A spectrum of equilibration processes in game theory

In game theory, four dynamic processes converging towards an equilibrium are distinguished and ordered by way of agents' decreasing cognitive capacities. In the eductive process, each player has enough information to simulate perfectly the others' behavior and gets immediately to the equilibrium. In epistemic learning, each player updates his beliefs about others' future strategies, with regard to their sequentially observed actions. In behavioral learning, each player modifies his own strategies according to the observed payoffs obtained from his past actions. In the evolutionary process, each agent has a fixed strategy and reproduces in proportion to the utilities obtained through stochastic interactions. All along the spectrum, longer term dynamics makes up for weaker rationality, and physical relations substitute for mental interactions. Convergence, if any, is towards an always stronger equilibrium notion and selection of an equilibrium state becomes more sensitive to context and history. The processes can be mixed if associated to different periods, agents or mechanisms and deepened if obtained by formal reasoning principles.     

Optimal limited authority for principal

This article studies a principal-agent problem where the only commitment for the uninformed principal is to restrict the set of decisions she makes following a report by the informed agent. We show that an ex ante optimal equilibrium for the principal corresponds to a finite partition of the state space, and each retained decision is ex post suboptimal for the principal, biased toward the agent?s preference. Generally an optimal equilibrium does not maximize the number of decisions the principal can credibly retain. Compared to no commitment, limited authority improves the quality of communication from the agent. As a result, it can give the principal a higher expected payoff than delegating the decision to the agent.     

Learning in society

In an individual experimentation problem a decision maker learns only from his own experience. It is well known that an optimal experimentation strategy for such problems sometimes results in the best alternative being dropped altogether, which is the so-called “Rothschild effect.” Many experimentation problems of interest, however, involve learning from both individual experience and the experience of others. This paper shows that learning in society can overcome the Rothschild effect. We consider an economy with a continuum of infinitely lived players in which each player faces a multi-armed bandit and in each period a player observes the action choice of another randomly chosen player. We show that social conformity always happens in the long run, and we use this fact to derive a condition on the distribution of prior beliefs that implies that the fraction of players who choose the best alternative always converges to one in the long run.     

Proper belief revision and equilibrium in dynamic games

We present a theory of rationality in dynamic games in which players, during the course of the game, may revise their beliefs about the opponents’ utility functions. The theory is based upon the following three principles: (1) the players’ initial beliefs about the opponents’ utilities should agree on some profile u of utility functions, (2) every player should believe, at each of his information sets, that his opponents are carrying out optimal strategies and (3) a player at information set h should not change his belief about an opponent's ranking of strategies a and b if both a and b could have led to h. Scenarios with these properties are called preference conjecture equilibria for the profile u of utility functions. We show that every normal form proper equilibrium for u induces a preference conjecture equilibrium for u, thus implying existence of preference conjecture equilibrium.     

How good are supply function equilibrium models: an empirical analysis of the ERCOT balancing market

We present an empirical analysis of a supply function equilibrium model in the Texas spot electricity market. We derive conditions for optimal bidding behavior in a spot market with ex ante bilaterally contracted sales. By estimating costs, we are able to derive a set of ex post-and ex ante-optimal supply functions and use a non-parametric behavioral model to compare our theoretically optimal supply functions to actual offers made. Our results show that with the exception of the largest generators, firms make offers with markups and markdowns far in excess of what a model of profit-maximizing behavior suggests.     

Sampling equilibrium, with an application to strategic voting

We suggest an equilibrium concept for a strategic model with a large number of players in which each player observes the actions of only a small number of the other players. The concept fits well situations in which each player treats his sample as a prediction of the distribution of actions in the entire population, and responds optimally to this prediction. We apply the concept to a strategic voting model and investigate the conditions under which a centrist candidate can win the popular vote although his strength in the population is smaller than the strengths of the right and left candidates.     

Arms Races and Negotiations

Two players simultaneously decide whether or not to acquire new weapons in an arms race game. Each player's type determines his propensity to arm. Types are private information, and are independently drawn from a continuous distribution. With probability close to one, the best outcome for each player is for neither to acquire new weapons (although each prefers to acquire new weapons if he thinks the opponent will). There is a small probability that a player is a dominant strategy type who always prefers to acquire new weapons. We find conditions under which the unique Bayesian–Nash equilibrium involves an arms race with probability one. However, if the probability that a player is a dominant strategy type is sufficiently small, then there is an equilibrium of the cheap-talk extension of the game where the probability of an arms race is close to zero.     

An experiment on learning with limited information: nonconvergence, experimentation cascades, and the advantage of being slow

We present the results of an experiment on learning in a continuous-time low-information setting. For a dominance solvable version of a Cournot oligopoly with differentiated products, we find little evidence of convergence to the Nash equilibrium. In an asynchronous setting, characterized by players updating their strategies at different frequencies, play tends toward the Stackelberg outcome which favors the slower player. Convergence is significantly more robust for a “serial cost sharing” game, which satisfies a stronger solution concept of overwhelmed solvability. As the number of players grows, this improved convergence tends to diminish, seemingly driven by frequent and highly structured experimentation by players leading to a cascading effect in which experimentation by one player induces experimentation by others. These results have implications both for traditional oligopoly competition and for a wide variety of strategic situations arising on the Internet.     

Instability in the implementation of Walrasian allocations

The existence of game forms which implement Walrasian allocations as Cournot (Nash) equilibrium outcomes is well known. However, if the equilibria are also required to be locally dynamically stable, at least for environments with unique Walrasian allocations, this paper shows that the requisite game forms do not exist. Our definition of a game form entails certain regularity conditions, and requires the Cournot equilibrium to be unique when the Walrasian equilibrium is unique. The main result is that for such a game form, there does not exist a continuous-time strategy adjustment process which ensures the local stability of Cournot equilibria throughout a certain class of environments having unique Walrasian equilibria. Each trader adjusts his strategy in response to his own characteristics and the observed current strategies of others; but the direction and magnitude of adjustments are not constrained by any behavioral assumptions. The definitions permit the inclusion of an artificial player, such as an auctioneer, so the well-known tatonnement instability emerges as a special case.     

Equilibrium in Justifiable Strategies: A Model of Reason-based Choice in Extensive-form Games

I explore the idea that people care about the justifiability of their decisions in the context of two-person extensive games. Each player justifies his strategy  s  with a belief  b  of the opponent's strategy, which is consistent with the play path and maximally plausible (according to some exogenous criterion). We say that  s  is justifiable if against the ex post criticism that some other strategy  s'  outperforms  s  against  b  , the player can argue that playing  s'  would have exposed him to similar criticism in the opposite direction. Under a simplicity-based plausibility criterion, this concept implies systematic departures from maximizing behaviour in familiar games.     

Indirect higher order beliefs and cooperation

This study experimentally examines the role of indirect higher order beliefs in sequential psychological games. We consider a three-player sequential game in which the first and third players do not interact sequentially, but only through an intermediary, the second player. We posit that the third player’s decision to cooperate depends on his indirect higher order belief, namely, his belief about what the first player believes the second player would choose. We employ pre-play communication between the first and third players as a way to influence the third player’s indirect higher order belief. The evidence demonstrates that communication can effectively induce cooperation from the third player by shaping his indirect higher order belief.     

Informational properties of the Nash solutions of two stochastic nonzero-sum games

Within the framework of stochastic two-person nonzero-sum games, we deal with two commonly used models in engineering and economics—namely, the LQG (Linear-Quadratic-Gaussian) and the duopoly problems. We investigate how variations in information available to either player affect the equilibrium Nash strategies for these two models, whose existence and uniqueness have been proven in the paper. We show that for the LQG model better information for either player results in lower average Nash costs for both players; whereas for the duopoly model better information for one player helps him alone to achieve a higher average Nash profit, and it hurts the other player in the sense that his average Nash profit decreases. We further relate these properties of the Nash solutions for these two games to some of the distinct features of zero-sum games and team problems.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

Information acquisition in conflicts

This paper considers incentives for information acquisition ahead of conflicts. First, we characterize the (unique) equilibrium of the all-pay auction between two players with one-sided asymmetric information where one player has private information about his valuation. Then, we use our results to study information acquisition prior to an all-pay auction. If the decision to acquire information is observable, but not the information received, one-sided asymmetric information can occur endogenously in equilibrium. Moreover, the cut-off values of the cost of information that determine equilibrium information acquisition are higher than those in the first best. Thus, information acquisition is excessive. In contrast, with open or covert information acquisition, the equilibrium cut-off values are as in the first best.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

On the convergence to the Nash bargaining solution for action-dependent bargaining protocols

We consider a non-cooperative multilateral bargaining game and study an action-dependent bargaining protocol, that is, the probability with which a player becomes the proposer in a round of bargaining depends on the identity of the player who previously rejected. An important example is the frequently studied rejector-becomes-proposer protocol. We focus on subgame perfect equilibria in stationary strategies which are shown to exist and to be efficient. Equilibrium proposals do not depend on the probability to propose conditional on the rejection by another player. We consider the limit, as the bargaining friction vanishes. In case no player has a positive probability to propose conditional on his rejection, each player receives his utopia payoff conditional on being recognized. Otherwise, equilibrium proposals of all players converge to a weighted Nash bargaining solution, where the weights are determined by the probability to propose conditional on one's own rejection.     

Closed-loop equilibrium in a multi-stage innovation race

Summary. We examine a multistage model of an R&D race where players have multiple projects. We also develop perturbation methods for general dynamic games that can be expressed as analytic operators in a Banach space. We apply these perturbation methods to solve races with a small prize. We compute second-order asymptotically valid solutions for equilibrium and socially optimal decisions to determine qualitative properties of equilibrium. We find that innovators invest relatively too much on risky projects. Strategic reactions are ambiguous in general; in particular, a player may increase expenditures as his opponent moves ahead of him. Received: January 3, 2002; revised version: June 14, 2002 RID="*" ID="*" This is the final version of Judd (1985). The author gratefully acknowledges the comments of anonymous referees, Paul Milgrom, seminar participants at Northwestern University, the University of Chicago, the 1984 Summer Meetings of the Econometric Society, University of California at Berkeley, Stanford University, and Yale University, and the financial support of the National Science Foundation (SES-8409786, SES-8606581)     

Do people plan?

We report the results of an experimental investigation of a key axiom of economic theories of dynamic decision making—namely, that agents plan. Inferences from previous investigations have been confounded with issues concerning the preference functionals of the agents. Here, we present an innovative experimental design which is driven purely by dominance: if preferences satisfy dominance, we can infer whether subjects are planning or not. We implement three sets of experiments: the first two (the Individual Treatments) in which the same player takes decisions both in the present and the future; and the third (the Pairs Treatment) in which different players take decisions at different times. The two Individual treatments differed in that, in one, the subjects played sequentially, while, in the other, the subjects had to pre-commit to their future move. In all contexts, according to economic theory, the players in the present should anticipate the decision of the player in the future. We find that over half the participants in all three experimental treatments do not appear to be planning ahead; moreover, their ability to plan ahead does not improve with experience, except possibly when we force subjects to pre-commit to their future decision. These findings identify an important lacuna in economic theories, both for individual behaviour and for behaviour in games.

---

Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.      

Non-cooperative bargaining in apex games and the kernel

This paper studies non-cooperative bargaining with random proposers in apex games. Two different protocols are considered: the egalitarian protocol, which selects each player to be the proposer with equal probability, and the proportional protocol, which selects each player with a probability proportional to his number of votes. Expected equilibrium payoffs coincide with the kernel for the grand coalition regardless of the protocol. The equilibrium is in mixed strategies and the indifference conditions can be reinterpreted in the language of the kernel.