Eco514--Game Theory Lecture 3: Nash equilibrium Marciano siniscalchi September 23. 1999 Introduction Nash equilibrium has undoubtedly proved to be the most influential idea in game theory Its development was a major intellectual achievement; what is perhaps more important, it enabled fundamental breakthroughs in economics and the social sciences Recent foundational research has emphasized the subtleties in the interpretation of Nash quilibrium. This lecture deals with the technical details of equilibrium analysis, but also with these interpretational issues. However, a more precise appraisal of the situation must be postponed until we develop a full-blown model of interactive beliefs Definitions Fix a game G=(N,(Ai, Ii, uiieN), where each(Ai, Ii) is a compact metrizable space(as usual, you can think of finite action spaces unless explicitly noted) and every u; is bounded and continuous Recall that we denote by ri: A(A-i, B(A-1))=A: Player i's best-reply correspondence, which associates with each belief (probability distribution on A-i, endowed with the borel gma-algebra) the set of The definition of Nash equilibrium involves best replies to beliefs concentrated on a single action profile. If the game is finite, these beliefs assign probability one to the specified profile, and probability zero to any other belief; if the game is infinite, these beliefs are Dirichlet measures: that is, for any a_i E A-i, we define Sa-i E A(A-i, B(T-i) by letting d(a-i(e)=l iff a-i E E for any E E B(T-i. All you need to know about this is that nilaiEco514—Game Theory Lecture 3: Nash Equilibrium Marciano Siniscalchi September 23, 1999 Introduction Nash equilibrium has undoubtedly proved to be the most influential idea in game theory. Its development was a major intellectual achievement; what is perhaps more important, it enabled fundamental breakthroughs in economics and the social sciences. Recent foundational research has emphasized the subtleties in the interpretation of Nash equilibrium. This lecture deals with the technical details of equilibrium analysis, but also with these interpretational issues. However, a more precise appraisal of the situation must be postponed until we develop a full-blown model of interactive beliefs. Definitions Fix a game G = (N,(Ai , Ti , ui)i∈N ), where each (Ai , Ti) is a compact metrizable space (as usual, you can think of finite action spaces unless explicitly noted) and every ui is bounded and continuous. Recall that we denote by ri : ∆(A−i , B(A−i)) ⇒ Ai Player i’s best-reply correspondence, which associates with each belief (probability distribution on A−i , endowed with the Borel sigma-algebra) the set of best replies to it. The definition of Nash equilibrium involves best replies to beliefs concentrated on a single action profile. If the game is finite, these beliefs assign probability one to the specified profile, and probability zero to any other belief; if the game is infinite, these beliefs are Dirichlet measures: that is, for any a−i ∈ A−i , we define δa−i ∈ ∆(A−i , B(T−i)) by letting δ(a−i)(E) = 1 iff a−i ∈ E for any E ∈ B(T−i). All you need to know about this is that Z A−i ui(ai , ·)dδa−i = ui(ai , a−i) 1