Mixed-strategy Equilibrium Research Articles

Desemprego e flexibilidade de salários em um contexto evolucionário

Neste artigo apresentamos um modelo de jogo evolucionário, por meio do qual discutimos as relações entre flexibilidade de salários e desemprego, em uma economia com barganhas salariais descentralizadas. As estratégias dos trabalhadores consistem em um número finito, porém arbitrariamente grande, salários nominais, e as firmas escolhem os níveis de emprego maximizadores de lucros. Mostramos que: 1) o modelo comporta múltiplos equilíbrios com homogeneidade de salários nominais (de estratégia pura) e com heterogeneidade de salários nominais (de estratégia mista); 2) o equilíbrio de médio prazo é selecionado por um processo de aprendizagem social, representado como uma dinâmica replicadora; e 3) conceber a variação dos salários nominais como um processo evolucionário, no qual os trabalhadores procuram a melhor estratégia, implica que a economia não tende necessariamente ao pleno emprego.

Learning and Information Aggregation in an Exit Game

We analyse information aggregation in a stopping game with uncertain pay-offs that are correlated across players. Players learn from their own private experiences as well as by observing the actions of other players. We give a full characterization of the symmetric mixed strategy equilibrium, and show that information aggregates in randomly occurring exit waves. Observational learning induces the players to stay in the game longer. The equilibria display aggregate randomness even for large numbers of players.

Set Identification in Models with Multiple Equilibria

We propose a computationally feasible way of deriving the identified features of models with multiple equilibria in pure or mixed strategies. It is shown that in the case of Shapley regular normal form games, the identified set is characterized by the inclusion of the true data distribution within the core of a Choquet capacity, which is interpreted as the generalized likelihood of the model. In turn, this inclusion is characterized by a finite set of inequalities and efficient and easily implementable combinatorial methods are described to check them. In all normal form games, the identified set is characterized in terms of the value of a submodular or convex optimization program. Efficient algorithms are then given and compared to check inclusion of a parameter in this identified set. The latter are illustrated with family bargaining games and oligopoly entry games.

Nash equilibria in quantum games

For any two-by-two game G \mathbf {G} , we define a new two-player game G Q \mathbf {G}^Q . The definition is motivated by a vision of players in game G \mathbf {G} communicating via quantum technology according to the protocol introduced by J. Eisert and M. Wilkins. In the game G Q \mathbf {G}^Q , each player’s (mixed) strategy set consists of the set of all probability distributions on the 3-sphere S 3 \mathbf {S}^3 . Nash equilibria in the game can be difficult to compute. Our main theorems classify all possible mixed-strategy equilibria. First, we show that up to a suitable definition of equivalence, any strategy that arises in equilibrium is supported on at most four points; then we show that those four points must lie in one of a small number of allowable geometric configurations.

Brown–Robinson method for interval matrix games

In this paper, two-person interval matrix games are considered, and by means of acceptability index, Brown–Robinson method to find a mixed-strategy equilibrium is adapted to interval matrix games. Numerical examples are also given.

Analysis of a dual-sourcing service provision game with symmetric and asymmetric vendors

This paper models a service provision game between two vendors under symmetric and asymmetric cost structures, who compete in first-period service quality levels with each other, with the aim of winning the larger share of the buyer’s fixed reward in the second period. This game differs from the previous studies in that the buyer maintains dual sourcing over two periods and thus has different characteristics. We show that this service provision game has distinct mixed-strategy equilibria with the vendors under symmetric and asymmetric cost structures. We find that the larger the winner’s share in the second period, the higher the vendors’ first-period service quality levels. However, increasing the winner’s share in the second period does not necessarily benefit the vendor with the lower marginal cost, but surely hurts the equilibrium profit of the vendor with the higher marginal cost.

All-pay auctions with discrete action spaces

This article describes all-pay auctions with discrete action spaces and shows that pure-strategy Nash equilibria may exist. Such equilibria are shown to be potentially more favourable to both buyers and sellers than the more standard mixed-strategy equilibria, thus suggesting that it can be in the interest of both sides to limit bidders' action spaces.

Information Policy in Contests with Little Noise

In this paper, we analyze a contest organizer’s decision to release intermediate information to the contestants. Contrary to the existing literature, we assume that there is little noise affecting the contest outcome so that there may be only mixed-strategy equilibria at the contest stage. Optimal information policy is found to depend on the prize spread and information may be revealed if contestants are very heterogeneous. This is different from results in the previous literature, which finds that the optimal information policy is independent of the prize structure and that information should be concealed from the contestants in case they are heterogeneous.

Bertrand-Edgeworth Auction with Multiple Asymmetric Bidders

Motivated by supply competitions in the service sector, we consider a version of the Bertrand-Edgeworth game where capacitated suppliers compete in prices to serve a deterministic demand and a price cap is imposed exogenously. We characterize the equilibrium structure for games with multiple suppliers that are asymmetric in cost and capacity. A pure-strategy equilibrium exists only in special cases, otherwise a mixed-strategy equilibrium prevails. For a mixed-strategy equilibrium, we show there exists an anchoring supplier whose pricing range covers those of the other suppliers, and we pin down the lower and upper bounds of that price range. We derive a canonical closed-form solution for a set of games, and for other more general cases, we demonstrate a numerical scheme for equilibrium calculation. In contrast to a well-known intuition established in the existing literature, we identify a new equilibrium pricing structure in games with N >= 3 suppliers who have similar costs but different capacities: A low-capacity supplier's price distribution is more concentrated than those of its high-capacity competitors, by spanning a smaller interval. We demonstrate an interesting strategy for a buyer with a competitive supply base: Subsidizing a cost-competitive incumbent may reduce the total procurement cost.

Existence of Equilibria in Auctions with Private Values

A first-price sealed-bid auction of an item for which bidders are risk-neutral and have privately known values is shown to have an equilibrium in mixed behavioral strategies if the joint distribution of bidders' values has a continuous density on a cubical support. Such an equilibrium has atomless distributions of bids and is not affected by the rule for resolving tied bids.

Existence of Equilibria in Auctions with Interdependent Values: Two Symmetic Bidders

For two symmetric bidders, weak monotonicity conditions are shown to imply existence of an equilibrium in mixed behavioral strategies for a sealed-bid first-price auction of an item for which each bidder's value depends on every bidder's observed signal. Such an equilibrium has atomless distributions of bids and thus is unaffected by how tied bids are resolved.

Imposing a turnout threshold in referendums

We construct a model of a yes/no referendum for which the outcome is valid only if the voter turnout rate is greater than a predetermined value. With such a turnout threshold, three types of equilibria exist, in each of which the voters who favor the status quo either go to the polls, abstain, or use mixed strategies. We show that abstaining is more likely to realize the status quo than going to the polls if the threshold is sufficiently high, whereas one of the mixed-strategy equilibria is least likely to realize the status quo for any level of threshold.

COSTLY BUYER SEARCH IN LABORATORY MARKETS WITH SELLER ADVERTISING*

In this experiment, sellers simultaneously choose prices and advertising strategies. Buyers either purchase at an advertised price or search sequentially for other prices. In the unique symmetric equilibrium, sellers charge a high unadvertised price or advertise a price chosen from a lower interval. Increases in search or advertising costs raise equilibrium prices and affect equilibrium advertising intensity. Empirical results are consistent with most comparative static predictions. Sellers, however, price much lower and advertise more intensely than predicted. Consequently, market outcomes more closely resemble a perfect information, Bertrand‐like equilibrium than the imperfect information, mixed strategy equilibrium with significant seller market power.

Technology Adoption in Critical Mass Games: Theory and Experimental Evidence

We analyze the choices between two technologies A and B that both exhibit network eects. We introduce a critical mass game in which coordination on either one of the stan- dards constitutes a Nash equilibrium outcome while coordination on standard B is assumed to be payo¤-dominant. We present a heuristic de…nition of a critical mass and show that the critical mass is inversely related to the mixed strategy equilibrium. We show that the critical mass is closely related to the risk dominance criterion, the global game theory, and the maximin criterion. We present experimental evidence that both the relative degree of payodominance and risk dominance explain players'choices. We …nally show that users' adoption behavior induces …rms to select a relatively unrisky technology which minimizes

Self-organized criticality in a dynamic game

We investigate conditions under which self-organized criticality (SOC) arises in a version of a dynamic entry game. In the simplest version of the game, there is a single location—a pool—and one agent is exogenously dropped into the pool every period. Payoffs to entrants are positive as long as the number of agents in the pool is below a critical level. If an agent chooses to exit, he cannot re-enter, resulting in a future payoff of zero. Agents in the pool decide simultaneously each period whether to stay in or not. We characterize the symmetric mixed strategy equilibrium of the resulting dynamic game. We then introduce local interactions between agents that occupy neighboring pools and demonstrate that, under our payoff structure, local interaction effects are necessary and sufficient for SOC and for an associated power law to emerge. Thus, we provide an explicit game-theoretic model of the mechanism through which SOC can arise in a social context with forward looking agents.

Mixed strategy equilibria in repeated games with one‐period memory

Infinitely repeated games is the pre‐dominant paradigm within which economists study long‐term strategic interaction. The standard framework allows players to condition their strategies on all past actions; that is, assumes that they have unbounded memory. That is clearly a convenient simplification that is at odds with reality. In this paper we restrict attention to one‐period memory and characterize all totally mixed equilibria. In particular, we focus on strongly mixed equilibria. We provide conditions that are necessary and sufficient for a game to have such an equilibrium. We further demonstrate the exact set of payoffs that can be generated by such equilibria.

Regulation of systemic liquidity risk

This paper provides a baseline model for regulatory analysis of systemic liquidity shocks. We show that banks may have an incentive to invest excessively in illiquid long-term projects. In the prevailing mixed-strategy equilibrium, the allocation is inferior from the investor’s point of view since some banks free ride on the liquidity provision due to their limited liability. The paper compares different regulatory mechanisms to cope with the externalities. We show that a combination of liquidity regulation ex ante and lender of last resort policy ex post can maximize investor payoff. In contrast, both “narrow banking” and imposing equity requirements as a buffer are inferior mechanisms for coping with systemic liquidity risk.

On game-theoretic characterisation of stochastic independence

Dropping the assumption of the stochastic independence of players' randomised choices in non-cooperative games, we introduce the notion of a type of dependence. It is proved that the stochastic independence is the unique type of dependence for which any finite non-cooperative game has a Nash equilibrium in mixed strategies.

Pass or Run: An Empirical Test of the Matching Pennies Game Using Data from the National Football League

This article examines play calling in the National Football League (NFL). It finds that a mixed strategy equilibrium game explains NFL play calling better than standard optimization techniques. When a quarterback is injured and replaced with a less capable backup, standard optimization theory suggests that the offense will run more often, passing less. Our game theoretic model predicts that the offense will not change its play calling because the defense will play against the run more often. Using every first half play from the 11 teams that had a starting quarterback miss action because of injury in the 2006 season, we find that the injury did not alter the likelihood that the offense would pass. We also find that coaches randomly mix passing and running plays, as the mixed strategy games predict.

Cycles versus equilibrium in evolutionary games

Mixed-strategy equilibria are typically rather unstable in evolutionary game theory. “Monocyclic” games, such as Rock–Paper–Scissors, have only mixed equilibria, some of which are “stable” in the sense that sequential best replies lead to them; yet, even these games are prone to stable cycles under discrete-time simultaneous best replies, giving an unusual equilibrium-selection problem. This article analyzes such games in a random-utility setting where changing strategies is costly, and the speed of the dynamic is, thus, endogenous. The stochastically stable outcome is determined by the cost of switching strategies; when switching costs are high, mixed equilibria are selected, whereas when switching costs are low, cycles are selected.