Quantal response equilibria for posted offer-markets

There is a growing body of data from game theory and industrial organization experiments that reveals systematic deviations from Nash equilibrium behavior. In this paper, the perfectly rational decision making embodied in Bertrand Nash equilibrium is generalized to allow for endogenously determined decision errors. Closed form solutions for equilibrium price distributions with endogenous errors are derived for several different models. In some of these models, the price distribution in a quantal response equilibrium, QRE, is affected by changes in structural variables although the Nash equilibrium remains unaltered. The quantal response approach is appealing since it thereby accounts for systematic deviations from the Bertrand Nash equilibrium.

Progress in Behavioral Game Theory

Is game theory meant to describe actual choices by people and institutions or not? It is remarkable how much game theory has been done while largely ignoring this question. The seminal book by von Neumann and Morgenstern, The Theory of Games and Economic Behavior, was clearly about how rational players would play against others they knew were rational. In more recent work, game theorists are not always explicit about what they aim to describe or advise. At one extreme, highly mathematical analyses have proposed rationality requirements that people and firms are probably not smart enough to satisfy in everyday decisions. At the other extreme, adaptive and evolutionary approaches use very simple models-mostly developed to describe nonhuman animals-in which players may not realize they are playing a game at all. When game theory does aim to describe behavior, it often proceeds with a disturbingly low ratio of careful observation to theorizing

Lost in Translation? Basis Utility and Proportionality in Games

Cooperative and noncooperative games have no representation of players's basis utilities. Basis utility is the natural reference point on a player's utility scale that enables the determination the marginal utility of any payoff or allocation. A player's basis utility can be determined by an observer and other players under standard rationality assumptions. Basis utility allows interpersonal comparison of proportional utility gains relative to the disagreement outcome. Proportional pure bargaining is the unique solution satisfying efficiency, symmetry, affine transformation invariance and monotonicity in pure bargaining games with basis utility. Characterization of the Nash (1950) bargaining solution requires the assumption of the irrelevance of basis utility in games with basis utility. All existing cooperative solution functions become translation invariant once proper account is taken of basis utility. The noncooperative rationality of these results is demonstrated with a proportional bargaining based on Gul (1988). Further noncooperative application is demonstrated by showing that quantal response equilibria with multiplicative error structures (Goeree, Holt and Palfrey (2004)) become translation invariant with specification of basis utility.Basis utility, equal split, Kalai-Smorodinsky solution, Nash bargaining, quantal response equilibria, proportional bargaining, translation invariance.

A cognitive hierarchy theory of one-shot games: Some preliminary results

Strategic thinking, best-response, and mutual consistency (equilibrium) are three key modelling principles in noncooperative game theory. This paper relaxes mutual consistency to predict how players are likely to behave in in one-shot games before they can learn to equilibrate. We introduce a one-parameter cognitive hierarchy (CH) model to predict behavior in one-shot games, and initial conditions in repeated games. The CH approach assumes that players use k steps of reasoning with frequency f (k). Zero-step players randomize. Players using k (≥ 1) steps best respond given partially rational expectations about what players doing 0 through k - 1 steps actually choose. A simple axiom which expresses the intuition that steps of thinking are increasingly constrained by working memory, implies that f (k) has a Poisson distribution (characterized by a mean number of thinking steps τ ). The CH model converges to dominance-solvable equilibria when τ is large, predicts monotonic entry in binary entry games for τ < 1:25, and predicts effects of group size which are not predicted by Nash equilibrium. Best-fitting values of τ have an interquartile range of (.98,2.40) and a median of 1.65 across 80 experimental samples of matrix games, entry games, mixed-equilibrium games, and dominance-solvable p-beauty contests. The CH model also has economic value because subjects would have raised their earnings substantially if they had best-responded to model forecasts instead of making the choices they did

Against Game Theory

People make choices. Often, the outcome depends on choices other people make. What mental steps do people go through when making such choices? Game theory, the most influential model of choice in economics and the social sciences, offers an answer, one based on games of strategy such as chess and checkers: the chooser considers the choices that others will make and makes a choice that will lead to a better outcome for the chooser, given all those choices by other people. It is universally established in the social sciences that classical game theory (even when heavily modified) is bad at predicting behavior. But instead of abandoning classical game theory, those in the social sciences have mounted a rescue operation under the name of “behavioral game theory.” Its main tool is to propose systematic deviations from the predictions of game theory, deviations that arise from character type, for example. Other deviations purportedly come from cognitive overload or limitations. The fundamental idea of behavioral game theory is that, if we know the deviations, then we can correct our predictions accordingly, and so get it right. There are two problems with this rescue operation, each of them is fatal. (1) For a chooser, contemplating the range of possible deviations, as there are many dozens, actually makes it exponentially harder to figure out a path to an outcome. This makes the theoretical models useless for modeling human thought or human behavior in general. (2) Modeling deviations are helpful only if the deviations are consistent, so that scientists (and indeed decision makers) can make predictions about future choices on the basis of past choices. But the deviations are not consistent. In general, deviations from classical models are not consistent for any individual from one task to the next or between individuals for the same task. In addition, people’s beliefs are in general not consistent with their choices. Accordingly, all hope is hollow that we can construct a general behavioral game theory. What can replace it? We survey some of the emerging candidates

Monotone concave operators: An application to the existence and uniqueness of solutions to the Bellman equation

We propose a new approach to the issue of existence and uniqueness of solutions to the Bellman equation, exploiting an emerging class of methods, called monotone map methods, pioneered in the work of Krasnosel’skii (1964) and Krasnosel’skii-Zabreiko (1984). The approach is technically simple and intuitive. It is derived from geometric ideas related to the study of fixed points for monotone concave operators defined on partially order spaces.Dynamic Programming; Bellman Equation; Unbounded Returns