The logit equilibrium: a perspective on intuitive behavioral anomalies.

• Author: Anderson, Simon P.

1. Introduction

   With the possible exception of supply and demand, the Nash equilibrium is the most widely used theoretical construct in economics today. Indeed, almost all developments in some fields such as industrial organization are based on a game-theoretic analysis. With Nash as its centerpiece, game theory is becoming more widely applied in other disciplines such as law, biology, and political science and it is arguably the closest thing there is to being a general theory of social science, a role envisioned early on by von Neumann and Morgenstern (1944). However, many researchers are uneasy about using a strict game-theoretic approach, given the widespread documentation of anomalies observed in laboratory experiments (Kagel and Roth 1995; Goeree and Holt 2001). This skepticism is particularly strong in psychology, where experimental methods are central. In relatively nonexperimental fields such as political science, the opposition to the use of the "rational choice" approach is based in part on doubts about the extrem e rationality assumptions that underlie virtually all "formal modeling" of political behavior. (1)

   This paper presents basic results for a relatively new approach to the analysis of strategic interactions, based on a relaxation of the assumption of perfect rationality used in standard game theory. By introducing some noise into behavior via a logit probabilistic choice function, the sharp-edged best reply functions that intersect at a Nash equilibrium become "smooth." The resulting logit equilibrium is essentially at the intersection of stochastic best-response functions (McKelvey and Palfrey 1995). The comparative statics properties of such a model are analogous to those obtained by shifting smooth demand and supply curves, as opposed to the constant price that results from a supply shift with perfectly elastic demand (or vice versa). Analogously, parameter changes that have no effect on a Nash equilibrium may move logit equilibrium predictions in a smooth and often intuitive manner. Many general properties and specific applications of this approach have been developed for models with a discrete set of ch oices (e.g., McKelvey and Palfrey 1995, 1998; Chen, Friedman, and Thisse 1996; McKelvey, Palfrey, and Weber 2000; Goeree and Holt 2000a, b, c; Goeree, Holt, and Palfrey 2000, in press). In contrast, this paper considers models with a continuum of decisions, of the type that naturally arise in standard economic models where prices, claims, efforts, locations, etc. are usually assumed to be continuous. With noisy behavior and a continuum of decisions, the equilibria are characterized by densities of decisions, and the properties of these models are studied by using techniques developed for the analysis of functions, that is, differential equations, stochastic dominance, fixed points in function space, etc. The contribution of this paper is to provide an easy-to-use tool kit of theoretical results (existence, uniqueness, symmetry, and comparative statics) that can serve as a foundation for subsequent applications. In addition, this paper gives a unified perspective on behavioral anomalies that have been reported in a series of laboratory experiments.

2. Stochastic Elements and Probabilistic Choice

   Regardless of whether randomness, or noise, is due to preference shocks, experimentation, or actual mistakes in judgment, the effect can be particularly important when players' payoffs are sensitive to others' decisions, for example, when payoffs are discontinuous as in auctions, or highly interrelated as in coordination games. Nor does noise cancel out when the Nash equilibrium is near a boundary of the set of feasible actions and noise pushes actions toward the interior, as in a Bertrand game in which the Nash equilibrium price equals marginal cost. In such games, small amounts of noise in decision making may have a large "snowball" effect when endogenous interactions are considered. (2) In particular, we will show that an amount of noise that only causes minor deviations from the optimal decision at the individual level may cause a dramatic shift in behavior in a game where one player's choice affects others' expected payoffs.

   The Nash equilibrium in the above-mentioned games is often insensitive to parameter changes that most observers would expect to have a large impact on actual behavior. In a minimum-effort coordination game, for example, a player's payoff is the minimum of all players' efforts minus the cost of the player's own effort. With simultaneous choices, both intuition and experimental evidence suggest that coordination on desirable, high-effort outcomes will be harder with more players and higher effort costs, despite the fact that any common effort level is a Nash equilibrium (Goeree and Holt 1999b). Another well-known example is the "Bertrand paradox" that the Nash equilibrium price is equal to marginal cost, regardless of the number of competitors, even though intuition and experimental evidence suggest otherwise (Dufwenberg and Gneezy 2000).

   The rationality assumption implicit in the Nash approach is that decisions are determined by the signs of the payoff differences, not by the magnitudes of the payoff gains or losses. But the losses for unilateral deviations from a Nash equilibrium are often highly asymmetric. In the minimum-effort coordination game, for example, a unilateral increase in effort above a common (Nash) effort level is not very risky if the marginal cost of effort is small, whereas a unilateral decrease would reduce the minimum and not save very much in terms of effort cost. Similarly, an effort increase is relatively more risky when effort costs are high. In each case, deviations in the less risky direction are more likely, and this is why effort levels observed in laboratory experiments are inversely related to effort cost.

   Many of the counterintuitive predictions of a Nash equilibrium disappear when some noise is introduced into the decision-making process, which is the approach taken in this paper. This randomness is modeled using a probabilistic choice function, that is, the probability of making a particular decision is a smoothly increasing function of the payoff associated with that decision. One attractive interpretation of probabilistic choice models is that the apparent noisiness is due to unobserved shocks in preferences, which cause behavior to appear more random when the observed payoffs become approximately equal. Of course, mistakes and trembles are also possible, and these presumably would also be more likely to have an effect when payoff differences are small, that is, when the cost of a mistake is small. In either case, probabilistic choice rules have the property that the probability of choosing the "best" decision is not one, and choice probabilities will be close to uniform when the other decisions are only s lightly less attractive.

   When a probabilistic choice function is used to analyze the interaction of strategic players, one has to model beliefs about others' decisions, since these beliefs determine expected payoffs. When prior experience with the game is available, beliefs will evolve as people learn. Learning slows down as observed decisions look more and more like prior beliefs, that is, as surprises are reduced. In a steady state, systematic learning ceases when beliefs are consistent with observed decisions. Following McKelvey and Palfrey (1995), the equilibrium condition used here has the consistency property that belief probabilities that determine expected payoffs match the choice probabilities that result from applying a probabilistic choice rule to those expected payoffs. In other words, players take into account the errors in others' decisions.

   Perhaps the most commonly used probabilistic choice function in empirical work is the logit model, in which the probability of choosing a decision is proportional to an exponential function of its expected payoff. This logit rule exhibits nice theoretical properties, such as having choice probabilities be unaffected by adding a constant to all payoffs. We have used the logit equilibrium extensively in a series of applications that include rent-seeking contests, price competition, bargaining, public goods games, coordination games, first-price auctions, and social dilemmas with continuous choices. (3) In the process, we noticed that many of the models share a common auctionlike structure with payoff functions that depend on rank, such as whether a player's decision is higher or lower than another's. In this paper, we offer general proofs of theoretical properties on the basis of characteristics of the expected payoff functions. Section 3 summarizes a logit equilibrium model of noisy behavior for games with ran k-based outcomes. Proofs of existence, uniqueness, and comparative statics follow in section 4. In section 5, we apply these results to a variety of models that represent many of the standard applications of game theory to economics and social science. Comparisons with learning theories and other ways of explaining behavioral anomalies are discussed in section 6, and the final section concludes.

3. An Equilibrium Model of Noisy Behavior in Auctionlike Games

   The standard way to motivate a probabilistic choice rule is to specify a utility function with a stochastic component. If decision i has expected payoff [[pi].sup.e.sub.i], then the person is assumed to choose the decision with the highest value of U(i) = [[pi].sup.e](i) + [mu][[member of].sub.i], where [mu] is a positive "error" parameter and [[member of].sub.i] is the realization of a random variable. When [mu] = 0, the decision with the highest expected payoff is selected, but high values of [mu] imply more noise relative to payoff maximization. This noise can be due to either (i) errors, for example, distractions, perception biases, or miscalculations that lead to nonoptimal decisions, or (ii) unobserved...