Mixed-strategy Equilibrium Research Articles

Private provision of discrete public goods

We investigate private provision of discrete public goods under refund and cost-sharing. If it is commonly believed that individuals may be warm-glow altruists and the group-size is a Poisson random variable, then the equilibrium distribution of collected contributions is uniquely determined. If composition uncertainty is very small and the expected group-size sufficiently large, the distribution of contributions can be described by concentrating in a symmetric mixed-strategy equilibrium. As the expected group-size increases, the probability a randomly selected player contributes and the associated expected number of contributions converge to the corresponding ones in the symmetric mixed-strategy equilibrium of the game with no uncertainty.

Multi-Robot Searching using Game-Theory Based Approach

This paper proposes a game-theory based approach in a multi–target searching using a multi-robot system in a dynamic environment. It is assumed that a rough priori probability map of the targets' distribution within the environment is given. To consider the interaction between the robots, a dynamic-programming equation is proposed to estimate the utility function for each robot. Based on this utility function, a cooperative nonzero-sum game is generated, where both pure Nash Equilibrium and mixed-strategy Equilibrium solutions are presented to achieve an optimal overall robot behaviors. A special consideration has been taken to improve the real-time performance of the game-theory based approach. Several mechanisms, such as event-driven discretization, one-step dynamic programming, and decision buffer, have been proposed to reduce the computational complexity. The main advantage of the algorithm lies in its real-time capabilities whilst being efficient and robust to dynamic environments.

Coin Assaying and Commodity Money

How beneficial is a fully recognizable money to the economy? This paper answers this question for the commodity money system. We build a model in which money is made of precious metal coins, yet its intrinsic quality is imperfectly recognizable by sellers. Based on historical evidence, we allow buyers to resort to a fixed cost coin certification technology - known as coin assaying - that fully reveals the quality of coins to sellers. We show that the two inefficiencies associated with imperfectly recognizable commodity money. Lower quantities traded or lower trading frequencies. are reduced thanks to this technology. We characterize pure and mixed strategy equilibria in which agents certify their coins, and show that certification coexists as a pure strategy equilibrium with the low-quantity inefficiency but does not coexist with the low-frequency inefficiency. The coin inspection technology is welfare improving when it removes the low-frequency inefficiency, but welfare deteriorating when it removes the low-quantity inefficiency.

Purification of a Mixed Strategy Equilibrium

Once two clever Athenian policemen were pursuing a Theban thief towards the city boundaries when they came upon a sign: “The Sign of the Grape. Thebans made welcome.” One said: “He will have taken refuge here.” “No,” cried the other, “this is just the place where he will expect us to look for him.” “Exactly,” rejoined the first, “so he will have decided to outwit us by entering.” They therefore searched the place thoroughly. Meanwhile, the Theban thief, who could not read, had run on to safety across the border.

Common Value Auctions with Asymmetric Bidder Information

We study common value auctions with two asymmetrically informed bidders using a simple binary model. A unique, generically asymmetric, mixed-strategy equilibrium exists for the first-price auction. Bidders get positive payoffs, with a superiorly informed bidder getting a higher payoff. One of the bidders submits a higher bid than the other on average: aggressive bidding is not necessarily associated with inferiority of information. A continuum of pure-strategy equilibria exist for the second-price auction. If bidders play the unique symmetric equilibrium strategies, second-price auctions are generically revenue-dominant. A change in the degree of asymmetry in general has an ambiguous effect on revenue.

Temporal and Spatial Allocation of Motor Preparation During a Mixed-Strategy Game

Adopting a mixed response strategy in competitive situations can prevent opponents from exploiting predictable play. What drives stochastic action selection is unclear given that choice patterns suggest that, on average, players are indifferent to available options during mixed-strategy equilibria. To gain insight into this stochastic selection process, we examined how motor preparation was allocated during a mixed-strategy game. If selection processes on each trial reflect a global indifference between options, then there should be no bias in motor preparation (unbiased preparation hypothesis). If, however, differences exist in the desirability of options on each trial then motor preparation should be biased toward the preferred option (biased preparation hypothesis). We tested between these alternatives by examining how saccade preparation was allocated as human subjects competed against an adaptive computer opponent in an oculomotor version of the game "matching pennies." Subjects were free to choose between two visual targets using a saccadic eye movement. Saccade preparation was probed by occasionally flashing a visual distractor at a range of times preceding target presentation. The probability that a distractor would evoke a saccade error, and when it failed to do so, the probability of choosing each of the subsequent targets quantified the temporal and spatial evolution of saccade preparation, respectively. Our results show that saccade preparation became increasingly biased as the time of target presentation approached. Specifically, the spatial locus to which saccade preparation was directed varied from trial to trial, and its time course depended on task timing.

CSR as Contractarian Model of Multi-Stakeholder Corporate Governance and the Game-Theory of its Implementation

Corporate Social Responsibility (CSR) is here defined as a multi-stakeholder model of corporate governance and fiduciary duties naturally emerging from a critical assessment of the incomplete contracts view of the firm based on concepts like as authority and residual rights of control. As far as the normative point of view is concerned, multi-stakeholder fiduciary duties are deduced from a theory of the firm’s stakeholders Social Contract. This provide for a clear cut and calculable objective function, a criterion for governance and strategic management no less able to set a bottom-line to the firm management than the profit maximization principle. The theory of co-operative bargaining games, and the Nash bargaining solution in particular, provides the key concepts. By the way this also answers some criticisms raised by Michael Jensen (2001) against the notion of stakeholders value. As far as implementation of the normative model is concerned, four roles of voluntary but explicit CSR norms or social standard are presented in terms of a non-cooperative game theory of implementation. It is shown that they allow the description of strategies and equilibria, even if multiple, in a game played under unforeseen contingencies. Secondly, a CSR norm permits the ex ante selection of the equilibrium point that meets the requirements of an impartial choice. An explicit agreement on a contractarian norm is moreover a way to introduce psychological conformist equilibria, and quite surprisingly to derive the significant result that mixed strategy equilibria are absent in a psychological repeated Trust Game. Lastly, a cognitive and predictive role is played by an agreed CSR norm as the appropriate starting point for an equilibrium selection mechanism that, from a state of predictive uncertainty about possible equilibria, generates a state of mutually consistent expectations consistent with the prediction that all players will converge on the psychological equilibrium fully conforming with the norm as the effective solution of the game. Keywords: corporate social responsibility,stakeholders, incomplete contracts, social norms, reputations, psychological games, conformity, equilibrium selection

Multiproduct price competition with heterogeneous consumers and nonconvex costs

This paper extends the oligopolistic model of price competition to environments with multiple goods, heterogeneous consumers, and arbitrary continuous cost functions. A Nash equilibrium in mixed strategies with an endogenous sharing rule is proven to exist. It is also shown that, in environments with fixed costs and constant marginal costs, all (symmetric and asymmetric) equilibria exhibit price dispersion across stores. Furthermore, the paper identifies scenarios in which prices will necessarily be random. In these markets, stores keep each other guessing because, given the fixed costs, they would incur a loss if their price strategies were anticipated and beaten by competitors. This is interpreted as an important economic feature that is possibly behind random price promotions such as weekly specials.

THE MIXED STRATEGY NASH EQUILIBRIUM OF THE TELEVISON NEWS SCHEDULING GAME

We characterize the unique mixed-strategy equilibrium of an extension of the television news sheduling game of Cancian, Bergstrom and Bills (1995) where viewers want to watch the first newscast broadcast after they return home. A fraction of the viewers record randomly one of the newscast to watch them in case they are too late. At equilibrium, neither of the two stations broadcasts its evening news in the first part of the evening and the density function is strictly decreasing.

Vickrey’s model of traffic congestion discretized

Vickrey’s seminal analysis of urban traffic congestion assumes a continuum of commuters acting selfishly and a continuous strategy space. We propose a discrete version of his model that assumes a finite number of commuters and a discrete strategy space. We then present an algorithm for numerically computing a symmetric mixed-strategy equilibrium solution for the discrete model and compare it with the deterministic equilibrium solution for the continuous model due to Arnott et al. [Arnott, R., de Palma, A., Lindsey, R., 1990. Economics of a bottleneck. Journal of Urban Economics 27(1), 111–130]. We report significant discrepancies in travel costs and distributions of departure time between the two solutions that slowly decrease as the number of commuters increases. We then propose and exemplify two extensions of the discrete model that allow for an uncertain number of commuters and an alternative mode of transportation not subject to congestion.

A NONCOOPERATIVE SHIPPING COURNOT DUOPOLY WITH LINEAR-QUADRATIC TRANSPORT COSTS AND CIRCULAR SPACE*

The mixed strategy equilibria in a location quantity duopoly model with circular markets are investigated. We find that a continuum of equilibria exists when the transport cost function is linear. However, if costs are strictly concave or convex, most strategies fail to qualify as equilibria. For any integer n, there are nondegenerate mixed strategy equilibria in which each firm locates at n possible locations with equal probability. This result explains possible outcomes in which firms’ on-path locations are not minimally or maximally differentiated.

Conhecimento imperfeito, custo de otimização e racionalidade limitada: uma dinâmica evolucionária de ajustamento nominal incompleto

Elabora-se um modelo dinâmico evolucionário que proporciona microfundamentos ao ajustamento nominal incompleto. Mostra-se a emergência de equilíbrios evolucionários tanto de estratégia pura (sobrevivência somente de estratégias de fixação de preço plenamente racionais ou de racionalidade limitada) como de estratégia mista (sobrevivência de ambas as estratégias). Nos equilíbrios de estratégia pura a moeda é neutra, enquanto no equilíbrio de estratégia mista a moeda pode não ser neutra. Além disso, analisa-se a possibilidade de bifurcações no número de - e nas propriedades de estabilidade dos - equilíbrios evolucionários geradas pela política monetária.

Capacity and entry issues in online exchanges

With the advent of open standards and Internet technologies, the number of sellers who can participate in online exchanges is greatly increased. We model the competition between identical sellers vying for the same business, and find that there exists a mixed-strategy equilibrium in prices. The results help us understand the dynamics between a seller’s capacity and his motivation to participate in an auction.

Auctions with Variable Supply: Uniform Price Versus Discriminatory

We examine an auction in which the seller determines the supply after observing the bids. We compare the uniform price and the discriminatory auction in a setting of supply uncertainty. Uncertainty is caused by the interplay of two factors: the seller's private information about marginal cost, and the seller's incentive to sell the profit-maximizing quantity given the received bids. In every symmetric mixed strategy equilibrium, bidders submit higher bids in the uniform price auction than in the discriminatory auction. In the two-bidder case this result extends to the set of rationalizable strategies. As a consequence, we find that the uniform price auction generates higher expected revenue for the seller and higher trade volume

Predicting Human Interactive Learning by Regret-Driven Neural Networks

Much of human learning in a social context has an interactive nature: What an individual learns is affected by what other individuals are learning at the same time. Games represent a widely accepted paradigm for representing interactive decision-making. We explored the potential value of neural networks for modeling and predicting human interactive learning in repeated games. We found that even very simple learning networks, driven by regret-based feedback, accurately predict observed human behavior in different experiments on 21 games with unique equilibria in mixed strategies. Introducing regret in the feedback dramatically improved the performance of the neural network. We show that regret-based models provide better predictions of learning than established economic models.

Approximate equilibria for Bayesian games

In this paper the problem of the existence of approximate equilibria in mixed strategies is central. Sufficient conditions are given under which approximate equilibria exist for non-finite Bayesian games. Further one possible approach is suggested to the problem of the existence of approximate equilibria for the class of multicriteria Bayesian games.

Large games and the law of large numbers

This paper introduces discrete large games where the set of players is a countable dense ‘grid’ with a finitely additive distribution. In these games any function from player names to mixed actions is a legitimate strategy profile. No extraneous continuity or measurability conditions are assumed. Randomness can be modeled explicitly and an exact law of large numbers holds. Equilibria enjoy a strong purification property: every realization of every mixed strategy equilibrium is a pure strategy equilibrium almost surely. Every continuum-player game has a discrete large game representation that preserves the original payoffs, strategy profiles and equilibria. It is argued that strategy profiles in continuum-player games have an ambiguous meaning because measurability requirements force the smoothing out of individual variations. These variations have clear strategic meaning in finite-player games and can be expressed in discrete large games, but not when the set of players is the continuum.

Cost differentials and mixed strategy equilibria in a Hotelling model

We introduce heterogeneity of production costs into the location-price Hotelling model discussed by d’Aspremont et al. (Econometrica 47:1145–1150, 1979). Maximum differentiation appears if the cost difference between two firms is small, whereas no pure strategy equilibrium exists if it is large. We examine the mixed strategy equilibria when no pure strategy equilibrium exists. We find that the following simple symmetric mixed strategy equilibrium exists, which never becomes an equilibrium if no cost differential exists: each firm chooses to locate at the two edges of the linear city randomly.

Statistical Prediction of the Outcome of a Noncooperative Game

Conventionally, game theory predicts that the joint mixed strategy of players in a noncooperative game will satisfy some concept. Relative probabilities of the joint strategies satisfying the concept are unspecified, and all strategies not satisfying it are assigned probability zero. As an alternative, we recast the prediction problem of game theory as statistically estimating the joint strategy, from data that consists of the game specification. This replaces the focus of game theory, on specifying a set of equilibrium mixed strategies, with a new focus, on specifying a probability density over all mixed strategies. We explore a Bayesian version of such a Predictive Game Theory (PGT). We show that for some games the peaks of the posterior over joint strategies approximate quantal response equilibria. We also show how PGT provides a best single prediction for any noncooperative game, i.e., a universal refinement. We also show how regulators can use PGT to make optimal decisions in situations where conventional game theory cannot provide advice.

All-Pay Auctions with Negative Prize Externalities: Theory and Experimental Evidence

The paper characterizes the mixed-strategy equilibria in all-pay auctions with endogenous prizes that depend positively on own effort and negatively on the effort of competitors. Such auctions arise naturally in the context of investment games, lobbying games, and promotion tournaments. We also provide an experimental analysis of a special case which captures the strategic situation of a two-stage game with investment preceding homogenous Bertrand competition. We obtain overinvestment both relative to the mixed-strategy equilibrium and the social optimum.