Pure-strategy Equilibrium Research Articles

To Adjust or Not to Adjust after a Cost-Push Shock? A Simple Duopoly Model with (and without) Resilience

We characterize the equilibrium in a homogeneous good Cournot duopoly in which firms have the choice to react to a cost-push shock by paying a lump-sum adjustment cost in order to offset the initial rise in marginal cost. Our results show that the size of the shock and the size of the adjustment cost jointly determine the nature and the number of the equilibria generated in the game. In particular, if the adjustment cost is high enough, at least one firm decides not to adjust at the pure-strategy equilibrium, and such a partial adjustment by the industry can be socially efficient as well. The main conclusions are robust to the endogenization of the adjustment cost as an increasing function of the shock size. Some implications of this partial equilibrium analysis about industry's resilience are outlined.

To adjust or not to adjust after a cost-push shock? A simple duopoly model with (and without) resilience

We characterize the equilibrium in a homogeneous good Cournot duopoly in which firms have the choice to react to a cost-push shock by paying a lump-sum adjustment cost in order to offset the initial rise in marginal cost. Our results show that the size of the shock and the size of the adjustment cost jointly determine the nature and the number of the equilibria generated in the game. In particular, if the adjustment cost is high enough, at least one firm decides not to adjust at the pure-strategy equilibrium, and such a partial adjustment by the industry can be socially efficient as well. The main conclusions are robust to the endogenization of the adjustment cost as an increasing function of the shock size. Some implications of this partial equilibrium analysis about industry's resilience are outlined.

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.

Psychologically-based voting with uncertainty

We analyse a psychologically-based model of voter turnout in an election with common value and uncertainty about who the best candidate is. Potential voters' behaviour is based on anticipated regret, where voters will experience regret if they fail to vote or vote for the wrong candidate. Furthermore, such regret is stronger when the margin of victory is smaller. We characterize mixed and pure-strategy equilibria, which can be desirable, where the right candidate is always elected in all states, or undesirable, where the wrong candidate is elected in some state. Our model yields distinctive comparative statics results. First, an increase in the proportion of informed citizens may cause the winning margin for the right candidate to either rise or fall, depending on the type of equilibrium. In addition, such an increase can have a non-monotonic effect on turnout of uninformed citizens. Second, as the prior becomes more favourable towards the ex ante favoured candidate, turnouts of informed and uninformed voters both change in a non-monotonic way. Furthermore, total turnout can be positively or negatively correlated with winning margins. The distinctive implications of our model could be tested using experimental or empirical studies. In addition, given its inclusion of uncertainty, our model can be used to study, in future research, information provision and acquisition as well as other policy questions.

User reward programs in online social media

Online social media (OSMs) have become a popular and growing Internet phenomenon, as exemplified by the millions of followers of websites like YouTube, Twitter, and Facebook. Given the Internet’s ease of access and the high degree of competition to attract users to these sites, a question arises as to whether OSMs should develop revenue-sharing programs as a way to reward their contributing users. We present an ex ante asymmetric duopoly OSM game, where heterogeneous users are either active or passive with respect to each OSM. The game includes two steps: First, the OSMs simultaneously announce their rewards for active users; and second, based on their preference, users choose their level of contribution with respect to each OSM. We show that this game has a unique Nash equilibrium in pure strategies, and we identify the conditions under which a symmetric equilibrium exists, despite the asymmetry between the OSMs. Moreover, at equilibrium, no user chooses to contribute content exclusively to the less favourable OSM, even when the more favourable firm shares a lower reward than the less favourable firm. Furthermore, in some circumstances, a higher asymmetry can diminish the net revenue of the more favourable firm and vice versa.

Seeking and Hiding Fundamental Information in Order Flows

We model the strategic interaction between fundamental informed trading and order-flow informed trading. Adding to a two-period Kyle (1985) model, a observes a signal of the fundamental informed investor's period-1 order after the order is filled. Learning from past order-flow information, the back-runner competes with the fundamental investor in period 2. If order-flow information is accurate, the fundamental investor hides her information by randomizing her period-1 trade, resulting in a mixed-strategy equilibrium. A pure-strategy equilibrium obtains if order-flow information is inaccurate. Back-running delays price discovery and reduces fundamental information acquisition. Recent evidence on high-frequency trading supports our theoretical predictions.

Lottery Allocations and Games for Public Rental Apartments

A lottery is often used to allocate public rental apartments in practice. The existing random assignments in the literature do not apply to public rental apartments because practical constraints are more restrictive. We provide a lottery-TTC algorithm to find a lottery allocation that has two desired properties of individual stability (IS) and coalitional stability (CS). The new IS notion is proposed to resolve the nonexistence of an envy-free allocation. Further more, we provide a lottery game and show that any IS allocation induces a Nash equilibrium in pure strategies, and vice versa. We also investigate a restrictive domain where a strict envy-free (SEF) allocation does exist. In that domain, we show that the SEF allocation is the unique IS and CS allocation and induces a random assignment that is Pareto optimal and equal to the pseudo market random assignment found by Hylland and Zeckhauser (1979). Our lottery-TTC algorithm provides a simple method to implement the pseudo market competitive mechanism in Hylland and Zeckhauser (1979) in Nash equilibrium in pure strategies on the restrictive domain.

Optimal entry timing

A player of privately known strength chooses when to enter a market, and an incumbent chooses whether to compete or concede. Information about the potential entrant's type is revealed publicly according to an exogenous news process and the timing of entry. I analyze stationary equilibria using the public belief as a state variable. No equilibria in pure strategies exist, and smooth-pasting conditions need not hold. Under both D1 and a novel refinement, the informed player has nondecreasing value functions and her strategy has the following structure: for high states, both types enter with certainty; for a possibly empty interval of intermediate states, no type enters; and for low states, the high type enters while the low type mixes. I obtain closed form solutions and analyze comparative statics for such equilibria. The welfare effects of the presence of news, relative to no news, depend on the starting belief; however, for a fixed equilibrium, a marginal increase in news quality always helps the informed player regardless of her type and always hurts total welfare.

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.

Competing over a finite number of locations

We consider a Hotelling location game where retailers can choose one of a finite number of locations. Consumers have strict preferences over the possible available store locations and retailers aim to attract the maximum number of consumers. We prove that a pure strategy equilibrium exists if the number of retailers is large enough. Moreover, as the number of retailers grows large, in equilibrium the distribution of retailers over the locations converges to the distribution of consumers’ preferences.

Competing Over a Finite Number of Locations

We consider a Hotelling location game where retailers can choose one of a finite number of locations. Consumers have strict preferences over the possible available store locations and retailers aim to attract the maximum number of consumers. We prove that a pure strategy equilibrium exists if the number of retailers is large enough. Moreover, as the number of retailers grows large, in equilibrium the distribution of retailers over the locations converges to the distribution of consumers' preferences.

Existence of Pure Strategy ε-Equilibria of Games with Incomplete Information

This paper is concerned with the existence of approximate pure strategy equilibria of games with incomplete information.

A Bayesian potential game to illustrate heterogeneity in cost/benefit characteristics

We employ the theory of Bayesian potential games to characterize pure-strategy equilibria of a Bayesian game with finite type structures, taking into account the cost/benefit features of agents. Building on a standard model of oligopolistic competition also applicable to environmental issues such as profit maximization in presence of an externality affecting all players, we rely on types to describe their heterogeneity. When the damage functions are additively separable, the potential provides necessary and sufficient conditions to ensure monotonicity of pure strategies.

Sequential Auctions, Price Trends, and Risk Preferences

We analyze sequential Dutch and Vickrey auctions where risk averse, or risk preferring, bidders may have heterogeneous risk exposures. We derive and characterize a pure strategy equilibrium of both auctions for arbitrary number of identical objects. A sufficient, and to certain extent necessary, condition for this result is that bidders' marginal utilities are log-submodular in income and type. We then show that when bidders are risk averse (preferring), the equilibrium price sequences should be downward (upward) drifting, and in each period the conditional expected revenue is higher (lower) in the Dutch than in the Vickrey sequential auctions. In particular, the declining price anomaly is perfectly consistent with nonincreasing absolute risk aversion when bidders have exposures to background risk.

A Game Theory Approach Using Excel

Understanding the behavior of companies in market conditions has been an important topic of studies for the economists and mathematicians. Many companies have treated their relationship of service providers as a game. Thus, the attitudes of the players can be endowed with rational thinking, which leads to the conclusion that the use of game theory as a tool to understand such behavior is of great importance. Based on the previous studies, this paper presents a study on the cooperative game theory, discussing the Nash equilibrium in pure and mixed strategies, treating solutions using the minimax John Von Neumann theorem and illustrating a mathematical modeling of a game theory problem. Furthermore, a solution of linear programming using Microsoft Excel is also proposed and presented. The methodology adopted to model the problem may help students to familiarize themselves to game theory.

Strong equilibrium in network congestion games: increasing versus decreasing costs

A network congestion game is played on a directed, two-terminal network. Every player chooses a route from his origin to his destination. The cost of a route is the sum of the costs of the arcs on it. The arc cost is a function of the number of players who use it. Rosenthal proved that such a game always has a Nash equilibrium in pure strategies. Here we pursue a systematic study of the classes of networks for which a strong equilibrium is guaranteed to exist, under two opposite monotonicity assumptions on the arc cost functions. Our main results are: (a) If costs are increasing, strong equilibrium is guaranteed on extension-parallel networks, regardless of whether the players’ origins and destinations are the same or may differ. (b) If costs are decreasing, and the players have the same origin but possibly different destinations, strong equilibrium is guaranteed on series-parallel networks. (c) If costs are decreasing, and both origins and destinations may differ, strong equilibrium is guaranteed on multiextension-parallel networks. In each case, the network condition is not only sufficient but also necessary in order to guarantee strong equilibrium. These results extend and improve earlier ones by Holzman and Law-Yone in the increasing case, and by Epstein et al. in the decreasing case.

Network topology and equilibrium existence in weighted network congestion games

Every finite game can be represented as a weighted network congestion game on some undirected two-terminal network. The network topology may reflect certain properties of the game. This paper solves the topological equilibrium-existence problem of identifying all networks on which every weighted network congestion game has a pure-strategy equilibrium.

Efficient Coalitional Bargaining with Noncontingent Offers II: Sequential Proposer Protocol

A new feature pertaining to proposer's ability to implement offers is introduced in the extensive form bargaining mechanism studied in Chatterjee et. al. (1993). This mechanism is used to analyze two classes of coalitional games with transferable utility. One class is that of strictly supermodular games; the other has the property that per capita value is increasing as a coalition adds to its members. The new feature in the mechanism is that the proposer has a choice to implement his proposal with any subset of responders who have accepted it. It is shown that for all coalitional games in either class and for all sufficiently high discount factors, there exists an order-independent efficient subgame perfect equilibrium in pure stationary strategies whose limiting outcome as players get more patient is the core-constrained Nash Bargaining Solution. For strictly supermodular games, Core is a binding constraint on Nash Bargaining Solution while for the other class it is not.

Multiagent Reinforcement Learning With Unshared Value Functions

One important approach of multiagent reinforcement learning (MARL) is equilibrium-based MARL, which is a combination of reinforcement learning and game theory. Most existing algorithms involve computationally expensive calculation of mixed strategy equilibria and require agents to replicate the other agents' value functions for equilibrium computing in each state. This is unrealistic since agents may not be willing to share such information due to privacy or safety concerns. This paper aims to develop novel and efficient MARL algorithms without the need for agents to share value functions. First, we adopt pure strategy equilibrium solution concepts instead of mixed strategy equilibria given that a mixed strategy equilibrium is often computationally expensive. In this paper, three types of pure strategy profiles are utilized as equilibrium solution concepts: pure strategy Nash equilibrium, equilibrium-dominating strategy profile, and nonstrict equilibrium-dominating strategy profile. The latter two solution concepts are strategy profiles from which agents can gain higher payoffs than one or more pure strategy Nash equilibria. Theoretical analysis shows that these strategy profiles are symmetric meta equilibria. Second, we propose a multistep negotiation process for finding pure strategy equilibria since value functions are not shared among agents. By putting these together, we propose a novel MARL algorithm called negotiation-based Q-learning (NegoQ). Experiments are first conducted in grid-world games, which are widely used to evaluate MARL algorithms. In these games, NegoQ learns equilibrium policies and runs significantly faster than existing MARL algorithms (correlated Q-learning and Nash Q-learning). Surprisingly, we find that NegoQ also performs well in team Markov games such as pursuit games, as compared with team-task-oriented MARL algorithms (such as friend Q-learning and distributed Q-learning).

Probabilistic procurement auctions

We analyse procurement auctions in which sellers are distinguished on the basis of the ratios of quality per unit of money that they offer. Sellers are privately informed on the offered quality of the technology or good. We assume that the procurer cannot perfectly identify the best offer. Thus, with positive and decreasing probability, the second, third, etc. best ratio offered is selected as the winner of the auction. We model this decision process as based on a general noisy ranking of offers. We show that, although the problem seems to be analytically intractable in general, there exists a simple symmetric, pure-strategy equilibrium in which everyone follows the simple heuristic to match the same ‘focal’ price–quality ratio.