Pure-strategy Equilibrium Research Articles

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.

Optimal product differentiation in a circular model

Since circular model was introduced in Salop (Bell J Econ 10:141–156, 1979), it has been the workhorse for analyzing spatial competition among differentiated firms. A common assumption in this literature is that firms are evenly spaced on the circle, even when entry is allowed. We characterize conditions for even spacing to be an equilibrium, using a two-stage (location-then-price) circular model with general transport cost function. Under duopoly competition, we characterize a mild sufficient condition—the first derivative of transport cost is concave (together with an assumption governing the transport cost difference to the two firms). If one only considers pure strategy equilibrium in prices, this sufficient condition is weakened to the first derivative of transport cost being $$-$$ 1-concave. These conditions ensure that firms’ profits are concave in their prices when firms are evenly spaced and that even spacing maximizes profits. Under oligopoly competition ( $$N\ge 2$$ firms), we characterize a necessary condition for even spacing to be an equilibrium. This necessary condition requires a firm’s profit to be concave in location at the symmetric location. It involves the third derivative of transport cost function, so having convex transport cost in general is neither necessary nor sufficient to determine equilibrium location choice. Our results have implications for studies employing circular models, especially in terms of welfare analysis which depends on firms’ location choices.

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.

A Bertrand–Edgeworth oligopoly with a public firm

We determine conditions under which a pure-strategy equilibrium of a mixed Bertrand–Edgeworth oligopoly exists. In addition, we determine its pure-strategy equilibrium whenever it exists and compare the equilibrium outcome with that of the standard Bertrand–Edgeworth oligopoly with only private firms.

Alternating-offers bargaining in one-to-many and many-to-many settings

Automating negotiations in markets where multiple buyers and sellers operate is a scientific challenge of extraordinary importance. One-to-one negotiations are classically studied as bilateral bargaining problems, while one-to-many and many-to-many negotiations are studied as auctioning problems. This paper aims at bridging together these two approaches, analyzing agents' strategic behavior in one-to-many and many-to-many negotiations when agents follow the alternating-offers bargaining protocol (Rubinstein Econometrica 50(1), 97---109, 33). First, we extend this protocol, proposing a novel mechanism that captures the peculiarities of these settings. Then, we analyze agents' equilibrium strategies in complete information bargaining and we find that for a large subset of the space of the parameters, the equilibrium outcome depends on the values of a narrow number of parameters. Finally, we study incomplete information bargaining with one-sided uncertainty regarding agents' reserve prices and we provide an algorithm based on the combination of game theoretic analysis and search techniques which finds agents' equilibrium in pure strategies when they exist.

Existence of Equilibria in Joint Admission and Power Control for Inelastic Traffic

This letter considers the problem of admission and discrete power control, in the interfering-multiple-access channel, with rate constraints on admitted links. This problem is formulated as a normal form noncooperative game. The utility function models inelastic demand. An example demonstrates that in the fading channel, in some networks, a pure strategy equilibrium does not exist with strictly positive probability. Hence, the probability of existence of an equilibrium is analyzed and bounds are computed. To this end, the problem of finding equilibria is transformed into a constraint satisfaction problem. Next the letter considers the incomplete information setting, with compact convex channel power gains. The resulting Bayesian game is proven to possess at least one pure Bayesian–Nash equilibrium in on–off threshold strategies. Numerical results are presented to corroborate the findings.

Strategic Experimentation on a Common Threshold

A dynamic experimentation game is examined in which players search for an unknown threshold. Players contribute to the rate of decline in a state variable, and the game ends with a costly breakdown once the state falls below the threshold. In the unique symmetric pure-strategy stationary Markov equilibrium, the state decreases gradually over time and settles asymptotically at a cutoff level, conditional on no breakdown. The cutoff depends on patience, the cost of the breakdown, and the prior distribution of the threshold, but not on the number of players. In a discrete-time version of the game, the equilibrium time path of the state converges to that of the continuous-time model when period length tends toward zero.

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.

Social Norms and Cooperation

A theoretical framework is introduced to model social norms using the tools of psychological game theory. In the Prisoner's Dilemma Game, a preference for norm compliance can lead to cooperation arising with a positive probability, and if this preference is sufficiently strong, occurring as a pure strategy equilibrium. Thus, cooperation can arise as a social norm in a one-shot Prisoner's Dilemma Game without repeated play, communication, or sanctions.

Existence of pure-strategy equilibria in Bayesian games: a sharpened necessity result

In earlier work, the authors showed that a pure-strategy Bayesian-Nash equilibria in games with uncountable action sets and atomless private information spaces may not exist if the information space of each player is not saturated. This paper sharpens this result by exhibiting a failure of the existence claim for a game in which the information space of only one player is not saturated. The methodology that enables this extension of the necessity theory is novel relative to earlier work, and its conceptual underpinnings may have independent interest.

The Max-Min Group Contest: Weakest-link (Group) All-Pay Auction

We investigate a group all-pay auction in which each group's effort is represented by the minimum among the effort levels exerted by the group members and the prize is a group-specific public good. We fully characterize the symmetric equilibria for two groups. There are four types of equilibria: the pure strategy equilibria in which all (active) players exert the same effort; the semi-pure strategy equilibria in which the players in a group play the same pure strategy whereas those in the other group play the same mixed strategy; the nondegenerate mixed strategy equilibria with continuous support; and the nondegenerate mixed strategy equilibria with discontinuous support. We then analyze a general contest with n groups.

Dynamic Contests With Bankruptcy: The Despair Effect

AbstractWe analyze a two-period contest in which agents may become bankrupt at the end of the first period. A bankrupt agent is excluded from the contest in the second period of the game. We investigate the existence of a subgame perfect equilibrium in pure strategies. We distinguish between a borrowing equilibrium in which at least one agent might be bankrupted and a non borrowing equilibrium in which no agent is bankrupted. We prove that the former occurs when the agent taking loans is relatively poor and the future does not matter very much. This action represents the Despair Effect, in which severely handicapped agents take actions that jeopardize their existence in the long run but are currently helpful. We find conditions under which borrowing and non borrowing equilibria overlap and do not overlap. We provide an example in which no equilibrium exists.

A note on auctions with compulsory partnership

We study a symmetric, profitt share, common value auction with a twist: One (fixed) Bidder, if not winning the auction, has to enter a partnership with the winner, sharing both expenses and revenue at rate (say) 0 < r < 1. We show that it doesn't have an equilibrium in pure strategies.

Is the Severity of the Penalty an Effective Deterrent? A Strategic Approach for the Crime of Tax Evasion

In order to analyze the severity of sentencing, and to show how the probabilistic interpretation of strategic behavior can be tricky, this paper uses the crime strategic model (inspection game) proposed by Tsebelis. This model shows that any attempts to increase the severity of punishment will alter the payoff of the individuals involved, leaving unchanged the frequency of violation at equilibrium. This result is misleading: payoffs are not independent and the crime game can not be simply read with mixed strategies. These are inconclusive on how the players act rationally. This is undeniably true for the crime of tax evasion, where the dishonest taxpayers are rational agents, motivated by the comparison of payoffs, considering the risk of breaking the law. Although an irreducible minimum of uncertainty remains, the Nash equilibrium in mixed strategies provides us with the necessary information on equilibria in pure strategies that will be played. In this context, tougher sentencing deters crime, although, as the Italian historical experience teaches, the necessary condition required is the certainty of punishment and the ability of the government to enforce it.

Communication Versus (Restricted) Delegation: An Experimental Comparison

This paper reports the results from a laboratory experiment investigating a manager's decision whether or not to delegate authority to a better informed worker whose interests are often, but not always, congruent. Keeping authority implies a loss of information, as the worker communicates his information strategically. Delegating authority leads to a loss of control. A key aspect of our design is that the manager can restrict the worker's choice set when delegating authority. We find that, in case of delegation, managers (as predicted) put tighter restrictions when interests are less aligned. Workers send more informative messages under communication than predicted by the pure strategy equilibria. This finding neither appears to be driven by lying aversion of workers nor by credulity of managers. Qualitatively, our results are in line with a mixed strategy equilibrium under communication, which strictly outperforms optimal restricted delegation and is relatively close to the optimal stochastic mechanism in our setting.

Pre-Play Communication with Forgone Costly Messages: Experimental Evidence on Forward Induction

We study communication in a two-player coordination game with Pareto-ranked equilibria. Prior research demonstrates that efficient coordination is difficult without communication but obtains regularly with (mandatory) costless pre-play messages. In a laboratory experiment, we modify communication by making the sending of messages optional and costly. Even small costs dramatically reduce message use, but efficient coordination of actions occurs with similar frequency to that observed under costless communication. Our results can be accounted for by Govindan and Wilson's formalization of forward induction (GW-FI), which selects, among the pure-strategy equilibrium outcomes, the one in which efficiency is achieved without communication. Consistent with the introspective character of GW-FI, the fraction of players who achieve efficient coordination by forgoing the use of reasonably costly optional messages is substantial from the first period, is remarkably stable at that level, and is not significantly affected by learning.

Nash equilibrium in discontinuous games

We provide several generalizations of the various equilibrium existence results in Reny (Econometrica 67:1029–1056, 1999), Barelli and Meneghel (Econometrica 81:813–824, 2013), and McLennan et al. (Econometrica 79:1643–1664, 2011). We also provide an example demonstrating that a natural additional generalization is not possible. All of the theorems yielding existence of pure-strategy Nash equilibria here are stated in terms of the players’ preference relations over joint strategies. Hence, in contrast to much of the previous work in the area, the present results for pure-strategy equilibria are entirely ordinal.

Privacy-Constrained Network Formation

With the increasing ease with which information can be shared in social media, the issue of privacy has become central for the functioning of various online platforms. In this paper, we consider how privacy concerns affect individual choices in the context of a network formation game (where links can be interpreted as friendships in a social network, connections over a social media platform or trading activity in online platform). In the model, each individual decides which other agents to “befriend”, i.e., form links with. Such links bring direct (heterogeneous) benefits from friendship and also lead to the sharing of information. But such information can travel over other linkages (e.g., shared by the party acquiring the information with others) through a percolation process over the equilibrium network. Privacy concerns are modeled as a disutility that individual suffers as a result of her private information being acquired by others, and imply that the individual has to take into account who the friends of her new friend (and who the friends of friends of her new friend etc.) are. We specify conditions under which pure-strategy equilibria exist and characterize both pure-strategy and mixed-strategy equilibria. Our two main results show that, as in many real-life examples, the resulting equilibrium networks feature clustered connections and homophily. Clustering emerges because if player a is friend with b and b is friend with c, then a’s information is likely to be shared indirectly with c anyway, thus making it less costly for a to befriend c. Homophily emerges because even an infinitesimal advantage in terms of direct benefits of friendship within a group makes linkages within that group more likely, and the travel of information within that group reduces the costs, and thus increases the likelihood, of further within-group links. Keywords— Network formation, Networks, Privacy. JEL Classification : D85 ∗We thank various numerous seminar and conference participants for useful suggestions. We gratefully acknowledge financial support from the Toulouse Network with Information Technology and Army Research Office, ARO MURI W911NF-12-1-0509. †Department of Economics, Massachusetts Institute of Technology, Cambridge, MA, 02139 . daron@mit.edu ‡Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, 02139 makhdoum@mit.edu §Rotman School of Management, University of Toronto, Toronto, ON, M5S 3E6 azarakhsh.malekian@rotman.utoronto.ca ¶Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, 02139 asuman@mit.edu

Competitive markdown timing for perishable and substitutable products

We model as a duopoly two firms selling their fixed stocks of two substitutable items over a selling season. Each firm starts with an initial price, and has the option to decrease the price once. The problem for each firm is to determine when to mark its price down in to maximize its revenue. We show that the existence and characterization of a pure-strategy equilibrium depend on the magnitude of the increase in the revenue rate of a firm when its competitor runs out of stock. When the increase is smaller than the change in the revenue rate of the price leader when both firms are in stock for all of the three possible scenarios, neither firm has the incentive to force its rival to run out of stock and if a firm marks its price down after the season starts, its inventory runs out precisely at the end of the season. When the increase is larger than the change of the price leader׳s revenue rate in one particular scenario, waiting until its rival runs out of inventory may be an equilibrium strategy for the larger firm even though this may lead to leftover inventory for itself. In other cases, there may be no pure-strategy equilibrium in the game. In certain regions of the parameter space, a firm׳s revenue may be decreasing in its starting inventory which shows that a firm may be better off if it can credibly salvage a portion of its inventory prior to the game. While most of our analysis is for open-loop strategies, in the final part of the paper, we show that the open-loop equilibrium survives as an equilibrium when we consider closed-loop strategies for an important subset of the parameter space.

ENDOGENOUS TIMING IN QUALITY CHOICES AND PRICE COMPETITION

The authors modify the price-setting version of the vertically differentiated duopoly model by Aoki (Effect of Credible Quality Investment with Bertrand and Cournot Competition, 2003) by introducing an extended game in which firms noncooperatively choose the timing of moves at the quality stage. Their results show that there are multiple equilibria in pure strategies, whereby firms always select sequential play at the quality stage. They also investigate the mixed-strategy equilibrium, revealing that the probability of generating out-of-equilibrium outcomes is higher than its complement to one. In the alternative case with full market coverage, the authors show that the quality stage is solved in dominant strategies and therefore the choice of roles becomes irrelevant as the Nash and Stackelberg solutions coincide.