Lecture Notes on Game Theory Set 3 – Mixed Strategy Equilibria
نویسنده
چکیده
A game in strategic form does not always have a Nash equilibrium in which each player deterministically chooses one of his strategies. However, players may instead randomly select from among these pure strategies with certain probabilities. Randomizing one’s own choice in this way is called a mixed strategy. A profile of mixed strategies is called a mixed equilibrium if no player can gain on average by unilateral deviation. Nash showed in 1951 that any finite strategic-form game has a mixed equilibrium (J. F. Nash (1951), Non-cooperative games. Annals of Mathematics 54, pp. 286–295). We will show how Nash proved this theorem in Section 3.7 below. Average (that is, expected) payoffs must be considered because the outcome of the game may be random. This requires that each payoff in the game represents an “expected utility”, in the sense that the payoffs can be weighted with probabilities in order to represent the player’s preference for a random outcome.
منابع مشابه
Lecture Notes on Game Theory
1. Extensive form games with perfect information 3 1.1. Chess 3 1.2. Definition of extensive form games with perfect information 4 1.3. The ultimatum game 5 1.4. Equilibria 5 1.5. The centipede game 6 1.6. Subgames and subgame perfect equilibria 6 1.7. Backward induction, Kuhn’s Theorem and a proof of Zermelo’s Theorem 7 2. Strategic form games 10 2.1. Definition 10 2.2. Nash equilibria 10 2.3....
متن کاملEvolutionary game theory and population dynamics
We begin these lecture notes by a crash course in game theory. In particular, we introduce a fundamental notion of a Nash equilibrium. To address the problem of the equilibrium selection in games with multiple equilibria, we review basic properties of the deterministic replicator dynamics and stochastic dynamics of finite populations. We show the almost global asymptotic stability of an efficie...
متن کاملCS364A: Algorithmic Game Theory Lecture #20: Mixed Nash Equilibria and PPAD-Completeness∗
Today we continue our study of the limitations of learning dynamics and polynomial-time algorithms for converging to and computing equilibria. Recall that we have sweeping positive results for coarse correlated and correlated equilibria, which are tractable in arbitrary games. We have only partial positive results for pure Nash equilibria of routing and congestion games, and last lecture we dev...
متن کاملAlgorithms for Computing Solution Concepts in Game Theory
These are notes for the first five lectures of the course on Algorithmic Game Theory, given (starting November 2008) in the Weizmann Institute jointly by Uriel Feige, Robert Krauthgamer and Moni Naor. The lecture notes are not intended to provide a comprehensive view of solution concepts in game theory, but rather discuss some of the algorithmic aspects involved. Hence some of the definitions w...
متن کاملOn the existence of equilibria in discontinuous games: three counterexamples
We study whether we can weaken the conditions given in Reny [4] and still obtain existence of pure strategy Nash equilibria in quasiconcave normal form games, or, at least, existence of pure strategy ε−equilibria for all ε > 0. We show by examples that there are: 1. quasiconcave, payoff secure games without pure strategy ε−equilibria for small enough ε > 0 (and hence, without pure strategy Nash...
متن کامل