Eco514-Game Theory ecture 7: Interactive Epistemology(2 Marciano siniscalchi October 7. 1999 Introduction This lecture presents the two main contributions of "interactive epistemology"to thethe- ory of normal-form games: a characterization of Nash equilibrium beliefs, and a full (i.e behavioral) characterization of rationalizability a review of the basic definitions For your convenience, I summarize the essential definitions pertaining to models of interactive beliefs; please consult the notes for Lecture 6 for details (92, (Ti, ai)ieN) such that, for every player i E/a). A frame for G is a tuple F We fix a simultaneous game G=(N, (Ai is a partition of Q2, and a; is a Ti- mesurable map a1:→A We denote by ti(w)the cell of the partition Ti containing w A model for G is a tuple M=(F, (piie), where F is a frame for G and each pi is a probability distribution on A() Given any belief a-i E A(A-i) for Player i, rila_i) is the set of best replies for i given -i. The first-order beliefs function a-i: S-A(A-i) extracts Player i's beliefs about A-i from her beliefs about n Armed with this notation, we can define the event, "Player i is rational "by R1={u∈9:a(u)∈r(a-(∞)} nd the event, "Every player is rational"by r=nien Ri Define the belief operator Bi: 23-2 by VECQ, Bi(E)=WwE: pi(eti(w=1Eco514—Game Theory Lecture 7: Interactive Epistemology (2) Marciano Siniscalchi October 7, 1999 Introduction This lecture presents the two main contributions of “interactive epistemology” to the the￾ory of normal-form games: a characterization of Nash equilibrium beliefs, and a full (i.e. behavioral) characterization of rationalizability. A review of the basic definitions For your convenience, I summarize the essential definitions pertaining to models of interactive beliefs; please consult the notes for Lecture 6 for details. We fix a simultaneous game G = (N,(Ai , ui)i∈N . 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 Ti￾mesurable map ai : Ω → Ai . We denote by ti(ω) the cell of the partition Ti containing ω. 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). Given any belief α−i ∈ ∆(A−i) for Player i, ri(α−i) is the set of best replies for i given α−i . The first-order beliefs function α−i : Ω → ∆(A−i) extracts Player i’s beliefs about A−i from her beliefs about Ω. Armed with this notation, we can define the event, “Player i is rational” by Ri = {ω ∈ Ω : ai(ω) ∈ ri(α−i(ω))} and the event, “Every player is rational” by R = T i∈N Ri . Define the belief operator Bi : 2Ω → 2 Ω by ∀E ⊂ Ω, Bi(E) = {ω ∈ Ω : pi(E|ti(ω)) = 1}. 1