Unique Pure-strategy Equilibrium Research Articles

An analysis of the equilibrium strategies for route-choosing customers in a two-station queueing system

We investigate a two-station queueing system where strategic customers must be sequentially serviced at both stations. We prove that an established property—that the distribution of the total time spent in a two-station system is independent of the chosen route when services times are exponentially distributed—is not a general one by providing a counterexample with deterministic service times. In doing so, we also prove a concomitant property—that any routing strategy is an equilibrium—is peculiar to the system with the particular assumption of the exponential service time distribution. Using simulations, we show that—depending on the distribution of service times—there can be (1) cases with three equilibria, (2) a case with a unique pure strategy equilibrium, and (3) cases with a unique mixed strategy equilibrium.

Imposing Choice on the Uninformed: The Case of Dynamic Currency Conversion

Over the course of the past two decades, it has become a common experience for consumers authorizing an international transaction via credit card to be invited to choose the currency in which they wish the transaction to be executed. While this choice, made feasible by a technology known as dynamic currency conversion (DCC), seems to foster competition, we argue that the opposite is the case. In fact, the unique pure-strategy equilibrium in a natural fee-setting game, with uninformed and possibly inattentive consumers, turns out to be highly asymmetric, entailing fees for the service provider that persistently exceed the monopoly level. Although losses in welfare may be substantial, a regulatory solution is unlikely to come about due to a global free-rider problem.

Voting with abstention

AbstractMost voting models in the literature neglect abstention, but is such a simplification justified? I investigate this question in a model with outside pressure on voters. For sequential voting (e.g., roll call votes), with and without an abstention option, there is a unique subgame perfect equilibrium, which implies that true majorities always succeed. Abstention can be an equilibrium strategy for some voters, in particular under complex decision rules (e.g., weighted voting, double majorities). Simultaneous voting often has a unique pure strategy equilibrium but also a plethora of mixed and pure/mixed strategy equilibria. Therefore, only with equilibrium selection, can we evaluate the consequences of neglecting abstention. For equal weight voting, equilibria selected by the procedure of Harsanyi and Selten change completely with an abstention option, even if abstention itself is not or rarely used. With small enough outside pressure, however, the selected equilibrium honors true majorities in both cases.

Two-Sided Competition Between On-Demand Service Platforms

On-demand service platforms compete for both workers and customers, forming a two-sided competition. In this paper, we study the game between two platforms implied by such a two-sided competition and examine how the characteristics of the embedded workers-customers subgame affect outcomes in equilibrium. Specifically, we consider two subgame settings: (i) workers and customers move sequentially and (ii) workers and customers move simultaneously. We find that these two settings yield different subgame outcomes. In the first setting, given any platform strategy profile, there is no subgame equilibrium such that both platforms have a positive market share. In the simultaneous-move setting, such subgame equilibria can exist, though not unique. In the presence of multiple equilibria, we show that under some refinement rules both settings (the sequential and simultaneous subgames) yield the same unique pure-strategy equilibrium. We discuss how these results shed light on the competitive dynamics of on-demand service platforms in practice.

Multiple equilibrium behaviors of auto travellers and a freight carrier under the cordon-based large-truck restriction regulation

The cordon-based large-truck restriction regulation (CBLTRR) makes large truck based cargo movement process interrupted because large trucks are deterred from moving into some specified areas within given time periods. As a result, cargoes carried by a large truck need to be transferred from the large truck to small ones at a freight distribution center (FDC) for the last few miles. This study aims to analyze the multiple equilibrium behaviors for auto travellers and one freight carrier under the CBLTRR. A link flow based generalized Nash equilibrium (GNE) model is first developed to characterize mixed routing behaviors of these two types of users. In the model, auto travellers follow first Wardropian principle to find their individual shortest paths, while the freight carrier minimizes its total freight cost. We then proceed to explore the existence and uniqueness of pure strategy equilibrium, and reduction to variational inequality. When the pure strategy equilibrium does not exist or is intractable, we propose an approach to construct a mixed equilibrium with restricted action space (i.e., the path choice set) (called restricted mixed equilibrium). The restricted action space is formed based on recurring best-responses of two types of users which could reflect their actual interactive decisions of a non-cooperative traffic pattern evolution. A solution algorithm is developed to find those equilibriums. The developed models and solution algorithms are evaluated and their potential applications are illustrated through numerical experiments.

When will party whips succeed? Evidence from almost symmetric voting games

The true preferences of some members of parliament sometimes imply voting against the party line, but these members fear sanctions. In the mostly unique pure strategy equilibrium of such a voting game, voters always follow the party line, even if their true preferences are strong and (opportunity) costs from sanctions are arbitrarily small. I qualify this disturbing result by applying the equilibrium selection theory of Harsanyi and Selten (1988). Pure strategy equilibria are selected only for large deviation costs. Otherwise, mixed strategy equilibria are selected which support majorities in true preferences (true majorities), for negligible costs almost with certainty.

Optimal dynamic auditing based on game theory

A dynamic model based on game-theory is proposed to address the problem of fraud detection in auditing under non-linear payoff functions. Non-linearity is introduced by incorporating learning and sympathy effects in the audit process. It is proven that the audit/fraud detection game between two new engagement parties has a unique mixed strategy equilibrium, between an experienced auditor and a client has a unique pure strategy equilibrium, whereas in the long-run the game converges to a pure strategy equilibrium. In addition to this, to ensure an acceptable level of quality in the audit process, a closed form formula used to estimate the optimal auditor’s replacement time is extracted. The validity of the proposed scheme is tested on empirical data and modeling results comply with the International Standard of Auditing that requires the key audit partner to be rotated after a predefined period.

Price competition when three are few and four are many

I examine price competition in a market for a homogeneous good when consumers observe prices subject to a random shock (perception error). When firms have symmetric costs, there exists a unique equilibrium in pure strategies, which is symmetric. When there are up to three sellers in the market, the sellers extract the entire consumer surplus. However, with at least four firms, assuming that the marginal cost is sufficiently low relative to consumers’ valuation, both consumers and producers may enjoy a positive surplus. The marginal-cost pricing is never observed in an equilibrium with finitely many firms. Potential policy implications are discussed.

Stochastic asymmetric Blotto games: An experimental study

We consider a model where two players compete for n items having different common values in a Blotto game. Players must decide how to allocate their common budgets across all n items. The winner of each item is determined stochastically using a lottery mechanism which yields a unique equilibrium in pure strategies. We analyze behavior under two competing payoff objectives found in the Blotto games literature that have not been previously compared: (i) players aim to maximize their total expected payoff and (ii) players maximize the probability of winning a majority value of all n items. We report results from an experiment where subjects face both payoff objectives and we find support for the differing theoretical predictions.

Two-sided competition with vertical differentiation

This paper studies duopoly in which two-sided platforms compete in differentiated products in a two-sided market. Direct competition on both sides leads to results that depart from much of the current literature. Under some conditions the unique equilibrium in pure strategies can be computed. It features discounts on one side and muted differentiation as the cross-market externality intensifies competition. Less standard, that equilibrium fails to exist when the externality is too powerful (that side becomes too lucrative). A mixed-strategy equilibrium always exists and is characterized. These results are robust to variations in the extensive form. The model may find applications in the media, internet trading platforms, search engine competition, social media or even health insurance (HMO/PPO).

Behavioral perfect equilibrium in Bayesian games

We develop the notion of perfect Bayesian Nash Equilibrium—perfect BNE—in general Bayesian games. We test perfect BNE against the criteria laid out by Kohlberg and Mertens (1986). We show that, for a focal class of Bayesian games, perfect BNE exists. Moreover, when payoffs are continuous, perfect BNE is limit undominated for almost every type.We illustrate the use of perfect BNE in the context of a second-price auction with interdependent values. Perfect BNE selects the unique pure strategy equilibrium in continuous strategies that separates types. Moreover, when valuations become independent, the equilibrium converges to the classical truthful dominant strategy equilibrium.We also show that less intuitive equilibria in which types are pooled are ruled out by our selection criterion. We further argue that standard selection criteria for second-price auctions have no bite here. Bidders have no dominant strategies, and the separating equilibrium is not sincere.

Rank Order Contests with Sabotage: Equilibrium Analysis and Optimal Design

We establish the existence and uniqueness of pure-strategy equilibrium in two-worker rank order contests with sabotage while allowing interdependent effects of productive and sabotage effort. We find that diverging marginal costs in workers' productive effort discourage sabotage activity. For symmetric workers, it is optimal to set a pay dispersion that induces positive sabotage if and only if sabotage is less effective and the marginal cost of sabotage is small. The optimal pay dispersion decreases with the effectiveness and increases with the marginal cost of sabotage, but it does not change monotonically with the effectiveness of productive effort.

Asymmetric Bertrand-Edgeworth-Chamberlin Competition with Linear Demand: A Pediatric Vaccine Pricing Model

Pricing strategies in the U.S. pediatric vaccines market are studied using a Bertrand-Edgeworth-Chamberlin price game. The game analyzes the competition between asymmetric manufacturers with limited production capacities and linear demand, producing differentiated products. The model completely characterizes the unique pure strategy equilibrium in the Bertrand-Edgeworth-Chamberlin competition in an oligopoly setting. In addition, the complete characterization of mixed strategy equilibrium is provided for a duopoly setting. The results indicate that the pure strategy equilibrium exists if the production capacity of a manufacturer is at their extreme. For the capacity regions where no pure strategy equilibrium exists, there exists a mixed strategy equilibrium. A duopoly setting provides the distribution functions of the mixed strategy equilibrium for both manufacturers. The proposed game is applied to the U.S. pediatric vaccine market, in which a few asymmetric vaccine manufacturers produce differentiated vaccines. The source of differentiation in the competing vaccines is the number of medically adverse events, the number of different antigens, and special advantages of those vaccines. The results indicate that the public sector prices of the vaccines are higher than the vaccine equilibrium prices. Furthermore, the situation when shortages of certain pediatric vaccines occur is studied. Market demand and degree of product differentiation are shown as two key factors in computing the equilibrium prices of the vaccines.

Rational exaggeration and counter-exaggeration in information aggregation games

We study an information aggregation game in which each of a finite collection of “senders” receives a private signal and submits a report to the center, who then makes a decision based on the average of these reports. The integration of three features distinguishes our framework from the related literature: players’ reports are aggregated by a mechanistic averaging rule, their strategy sets are intervals rather than binary choices, and they are ex ante heterogeneous. In this setting, players engage in a “tug-of-war,” as they exaggerate and counter-exaggerate in order to manipulate the center’s decision. While incentives to exaggerate have been studied extensively, the phenomenon of counter-exaggeration is less well understood. Our main results are as follows. First, the cycle of counter-exaggeration can be broken only by the imposition of exogenous bounds on the space of admissible sender reports. Second, in the unique pure-strategy equilibrium, all but at most one player is constrained with positive probability by one of the report bounds. Our third and fourth results hold for a class of “anchored” games. We show that if the report space is strictly contained in the signal space, then welfare is increasing in the size of the report space, but if the containment relation is reversed, welfare is independent of the size of the space. Finally, the equilibrium performance of our heterogeneous players can be unambiguously ranked: a player’s equilibrium payoff is inversely related to the probability that her exaggeration will be thwarted by the report bounds.

A symmetric capacity-constrained differentiated oligopoly model for the United States pediatric vaccine market with linear demand

The United States pediatric vaccine market is examined using Bertrand–Edgeworth–Chamberlin price competition. The proposed game captures interactions between symmetric, capacity-constrained manufacturers in a differentiated, single-product market with linear demand. Results indicate that a unique pure strategy equilibrium exists in the case where the capacities of the manufacturers are at their extreme. For the capacity region where no pure strategy equilibrium exists, there exists a mixed strategy equilibrium where the distribution function, its support, and the expected profit of the manufacturers are characterized. Three game instances are introduced to model the United States pediatric vaccine market. In each instance, the manufacturers are assumed to have equal capacity in producing vaccines. Vaccines are differentiated based upon the number of reported adverse medical events for that vaccine. Using a game-theoretic model, equilibrium prices are computed for each monovalent vaccine. Results indicate that the equilibrium prices for monovalent vaccines are lower than the federal contract prices. The numerical results provide both a lower and upper bound for the vaccine equilibrium prices in the public sector, based on the capacity of the vaccine manufacturers. Results illustrate the importance of several model parameters such as market demand and vaccine adverse events on the equilibrium prices. Supplementary materials are available for this article. Go to the publisher’s online edition of IIE Transactions for datasets, additional tables, detailed proofs, etc.

A New Approach to Solving Production-Appropriation Games with Many Heterogeneous Players

We set out a model of production and appropriation involving many players, who differ with respect to both resource endowments and productivities. We write down the model in a novel way that permits our analysis to avoid the proliferation of dimensions associated with the best response function approach as the number of heterogeneous players increases. We establish existence of a unique equilibrium in pure strategies, and carry out some comparative static exercises.

Equilibrium play and best response to (stated) beliefs in normal form games

We report experimental results on a series of ten one-shot two-person 3 × 3 normal form games with unique equilibrium in pure strategies played by non-economists. In contrast to previous experiments in which game theory predictions fail dramatically, a majority of actions taken coincided with the equilibrium prediction (70.2%) and were best-responses to subjects' stated beliefs (67.2%). In constant-sum games, 78% of actions taken were predicted by the equilibrium model, outperforming simple K-level reasoning models. We discuss how non-trivial game characteristics related to risk aversion, efficiency concerns and social preferences may affect the predictive value of different models in simple normal form games.

A Duopoly Location Toolkit: Consumer Densities Which Yield Unique Spatial Duopoly Equilibria

Anderson, Goeree and Ramer (1997) observe that although the Uniform consumer density is almost universally assumed in Hotelling style location games, it is more realistic to assume non-uniform distributions. Using Anderson et al.'s (1997) sufficient conditions for the existence of a unique pure strategy equilibrium, we extend the list of known distributions with characterised equilibria from two (Uniform and Triangular) to seven. Our extension includes the Normal and Logistic distributions which are commonly used in empirical research. We also analyse the effects of density choice on equilibrium outcomes. Holding the mean and dispersion of consumer tastes constant, we find that the Uniform distribution inflates differentiation effects.

Bertrand–Edgeworth equilibrium with a large number of firms

We examine a model of price competition with strictly convex costs where the firms simultaneously decide on both price and quantity, are free to supply less than the quantity demanded, and there is discrete pricing. If firms are symmetric then, for a large class of residual demand functions, there is a unique equilibrium in pure strategies whenever, for a fixed grid size, the number of firms is sufficiently large. Moreover, this equilibrium price is within a grid-unit of the competitive price. The results go through to a large extent when the firms are asymmetric, or they are symmetric but play a two-stage game and the tie-breaking rule is ‘weakly manipulable’.

A note on renegotiation in repeated Bertrand duopolies

Weak Renegotiation-Proofness (WRP) singles out marginal cost pricing as a unique pure-strategy equilibrium of the infinitely repeated Bertrand duopoly. We show that, with a discrete strategy space, WRP does not eliminate any relevant subgame perfect equilibrium outcome.