Existence of Perfect Equilibria: A Direct Proof

We formulate and prove a modification of Eilenberg-Montgomery fixed-point theorem, which is a generalization of Kakutani’s theorem. It enables us to provide a direct proof of the existence of perfect equilibria in finite normal form games and extensive games with perfect recall. We construct a correspondence whose fixed points are precisely the perfect equilibria of a given finite game. Existence of a fixed point is secured by the modified version of Eilenberg-Montgomery theorem.Perfect equilibrium, best response correspondence, unit simplex, absolute neighborhood retract, deformation retract, fixed point

A Refinement of Perfect Equilibria Based On Substitute Sequences

We propose an equilibrium refinement of strict perfect equilibrium for the finite normal form games, which is not known in the literature. Okada came up with the idea of strict perfect equilibrium by strengthening the main definition of a perfect equilibrium, due to Selten [14]. We consider the alternative (and equivalent) definition of perfect equilibrium, based on the substitute sequences, as appeared in Selten [14]. We show that by strengthening and modifiyng this definition slightly, one can obtain a refinement stronger than strict perfectness. We call the new refinement strict substitute perfect equilibrium. The main advantage of this solution concept is that it reflects the local dominance of an equilibrium point. An example is provided to show that a strict perfect equilibrium may fail to be strict substitute perfect.Perfect equilibrium, strictly perfect equilibrium, substitute sequence, substitute perfect equilibrium, unit simplex

Continuity and Equilibrium Stability

This paper discusses the problem of stability of equilibrium points in normal form games in the tremling-hand framework. An equilibrium point is called perffect if it is stable against at least one seqence of trembles approaching zero. A strictly perfect equilibrium point is stable against every such sequence. We give a sufficient condition for a Nash equilibrium point to be strictly perfect in terms of the primitive characteristics of the game (payoffs and strategies), which is new and not known in the literature. In particular, we show that continuity of the best response correspondence (which can be stated in terms of the primitives of the game) implies strict perfectness; we prove a number of other useful theorems regarding the structure of best responce correspondence in normal form games.Strictly perfect equilibrium, best responce correspondence, unit simplex, face of a unit simplex

Best-shot versus weakest-link in political lobbying: an application of group all-pay auction

We analyze a group political lobbying all-pay auction with a group specific public good prize, in which one group follows a weakest-link and the other group follows a best-shot impact function. We completely characterize all semi-symmetric equilibria. There are two types of equilibria: (1) each player in the best-shot group puts mass at the upper bound of the support, whereas each player in the other group puts mass at the lower bound of the support; (2) players in the best-shot group put masses at both the lower and the upper bounds, while the other group randomizes without a mass point. An earlier and longer version of this study was circulated under the title “The Group All-pay Auction with Heterogeneous Impact Functions.” We appreciate the comments of an Associate Editor and two anonymous referees, Kyung Hwan Baik, Walter Enders, Matt Van Essen, Paan Jindapon, David Malueg, Paul Pecorino, Seth Streitmatter, Ted Turocy, the participants at the 2015 conference of ‘Contest: Theory and Evidence’ at the University of East Anglia, and the seminar participants at the University of Alabama and Korea University. Iryna Topolyan gratefully acknowledges the support from the Charles Phelps Taft Research Center. Any remaining errors are our own

Trembling-hand myopia and trembling-hand perfection

In 1975 Selten introduced trembling-hand perfection, however it was implicitly assumed that specific values of rationality imperfections are common knowledge. Relaxing this assumption, we develop the notion of trembling-hand myopia, and completely characterize equilibria which are robust in this setting.Imperfect rationality Common knowledge Perfect equilibrium Substitute sequence Substitute perfect equilibrium

The attack-and-defense conflict with the gun-and- butter dilemma

We analyze a general equilibrium model of attack and defense with production and conflict. One attacker and one defender allocate their fixed endowments either to produce gun or to produce butter, and the volume of guns produced determines the winner in the conflict. If the attacker wins, then it appropriates all the butter produced in the economy; otherwise, each consume only their own butter. We characterize the unique interior and unique corner equilibrium for this game. We find that (i) the defender may spend more resources on conflict than the attacker even without loss aversion or other behavioral biases, (ii) the attacker may expend all their resources only in conflict, and (iii) the interior and the corner equilibria cannot coexist

A proof of the existence of the minimax point of a strategic game

We present some proofs of the existence of the minimax point of a strategic game that does not seem to be available in the literature. The existence of the minimax point plays an important role in the theory of games.Strategic game Minimax point Attainable minimax points