Nash Equilibrium Predictions Research Articles

Behavioral convergence properties of Cournot and Bertrand markets: An experimental analysis

This paper reports an experiment that examines the relative convergence properties of differentiated-product Cournot and Bertrand oligopolies. Overall, Bertrand markets tend to converge to Nash equilibrium predictions more quickly and more completely than Cournot markets. Further, when products are close substitutes Bertrand markets respond more quickly to an announced nominal shock. As products become weaker substitutes, however, an increased tendency for tacit collusion degrades convergence in Bertrand markets. This effect is particularly pronounced following a nominal shock. Our results suggest that in an oligopoly context variations in decision error costs dominate a ‘Strategic Substitutes Effect’ isolated in previous experimental research.

On the Effects of Joint Bidding in Independent Private Value Auctions: An Experimental Study

This article explores the effect of a subset of symmetric bidders joining to bid together. Possible applications include mergers, collusion and legal joint-bidding arrangements. The change produces a party with a more advantageous value distribution than the remaining bidder(s). The predicted effects include inefficiency, a decrease in the seller's revenue, and higher bidders' payoffs. When the bidders are risk neutral, the members of the strong party are expected to benefit less than the weak bidders. The prediction is reversed when the bidders are sufficiently risk averse. These hypotheses are tested experimentally. The Nash equilibrium with risk aversion is consistent with some features of the data. Contrary to the theory, joint-bidding increases efficiency and the seller's revenue decreases by less than expected. Strong bidders benefit more than weak bidders indicating that the incentives to bid jointly may be greater than hypothesized. Additionally, the experiment assesses the effect of group decision making. A Nash equilibrium prediction for individual-group differences based on differences in risk attitudes is not supported by the data.

A Note on Imitation-Based Competition in Common-Pool Resources

The Tragedy of the Commons refers to the dissipation of common ressources when several appropriators have access to it. The Nash equilibrium in common-pool resource games describes such an overexploitation phenomena. However, some empirical facts and experimental results suggest that the degradation may exceed the Nash equilibrium prediction. Rapid exhaustion of resources can then be explained by learning processes based on imitation behavior. Indeed, when the number of players is limited, the evolutionary outcome implies a higher level of dissipation than the Nash equilibrium.

Varying the intensity of competition in a multiple prize rent seeking experiment

We experimentally test a rent seeking model under five levels of competition. At one extreme, a subject’s probability of winning a prize is equal to her share of the total expenditures. At lower levels of competition, a subject’s probability of winning is affected more by her own expenditures than by the expenditures of others. Predicted expenditure levels are positively associated with higher levels of competition. Consistent with previous rent seeking experiments, we find that subjects spend significantly more than the Nash equilibrium prediction at all levels of competition. However, expenditure patterns generally follow the Nash prediction; expenditures decrease as the level of competition decreases. Our experimental design also includes a lottery choice experiment to control for subjects’ risk preference. We find that subjects who are more risk averse spend significantly less in the contest and this effect is particularly strong for female subjects.

Bidding ‘as if’ risk neutral in experimental first price auctions without information feedback

Experimental research on first price sealed bid auctions has usually involved repeated settings with information feedback on winning bids and payoffs after each auction round. Relative to the risk neutral Nash equilibrium, significantly higher bidding has been reported. The present paper reports the results of experimental first price auctions with n=7 where feedback on payoffs and winning bids is withheld. Under these conditions, average bidding is below the risk neutral Nash equilibrium prediction but converges to it with repetition.

Auction choice for ambiguity-averse sellers facing strategic uncertainty

The robustness of the Bayes–Nash equilibrium prediction for seller revenue in auctions is investigated. In a framework of interdependent valuations generated from independent signals, seller expected revenue may fall well below the equilibrium prediction, even though the individual payoff consequences of suboptimal bidding may be small for each individual bidder. This possibility would be relevant to a seller who models strategic uncertainty as ambiguity, and who is ambiguity-averse in the sense of Gilboa and Schmeidler. It is shown that the second-price auction is more exposed than the first-price auction to lost revenue from the introduction of bidder behavior with small payoff errors.

The Paradox of Voter Participation? A Laboratory Study

It is widely believed that rational choice theory is grossly inconsistent with empirical observations about voter turnout. We report the results of an experiment designed to test the voter turnout predictions of the rational choice Palfrey–Rosenthal model of participation with asymmetric information. We find that the three main comparative statics predictions are observed in the data: the size effect, whereby turnout goes down in larger electorates; the competition effect, whereby turnout is higher in elections that are expected to be close; and the underdog effect, whereby voters supporting the less popular alternative have higher turnout rates. We also compare the quantitative magnitudes of turnout to the predictions of Nash equilibrium. We find that there is undervoting for small electorates and overvoting for large electorates, relative to Nash equilibrium. These deviations from Nash equilibrium are consistent with the logit version of Quantal Response Equilibrium, which provides a good fit to the data, and can also account for significant voter turnout in very large elections.

Game theory and human evolution: A critique of some recent interpretations of experimental games

Economists and psychologists have been testing Nash equilibrium predictions of game theory models of human behavior. In many instances, humans do not conform to the predictions. These results are of great interest to biologists because they also raise questions about well-known ESS models of cooperation. Cooperation in certain one-shot, anonymous interactions, and a willingness to punish others at a net cost to oneself are some of the most intriguing deviations from standard theory. One proposed explanation for these results that is receiving increasing attention invokes the cultural group selection of ‘other regarding’ social norms. We critically review this explanation. We conclude that experimental results reveal limits in two implicit models of cognitive structure commonly employed by economists and evolutionary biologists.

Individual behavior of first-price auctions: The importance of information feedback in computerized experimental markets

This article reports the results of an individual choice experiment designed to test the Nash equilibrium predictions of the first-price sealed-bid auction. A subject faced in 100 auctions always the same resale value and competed with computer-simulated bids. The design used between-subjects variation and involved information feedback as the treatment variable. Earlier experimental work on first price auctions has frequently reported an overbidding relative to the risk neutral Nash equilibrium. Our data provide evidence that overbidding can be fostered by the standard information feedback in auction experiments, which, after each auction, reveals the winning bid only. By means of learning direction theory we explain the individual bidding dynamics in our experiment. Finally we apply impulse balance theory and make long run predictions of individual bidding behavior.

Can Genetic Algorithms Explain Experimental Anomalies?

In experimental data, it is common to find persistent oscillations in the aggregate outcomes and high levels of heterogeneity in individual behavior. Furthermore, it is not unusual to find significant deviations from aggregate Nash equilibrium predictions. In this paper, we employ an evolutionary model with boundedly rational agents to explain these findings. We use data from common property resource experiments (Casari and Plott, 2003). Instead of positing individual-specific utility functions, we model decision makers as selfish and identical. Agent interaction is simulated using an individual learning genetic algorithm (GA), where agents have constraints in their working memory, a limited ability to maximize, and experiment with new strategies. We show that the model replicates most of the patterns that can be found in common property resource experiments.

Effect of monitor-subject cheap talk on ultimatum game offers

We explore the influence of experimental monitor cheap talk on offers in single-shot ultimatum bargaining games. We find that behavior more closely resembles Nash equilibrium predictions when subjects are provided information about the theoretical money maximizing solution in addition to information about why behavior may deviate from this equilibrium. Among other things, results suggest subjects fail to behave consistently with the Nash equilibrium prediction because of a lack of knowledge of the theoretical money maximizing solution or because of a lack of common expectations about the rationality and beliefs of other players.

Explaining Overbidding in First Price Auctions Using Controlled Lotteries

In this paper, we study the behavior of individuals when facing two different, but incentive-wise identical, institutions. We pair the first price auction with an equivalent lottery. Once a subject is assigned a value for the auctioned object, the first price auction can be modeled as a lottery in which the individual faces a given probability of winning a certain payoff. This set up allows us to explore to what extent the misperception of the probability of winning in the auction is responsible for bidders in a first price auction to bidding above the risk neutral Nash equilibrium prediction. The first result we obtain is that individuals, even though facing the same choice over probability/payoff pairs, behave differently depending on the type of choice they are called to make. When facing an auction, subjects with high values tend to bid significantly above the bid they choose in the corresponding lottery environment. We further find that in both the lottery and the auction environments, subjects tend to bid in excess of the bid predicted by the risk neutral model, at least for intermediate range values. Finally, we find that the difference between the lottery behavior and the auction behavior is substantially, but not totally, eliminated by showing the subjects the probability of winning the auction.

The Logit Equilibrium: A Perspective on Intuitive Behavioral Anomalies

This paper considers a class of models in which rank‐based payoffs are sensitive to “noise” in decision making. Examples include auctions, price competition, coordination, and location games. Observed laboratory behavior in these games is often responsive to the asymmetric costs associated with deviations from the Nash equilibrium. These payoff‐asymmetry effects are incorporated in an approach that introduces noisy behavior via a logit probabilistic choice function. In the resulting logit equilibrium, behavior is characterized by a probability distribution that satisfies a “rational expectations” consistency condition: The beliefs that determine players' expected payoffs match the decision distributions that arise from applying the logit rule to those expected payoffs. We prove existence of a unique, symmetric logit equilibrium and derive comparative statics results. The paper provides a unified perspective on many recent laboratory studies of games in which Nash equilibrium predictions are inconsistent with both intuition and experimental evidence.

The Logit Equilibrium: A Perspective on Intuitive Behavioral Anomalies

costs associated with deviations from the Nash equilibrium. These payoff-asymmetry effects are incorporated in an approach that introduces noisy behavior via a logit probabilistic choice function. In the resulting logit equilibrium, behavior is characterized by a probability distribution that satisfies a rational expectations consistency condition: The beliefs that determine players' expected payoffs match the decision distributions that arise from applying the logit rule to those expected payoffs. We prove existence of a unique, symmetric logit equilibrium and derive comparative statics results. The paper provides a unified perspective on many recent laboratory studies of games in which Nash equilibrium predictions are inconsistent with both intuition and experimental evidence.

Economic inequality and burden-sharing in the provision of local environmental quality

A large, but inconclusive, literature addresses how economic heterogeneity affects the use of local resources and local environmental quality. One line of thought, which derives from Nash equilibrium provision of public goods, suggests that in contexts in which individual actions degrade local environmental quality, wealthier people in a community will tend to do more to protect environmental quality. In this paper we report on experiments performed in rural Colombia that were designed to explore the role that economic inequality plays in the ‘provision’ of local environmental quality. Subjects were asked to decide how much time to devote to collecting firewood from a local forest, which degrades local water quality, and how much to unrelated pursuits. Economic heterogeneity was introduced by varying the private returns to these alternative pursuits. Consistent with the Nash equilibrium prediction, we found that the players with more valuable alternative options put less pressure on local water quality. However, the subjects with less valuable alternative options showed significantly more restraint relative to their pure Nash strategies. Furthermore, they were willing to bear significantly greater opportunity costs to move their groups to outcomes that yielded higher average payoffs and better water quality than the Nash equilibrium outcome.

Voluntary Participation and Spite in Public Good Provision Experiments: an International Comparison

This paper studies voluntary public good provision in the laboratory, in a cross-cultural experiment conducted in the United States and Japan. Our environment differs from the standard voluntary contribution mechanism because subjects first decide whether or not to participate in providing this non-excludable public good. This participation decision is conveyed to the other subject prior to the subjects' contribution decisions. We find that only the American data are consistent with the evolutionary-stable-strategy Nash equilibrium predictions, and that behavior is significantly different across countries. Japanese subjects are more likely to act spitefully in the early periods of the experiment, even though our design changes subject pairings each period so that no two subjects ever interact twice. Surprisingly, this spiteful behavior eventually leads to more efficient public good contributions for Japanese subjects than for American subjects.

Reinforcement-based vs. Belief-based Learning Models in Experimental Asymmetric-information Games

This paper examines the abilities of learning models to describe subject behavior in experiments. A new experiment involving multistage asymmetric-information games is conducted, and the experimental data are compared with the predictions of Nash equilibrium and two types of learning model: a reinforcement-based model similar to that used by Roth and Erev (1995), and belief-based models similar to the ‘cautious fictitious play’ of Fudenberg and Levine (1995, 1998) These models make predictions that are qualitatively similar cycling around the Nash equilibrium that is much more apparent than movement toward it. While subject behavior is not adequately described by Nash equilibrium, it is consistent with the qualitative predictions of the learning models. We examine several criteria for quantitatively comparing the predictions of alternative models. According to almost all of these criteria, both types of learning model outperform Nash equilibrium. According to some criteria, the reinforcement-based model performs better than any version of the belief-based model; according to others, there exist versions of the belief-based model that outperform the reinforcement-based model. The abilities of these models are further tested with respect to the results of other published experiments. The relative performance of the two learning models depends on the experiment, and varies according to which criterion of success is used. Again, both models perform better than equilibrium in most cases.

An experimental study of information and mixed-strategy play in the three-person matching-pennies game

Recent experiments on mixed-strategy play in experimental games reject the hypothesis that subjects play a mixed strategy even when that strategy is the unique Nash equilibrium prediction. However, in a three-person matching-pennies game played with perfect monitoring and complete payoff information, we cannot reject the hypothesis that subjects play the mixed-strategy Nash equilibrium. Given this support for mixed-strategy play, we then consider two qualitatively different learning theories (sophisticated Bayesian and naive Bayesian) which predict that the amount of information given to subjects will determine whether they can learn to play the predicted mixed strategies. We reject the hypothesis that subjects play the symmetric mixed-strategy Nash equilibrium when they do not have complete payoff information. This finding suggests that players did not use sophisticated Bayesian learning to reach the mixed-strategy Nash equilibrium.

Location, cooperation and communication: An experimental examination

This study examines freeform non-binding communication and simplification as devices to facilitate cooperative outcomes using a fixed-price location choice duopoly in the spirit of Hotelling[Hotelling, H., 1929. Stability in competition, Economic Journal 39, 41–57]. We extend Brown-Kruse et al. [Brown-Kruse, J., Cronshaw, M.B., Schenk, D.J., 1993. Theory and experiments on spatial competition, Economic Inquiry 31(1), 139–165] which studied freeform communication for a uniform customer distribution. This work tests two more distributions that make collusion more or less risky. We also test a simple 2×2 model with similar Nash equilibrium predictions. When subjects cannot communicate, cooperation is easier to achieve and sustain in the 2×2 model. However, freeform communication is a robust facilitator of joint profit maximizing outcomes regardless of risk in cooperating.

Equilibrium and reinforcement learning in private-information games: An experimental study

Abstract This paper examines the abilities of standard game-theoretic solution concepts (static Nash equilibrium) and a simple model of individual learning to describe subject behavior in an experiment involving private-information games with mixed-strategy Nash equilibrium outcomes. The experimental data are compared in several ways with the predictions of static Nash equilibrium and the quite different predictions of the learning model, which is based on the models of Roth and Erev (1995) , (Games and Economic Behavior, Special Issue: Nobel symposium, vol. 8, pp. 164–212). Nash equilibrium does not adequately describe subject behavior, whereas the learning model performs well in describing aspects of the evolution in behavior over time.