Ultimatum Bargaining Behavior in Israel, Japan, Slovenia, and the United States: A Social Utility Analysis

We explain the main features of the results of the four-country ultimatum bargaining experiments of Roth et al. (1991), Amer. Econom. Rev.81, 1068–1095) by a social utility model. The specification of social utility of a player has two parts: a linear combination of the monetary payoffs of the proposer and the responder and payoff uncertainty. We find that, on average, responders have negative regard for proposers' earnings in all countries. Proposers have negative regard for responders' monetary earnings in countries where responders have high negative regard for proposers' earnings (USA and Slovenia). In countries where responders have low negative regard for proposers' earnings (Israel and Japan), proposers are expected payoff maximizers. Journal of Economic Literature Classification Numbers: A13, C19, C44, C72, C92, D63, D64.     

Commuters route choice behaviour

The paper reports laboratory experiments with a two route choice scenario. In each session 18 subjects had to choose between a main road M and a side road S. The capacity of M was larger. Feedback was given in treatment I only on the subjects' own travel time and in treatment II on travel time for M and S. The main results are as follows:

• Mean numbers on M and S are near to pure equilibrium.

• Fluctuations persist until the end of the sessions.

• The total number of changes is significantly greater in treatment I.

• Subjects' road changes and payoffs are negatively correlated.

• A direct response mode results in more changes for bad payoffs whereas a contrary response mode shows opposite reactions.

• Simulations of an extended payoff sum learning model fits the main results of the statistical evaluation of the data.

Coordination and Information in Critical Mass Games: An Experimental Study

We present experimental results on a repeated coordination game with Pareto-ranked equilibria in which a payoff from choosing an action is positive only if a critical mass of players choose that action. We design a baseline version of the game in which payoffs remain constant for values above the critical mass, and an increasing returns version in which payoffs keep increasing for values above the critical mass. We test the predictive power of security and payoff-dominance under different information treatments. Our results show that convergence to the payoff-dominant equilibrium is the modal limit outcome when players have full information about others' previous round choices, while this outcome never occurs in the remaining treatments. The paths of play in some groups reveal a tacit dynamic coordination by which groups converge to the efficient equilibrium in a step-like manner. Moreover, the frequency and speed of convergence to the payoff-dominant equilibrium are higher, ceteris paribus, when increasing returns are present. Finally, successful coordination seems to crucially depend on players' willingness to signal to others the choice of the action supporting the efficient equilibrium.     

Evolution and Mixed Strategies

Selten (1980, J. Theoret. Biol., 84, 93–101) showed that no mixed equilibria are evolutionarily stable when players can condition their strategies on their roles in a game. Alternatively, Harsanyi's (1973, Int. J. Game Theory, 2, 1–23) purification argument implies that all mixed equilibria are approximations of strict, and hence evolutionarily stable, equilibria of games with slightly perturbed payoffs. This paper reconciles these results: Approximations of mixed equilibria have high invasion barriers, and hence are likely to persist, when payoff perturbations are relatively important and role identification is relatively noisy. When payoff perturbations are unimportant and role identification is precise, approximations of mixed equilibria will have small invasion barriers and are unlikely to persist. Journal of Economic Literature Classification Numbers: C70, C78.     

The power of ESS: An experimental study

Abstract. Our experimental design mimics a traditional evolutionary game framework where players are matched pairwise to play a symmetric 33 bimatrix game that has two Nash equilibria. One equilibrium is an evolutionary stable state, or ESS; the other is an equilibrium in dominated strategies. Our primary experimental result is the observation that the ESS becomes extremely attractive when subjects have minimal information about the payoff functions, although the dominated equilibrium assures the highest equilibrium payoff. The attractiveness of the ESS is only moderate when players are completely informed about the 33 payoff matrix. Correspondence to: S.K. Berninghaus     

Computationally restricted unmediated talk under incomplete information

Summary. We show the role of unmediated talk with computational complexity bounds as both an information transmission and a coordination device for the class of two-player games with incomplete information and rational parameters. We prove that any communication equilibrium payoff of such games can be reached as a Bayesian-Nash equilibrium payoff of the game extended by a two phase universal mechanism of interim computationally restricted pre-play communication. The communication protocols are designed with the help of modern cryptographic tools. A familiar context in which our results could be applied is bilateral trading with incomplete information.Received: 9 September 2002, Revised: 14 March 2003, JEL Classification Numbers: C72. Correspondence to: Amparo UrbanoWe wish to thank financial aid from the Valencian Institute of Economic Research (IVIE) and partial support by DIGCYT under project PB95 - 1074. A previous version of this work appears as IVIE Working Paper WP-AD 99-07, under the title: "Unmediated talk under incomplete information".     

Learning to Play Limited Forecast Equilibria

This paper provides a learning justification for limited forecast equilibria, i.e., strategy profiles such that (1) players choose their actions in order to maximize the discounted average payoff over their horizon of foresight as given by their forecasts and (2) forecasts are correct on and off the equilibrium path. The limited forecast equilibria appear to be the stochastically stable outcomes of a simple learning process involving (vanishing) trembles.Journal of Economic LiteratureClassification Numbers: C72, D83.     

Sowing Doubt Optimally in Two-Person Repeated Games

Consider a two-person repeated game, where one of the players, P1, can sow doubt, in the mind of his opponent, as to what P1's payoffs are. This results in a two-person repeated game with incomplete information. By sowing doubt, P1 can sometimes increase his minimal equilibrium payoff in the original game. We prove that this minimum is maximal when only one payoff matrix, the negative of the payoff matrix of the opponent, is added (the opponent thus believes that he might play a zero-sum game). We obtain two formulas for calculating this maximal minimum payoff. Journal of Economic Literature Classification Numbers: C7, D8.     

Correlated Equilibrium in Stochastic Games

We study the existence of uniform correlated equilibrium payoffs in stochastic games. The correlation devices that we use are either autonomous (they base their choice of signal on previous signals, but not on previous states or actions) or stationary (their choice is independent of any data and is drawn according to the same probability distribution at every stage). We prove that any n-player stochastic game admits an autonomous correlated equilibrium payoff. When the game is positive and recursive, a stationary correlated equilibrium payoff exists. Journal of Economic Literature Classification Numbers: C72, C73.     

THE EXPECTED NUMBER OF COOPERATORS IN THE N‐PERSON PRISONERS’ DILEMMA WITH RANDOM ORDER MOVES*

The present paper examines the possibility of cooperation occurring in the N‐person prisoners’ dilemma, played with randomly ordered moves and imperfect information. To take into account imperfect cooperation, the degree of cooperation is evaluated in terms of the expected number of cooperators. It is shown that: (i) the expected number of cooperators is maximized at an equilibrium, a combination of strategies each of which is a type of trigger strategy; (ii) the maximum expected number of cooperators is attained in a state of imperfect cooperation for a range of payoff functions; and (iii) the maximum expected number of cooperators is non‐decreasing when the size of players’ group increases.     

Sequential Equilibrium and Competition in a Diamond–Dybvig Banking Model

Within the framework of a Diamond–Dybvig model [J. Polit. Econ.91(1983), 401–419], but with explicitly modelling the autarky choice during the planning period, we demonstrate that a mixed strategy bank run equilibrium that does not rely on sunspots may coexist with the sunspot run equilibrium previously studied in the literature. In a version of the model with multiple banks, there exist sequential equilibria that imply positive profits. However, the zero-profit contract in which runs never occur can be supported as the unique equilibrium outcome if the agents play pure strategies only and their beliefs are restricted to be consistennt with a forward induction argument.Journal of Economic LiteratureClassification Numbers: C72, G21     

Evolutionary Learning in Signalling Games

We study equilibrium selection by evolutionary learning in monotone two-type signalling games. The learning process we study extends that introduced by Young (1993, Econometrica61, 57–84) to deal with incomplete information and sequential moves; it thus involves stochastic trembles. For vanishing trembles the process gives rise to strong selection among sequential equilibria: if the game has separating equilibria, then in the long run only play according to the so-called Riley equilibrium will be observed frequently. Journal of Economic Literature Classification Number: C72.     

On the Rate of Convergence of Continuous-Time Fictitious Play

This paper shows, first, that continuous-time fictitious play converges (in both payoff and strategy terms) uniformly at ratet ^{− 1}in any finite two-person zero-sum game. The proof is, in essence, a simple Lyapunov-function argument. The convergence of discrete-time fictitious play is a straightforward corollary of this result. The paper also shows that continuous-time fictitious play converges in all finite weighted-potential games. In this case, the convergence is not uniform. It is conjectured, however, that any given continuous-time fictitious play of a finite weighted-potential game converges (in both payoff and strategy terms) at ratet ^{− 1}.Journal of Economic LiteratureClassification Numbers: C6, C7.     

Starting Small and Commitment

I study a model of a long-term partnership with two-sided incomplete information. The partners jointly determine the stakes of their relationship and individually decide whether to cooperate with or betray each other over time. I characterize the extremal—interim incentive efficient—equilibria. In these equilibria, the partners generally “start small,” and the level of interaction grows over time. The types of players separate quickly. Further, cooperation between “good” types is viable regardless of how pessimistic the players are about each other initially. The quick nature of separation in an extremal equilibrium contrasts with the outcome selected by a strong renegotiation criterion (as studied in Watson (1999, J. Econ. Theory85, 52–90). Journal of Economic Literature Classification Numbers: C72, C73, D74.     

The Role of Absolute Continuity in “Merging of Opinions” and “Rational Learning”

D. Blackwell and L. Dubins (1962, Ann. Math. Statist.38, 882–886) showed that opinions merge when priors are absolutely continuous. E. Kalai and E. Lehrer (1993, Econometrica61, 1019–1045) use this result to show that players in a repeated game eventually play like a Nash equilibrium. We provide an alternative proof of merging of opinions that clarifies the role of absolute continuity while casting doubt on the relevance of the result. Persistent disagreement, the opposite of merging, allows the construction of a sequence of mutually favorable “bets.” By a law of large numbers, both agents are certain they will win these bets on average. This certain disagreement violates absolute continuity. Journal of Economic Literature Classification Numbers: C11, C69, C72, D83.     

Testing for Ongoing Convergence in Transition Economies, 1970 to 1998

In this paper, the authors use a time-varying parameters procedure to test for a common growth path in the ex-Communist bloc, both pre- and postreform. They test whether there has been convergence within the bloc or between the bloc as a group and the West. Surprisingly, there is little evidence of convergence within the bloc, which brings into question the effectiveness of policies to reduce differentials in income per capita under the Communists. There is also little evidence of convergence with respect to the West, either during the period from 1970 to 1990 or if the reform years are included (i.e., 1970 to 1998). J. Comp. Econ., December 2001, 29(4), pp. 677–691. © 2001 Elsevier ScienceJournal of Economic Literature Classification Numbers: O40, C22, C23, C15.     

Delay and information aggregation in stopping games with private information

We consider equilibrium timing decisions in a model with a large number of players and informational externalities. The players have private information about a common payoff parameter that determines the optimal time to invest. They learn from each other in real time by observing past investment decisions. We develop new methods of analysis for such large games, and we give a full characterization of symmetric equilibria. We show that the equilibrium statistical inferences are based on an exponential learning model. Although the beliefs converge to truth, learning takes place too late. Ex-ante welfare is strictly between that without observational learning and that with full information.     

Monotone imitation

We analyze the social learning process of a group of individuals who have limited information about the payoff distributions of each action. We say that a behavioral rule is first-order monotone (FOM) if the number of individuals who play actions with first-order stochastic dominant payoff distributions is expected to increase in any environment. We provide a characterization of FOM rules. Both Imitate if Better and Schlag’s (J Econ Theory 78:130–156, 1998) Proportional Imitation rule are FOM. No FOM rule is dominant in the sense of having the best performance in every environment.     

Learning to collude: An experiment in convergence and equilibrium selection in oligopoly

The paper considers a simple oligopoly model where firms know their own and the average pay-off in the industry. Firms choose decision rules for trading. The theory predicts that there are three types of Nash equilibria in this game (collusive, Cournot and Stackelberg). Our experiments test the selection process. We find that there is clear evidence of convergence to an equilibrium, and whilst both Cournot and collusive outcomes were selected, the collusive equilibrium is more common. The experimental results also give insights into the process of individual learning, confirming that subjects follow aspiration rules rather than reinforcement rules.     

Bertrand–Edgeworth Duopoly with Buyouts or First Refusal Contracts

In this game the players are firms involved in a Bertrand–Edgeworth duopoly market. Payoffs to the low priced firm depend only on the own price, whereas the payoff to the high priced firm depends on both its own price and the price of the opponent. The price of the opponent enters the payoff function of the high priced firm through buyout or a first refusal contract. Only when the total capacity in the market is less than the output in a monopoly situation, there is an equilibrium in pure strategies.Journal of Economic LiteratureClassification Numbers: C72, D43, L12.