Symmetric Pure-strategy Equilibrium Research Articles

Cheap Talk, Monitoring and Collusion

Many collusive agreements involve the exchange of self-reported sales data between competitors, which use them to monitor compliance with a target market share allocation. Such communication may facilitate collusion even if it is unverifiable cheap talk and the underlying information becomes publicly available with a delay. The exchange of sales information may allow firms to implement incentive-compatible market share reallocation mechanisms after unexpected swings, limiting the recourse to price wars. Such communication may allow firms to earn profits that could not be earned in any collusive, symmetric pure-strategy equilibrium without communication.

Multi-prize contests with risk-averse players

This paper studies a multi-prize imperfectly discriminatory contest with symmetric risk-averse contestants. Adopting a multi-winner nested Tullock contest model, we first establish the existence and uniqueness of a symmetric pure-strategy equilibrium under plausible conditions. We then investigate the optimal prize allocation in the contest. Our analysis provides a formal account of the incentive effects triggered by a variation in the prevailing prize structure when contestants are risk averse. We demonstrate that contestants' incentive subtly depends not only on the degree of a contestant's risk aversion but also that of his prudence. The former affects the marginal benefit of effort, while the latter affects the marginal cost. We derive sufficient conditions under which a single-(multi-)prize contest would be optimal when the contest designer aims to maximize total effort. We also discuss in depth the roles played by risk aversion and prudence in optimal prize allocation.

Competitive price discrimination, imperfect information, and consumer search

Price discrimination in real-world settings is likely based on imperfect information. I analyse a homogenous goods framework where firms receive binary and noisy signals about consumer valuations and consumers engage in sequential search. Firms have no information about consumers' search histories. In this framework, the existence of on-path search can be understood as an imperfect screening device that firms employ to the detriment of consumers. Firm profits and equilibrium prices are highest in the unique symmetric pure-strategy equilibrium with search on the equilibrium path, as compared to any other symmetric pure-strategy equilibrium. The equilibrium with on-path search can only be sustained when search costs are at an intermediate level. At low search costs, an equilibrium is played in which there is no on-path search, but consumers use the threat of searching to ensure low prices. High levels of signal precision are detrimental to consumers by facilitating existence of the equilibrium with on-path search.

Strategic advance sales, demand uncertainty and overcommitment

We study a game in which producers can sell both before and after the realization of a random demand. Models of strategic forward trading and of advance sales to intermediaries or consumers share this structure. Demand uncertainty and committed advance sales imply that final-stage net residual demand may be so low that producers find additional sales unprofitable, or even so low that the final period price is zero, introducing some convexity into producers’ payoffs. If such an ex-post overcommitment occurs on the equilibrium path, producers reduce their advance sales, muting the pro-competitive effects found under deterministic demand. We prove existence of a unique symmetric pure-strategy equilibrium, whose nature depends on the support of the demand distribution relative to the marginal cost of production. For a narrow support, existence is assured for any distribution, while for wider support we establish a sufficient condition for existence. Our results provide a precise characterization of “minor” uncertainty, in which only the expected value of demand affects the equilibrium which is otherwise qualitatively similar to the deterministic case.

The Preventive and Treatment

The preventives are taken before getting ill, while the treatments are taken after getting ill. We consider the equilibrium decision of competitive firms to offer the preventive or the treatment and the inefficiency. We show that the profit of the preventive firm could be higher than the treatment firm under certain conditions. For the homogenous model, there is one symmetric pure strategy equilibrium and one mixed strategy equilibrium. For the Hotelling model, there could be different equilibria with different equilibrium decision of the preventive or treatment. The equilibrium could be that both firms offer the preventive or both firms offer the treatment. Moreover, we characterize the equilibrium efficacy decision of firms. Social welfare comparison is performed. Inefficiency and government regulation guidance is provided. Further, we consider the R&D and the side effects of the preventive and treatment. To the best of our knowledge, we are the first to consider these competitive economic effects of the preventive and treatment.

All-Pay Auctions with a Buy-Price Option

This paper studies all-pay auctions for a product in which there is a buy-price option for bidders to guarantee purchases at a seller-specified price. We analyze symmetric pure-strategy equilibria in the first- and second-price all-pay auctions with the buy-price option. Under these equilibria, the buy-price option will affect bidders' bidding behavior: while it has no impact on low-value bidders, it makes middle-value bidders bid more aggressively and high-value bidders less aggressively. Under the optimal buy-price, the first- and second-price all-pay auctions maintain the same expected profit while the buy-price in the second-price is higher than those in the first-price all-pay auction. Compared to the uniform posted-price selling mechanism, all-pay auctions with the buy-price cannot attain more expected profit under the fixed number of consumers. However, if there is an endogenous entry process, they can attract more consumers and then reach more profit in expectation.

Stationary, completely mixed and symmetric optimal and equilibrium strategies in stochastic games

In this paper, we address various types of two-person stochastic games—both zero-sum and nonzero-sum, discounted and undiscounted. In particular, we address different aspects of stochastic games, namely: (1) When is a two-person stochastic game completely mixed? (2) Can we identify classes of undiscounted zero-sum stochastic games that have stationary optimal strategies? (3) When does a two-person stochastic game possess symmetric optimal/equilibrium strategies? Firstly, we provide some necessary and some sufficient conditions under which certain classes of discounted and undiscounted stochastic games are completely mixed. In particular, we show that, if a discounted zero-sum switching control stochastic game with symmetric payoff matrices has a completely mixed stationary optimal strategy, then the stochastic game is completely mixed if and only if the matrix games restricted to states are all completely mixed. Secondly, we identify certain classes of undiscounted zero-sum stochastic games that have stationary optima under specific conditions for individual payoff matrices and transition probabilities. Thirdly, we provide sufficient conditions for discounted as well as certain classes of undiscounted stochastic games to have symmetric optimal/equilibrium strategies—namely, transitions are symmetric and the payoff matrices of one player are the transpose of those of the other. We also provide a sufficient condition for the stochastic game to have a symmetric pure strategy equilibrium. We also provide examples to show the sharpness of our results.

Innovation contests with entry auction

We consider innovation contests for the procurement of an innovation under moral hazard and adverse selection. Innovators have private information about their abilities, and choose unobservable effort in order to produce innovations of random quality. Innovation quality is not contractible. We compare two procurement mechanisms—a fixed prize and a first-price auction. Before the contest, a fixed number of innovators is selected in an entry auction, in order to address the adverse selection problem. We find that–if effort and ability are perfect substitutes–both mechanisms implement the same innovations in symmetric pure-strategy equilibrium, regardless of whether the innovators’ private information is revealed or not. These equilibria are efficient if the procurer is a welfare-maximizer.

Three firms on a unit disk market: intermediate product differentiation

Irmen and Thisse (1998) demonstrate that two firms competing with multicharacteristic products differentiate them in one characteristic completely, while keeping them identical in all others. This paper shows that their min- … -min-max differentiation result is not robust with respect to the number of firms. A market setting that replicates their result in a duopoly, but fails to do so in a three firm oligopoly is identified. Symmetric pure strategy equilibrium with three firms differentiating their products in two dimensions, but not completely in either of them, is a novel medium-medium differentiation result.

The optimal prize structure of symmetric Tullock contests

We show that the optimal prize structure of symmetric n-player Tullock tournaments assigns the entire prize pool to the winner, provided that a symmetric pure strategy equilibrium exists. If such an equilibrium fails to exist under the winner-take-all structure, we construct the optimal prize structure which improves existence conditions by dampening efforts. If no such optimal equilibrium exists, no symmetric pure strategy equilibrium induces positive efforts.

Endogenous Investment and Pricing under Uncertainty

I analyze investment and pricing incentives in a differentiated products framework with uncertain demand. Firms choose production capacities before observing demand and choose prices after demand is realized. Unlike previous models, when firms are identical, symmetric pure-strategy equilibria exist, even in the presence of very low capacity costs. The equilibrium outcomes are significantly different from the equivalent Cournot model. Firms choose to underutilize their capacity at times of low demand, and hold more capacity than predicted by Cournot. I show with a simple policy example that even the sign of the comparative statics may differ between the two models.

Competing auctions: Finite markets and convergence

The literature on competing auctions offers a model where sellers compete for buyers by setting reserve prices. An outstanding conjecture (e.g., Peters and Severinov 1997) is that the sellers post prices close to their marginal costs when the market becomes large. This conjecture is confirmed in this paper: we show that if all sellers have zero costs, then the equilibrium reserve price converges to 0 in distribution. Under further conditions there is a symmetric pure strategy equilibrium. In this equilibrium, if the ratio of buyers to sellers increases, then the equilibrium reserve price increases, and the reserve price is decreasing in the size of the market. Convergence of reserve prices occurs at the fast rate of 1/n if the ratio of buyers to sellers is held constant.

The Optimal Prize Structure of Symmetric Tullock Contests

We show that the optimal prize structure of symmetric n-player Tullock tournaments assigns the entire prize pool to the winner, provided that a symmetric pure strategy equilibrium exists. If such an equilibrium fails to exist under the winner-take-all structure, we construct the optimal prize structure which improves existence conditions by dampening efforts. If no such optimal equilibrium exists, no symmetric pure strategy equilibrium induces positive efforts.

Sequential Auctions with Information About Future Goods

When capacity-constrained bidders have information about a good sold in a future auction, they need to take the information into account in forming today's bids. The capacity constraint makes even otherwise unrelated objects substitutes and creates an equilibrium link between future competition and current bidding strategy. This paper proves the existence and uniqueness of a symmetric pure-strategy equilibrium under mild conditions on the population distribution of valuations, characterizes general properties of the equilibrium bidding strategy, and provides a simple technique for numerically approximating the bidding strategy for arbitrary valuation distributions. The key property of the equilibrium is that almost all bidders submit positive bids in the first stage, thereby ensuring trade with probability one. Even bidders who strongly prefer the second object submit a positive bid in the first auction, because losing the first auction is informative about the remaining competitors who also lost, and losing with a low bid indicates that these competitors are quite strong. Because of the guaranteed trade, the sequential auction with information about future goods is a very efficient trading mechanism, achieving more than 98 percent of the potential gains from trade across a wide variety of settings.

Evolution with Delay

This paper analyses evolutionary models with delays in players’ observations. I apply best response dynamics to symmetric 2 × 2 games, in particular the coordination game and the game with no symmetric pure strategy equilibrium. When the dynamics have no delay, their solutions converge to symmetric Nash equilibria monotonically. However when I introduce delays to the dynamics, although these dynamics represented by differential difference equations are seemingly simple, it shows that they have periodic solutions. JEL Classification Number: C79.

A Comment on "Learning, Mutation, and Long-Run Equilibria in Games"

mutation.) Kandori, Mailath, and Rob (henceforth KMR) first provide a useful general theorem concerning the stationary distribution of strategies under Darwinian dynamics. They then divide the analysis of the 2 x 2 game into three cases: dominant strategy games (e.g., prisoners' dilemma), coordination games, and games with no symmetric pure strategy equilibrium (e.g., battle of the sexes). We refer to these as DS, C, and NP games. In each case, KMR claim that, as the rate of mutation vanishes, the stationary distribution of strategies converges to a symmetric Nash equilibrium. They emphasize C games, which have two symmetric Nash equilibria, and characterize the conditions under which the distribution converges to the risk dominant equilibrium. In this note, we argue that while their formal conclusions for C games are correct, their results for DS and NP games are valid only for large populations of players. In small populations, Darwinian dynamics may produce non-Nash outcomes in these two cases. Section 1 summarizes the KMR model, and Section 2 provides examples of 2 x 2 games in which Darwinian dynamics generate non-Nash outcomes. A theorem in Section 3 describes the Darwinian equilibrium of any 2 x 2 game.

Gambling in a Malthusian Universe

Prospect choice is generally viewed as a game against nature. This article models prospect choice as an n- person game where each subject assumes that n- 1 others will be exposed to the same decision problem (prospect choice set) as self; the goal is not to “beat nature” but to do relatively better than rivals exposed to the same problem. Preference becomes strategy choice in n- person Nash equilibrium. When symmetric pure strategy equilibria do not exist, choice is a symmetric randomized equilibrium; here, uncertainty (probabilistic response) becomes a method of dealing with uncertainty in nature. The approach produces, qualitatively, several empirical expected utility paradoxes (the certainty effect, intransitive cycles, and one form of reflection), and an evolutionary game-theoretic extension accounts for all the phenomena revealed by the research of Kahneman and Tversky.