Dominance Solvable Games Research Articles

Gender differences in equilibrium play and strategic sophistication variability

We investigate the existence of gender differences in strategic sophistication in two weakly dominance solvable games where a prize is at stake. The first one is the two-person beauty contest, where strategies are numbers and players must perform mathematical operations. The second is the novel “gaze coach game”, where strategies are photographs of the eye region and the two players must assign emotional states to these images. We observe that females follow equilibrium play less often in the former game but not in the latter. Males display greater strategic sophistication variability. As a result, females are underrepresented among top performers in both games.

Bounded Rationality and Strategic Uncertainty in a Simple Dominance Solvable Game

How much of the failures to achieve Pareto efficient outcome observed in a simple 2 x 2 dominance solvable game can be attributed to strategic uncertainty and how much is actually due to individual bounded rationality? We address this question by conducting a set of experiments involving two main treatments: one in which two human subjects interact, and another in which one human subject interacts with a computer program whose behavior is known. By making the behavior of the computer opponent perfectly predictable, the latter treatment eliminates strategic uncertainty. Our results suggest that observed coordination failures can be attributed equally to individual bounded rationality and strategic uncertainty.

The path to equilibrium in sequential and simultaneous games: A mousetracking study

We study in the laboratory three-, four- and six-player, dominance solvable games of complete information. We consider sequential and simultaneous versions of games that have the same, unique Nash equilibrium, and we use mousetracking to understand the decision making process of subjects. We find more equilibrium choices in the sequential version than in the simultaneous version of the game and we highlight the importance of attentional measures. Indeed, depending on the treatment, equilibrium behavior is 30 to 80 percentage points higher for subjects who look at all the payoffs necessary to compute the Nash equilibrium and for those who look at payoffs in the order predicted by the sequential elimination of dominated strategies than for subjects who do not. Finally, the sequence of lookups reveals that subjects have an easier time finding the player with a dominant strategy in the sequential timing than in the simultaneous timing. However, once this player is found, the unraveling logic of iterated elimination of dominated strategies is performed (equally) fast and efficiently under both timings.

How does communication affect beliefs in one-shot games with complete information?

This paper experimentally studies unilateral communication of intentions in eight different two-player one-shot normal form games with complete information. We find that communication is used both to coordinate and to deceive, and that messages have a significant impact on beliefs and behavior even in dominance solvable games. Nash equilibrium and cognitive hierarchy jointly account for many regularities, but not all of the evidence. Sophisticated sender behavior is especially difficult to reconcile with existing models.

Fluid Intelligence and Cognitive Reflection in a Strategic Environment: Evidence from Dominance-Solvable Games.

Dominance solvability is one of the most straightforward solution concepts in game theory. It is based on two principles: dominance (according to which players always use their dominant strategy) and iterated dominance (according to which players always act as if others apply the principle of dominance). However, existing experimental evidence questions the empirical accuracy of dominance solvability. In this study, we study the relationships between the key facets of dominance solvability and two cognitive skills, cognitive reflection, and fluid intelligence. We provide evidence that the behaviors in accordance with dominance and one-step iterated dominance are both predicted by one's fluid intelligence rather than cognitive reflection. Individual cognitive skills, however, only explain a small fraction of the observed failure of dominance solvability. The accuracy of theoretical predictions on strategic decision making thus not only depends on individual cognitive characteristics, but also, perhaps more importantly, on the decision making environment itself.

Stochastic Learning Dynamics and Speed of Convergence in Population Games

We study how long it takes for large populations of interacting agents to come close to Nash equilibrium when they adapt their behavior using a stochastic better reply dynamic. Prior work considers this question mainly for 2 × 2 games and potential games; here we characterize convergence times for general weakly acyclic games, including coordination games, dominance solvable games, games with strategic complementarities, potential games, and many others with applications in economics, biology, and distributed control. If players' better replies are governed by idiosyncratic shocks, the convergence time can grow exponentially in the population size; moreover, this is true even in games with very simple payoff structures. However, if their responses are sufficiently correlated due to aggregate shocks, the convergence time is greatly accelerated; in fact, it is bounded for all sufficiently large populations. We provide explicit bounds on the speed of convergence as a function of key structural parameters including the number of strategies, the length of the better reply paths, the extent to which players can influence the payoffs of others, and the desired degree of approximation to Nash equilibrium.

Eye Movements in Strategic Choice.

In risky and other multiattribute choices, the process of choosing is well described by random walk or drift diffusion models in which evidence is accumulated over time to threshold. In strategic choices, level‐k and cognitive hierarchy models have been offered as accounts of the choice process, in which people simulate the choice processes of their opponents or partners. We recorded the eye movements in 2 × 2 symmetric games including dominance‐solvable games like prisoner's dilemma and asymmetric coordination games like stag hunt and hawk–dove. The evidence was most consistent with the accumulation of payoff differences over time: we found longer duration choices with more fixations when payoffs differences were more finely balanced, an emerging bias to gaze more at the payoffs for the action ultimately chosen, and that a simple count of transitions between payoffs—whether or not the comparison is strategically informative—was strongly associated with the final choice. The accumulator models do account for these strategic choice process measures, but the level‐k and cognitive hierarchy models do not. © 2015 The Authors. Journal of Behavioral Decision Making published by John Wiley & Sons Ltd.

On the Structure of Weakly Acyclic Games

The class of weakly acyclic games, which includes potential games and dominance-solvable games, captures many practical application domains. In a weakly acyclic game, from any starting state, there is a sequence of better-response moves that leads to a pure Nash equilibrium; informally, these are games in which natural distributed dynamics, such as better-response dynamics, cannot enter inescapable oscillations. We establish a novel link between such games and the existence of pure Nash equilibria in subgames. Specifically, we show that the existence of a unique pure Nash equilibrium in every subgame implies the weak acyclicity of a game. In contrast, the possible existence of multiple pure Nash equilibria in every subgame is insufficient for weak acyclicity in general; here, we also systematically identify the special cases (in terms of the number of players and strategies) for which this is sufficient to guarantee weak acyclicity.

Informed-user Algorithms that Converge to Nash Equilibrium in Traffic Games

This paper considers a group of self-interested agents (drivers) trying to optimize their utility by choosing the route with the least travel time and proposes an algorithm that converges to a pure Nash equilibrium almost surely in traffic games. Weakly acyclic games, which generalize potential and dominance solvable games, are closely related to multi-agent systems through the existence of a global objective function and its alignment to the local utilities of each agent. We show that in a multi-agent distributed traffic routing problem with both linear and non-linear link cost functions, in the form of a congestion game, the achievement of pure Nash equilibrium is possible even if the agents use only the utility information of the previous action. We propose a fast and adaptive algorithm for the informed-user problem that provides almost sure convergence to a pure Nash equilibrium in any weakly acyclic game.

Extrapolation in Games of Coordination and Dominance Solvable Games

We study extrapolation between games in a laboratory experiment. Participants in our experiment first play either the dominance solvable guessing game or a Coordination version of the guessing game for five rounds. Afterwards they play a 3x3 normal form game for ten rounds with random matching which is either a game solvable through iterated elimination of dominated strategies (IEDS), a pure Coordination game or a Coordination game with pareto ranked equilibria. We find strong evidence that participants do extrapolate between games. Playing a strategically different game hurts compared to the control treatment where no guessing game is played before and in fact impedes convergence to Nash equilibrium in both the 3x3 IEDS and the Coordination games. Playing a strategically similar game before leads to faster convergence to Nash equilibrium in the second game. In the Coordination games some participants try to use the first game as a Coordination device. Our design and results allow us to conclude that participants do not only learn about the population and/or successful actions, but that they are also able to learn structural properties of the games.

Higher cognitive ability is associated with lower entries in a p-beauty contest

“Beauty contests” are well-studied, dominance-solvable games that generate two interesting results. First, most behavior does not conform to the unique Nash equilibrium. Second, there is considerable unexplained heterogeneity in behavior. In this work, we explore the relationship between beauty contest behavior and cognitive ability. We find that subjects with high cognitive ability exhibit behavior that is closer to the Nash equilibrium.

Correlated equilibria, incomplete information and coalitional deviations

This paper proposes new concepts of strong and coalition-proof correlated equilibria where agents form coalitions at the interim stage and share information about their recommendations in a credible way. When players deviate at the interim stage, coalition-proof correlated equilibria may fail to exist for two-player games. However, coalition-proof correlated equilibria always exist in dominance-solvable games and in games with positive externalities and binary actions.

Iterated strict dominance in general games

We offer a definition of iterated elimination of strictly dominated strategies (IESDS *) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS * is always a well-defined order independent procedure that can be used to solve Nash equilibrium in dominance-solvable games. We characterize IESDS * by means of a “stability” criterion, and offer a sufficient and necessary epistemic condition for IESDS *. We show by an example that IESDS * may generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient/necessary conditions under which IESDS * preserves the set of Nash equilibria.

Iterated Strict Dominance in General Games

We offer a definition of iterated elimination of strictly dominated strategies (IESDS) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS is always a well-defined order independent procedure that can be used to solve Nash equilibrium in dominance-solvable games. We characterize IESDS by means of a stability criterion, and offer a sufficient and necessary epistemic condition for IESDS. We show by an example that IESDS may generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient/necessary conditions under which IESDS preserves the set of Nash equilibria.

Cooperation in Experimental Games of Strategic Complements and Substitutes

Results are reported of a laboratory experiment aimed at examining whether strategic substitutability and strategic complementarity have an impact on the tendency to cooperate in two-player dominance-solvable games with a Pareto-inefficient Nash equilibrium. We find that there is significantly more cooperation when actions exhibit strategic complementarities than in case of strategic substitutes.

Tacit Cooperation, Strategic Uncertainty, and Coordination Failure: Evidence from Repeated Dominance Solvable Games

This paper reports an experiment designed to discover how the prospect of future interaction influences people's ability to tacitly cooperate in repeated dominance solvable games. The experiment varies two treatment variables: whether the constituent game is solvable by strict or iterated dominance and whether prospective interaction is finitely or randomly terminated. Specifically, we introduce a special repeated matching protocol consisting of an initial phase terminated randomly and a terminal phase of T periods. We call this protocol T-death. The T-death protocol allows us to observe a pair's behavior in both a sequence of infinite continuation games and a sequence of finite continuation games. Journal of Economic Literature Classification Numbers: C720, C920, 1120, 1400.

Coalition-Proof Equilibrium

We characterize the agreements that the players of a noncooperative game may reach when they can communicate prior to play, but they cannot reach binding agreements: Acoalition-proof equilibriumis a correlated strategy from which no coalition has an improving and self-enforcing deviation. We show that any correlated strategy whose support is contained in the set of actions that survive the iterated elimination of strictly dominated strategies and weakly Pareto dominates every other correlated strategy whose support is contained in that set, is a coalition-proof equilibrium. Consequently, the unique equilibrium of a dominance solvable game is coalition-proof.Journal of Economic LiteratureClassification Numbers: C72, D82.

“Evolutionary” selection dynamics in games: Convergence and limit properties

This paper discusses convergence properties and limiting behavior in a class of dynamical systems of which the replicator dynamics of (biological) evolutionary game theory are a special case. It is known that such dynamics need not be well-behaved for arbitrary games. However, it is easy to show that dominance solvable games are convergent for any dynamics in the class and, what is somewhat more difficult to establish, weak dominance solvable games are as well, provided they are “small” in a sense to be made precise in the text. The paper goes on to compare dynamical solutions with standard solution concepts from noncooperative game theory.

On dominance solvable auctions in the general symmetric model

A new class, of second-price auctions of the general symmetric model, is shown to be a dominance solvable game. Illuminating examples are provided.

A Class of Dominance Solvable Common-Value Auctions

Dominant strategies seldom exist in non-cooperative games. Moulin's concept of a dominance solvable game generalizes, dominant strategy without dramatic loss in appeal. We consider a class of common-value auctions characterized by the property that the maximum of a collection of informative signals is a sufficient statistic for the entire collection. We demonstrate that this class of second-price auctions is dominance solvable.