Robustness to strategic uncertainty (original) (raw)
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Robustness to Strategic Uncertainty (Revision of DP 2010-70)
Amer Math Mon, 2010
We model a player's uncertainty about other players' strategy choices as smooth probability distributions over their strategy sets. We call a strategy profile (strictly) robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence (all sequences) of strategy profiles, in each of which every player's strategy is optimal under under his or her uncertainty about the others. We derive general properties of such robustness, and apply the definition to Bertrand competition games and the Nash demand game, games that admit infinitely many Nash equilibria. We show that our robustness criterion selects a unique Nash equilibrium in the Bertrand games, and that this agrees with recent experimental findings. For the Nash demand game, we show that the less uncertain party obtains the bigger share.
Robustness to strategic uncertainty in the Nash demand game
Mathematical Social Sciences, 2018
This paper studies the role of strategic uncertainty in the Nash demand game. A player's uncertainty about another player's strategy is modelled as an atomless probability distribution over that player's strategy set. A strategy profile is robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others (Andersson, Argenton and Weibull, 2014). In the context of the Nash demand game, we show that robustness to symmetric (asymmetric) strategic uncertainty singles out the (generalized) Nash bargaining solution. The least uncertain party obtains the bigger share.
Strategic uncertainty and the ex post Nash property in large games
2015
This paper elucidates the conceptual role that independent randomization plays in non-cooperative game theory. In the context of large (atomless) games in nor-mal form, we present precise formalizations of the notions of a mixed strategy equilibrium (MSE) and of a randomized strategy equilibrium in distributional form (RSED). We offer a resolution of two longstanding open problems and show that (i) any MSE induces a RSED and any RSED can be lifted to a MSE, and (ii) a mixed strategy profile is a MSE if and only if it has the ex post Nash property. Our substantive results are a direct consequence of an exact law of large numbers that can be formalized in the analytic framework of a Fubini extension. We discuss how the “measurability ” problem associated with a MSE of a large game is auto-matically resolved in such a framework. We also present an approximate result pertaining to a sequence of large but finite games.
Strategic approximations of discontinuous games
2011
Abstract An infinite game is approximated by restricting the players to finite subsets of their pure strategy spaces. A strategic approximationof an infinite game is a countable subset of pure strategies with the property that limits of all equilibria of all sequences of approximating games whose finite strategy sets eventually include each member of the countable set must be equilibria of the infinite game. We provide conditions under which infinite games admit strategic approximations.
Robustness to strategic uncertainty in price competition
RePEc: Research Papers in Economics, 2010
We model a player's uncertainty about other players' strategy choices as probability distributions over their strategy sets. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in each of which every player's strategy is optimal under under his or her uncertainty about the others. We apply this definition to Bertrand games with a continuum of equilibrium prices and show that our robustness criterion selects a unique Nash equilibrium price. This selection agrees with recent experimental findings.
On the Existence of Pure Strategy Equilibria in Games with a Continuum of Players
Journal of Economic Theory, 1997
We present results on the existence of pure strategy Nash equilibria in nonatomic games. We also show by means of counterexamples that the stringent conditions on the cardinality of action sets cannot be relaxed, and thus resolve questions which have remained open since Schmeidler's 1973 paper. Journal of Economic Literature Classification Number: C72.
2010
We model a player's uncertainty about other players' strategy choices as probability distributions over their strategy sets. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in each of which every player's strategy is optimal under under his or her uncertainty about the others. We apply this definition to Bertrand games with a continuum of equilibrium prices and show that our robustness criterion selects a unique Nash equilibrium price. This selection agrees with recent experimental findings.
On the existence of equilibria in discontinuous games: three counterexamples
International Journal of Game Theory, 2005
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 equilibria), 2. quasiconcave, reciprocally upper semicontinuous games without pure strategy ε−equilibria for small enough ε > 0, and 3. payoff secure games whose mixed extension is not payoff secure. The last example, due to Sion and Wolfe [6], also shows that nonquasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.