∈A ui(ai , a−i)α(ai , a−i) ≥ X (ai,a−i)∈A ui(di(ai), a−i)α(ai , a−i) The above is the standard definition of correlated equilibrium. However: Proposition 0.2 Fix a game G = (N,(Ai , ui)i∈N ) and a probability distribution α ∈ ∆(A). Then α is a correlated equilibrium of G iff, for any player i ∈ N and action ai ∈ Ai such that α({ai} × A−i) > 0, and for all a 0 i ∈ Ai , X a−i∈A−i ui(ai , a−i)α(a−i |ai) ≥ X a−i∈A−i ui(a 0 i , a−i)α(a−i |ai) where α(a−i |ai) = α({ai , a−i}|{ai} × A−i). Proof: Fix a player i ∈ N. Observe first that, for any function f : Ai → Ai , X (ai,a−i)∈A ui(f(ai), a−i)α(ai , a−i) = X ai∈Ai X a−i∈A−i ui(f(ai), a−i)α(ai , a−i) = = X ai:α({ai}×A−i)>0 α({ai} × A−i) X a−i∈A−i ui(f(ai), a−i)α(a−i |ai) P Suppose first that there exists an action a¯i ∈ Ai with α({a¯i} × A−i) > 0 such that a−i∈A−i ui(¯ai , a−i)α(a−i |a¯i) < P a−i∈A−i ui(a 0 i , a−i)α(a−i |a¯i). Then the function di : Ai → Ai defined by di(¯ai) = a 0 i and di(ai) = ai for all ai 6= ¯a constitutes a profitable ex-ante deviation (see the above observation), so α cannot be a correlated equilibrium. Conversely, suppose that the above inequality holds for all ai and a 0 i as in the claim. Now consider any function di : Ai → Ai : by assumption, P a−i∈A−i ui(ai , a−i)α(a−i |ai) ≥ P a−i∈A−i ui(di(ai), a−i)α(a−i |ai) for any ai such that α({ai} × A−i) > 0. The claim follows from our initial observation. 5