Bargaining,Reputation, and Equilibrium Selection in Repeated Games with Contracts

Consider a two‐person intertemporal bargaining problem in which players choose actions and offers each period, and collect payoffs (as a function of that period's actions) while bargaining proceeds. This can alternatively be viewed as an infinitely repeated game wherein players can offer one another enforceable contracts that govern play for the rest of the game. Theory is silent with regard to how the surplus is likely to be split, because a folk theorem applies. Perturbing such a game with a rich set of behavioral types for each player yields a specific asymptotic prediction for how the surplus will be divided, as the perturbation probabilities approach zero. Behavioral types may follow nonstationary strategies and respond to the opponent's play. In equilibrium, rational players initially choose a behavioral type to imitate and a war of attrition ensues. How much should a player try to get and how should she behave while waiting for the resolution of bargaining? In both respects she should build her strategy around the advice given by the “Nash bargaining with threats” (NBWT) theory developed for two‐stage games. In any perfect Bayesian equilibrium, she can guarantee herself virtually her NBWT payoff by imitating a behavioral type with the following simple strategy: in every period, ask for (and accept nothing less than) that player's NBWT share and, while waiting for the other side to concede, take the action Nash recommends as a threat in his two‐stage game. The results suggest that there are forces at work in some dynamic games that favor certain payoffs over all others. This is in stark contrast to the classic folk theorems, to the further folk theorems established for repeated games with two‐sided reputational perturbations, and to the permissive results obtained in the literature on bargaining with payoffs as you go.     

Axiomatic Equilibrium Selection for Generic Two‐Player Games

For a finite game with perfect recall, a refinement of its set of Nash equilibria selects closed connected subsets, called solutions. Assume that each solution's equilibria use undominated strategies and some of its equilibria are quasi‐perfect, and that all solutions are immune to presentation effects; namely, if the game is embedded in a larger game with more pure strategies and more players such that the original players' feasible mixed strategies and expected payoffs are preserved regardless of what other players do, then the larger game's solutions project to the original game's solutions. Then, for a game with two players and generic payoffs, each solution is an essential component of the set of equilibria that use undominated strategies, and thus a stable set of equilibria as defined by Mertens (1989).     

Strategic Experimentation with Exponential Bandits

We analyze a game of strategic experimentation with two‐armed bandits whose risky arm might yield payoffs after exponentially distributed random times. Free‐riding causes an inefficiently low level of experimentation in any equilibrium where the players use stationary Markovian strategies with beliefs as the state variable. We construct the unique symmetric Markovian equilibrium of the game, followed by various asymmetric ones. There is no equilibrium where all players use simple cut‐off strategies. Equilibria where players switch finitely often between experimenting and free‐riding all yield a similar pattern of information acquisition, greater efficiency being achieved when the players share the burden of experimentation more equitably. When players switch roles infinitely often, they can acquire an approximately efficient amount of information, but still at an inefficient rate. In terms of aggregate payoffs, all these asymmetric equilibria dominate the symmetric one wherever the latter prescribes simultaneous use of both arms.     

Repeated Games with Differential Time Preferences

When players have identical time preferences, the set of feasible repeated game payoffs coincides with the convex hull of the underlying stage- game payoffs. Moreover, all feasible and individually rational payoffs can be sustained by equilibria if the players are sufficiently patient. Neither of these facts generalizes to the case of different time preferences. First, players can mutually benefit from trading payoffs across time. Hence, the set of feasible repeated game payoffs is typically larger than the convex hull of the underlying stage-game payoffs. Second, it is not usually the case that every trade plan that guarantees individually rational payoffs can be sustained by an equilibrium, no matter how patient the players are. This paper provides a simple characterization of the sets of Nash and of subgame perfect equilibrium payoffs in two-player repeated games.     

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.     

Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information

This paper analyzes a class of games of incomplete information where each agent has private information about her own type, and the types are drawn from an atomless joint probability distribution. The main result establishes existence of pure strategy Nash equilibria (PSNE) under a condition we call the single crossing condition (SCC), roughly described as follows: whenever each opponent uses a nondecreasing strategy (in the sense that higher types choose higher actions), a player's best response strategy is also nondecreasing. When the SCC holds, a PSNE exists in every finite‐action game. Further, for games with continuous payoffs and a continuum of actions, there exists a sequence of PSNE to finite‐action games that converges to a PSNE of the continuum‐action game. These convergence and existence results also extend to some classes of games with discontinuous payoffs, such as first‐price auctions, where bidders may be heterogeneous and reserve prices are permitted. Finally, the paper characterizes the SCC based on properties of utility functions and probability distributions over types. Applications include first‐price, multi‐unit, and all‐pay auctions; pricing games with incomplete information about costs; and noisy signaling games.     

Simple Bounds on the Value of a Reputation

We introduce entropy techniques to study the classical reputation model in which a long‐run player faces a series of short‐run players. The long‐run player's actions are possibly imperfectly observed. We derive explicit lower and upper bounds on the equilibrium payoffs to the long‐run player.     

RISK AVERSION AND EXPECTED UTILITY THEORY: AN EXPERIMENT WITH LARGE AND SMALL STAKES

We employ a novel data set to estimate a structural econometric model of the decisions under risk of players in a game show where lotteries present payoffs in excess of half a million dollars. The decisions under risk of players in the presence of large payoffs allow us to estimate the parameters of the curvature of the von Neumann–Morgenstern utility function—not only locally, as in previous studies in the literature, but also globally. Our estimates of relative risk aversion indicate that a constant relative risk aversion parameter of about 1 captures the average of the sample population. We also find that individuals are practically risk neutral at small stakes and risk averse at large stakes—a necessary condition, according to Rabin’s calibration theorem, for expected utility to provide a unified account of individuals’ attitudes toward risk. Finally, we show that for lotteries characterized by substantial stakes, nonexpected utility theories fit the data equally as well as expected utility theory.     

Bargaining with Interdependent Values

A seller and a buyer bargain over the terms of trade for an object. The seller receives a perfect signal that determines the value of the object to both players, whereas the buyer remains uninformed. We analyze the infinite‐horizon bargaining game in which the buyer makes all the offers. When the static incentive constraints permit first‐best efficiency, then under some regularity conditions the outcome of the sequential bargaining game becomes arbitrarily efficient as bargaining frictions vanish. When the static incentive constraints preclude first‐best efficiency, the limiting bargaining outcome is not second‐best efficient and may even perform worse than the outcome from the one‐period bargaining game. With frequent buyer offers, the outcome is then characterized by recurring bursts of high probability of agreement, followed by long periods of delay in which the probability of agreement is negligible.     

Formalizing the Precautionary Principle Accounting for Strategic Interaction,Natural Factors,and Technological Factors

Four dimensions of the precautionary principle (PP), involving threat, uncertainty, action, and command, are formalized at the level of set theory and the level of individual players and natural and technological factors. Flow and decision diagrams with a feedback loop are developed to open up a new research agenda. The role of strategic interaction and games in the PP is underdeveloped or nonexistent in today's literature. To rectify this deficiency, six kinds of games are identified in the four PP dimensions. The games can be interlinked since player sets can overlap. Characteristics are illustrated. Accounting for strategic interaction, the article illustrates uncertainty in the PP regarding which game is played, which players participate in which game, strategy sets, payoffs, incomplete information, risk attitudes, and bounded rationality. The insurance and lottery games analyzed earlier for the safe minimum standard (SMS) for species extinction are revisited and placed into a broader context illustrating strategic interaction. Uncertainty about payoffs illustrates transformations back and forth between the chicken game, battle of the sexes, assurance game, and prisoner's dilemma.     

Game Theory and Risk Analysis

Risk analysts often analyze adversarial risks from terrorists or other intelligent attackers without mentioning game theory. Why? One reason is that many adversarial situations—those that can be represented as attacker‐defender games, in which the defender first chooses an allocation of defensive resources to protect potential targets, and the attacker, knowing what the defender has done, then decides which targets to attack—can be modeled and analyzed successfully without using most of the concepts and terminology of game theory. However, risk analysis and game theory are also deeply complementary. Game‐theoretic analyses of conflicts require modeling the probable consequences of each choice of strategies by the players and assessing the expected utilities of these probable consequences. Decision and risk analysis methods are well suited to accomplish these tasks. Conversely, game‐theoretic formulations of attack‐defense conflicts (and other adversarial risks) can greatly improve upon some current risk analyses that attempt to model attacker decisions as random variables or uncertain attributes of targets (“threats”) and that seek to elicit their values from the defender's own experts. Game theory models that clarify the nature of the interacting decisions made by attackers and defenders and that distinguish clearly between strategic choices (decision nodes in a game tree) and random variables (chance nodes, not controlled by either attacker or defender) can produce more sensible and effective risk management recommendations for allocating defensive resources than current risk scoring models. Thus, risk analysis and game theory are (or should be) mutually reinforcing.     

Linear programming approach to solve interval-valued matrix games

Matrix game theory is concerned with how two players make decisions when they are faced with known exact payoffs. The aim of this paper is to develop a simple and an effective linear programming method for solving matrix games in which the payoffs are expressed with intervals. Because the payoffs of the matrix game are intervals, the value of the matrix game is an interval as well. Based on the definition of the value for matrix games, the value of the matrix game may be regarded as a function of values in the payoff intervals, which is proven to be non-decreasing. A pair of auxiliary linear programming models is formulated to obtain the upper bound and the lower bound of the value of the interval-valued matrix game by using the upper bounds and the lower bounds of the payoff intervals, respectively. By the duality theorem of linear programming, it is proven that two players have the identical interval-type value of the interval-valued matrix game. Also it is proven that the linear programming models and method proposed in this paper extend those of the classical matrix games. The linear programming method proposed in this paper is demonstrated with a real investment decision example and compared with other similar methods to show the validity, applicability and superiority.     

Gambling Reputation: Repeated Bargaining With Outside Options

We study the role of incomplete information and outside options in determining bargaining postures and surplus division in repeated bargaining between a long‐run player and a sequence of short‐run players. The outside option is not only a disagreement point, but reveals information privately held by the long‐run player. In equilibrium, the uninformed short‐run players' offers do not always respond to changes in reputation and the informed long‐run player's payoffs are discontinuous. The long‐run player invokes inefficient random outside options repeatedly to build reputation to a level where the subsequent short‐run players succumb to his extraction of a larger payoff, but he also runs the risk of losing reputation and relinquishing bargaining power. We investigate equilibrium properties when the discount factor goes to 1 and when the informativeness of outside options diffuses. In both cases, bargaining outcomes become more inefficient and the limit reputation‐building probabilities are interior.     

Social Learning in One‐Arm Bandit Problems

We study a two‐player one‐arm bandit problem in discrete time, in which the risky arm can have two possible types, high and low, the decision to stop experimenting is irreversible, and players observe each other's actions but not each other's payoffs. We prove that all equilibria are in cutoff strategies and provide several qualitative results on the sequence of cutoffs.     

Contract,Mechanism Design,and Technological Detail

This paper develops a theoretical framework for studying contract and enforcement in settings with nondurable trading opportunities and complete but unverifiable information. The framework explicitly accounts for the parties' individual trade actions. The sets of implementable state‐contingent payoffs, under various assumptions about renegotiation opportunities, are characterized and compared. The results indicate the benefit of modeling trade actions as individual, rather than as public, and they highlight the usefulness of a structured game‐theoretic framework for applied research.     

The Coalitional Nash Bargaining Solution

The coalitional Nash bargaining solution is defined to be the core allocation for which the product of players' payoffs is maximal. We consider a non‐cooperative model with discounting in which one team may form and every player is randomly selected to make a proposal in every period. The grand team, consisting of all players, generates the largest surplus. But a smaller team may form. We show that as players get more patient if an efficient and stationary equilibrium exists, it must deliver payoffs that correspond to the coalitional Nash bargaining solution. We also characterize when an efficient and stationary equilibrium exists, which requires conditions that go beyond the nonemptiness of the core.     

Repeated Games Where the Payoffs and Monitoring Structure Are Unknown

This paper studies repeated games with imperfect public monitoring where the players are uncertain both about the payoff functions and about the relationship between the distribution of signals and the actions played. We introduce the concept of perfect public ex post equilibrium (PPXE), and show that it can be characterized with an extension of the techniques used to study perfect public equilibria. We develop identifiability conditions that are sufficient for a folk theorem; these conditions imply that there are PPXE in which the payoffs are approximately the same as if the monitoring structure and payoff functions were known. Finally, we define perfect type‐contingently public ex post equilibria (PTXE), which allows players to condition their actions on their initial private information, and we provide its linear programming characterization.     

On Forward Induction

A player's pure strategy is called relevant for an outcome of a game in extensive form with perfect recall if there exists a weakly sequential equilibrium with that outcome for which the strategy is an optimal reply at every information set it does not exclude. The outcome satisfies forward induction if it results from a weakly sequential equilibrium in which players' beliefs assign positive probability only to relevant strategies at each information set reached by a profile of relevant strategies. We prove that if there are two players and payoffs are generic, then an outcome satisfies forward induction if every game with the same reduced normal form after eliminating redundant pure strategies has a sequential equilibrium with an equivalent outcome. Thus in this case forward induction is implied by decision‐theoretic criteria.     

Bounded Memory and Biases in Information Processing

Before choosing among two actions with state‐dependent payoffs, a Bayesian decision‐maker with a finite memory sees a sequence of informative signals, ending each period with fixed chance. He summarizes information observed with a finite‐state automaton. I characterize the optimal protocol as an equilibrium of a dynamic game of imperfect recall; a new player runs each memory state each period. Players act as if maximizing expected payoffs in a common finite action decision problem. I characterize equilibrium play with many multinomial signals. The optimal protocol rationalizes many behavioral phenomena, like “stickiness,” salience, confirmation bias, and belief polarization.     

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.