A relation between perfect equilibria in extensive form games and proper equilibria in normal form games

Abstract

Abstract:The concept of quasi-perfect equilibria for games in extensive form is introduced. It is shown that a proper equilibrium of a normal form game induces a quasi-perfect equilibrium in every extensive form game having this normal form.

Computing Proper Equilibria of Zero-Sum Games

“…In a recent paper [12], we suggested that the notion of equilibrium refinements from game theory would be a natural vehicle for sorting out the insensible equilibria from the sensible ones, also for the application of computing prescriptive strategies for extensive-form zero-sum games, to be used by game playing software. We showed how to modify the Koller-Megiddo-von Stengel algorithm so that a quasi-perfect equilibrium (an equilibrium refinement due to van Damme [15]) is computed, and we showed how computing such an equilibrium would eliminate the insensible behavior in the computed strategy alluded to in Selby's poker example and in many other examples as well.…”

Uniform Sampling of the Infinite Noncooperative Game on Unit Hypercube and Reshaping Ultimately Multidimensional Matrices of Player’s Payoff Values

“…However, NE-solutions render a lot of the refined or modified principles of optimality, allowing to smooth differences in utility and equity [2], [10], [11]. Mainly, they are principles of Pareto equilibrium [2], [6], [8], [10], [13], [14], Mertens-stable equilibrium [15], trembling hand perfect equilibrium [16], proper equilibrium [17], [18], correlated equilibrium [19], sequential equilibrium [20], [21], quasi-perfect equilibrium [18], [22], [23], perfect Bayesian equilibrium [18], [20], [24], [25], quantal response equilibrium [26], [27], self-confirming equilibrium [28], [29], strong Nash equilibrium [30], [31], Markov perfect equilibrium [32], [33]. The question is only to find NEsolutions as fast as possible.…”

Equilibrium Selection in Sequential Games with Imperfect Information

“…It also follows that we can compare k-proper equilibria to extensive form solution concepts: Proposition 1 in Reny (1992) together with Remark 1 imply that given any game in extensive form, and given any k-proper equilibrium 2 (S) of the normal form associated with such extensive form game, there exists a system of beliefs such that and this system of beliefs satisfy weak sequential rationality; while Proposition 1 in Mailath, Samuelson and Swinkels (1997) together with Remark 1 imply that any strategy pro…le that is quasiperfect (van Damme 1984) in the extensive form is k-proper in the associated normal form game.…”