Competition for a Majority

We define the class of two‐player zero‐sum games with payoffs having mild discontinuities, which in applications typically stem from how ties are resolved. For such games, we establish sufficient conditions for existence of a value of the game, maximin and minimax strategies for the players, and a Nash equilibrium. If all discontinuities favor one player, then a value exists and that player has a maximin strategy. A property called payoff approachability implies existence of an equilibrium, and that the resulting value is invariant: games with the same payoffs at points of continuity have the same value and ɛ‐equilibria. For voting games in which two candidates propose policies and a candidate wins election if a weighted majority of voters prefer his proposed policy, we provide tie‐breaking rules and assumptions about voters' preferences sufficient to imply payoff approachability. These assumptions are satisfied by generic preferences if the dimension of the space of policies exceeds the number of voters; or with no dimensional restriction, if the electorate is sufficiently large. Each Colonel Blotto game is a special case in which each candidate allocates a resource among several constituencies and a candidate gets votes from those allocated more than his opponent offers; in this case, for simple‐majority rule we prove existence of an equilibrium with zero probability of ties.     

EXPLAINING GIFT‐EXCHANGE—THE LIMITS OF GOOD INTENTIONS

This paper explores the limitations of intention‐based social preferences as an explanation of gift‐exchange between a firm and a worker. In a framework with one self‐interested and one reciprocal player, gift‐giving never arises in equilibrium. Instead, any equilibrium in a large class of multistage games must involve mutually unkind behavior of both players. Besides gift‐exchange, this class of games also includes moral hazard models and the rotten kid framework. Even though equilibrium behavior may appear positively reciprocal in some of these games, the self‐interested player never benefits from reciprocity. We discuss the relation of these results to the theoretical and empirical literature on gift‐exchange in employment relations.     

Tenable Strategy Blocks and Settled Equilibria

When people interact in familiar settings, social conventions usually develop so that people tend to disregard alternatives outside the convention. For rational players to usually restrict attention to a block of conventional strategies, no player should prefer to deviate from the block when others are likely to act conventionally and rationally inside the block. We explore two set‐valued concepts, coarsely and finely tenable blocks, that formalize this notion for finite normal‐form games. We then identify settled equilibria, which are Nash equilibria with support in minimal tenable blocks. For a generic class of normal‐form games, our coarse and fine concepts are equivalent, and yet they differ from standard solution concepts on open sets of games. We demonstrate the nature and power of the solutions by way of examples. Settled equilibria are closely related to persistent equilibria but are strictly more selective on an open set of games. With fine tenability, we obtain invariance under the insertion of a subgame with a unique totally mixed payoff‐equivalent equilibrium, a property that other related concepts have not satisfied.     

The Timing of Capacity Investment with Lead Times: When Do Firms Act in Unison?

We study competitive capacity investment for the emergence of a new market. Firms may invest either in capacity leading demand or in capacity lagging demand at different costs. We show how the lead time and other operational factors including volume flexibility, existing capacity, and demand uncertainty impact equilibrium outcomes. Our results indicate that a type of bandwagon behavior is the most likely equilibrium outcome: if both firms are going to invest, then they are most likely to act in unison. Contrary to much received wisdom, we show that leader–follower behavior is very uncommon in equilibrium where firms do not have volume flexibility, and will not occur at all if lead times are sufficiently short. On the other hand, if there is volume flexibility in production, then the likelihood of this sequential investment behavior increases. Our findings underscore the importance of operational characteristics in determining the competitive dynamics of capacity investment timing.     

Investment and Competitive Matching

We study markets in which agents first make investments and are then matched into potentially productive partnerships. Equilibrium investments and the equilibrium matching will be efficient if agents can simultaneously negotiate investments and matches, but we focus on markets in which agents must first sink their investments before matching. Additional equilibria may arise in this sunk‐investment setting, even though our matching market is competitive. These equilibria exhibit inefficiencies that we can interpret as coordination failures. All allocations satisfying a constrained efficiency property are equilibria, and the converse holds if preferences satisfy a separability condition. We identify sufficient conditions (most notably, quasiconcave utilities) for the investments of matched agents to satisfy an exchange efficiency property as well as sufficient conditions (most notably, a single crossing property) for agents to be matched positive assortatively, with these conditions then forming the core of sufficient conditions for the efficiency of equilibrium allocations.     

Quadratic Concavity and Determinacy of Equilibrium

One of the central features of classical models of competitive markets is the generic determinacy of competitive equilibria. For smooth economies with a finite number of commodities and a finite number of consumers, almost all initial endowments admit only a finite number of competitive equilibria, and these equilibria vary (locally) smoothly with endowments; thus equilibrium comparative statics are locally determinate. This paper establishes parallel results for economies with finitely many consumers and infinitely many commodities. The most important new condition we introduce, quadratic concavity, rules out preferences in which goods are perfect substitutes globally, locally, or asymptotically. Our framework is sufficiently general to encompass many of the models that have proved important in the study of continuous‐time trading in financial markets, trading over an infinite time horizon, and trading of finely differentiated commodities.     

The Theory of Global Games on Test: Experimental Analysis of Coordination Games with Public and Private Information

The theory of global games has shown that coordination games with multiple equilibria may have a unique equilibrium if certain parameters of the payoff function are private information instead of common knowledge. We report the results of an experiment designed to test the predictions of this theory. Comparing sessions with common and private information, we observe only small differences in behavior. For common information, subjects coordinate on threshold strategies that deviate from the global game solution towards the payoff‐dominant equilibrium. For private information, thresholds are closer to the global game solution than for common information. Variations in the payoff function affect behavior as predicted by comparative statics of the global game solution. Predictability of coordination points is about the same for both information conditions.     

Strongly Consistent Self‐Confirming Equilibrium

Fudenberg and Levine (1993a) introduced the notion of self‐confirming equilibrium, which is generally less restrictive than Nash equilibrium. Fudenberg and Levine also defined a concept of consistency, and claimed in their Theorem 4 that with consistency and other conditions on beliefs, a self‐confirming equilibrium has a Nash equilibrium outcome. We provide a counterexample that disproves Theorem 4 and prove an alternative by replacing consistency with a more restrictive concept, which we call strong consistency. In games with observed deviators, self‐confirming equilibria are strongly consistent self‐confirming equilibria. Hence, our alternative theorem ensures that despite the counterexample, the corollary of Theorem 4 is still valid.     

Cognition and Behavior in Normal‐Form Games: An Experimental Study

This paper reports experiments designed to study strategic sophistication, the extent to which behavior in games reflects attempts to predict others’ decisions, taking their incentives into account. We study subjects’ initial responses to normal‐form games with various patterns of iterated dominance and unique pure‐strategy equilibria without dominance, using a computer interface that allowed them to search for hidden payoff information, while recording their searches. Monitoring subjects’ information searches along with their decisions allows us to better understand how their decisions are determined, and subjects’ deviations from the search patterns suggested by equilibrium analysis help to predict their deviations from equilibrium decisions.     

Time Horizon and Cooperation in Continuous Time

We study social dilemmas in (quasi‐) continuous‐time experiments, comparing games with different durations and termination rules. We discover a stark qualitative contrast in behavior in continuous time as compared to previously studied behavior in discrete‐time games: cooperation is easier to achieve and sustain with deterministic horizons than with stochastic ones, and end‐game effects emerge, but subjects postpone them with experience. Analysis of individual strategies provides a basis for a simple reinforcement learning model that proves to be consistent with this evidence. An additional treatment lends further support to this explanation.     

Experientia Docet: Professionals Play Minimax in Laboratory Experiments

We study how professional players and college students play zero‐sum two‐person strategic games in a laboratory setting. We first ask professionals to play a 2 × 2 game that is formally identical to a strategic interaction situation that they face in their natural environment. Consistent with their behavior in the field, they play very close to the equilibrium of the game. In particular, (i) they equate their strategies' payoffs to the equilibrium ones and (ii) they generate sequences of choices that are serially independent. In sharp contrast, however, we find that college students play the game far from the equilibrium predictions. We then study the behavior of professional players and college students in the classic O'Neill 4 × 4 zero‐sum game, a game that none of the subjects has encountered previously, and find the same differences in the behavior of these two pools of subjects. The transfer of skills and experience from the familiar field to the unfamiliar laboratory observed for professional players is relevant to evaluate the circumstances under which behavior in a laboratory setting may be a reliable indicator of behavior in a naturally occurring setting. From a cognitive perspective, it is useful for research on recognition processes, intuition, and similarity as a basis for inductive reasoning.     

CONVERGENCE: AN EXPERIMENTAL STUDY OF TEACHING AND LEARNING IN REPEATED GAMES

Nash equilibrium can be interpreted as a steady state where players hold correct beliefs about the other players’ behavior and act rationally. We experimentally examine the process that leads to this steady state. Our results indicate that some players emerge as teachers—those subjects who, by their actions, try to influence the beliefs of their opponent and lead the way to a more favorable outcome—and that the presence of teachers appears to facilitate convergence to Nash equilibrium. In addition to our experiments, we examine games, with different properties, from other experiments and show that teaching plays an important role in these games. We also report results from treatments in which teaching is made more difficult. In these treatments, convergence rates go down and any convergence that does occur is delayed.     

A Dynamic Reinterpretation of Nash Bargaining With Endogenous Threats

This paper concerns the two‐stage game introduced in Nash (1953). It formalizes a suggestion made (but not pursued) by Nash regarding equilibrium selection in that game, and hence offers an arguably more solid foundation for the “Nash bargaining with endogenous threats” solution. Analogous reasoning is then applied to an infinite horizon game to provide equilibrium selection in two‐person repeated games with contracts. In this setting, issues about enforcement of threats are much less problematic than in Nash's static setting. The analysis can be extended to stochastic games with contracts.     

Computing Normal Form Perfect Equilibria for Extensive Two‐Person Games

This paper presents an algorithm for computing an equilibrium of an extensive two‐person game with perfect recall. The method is computationally efficient by virtue of using the sequence form, whose size is proportional to the size of the game tree. The equilibrium is traced on a piecewise linear path in the sequence form strategy space from an arbitrary starting vector. If the starting vector represents a pair of completely mixed strategies, then the equilibrium is normal form perfect. Computational experiments compare the sequence form and the reduced normal form, and show that only the sequence form is tractable for larger games.     

Continuity,Inertia, and Strategic Uncertainty: A Test of the Theory of Continuous Time Games

The theory of continuous time games (Simon and Stinchcombe (1989), Bergin and MacLeod (1993)) shows that continuous time interactions can generate very different equilibrium behavior than conventional discrete time interactions. We introduce new laboratory methods that allow us to eliminate natural inertia in subjects' decisions in continuous time experiments, thereby satisfying critical premises of the theory and enabling a first‐time direct test. Applying these new methods to a simple timing game, we find strikingly large gaps in behavior between discrete and continuous time as the theory suggests. Reintroducing natural inertia into these games causes continuous time behavior to collapse to discrete time‐like levels in some settings as predicted by subgame perfect Nash equilibrium. However, contra this prediction, the strength of this effect is fundamentally shaped by the severity of inertia: behavior tends towards discrete time benchmarks as inertia grows large and perfectly continuous time benchmarks as it falls towards zero. We provide evidence that these results are due to changes in the nature of strategic uncertainty as inertia approaches the continuous limit.     

JOB MARKET SIGNALING OF RELATIVE POSITION,OR BECKER MARRIED TO SPENCE

This paper considers a matching model of the labor market where workers, who have private information on their quality, signal to firms that also differ in quality. Signals allow assortative matching in which the highest‐quality workers send the highest signals and are hired by the best firms. Matching is considered both when wages are rigid (nontransferable utility) and when they are fully flexible (transferable utility). In both cases, equilibrium strategies and payoffs depend on the distributions of worker and firm types. This is in contrast to separating equilibria of the standard model, which do not respond to changes in supply or demand. With sticky wages, despite incomplete information, equilibrium investment in education by low‐ability workers can be inefficiently low, and this distortion can become worse in a more competitive environment. In contrast, with flexible wages, greater competition improves efficiency.     

A Competitive Model of Customization with Lead‐Time Effects

In this article, we study the competitive interactions between a firm producing standard products and a firm producing custom products. Consumers with heterogeneous preferences choose between n standard products, which may not meet their preferences exactly but are available immediately, and a custom product, available only after a certain lead time l. Standard products incur a variety cost that increases with n and custom products incur a lead time cost that is decreasing in the lead time l. We consider a two‐stage game wherein at stage 1, the standard product firm chooses the variety and the custom firm chooses the lead time and then both firms set prices simultaneously. We characterize the subgame‐perfect Nash equilibrium of the game. We find that both firms can coexist in equilibrium, either sharing the market as local monopolists or in a price‐competitive mode. The standard product firm may offer significant or minimal variety depending on the equilibrium outcome. We provide several interesting insights on the variety, lead time, and prices of the products offered and on the impact of problem parameters on the equilibrium outcomes. For instance, we show that the profit margin and price of the custom product are likely to be higher than that of standard products in equilibrium under certain conditions. Also, custom firms are more likely to survive and succeed in product markets with larger potential market sizes. Another interesting insight is that increased consumer sensitivity to product fit may result in lower lead time for the custom product.     

The Culture–Performance Relationship in M&A: From Yes/No to How

Do cultural differences have an impact on the performance of M&A? Despite the widely accepted myth that they do, and in a negative way, a review of extant research provides contradictory findings. In this article, we explore reasons for this contradiction and propose solutions in the form of propositions and a theoretical framework. We begin with a brief overview of extant research on the culture‐performance relationship in M&A. In light of the contradictions emerging from this review, we move on to identifying three areas of complexity explaining this confusion, and for each one, we suggest propositions to guide future research. We then summarize our argument using a theoretical framework. Because of the long‐term and dynamic nature of the M&A process, we argue that instead of studying the simple performance impact of cultural differences in M&A, we should move on to thinking how cultural differences impact on the M&A process and its outcome.     

Multiperson Bargaining and Strategic Complexity

We investigate the effect of introducing costs of complexity in the n‐person unanimity bargaining game. As is well‐known, in this game every individually rational allocation is sustainable as a Nash equilibrium (also as a subgame perfect equilibrium if players are sufficiently patient and if n & 2). Moreover, delays in agreement are also possible in such equilibria. By limiting ourselves to a plausible notion of complexity that captures length of memory, we find that the introduction of complexity costs (lexicographically with the standard payoffs) does not reduce the range of possible allocations but does limit the amount of delay that can occur in any agreement. In particular, we show that in any n‐player game, for any allocation z, an agreement on z at any period t can be sustained as a Nash equilibrium of the game with complexity costs if and only if t≤n. We use the limit on delay result to establish that, in equilibrium, the strategies implement stationary behavior. Finally, we also show that ‘noisy Nash equilibrium’ with complexity costs sustains only the unique stationary subgame perfect equilibrium allocation.     

Stochastic Learning Dynamics and Speed of Convergence in Population Games

We study how long it takes for large populations of interacting agents to come close to Nash equilibrium when they adapt their behavior using a stochastic better reply dynamic. Prior work considers this question mainly for 2 × 2 games and potential games; here we characterize convergence times for general weakly acyclic games, including coordination games, dominance solvable games, games with strategic complementarities, potential games, and many others with applications in economics, biology, and distributed control. If players' better replies are governed by idiosyncratic shocks, the convergence time can grow exponentially in the population size; moreover, this is true even in games with very simple payoff structures. However, if their responses are sufficiently correlated due to aggregate shocks, the convergence time is greatly accelerated; in fact, it is bounded for all sufficiently large populations. We provide explicit bounds on the speed of convergence as a function of key structural parameters including the number of strategies, the length of the better reply paths, the extent to which players can influence the payoffs of others, and the desired degree of approximation to Nash equilibrium.