Weakly Dominated Research Articles

Weak belief and permissibility

We provide epistemic foundations for permissibility (Brandenburger, 1992), a strategic-form solution concept for finite games which coincides with the Dekel-Fudenberg procedure, i.e., the elimination of all weakly dominated strategies, followed by the iterated elimination of strictly dominated strategies. We show that permissibility characterizes the behavioral implications of “cautious rationality and common weak belief of cautious rationality” in the canonical, universal type structure for lexicographic beliefs. For arbitrary type structures, we show that the behavioral implications of these epistemic assumptions are characterized by the solution concept of full weak best response set, a weak dominance analogue of best response set (Pearce, 1984).

Iterated elimination procedures

We study the existence and uniqueness (i.e., order independence) of any arbitrary form of iterated elimination procedures in an abstract environment. By allowing for a transfinite elimination, we show a general existence of the iterated elimination procedure. Inspired by the seminal work of Gilboa et al. (OR Lett 9:85–89, 1990), we identify a fairly weak sufficient condition of Monotonicity* for the order independence of iterated elimination procedure. Monotonicity* requires a Monotonicity property along any elimination path. Our approach is applicable to different forms of iterated elimination procedures used in (in)finite games, for example iterated elimination of strictly dominated strategies, iterated elimination of weakly dominated strategies, rationalizability, and so on. We introduce a notion of CD* games, which incorporates Jackson’s (Rev Econ Stud 59:757–775, 1992) idea of “boundedness,” and show the iterated elimination procedure is order independent in the class of CD* games. In finite games, we also formulate and show an “outcome” order-independence result suitable for Marx and Swinkles’s (Games Econ Behav 18:219–245, 1997) notion of nice weak dominance.

Iterated weak dominance and interval-dominance supermodular games

This paper extends Milgrom and Robert's treatment of supermodular games in two ways. It points out that their main characterization result holds under a weaker assumption. It refines the arguments to provide bounds on the set of strategies that survive iterated deletion of weakly dominated strategies. I derive the bounds by iterating the best-response correspondence. I give conditions under which they are independent of the order of deletion of dominated strategies. The results have implications for equilibrium selection and dynamic stability in games.

Lexicographic beliefs and assumption

Foundations for iterated admissibility (i.e., the iterated removal of weakly dominated strategies) need to confront a fundamental challenge. On the one hand, admissibility requires that a player consider every strategy of their opponents possible. On the other hand, reasoning that the opponents are rational requires ruling out certain strategies. Brandenburger, Friedenberg, Keisler's (BFK, Econometrica, 2008) foundations for iterated admissibility address this challenge with two ingredients: lexicographic beliefs and the concept of “assumption.” However, BFK restrict attention to lexicographic beliefs whose supports are essentially disjoint. This restriction does not have a compelling behavioral rationale, or a clear intuitive interpretation. At the same time, it plays a crucial role in BFK's foundations for iterated admissibility—specifically, in their analysis of assumption. We provide an alternate characterization of assumption, which applies to all lexicographic beliefs. We also characterize two variants of assumption, based on two extensions of ‘weak dominance’ to infinite state spaces. These notions of assumption coincide with BFK's notion when the state space is finite and lexicographic beliefs have disjoint support; but they are different in more general settings. Leveraging these characterization results, we show that disjoint supports do not play a role in the foundations for iterated admissibility.

On the Epistemic Foundation for Iterated Weak Dominance: An Analysis in a Logic of Individual and Collective attitudes

This paper proposes a logical framework for representing static and dynamic properties of different kinds of individual and collective attitudes. A complete axiomatization as well as a decidability result for the logic are given. The logic is applied to game theory by providing a formal analysis of the epistemic conditions of iterated deletion of weakly dominated strategies (IDWDS), or iterated weak dominance for short. The main difference between the analysis of the epistemic conditions of iterated weak dominance given in this paper and other analysis is that we use a semi-qualitative approach to uncertainty based on the notion of plausibility first introduced by Spohn, whereas other analysis are based on a quantitative representation of uncertainty in terms of probabilities.

Common knowledge of payoff uncertainty in games

Using epistemic logic, we provide a non-probabilistic way to formalise payoff uncertainty, that is, statements such as ‘player i has approximate knowledge about the utility functions of player j.’ We show that on the basis of this formalisation common knowledge of payoff uncertainty and rationality (in the sense of excluding weakly dominated strategies, due to Dekel and Fudenberg (1990)) characterises a new solution concept we have called ‘mixed iterated strict weak dominance.’

Iterated weak dominance and subgame dominance

In this paper, we consider finite normal form games satisfying transference of decisionmaker indifference. We show that any set of strategies surviving k rounds of elimination of some weakly dominated strategies can be reduced to a set of strategies equivalent to the set of strategies surviving k rounds of elimination of all weakly dominated strategies in every round by (at most k) further rounds of elimination of weakly dominated strategies. The result develops work by Gretlein [Gretlein, R., 1983. Dominance elimination procedures on finite alternative games. International Journal of Game Theory 12, 107–113]. We then consider applications and demonstrate how we may obtain a unified approach to the work by Gretlein and recent results by Ewerhart [Ewerhart, C., 2002. Iterated weak dominance in strictly competitive games of perfect information. Journal of Economic Theory 107, 474-482] and Marx and Swinkels [Marx, L.M., Swinkels, J.M., 1997. Order independence for iterated weak dominance. Games and Economic Behavior 18, 219-245].

On the equivalence of weak dominance and sequential best response

In this paper, we identify the conditions under which a strategy is weakly dominated if and only if it is not sequential best response. In addition, we explore the implication for extensive form rationalizability.

Adaptive Learning and Iterated Weak Dominance

This article addresses the idea that rational players should not play iteratively weakly dominated strategies by showing that when a particular type of adaptive learning process converges, then players must have learned to play strategy profiles equivalent to those that survive iterated nice weak dominance and, for certain games, equivalent to those that survive iterated weak dominance. For games satisfying the weak single crossing condition, the set of strategies that survive iterated weak dominance is small in that its bounds are pure strategy Nash equilibria. The results hold regardless of the order in which dominated strategies are eliminated. Journal of Economic LiteratureClassification Number: C72.

Order Independence for Iterated Weak Dominance

In general, the result of the elimination of weakly dominated strategies depends on order. We definenice weak dominance. Under nice weak dominance, order does not matter. We identify an important class of games under which nice weak dominance and weak dominance are equivalent, and so the order under weak dominance does not matter. For all games, the result of iterative nice weak dominance is an upper bound on the result from any order of weak dominance. The results strengthen the intuitive relationship between backward induction and weak dominance and shed light on some computational problems relating to weak dominance.Journal of Economic LiteratureClassification Number: C72.

Weak Dominance and Approximate Common Knowledge

Considering finite normal-form games, we assume that players maximise expected utility, that they hold beliefs with full support, and that these facts are "approximate common knowledge" among players. We show that players satisfy these assumptions if and only if they choose strategies that survive the following procedure: first, all weakly dominated strategies are eliminated, and then strongly dominated strategies are iteratively eliminated. This procedure is due to E. Dekel and D. Fudenberg [ J. Econ. Theory 52 (1990), 243-267]. We compare our justification for this procedure to theirs. Journal of Economic Literature Classification Number: C72.

?Cautious? utility maximization and iterated weak dominance

This paper examines the implications of assuming that “rational” players in a two-player noncooperative game maximize expected utility using “cautious” beliefs; i.e., beliefs that give positive probability to all “rational” strategies of the other players. The result is a solution concept that captures some of the flavor of but is not equivalent to the iterated elimination of weakly dominated strategies. The solution concept yields attractive results for some games, but suffers from nonexistence problems in other games. We interpret the nonexistence problems as showing that it is logically inconsistent to assume that rational players analyze all games on the basis of cautious expected utility maximization.