Symmetric Mixed-strategy Equilibrium Research Articles

Efficient equilibria in common interest voting games

We develop a unified derivation of asymmetric pure strategy equilibria and their optimality in the canonical common interest voting model of Austen-Smith and Banks (Am Polit Sci Rev 90(1):34–45, 1996). We also study the relationship between the most efficient equilibria, which have a remarkably simple and intuitive structure, and the symmetric mixed strategy equilibrium that has been commonly studied in the literature. In particular, while the efficiency in the symmetric mixed strategy equilibrium under unanimity rule is known to be decreasing in the number of voters, the efficiency does not depend on the number of voters above a threshold in the most efficient equilibria.

Mixed-strategy equilibrium of the symmetric production in advance game: The missing case

The mixed-strategy equilibrium of the symmetric production-in-advance type capacity-constrained Bertrand-Edgeworth duopoly game has not been derived analytically over the entire range of intermediate capacities in the literature. In this paper we derive for the missing region a symmetric mixed-strategy equilibrium analytically.

Information sharing decisions in all-pay auctions with correlated types

In many real-life competitions, contestants may not be aware of their own type (e.g., value or ability) prior to the contest. Furthermore, contestants’ types, which are observed privately after entering the competition, are frequently correlated with one another. We examine a two-stage competition that involves two players with correlated (binary) types. In the first stage, players decide simultaneously or sequentially on the probabilities they use to disclose or conceal their type, which will become their private information later on. In the second stage, each player privately observes their own type and commits to disclosing or concealing it, after which they compete in an all-pay auction. We discover that information sharing does not occur when players’ types are negatively correlated. However, when players’ types are positively correlated, information is partially shared in all equilibria examined in this study. In an asymmetric pure strategy equilibrium, one player shares his information with probability one and the other player with probability zero. In a symmetric mixed strategy equilibrium, each player shares his information with the same positive probability.

Implicit coordination in an <inline-formula><tex-math id="M1">$ n $</tex-math></inline-formula>-player target selection game

We show that a class of target selection games has a unique symmetric mixed strategy Nash equilibrium $ x^\ast $. We also present a simple recursion that computes $ x^\ast $ within a finite number of iterations. Along the way, we introduce the concept of value compactness that points directly to the properties of the unique symmetric mixed strategy equilibrium, including the fact that the unique equilibrium does not force any player to make an arbitrary decision about associating positive probability with subsets of targets having a particular utility– a property that we refer to as implicit coordination.

Production in advance versus production to order: Equilibrium and social surplus

The mixed-strategy equilibrium of the production-in-advance type capacity-constrained Bertrand–Edgeworth duopoly game has not been derived analytically for the case of intermediate capacities in the literature. As in the case of the production-to-order version of the same game, the case of intermediate capacities turned out to be the most difficult one compared with the cases of small and large capacities. In this paper we derive analytically a symmetric mixed-strategy equilibrium of the production-in-advance version of this game for a large region of intermediate capacities. Nevertheless we show that in general the economic surplus within the production-to-order type environment is higher than in the respective production-in-advance type one, and, therefore, production-to-order should be preferred to production-in-advance if the mode of production can be influenced by the government.

A complete-information partnership contest with negative productivity spillovers

This note examines a complete-information contest with non-linear rank-order effects. The players’ payoffs incorporate a multiplicative term that allows for negative productivity spillovers. Under certain conditions, our game has a symmetric mixed-strategy equilibrium. To characterize this equilibrium, we adapt a method originally developed for linear settings. We obtain closed-form solutions for the mixing distribution, as well as for the expected efforts and payoffs. The note also explores the implications of negative productivity spillovers for optimal contest design. We show that a planner who seeks to maximize the participants’ expected payoffs may choose interior values for her instruments. We derive simple expressions for these values.

A model of sales with differentiated and homogeneous goods

This paper extends the classical model of sales (Varian, 1980; Rosenthal, 1980) by adding product differentiation. Instead of uninformed (i.e. loyal) customers, our setting features “variety seekers”. These consumers regard the products as imperfect substitutes. As in the original model, the firms also serve “bargain seekers” who buy the cheapest product. The discontinuous demand structure precludes any pure-strategy equilibria. We characterize the symmetric mixed-strategy equilibrium of the modified game. In contrast to the original model, the upper bound on prices and the equilibrium expected profits are decreasing in the mass of bargain seekers.

Characterizing the Symmetric Mixed-Strategy Equilibrium in Rent-Seeking Contests: A Real Analysis Approach

We take a real analysis approach to characterize the symmetric mixed-strategy equilibrium (SMSE) of rent-seeking contests for any intermediate value of the decisiveness parameter, allowing for both the ratio-form and the difference-form of contest success functions (CSF) that are axiomatized in Skaperdas (1996). For the ratio-form Tullock contest, we reproduce the results obtained by Ewerhart (2015) by invoking sophisticated complex analysis that SMSE is supported by a countably infinite set of pure strategies that has zero bid as an accumulation point and that the rent is completely dissipated. For the Hirshleifer-style difference-form of CSF, we show that the support of SMSE is a finite set of pure strategies including zero bid as one mass point and that the rent is partially dissipated. The similarity and differences in the profile of SMSE between the two canonical forms of CSF in the literature of contests are thereby illuminated.

Horizontal Product Differentiation in Varian's Model of Sales

We consider the explicit introduction of firms' choice of location to Varian's model of sales for a two-stage spatial competition model based on a standard Hotelling's linear city model. This model is the formalization of Varian's model of sales in the context of Hotelling's spatial competition. We obtain three main results. First, we show that there exists a subgame perfect equilibrium in which each firm chooses a symmetric mixed strategy equilibrium profile. This equilibrium includes symmetric location pairs and asymmetric location pairs. Second, the equilibrium behaviors in our model are randomized at both location and price stages. Third, we show that expected profits in a subgame perfect equilibrium are equal to the maximum monopoly profit from an uninformed market. Thus, even when product differentiation is explicitly introduced into a Varian-type model, Varian's implication can be retained; the opportunity for profit in an informed market is lost with competition.

WHY ZERO TOLERANCE OF MISCONDUCT IS UNDESIRABLE IN CONTESTS

Competition can foster misconduct in any circumstance where the organizer of a contest cannot perfectly monitor contestants' actions or when doing so is prohibitively costly. Although misconduct comprises all actions that are contrary to the interest of the organizer, it is not necessarily the case that it is optimal to prohibit all such behavior. In this paper, we determine the equilibrium level of misconduct chosen by players in a symmetric rank‐order tournament between two competitors in which the organizer tolerates some level of misconduct. In addition to showing that zero tolerance may not minimize the level of misconduct in equilibrium, we show that there exists a range of tolerated misconduct where a symmetric mixed strategy equilibrium exists with players cheating (i.e., misconduct above the tolerated level) with some probability. When the gain from misconduct is uncertain and unknown the contest organizer faces a tradeoff: tolerating more misconduct will reduce such behavior when the state of temptation is high, but increase it when temptation is low. (JEL J33, K42)

Screening and Adverse Selection in Frictional Markets

We develop a tractable framework for analyzing adverse selection economies with imperfect competition. In our environment, uninformed buyers offer a general menu of screening contracts to privately informed sellers. Some sellers receive offers from multiple buyers while others receive offers from only one buyer, as in Burdett and Judd (1983). This specification allows us to smoothly vary the degree of competition, nesting monopsony and perfect competition a la Rothschild and Stiglitz (1976) as special cases. We show that the unique symmetric mixed-strategy equilibrium exhibits a strict rank-preserving property, in that different types of sellers have an identical ranking over the various menus offered in equilibrium. These menus can be all separating, all pooling, or a mixture of both, depending on the distribution of types and the degree of competition in the market. This calls into question the practice of using the incidence of separating contracts as evidence of adverse selection without controlling for market structure. We examine the relationship between exante welfare and the degree of competition, and show that in some cases an interior level of frictions maximizes welfare, while in other cases competition is unambiguously bad for welfare. Finally, we study the effects of various policy interventions — such as disclosure and non-discrimination requirements — and show that our model generates new, and perhaps counter-intuitive insights.

Generalising Bertrand competition: a special case of the Hotelling model

In the classic Bertrand duopoly, a firm need only undercut its competitor by an arbitrarily small amount in order to sell to all the consumers. This paper generalises the Bertrand model so that a firm must be more than \(\varepsilon \) cheaper than its competitor in order to take the market. I characterise the unique symmetric mixed-strategy equilibrium. Firms earn strictly positive expected profits in equilibrium which are increasing in \(\varepsilon \). The model is also a special case of the price-setting stage of the Hotelling model with a non-uniform distribution of consumers. The model can also be formulated as one of sequential search without replacement from a discrete distribution of prices.

Mixed Equilibrium in a Pure Location Game: The Case of n≥4$$n \ge 4$$ Firms

AbstractThe Hotelling game of pure location allows interpretations in spatial competition, political theory, and strategic forecasting. In this paper, the doubly symmetric mixed-strategy equilibrium for

Mixed equilibria in Tullock contests

Any symmetric mixed-strategy equilibrium in a Tullock contest with intermediate values of the decisiveness parameter (“\(2<R<\infty \)”) has countably infinitely many mass points. All probability weight is concentrated on those mass points, which have the zero bid as their sole point of accumulation. With contestants randomizing over a non-convex set, there is a cost of being “halfhearted,” which is absent from both the lottery contest and the all-pay auction. Numerical bid distributions are generally negatively skewed and exhibit, for some parameter values, a higher probability of ex-post overdissipation than the all-pay auction.

Mixed Equilibrium in a Pure Location Game: The Case of N 4 Firms

The Hotelling game of pure location allows interpretations in spatial competition, political theory, and professional forecasting. In this paper, the doubly symmetric mixed-strategy equilibrium for n ≥ 4 firms is characterized as the solution of a well-behaved boundary value problem. The analysis suggests that, in contrast to the cases n = 3 and n → ∞ , the equilibrium for a finite number of n ≥ 4 firms tends to overrepresent locations at the periphery of its support interval. Moreover, in the class of examples considered, an increase in the number of firms universally leads to a wider range of location choices and to a more dispersed distribution of individual locations. The results are used to comment on the potential benefit of competition in forecasting markets.

The Minority of Three-Game: An Experimental and Theoretical Analysis

We report experimental results on the minority of three-game, where three players choose one of two alternatives and the most rewarding alternative is the one chosen by a single player. This coordination game has many asymmetric equilibria in pure strategies that are non-strict and payoff-asymmetric and a unique symmetric mixed strategy equilibrium in which each player’s behavior is based on the toss of a fair coin. This straightforward behavior is predicted by equilibrium selection, impulse-balance equilibrium, and payoff-sampling equilibrium. Experimental participants rely on various decision rules, and only a quarter of them perfectly randomize.

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.

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.

Endogenous Market Transparency and Price Competition

We introduce a model of price competition with endogenous market transparency, where it is costly for consumers to get informed about the announced prices. We show that there is symmetric mixed strategy equilibrium with a monotonic relationship between the degree of transparency and intensity of competition. Interestingly, we find that there are multiple equilibria with zero, low and high levels of market transparency with the high level of market transparency being the stable equilibrium. Once comparing the stable equilibrium with the welfare maximizing one we find that the private market solution entails excessive investments in information acquisition. That is, at high levels of transparency the gain of consumers from increased transparency is smaller than the loss in the firms’ profits leading to decrease in total welfare.

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, where 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 a higher expected revenue for the seller and a higher trade volume.