Introduction to Repeated Games with Private Monitoring

We present a brief overview of recent developments in discounted repeated games with (imperfect) private monitoring. The literature explores the possibility of cooperation in a long-term relationship, where each agent receives imperfect private information about the opponents' actions. Although this class of games admits a wide range of applications such as collusion under secret price-cutting, exchange of goods with uncertain quality, and observation errors, it has fairly complex mathematical structure due to the lack of common information shared by players. This is in sharp contrast to the well-explored case of repeated games under public information (with the celebrated Folk Theorems), and until recently little had been known about the private monitoring case. However, rapid developments in the past few years have revealed the possibility of cooperation under private monitoring for some class of games. Journal of Economic Literature Classification Numbers: C72, C73, D43, D82, L13, L41.     

Evolutionary Theory and Economic Policy with Reference to Sustainability

Synopsis The policy recommendations of most economists are driven by a view of economic reality embodied in Walrasian general equilibrium theory. Ironically, the Walrasian system has been all but abandoned by leading economic theorists. It has been demonstrated to be theoretically untenable, its basic assumptions about human decision making have been empirically falsified, and it consistently makes poor predictions of economic behavior. The current revolution in welfare economics offers opportunities on two related fronts for an evolutionary perspective on human behavior to reshape economic theory and policy. The first opportunity is to incorporate empirically-based information about human behavior to the study of human wants and their formation. This includes information about the evolution of the genetic component of decision making as well as the cultural dimensions of behavior. Expanding the role of economic analysis beyond stylized market behavior to focus on well-being (real utility) has far-reaching consequences for microeconomic policy. Secondly, abandoning the Walrasian model also means rethinking the microfoundations approach to the economic analysis of sustainability. This opens the door for economists to engage with the growing body of research on the evolution of whole societies. One link between the evolution of human behavior and the evolution of human societies is the psychological phenomenon of considering sunk costs. Understanding and overcoming the sunk cost fallacy may be the key to creating a sustainable society.     

Second chance offers versus sequential auctions: theory and behavior

Second chance offers in online marketplaces involve a seller conducting an auction for a single object and then using information from the auction to offer a losing bidder a take-it-or-leave-it price for another unit. We theoretically and experimentally investigate this practice and compare it to two sequential auctions. We show that the equilibrium bidding strategy in the second chance offer mechanism only exists in mixed strategies, and we observe that this mechanism generates more profit for the auctioneer than two sequential auctions. We also observe virtually no rejections of profitable offers in the ultimatum bargaining stage.      

Implementation by Iterative Dominance and Backward Induction: An Experimental Comparison

We report experimental results on the relative performance of simultaneous and sequential versions of the Abreu-Matsushima mechanism. Under the simultaneous version, subjects typically use undominated strategies, but apply only a limited number of iterations of dominance. Consequently the unique strategy surviving iterative elimination of strictly dominated strategies is rarely observed. Under the sequential version, subjects also typically use undominated strategies, but apply only a limited number of steps of backward induction. Thus the backward induction outcome is also rarely observed. The sequential version results in fewer observed outcomes corresponding to the predicted outcome than the simultaneous version. Journal of Economic Literature Classification Number(s): C72, C92.     

Common belief of rationality in games of perfect information

Aumann (1995) showed that for games with perfect information common knowledge of substantive rationality implies backward induction. Substantive rationality is defined in epistemic terms, that is, in terms of knowledge. We show that when substantive rationality is defined in doxastic terms, that is, in terms of belief, then common belief of substantive rationality implies backward induction. Aumann (1998) showed that material rationality implies backward induction in the centipede game. This result does not hold when rationality is defined doxastically. However, if beliefs are interpersonally consistent then common belief of material rationality in the centipede game implies common belief of backward induction.     

Strategic delay and endogenous offers in bargaining games with private information

This paperendogenously determines the order of offers and the duration of delay in reaching agreement between buyers and sellers in a continuous-time bargaining game in which a seller wishes to vend an object of known cost to a buyer, to whom the value of the good is private information, and in which each player can choose to strategically delay a response to a previous offer or to interrupt the delay of his rival. Both buyers and sellers are shown to prefer to move first in a model of bargaining in which: (1) either player can make the first offer; (2) after the minimum time has elapsed from the previous offer, either player can make an offer; and (3) players can choose to strategically delay and refrain from making an offer after the previous offer. When the buyer moves first, the equilibrium response for the seller is to accept the offer immediately. When the seller moves first the equilibrium is characterized by the seller making all but the last offer, with minimal feasible delay between successive offers. Observable endogenous delay in reaching an agreement in such equilibria approaches zero as the minimal feasible delay between offers approaches zero. This indicates that in noncooperative bargaining models with private information, where players can strategically delay their offers, endogenizing the order in which players make offers removes the ability of informational asymmetries to generate equilibria exhibiting endogenous delay in reaching an agreement.     

Level-k analysis of experimental centipede games

As one of the best-known examples of the paradox of backward induction, centipede games have prompted a host of studies with various approaches and explanations (McKelvey and Palfrey, 1992, Fey et al., 1996, Nagel and Tang, 1998, Rapoport et al., 2003, Palacios-Huerta and Volij, 2009). Focusing on initial plays observed in experiments, this paper attempts to offer another explanation based on thorough study of level-k models as applied to these games. Borrowing ideas from the cognitive hierarchy model (Camerer et al., 2004), the authors constructed a group of models based on levels of rationality, and also tested for various assumptions on the play of the most naïve player type in these models. It was found that level-k models generally perform better than the agent quantal response equilibrium (AQRE) model and its variant with altruistic player types for increasing-pie centipede games, while the AQRE model with altruistic player types performs better in constant-pie games.     

Strong Belief and Forward Induction Reasoning

We provide a unified epistemic analysis of some forward-induction solution concepts in games with complete and incomplete information. We suggest that forward induction reasoning may be usefully interpreted as a set of assumptions governing the players' belief revision processes, and define a notion of strong belief to formalize these assumptions. Building on the notion of strong belief, we provide an epistemic characterization of extensive-form rationalizability and the intuitive criterion, as well as sufficient epistemic conditions for the backward induction outcome in generic games with perfect information. Journal of Economic Literature Classification Numbers: C72, D82.     

Evolutionary dynamics and backward induction

The backward induction (or subgame-perfect) equilibrium of a perfect information game is shown to be the unique evolutionarily stable outcome for dynamic models consisting of selection and mutation, when the mutation rate is low and the populations are large.     

On the Econ within

This essay examines the critique of behavioral economics that Infante, Lecouteaux and Sugden (ILS) offer in:"Preference Purification and the Inner Rational Agent.” It identifies and questions three main criticisms that ILS make: (i) a methodological criticism, alleging that there is no psychological basis for the attribution of purified preferences, (ii) an epistemological criticism, alleging that there is little evidence for claims about purified preferences, and (iii) a normative criticism, arguing that policies should aim to facilitate people’s choices rather than to satisfy purified preferences. The essay also distinguishes the view of welfare economics defended in Preference, Value, Choice, and Welfare from the claims of behavioral economists ILS criticize.     

Social Comparisons in Ultimatum Bargaining

Experiments are used to examine the effects of social comparisons in ultimatum bargaining. We inform responders about the average offer before they decide whether to accept or reject their specific offer. This significantly increases offers and offer‐specific rejection probabilities. For comparison, we consider another change in informational conditions: telling responders the total pie is $30—ex ante it was either $15 or $30—affects offers and rejection probabilities roughly as much. Our results are consistent with people’s dislike for deviations from the norm of equity but inconsistent with fairness theories, where people dislike income disparity between themselves and their referents.     

A Backward Induction Experiment

This paper reports experiments with one-stage and two-stage alternating-offers bargaining games. Payoff-interdependent preferences have been suggested as an explanation for experimental results that are commonly inconsistent with players' maximizing their monetary payoffs and performing backward induction calculations. We examine whether, given payoff-interdependent preferences, players respect backward induction. To do this, we break backward induction into its components, subgame consistency and truncation consistency. We examine each by comparing the outcomes of two-stage bargaining games with one-stage games with varying rejection payoffs. We find and characterize systematic violations of both subgame and truncation consistency. Journal of Economic Literature Classification Numbers: C70, C78.     

Cycles of Learning in the Centipede Game

This paper studies the evolutionary properties of the Centipede Game. For this game, the use of backward induction as a model of rational behavior has been consistently challenged by the experimental evidence. Our claim is that backward induction can still accurately predict the players' behavior, provided that they are given time enough to appreciate the strategic environment in which they operate. We support this claim by proving convergence to the backward induction solution for all continuous-time monotonic selection dynamics. However, we also show that this solution is intrinsically unstable, and how this instability is positively related to the length of the game. Journal of Economic Literature Classification Numbers: C72, C79.     

Signaling bargaining power: Strategic delay versus restricted offers

Summary. I study the first-round separating equilibrium of a buyer-seller bargaining game, extended to allow for asymmetric information, strategically delayed offers and offers restricted to a portion of the good. When bargaining is over a consumption good, in equilibrium the “strong” buyer uses a restricted offer if his optimal consumption path is conservative relative to the “weak” buyer. A pure restricted offer may even be a costless, efficient signal. When the good is durable, a pure strategic delay is involved in signaling a strong bargaining position if the discount factor is high. Received: June 24, 1998; revised version: May 30, 1999     

An Experiment to Evaluate Bayesian Learning of Nash Equilibrium Play

We report on an experiment designed to evaluate the empirical implications of Jordan's model of Bayesian learning in games of incomplete information. A finite example is constructed in which the model generates unique predictions of subjects' choices in nearly all periods. When the “true” game defined by players' private information was one with a unique equilibrium in pure strategies, the experimental subjects' play converged to the equilibrium, as Jordan's theory predicts, even when the subjects had not attained complete information about one another. But when there were two pure strategy equilibria, the theory's predictions were not consistent with observed behavior. Journal of Economic Literature Classification numbers: D83, C72, C92.     

Negotiation and take it or leave it in common agency

This paper contrasts models of common agency in which principals compete in incentive contracts (that is, they make take it or leave it offers) with models where principals offer agents menus of incentive contracts from which the final contract is negotiated. It is shown that pure strategy equilibria in incentive contracts are robust to the possibility that principals might offer menus. In addition, a no-externalities condition is given such that any pure strategy equilibrium allocation with menus can be supported with competition in incentive contracts. The no-externalities condition is restrictive, but it is shown that it applies in most well-known common agency problems, including, for example, the Bertrand pricing problem.     

Endogenizing market institutions: An experimental approach

We study experimental two-sided markets in which the information structure is endogenous. When submitting an offer, a trader decides which other traders will be informed about the offer. This setup allows both a decentralized bargaining market (Chamberlin, J. Polit. Econ. 56 (1948) 95), and a double auction market (Smith J. Polit. Econ. 70 (1962) 111) as special cases. The results show that offers are typically directed to all traders of the other side of the market, but to none of the traders of the same side of the market. Even though traders receive much less information, the resulting market institution leads to the same outcomes in terms of prices and efficiency as a double auction market. In two additional treatments we examine the robustness of these results. First, it is found that the market institution adapts predictably, but not necessarily efficiently, to the imposition of transaction costs. Second, we find that the preference of sellers to conceal offers from competitors is strict. At the same time, sellers benefit collectively when they reveal offers to each other.     

Bargaining and Search with Incomplete Information about Outside Options

This paper considers a model of bargaining in which the seller makes offers and the buyer can search (at a cost) for an outside option; the outside option cannot be credibly communicated, and the seller's offer is recallable by the buyer for one period. There are essentially two equilibrium regimes. For sufficiently high search cost, the game ends immediately; otherwise the search occurs in equilibrium. Compared to the case where the buyer can communicate his outside option, the seller is worse off, and the game results in search for a smaller set of values of the search cost, i.e., less equilibrium delay.C72.     

Secrecy and fairness in plea bargaining with multiple defendants

I consider a model of plea bargaining with multiple codefendants. I mainly compare the equilibrium outcomes under joint negotiations whereby both defendants can observe both plea offers and under separate (secret) negotiations whereby they can observe only their own respective offer. Contrary to the widespread perceptions, the prosecutor is made worse off under secret negotiations or at best as well off as under joint negotiations. I also discuss the implication of equilibrium offers on fairness.      

Bargaining games with a random sequence of who makes the offers

This note relaxes the rigid assumption in recent bargaining theory of a strictly alternating sequence of offers, and instead considers the opposite extreme: at each bargaining round a random event determines who gets to make an offer, irrespective of who made the previous offer. The equilibrium of this game is practically identical to the equilibrium of the alternating offers game if the horizon is infinite. With a finite horizon the properties of the equilibrium are more satisfying than the properties of the equilibrium of the alternating offers game.