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.     

Multiperson Bargaining and Strategic Complexity

We investigate the effect of introducing costs of complexity in the n‐person unanimity bargaining game. As is well‐known, in this game every individually rational allocation is sustainable as a Nash equilibrium (also as a subgame perfect equilibrium if players are sufficiently patient and if n & 2). Moreover, delays in agreement are also possible in such equilibria. By limiting ourselves to a plausible notion of complexity that captures length of memory, we find that the introduction of complexity costs (lexicographically with the standard payoffs) does not reduce the range of possible allocations but does limit the amount of delay that can occur in any agreement. In particular, we show that in any n‐player game, for any allocation z, an agreement on z at any period t can be sustained as a Nash equilibrium of the game with complexity costs if and only if t≤n. We use the limit on delay result to establish that, in equilibrium, the strategies implement stationary behavior. Finally, we also show that ‘noisy Nash equilibrium’ with complexity costs sustains only the unique stationary subgame perfect equilibrium allocation.     

基于共识程度的双体博弈纳什均衡拓展研究

首先对一般双体博弈的策略进行拓展,提出共识纳什均衡的概念,并证明其存在性;根据共识程度的降低提出其他3种拓展形式——近似共识纳什均衡、局部共识纳什均衡和局部近似共识纳什均衡,并举例进行说明;然后,对各种均衡形式下的合作效率进行分析和比较。研究表明,共识纳什均衡的利益分配是基于信息本身——竞争环境的特殊形式,它是具有效率的。其他3种形式均衡的效率低于共识纳什均衡,但仍高于双体博弈纳什均衡。共识纳什均衡和其他3种形式的均衡为参与人组织更具效率的合作提供了渠道和理论解释。     

Implementation with Near‐Complete Information

Many refinements of Nash equilibrium yield solution correspondences that do not have closed graph in the space of payoffs or information. This has significance for implementation theory, especially under complete information. If a planner is concerned that all equilibria of his mechanism yield a desired outcome, and entertains the possibility that players may have even the slightest uncertainty about payoffs, then the planner should insist on a solution concept with closed graph. We show that this requirement entails substantial restrictions on the set of implementable social choice rules. In particular, when preferences are strict (or more generally, hedonic), while almost any social choice function can be implemented in undominated Nash equilibrium, only monotonic social choice functions can be implemented in the closure of the undominated Nash correspondence.     

Berk–Nash Equilibrium: A Framework for Modeling Agents With Misspecified Models

We develop an equilibrium framework that relaxes the standard assumption that people have a correctly specified view of their environment. Each player is characterized by a (possibly misspecified) subjective model, which describes the set of feasible beliefs over payoff‐relevant consequences as a function of actions. We introduce the notion of a Berk–Nash equilibrium: Each player follows a strategy that is optimal given her belief, and her belief is restricted to be the best fit among the set of beliefs she considers possible. The notion of best fit is formalized in terms of minimizing the Kullback–Leibler divergence, which is endogenous and depends on the equilibrium strategy profile. Standard solution concepts such as Nash equilibrium and self‐confirming equilibrium constitute special cases where players have correctly specified models. We provide a learning foundation for Berk–Nash equilibrium by extending and combining results from the statistics literature on misspecified learning and the economics literature on learning in games.     

Markov Perfect Industry Dynamics With Many Firms

We propose an approximation method for analyzing Ericson and Pakes (1995)‐style dynamic models of imperfect competition. We define a new equilibrium concept that we call oblivious equilibrium, in which each firm is assumed to make decisions based only on its own state and knowledge of the long‐run average industry state, but where firms ignore current information about competitors' states. The great advantage of oblivious equilibria is that they are much easier to compute than are Markov perfect equilibria. Moreover, we show that, as the market becomes large, if the equilibrium distribution of firm states obeys a certain “light‐tail” condition, then oblivious equilibria closely approximate Markov perfect equilibria. This theorem justifies using oblivious equilibria to analyze Markov perfect industry dynamics in Ericson and Pakes (1995)‐style models with many firms.     

The Democratic Political Economy of Progressive Income Taxation

Why do both left and right political parties typically propose progressive income taxation schemes in political competition? Analysis of this problem has been hindered by the two-dimensionality of the issue space. To give parties a choice over a domain that contains both progressive and regressive income tax policies requires an issue space that is at least two-dimensional. Nash equilibrium in pure strategies of the standard two-party game, whose players have complete preferences over a two-dimensional policy space, generically fails to exist. I introduce a new equilibrium concept for political games, based on the fact of factional conflict within parties. Each party is supposed to consist of reformists, militants, and opportunists: each faction has a complete preference order on policy space, but together they can only agree on a partial order. Nash equilibria of the two-party game, where the policy space consists of all quadratic income tax functions, and each party is represented by its partial order, exist, and it is shown that, in such equilibria, both parties propose progressive income taxation.     

Optimal Contracts when Enforcement is a Decision Variable

This paper analyzes choice‐theoretic costly enforcement in an intertemporal contracting model with a differentially informed investor and entrepreneur. An intertemporal contract is modeled as a mechanism in which there is limited commitment to payment and enforcement decisions. The goal of the analysis is to characterize the effect of choice‐theoretic costly enforcement on the structure of optimal contracts. The paper shows that simple debt is the optimal contract when commitment is limited and costly enforcement is a decision variable (Theorem 1). In contrast, stochastic contracts are optimal when agents can commit to the ex‐ante optimal decisions (Theorem 2). The paper also shows that the costly state verification model can be viewed as a reduced form of an enforcement model in which agents choose payments and strategies as part of a perfect Bayesian Nash equilibrium.     

On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms

Consider nonempty finite pure strategy sets S₁,…,S_n, let S=S₁×⋅⋅⋅×S_n, let Ω be a finite space of “outcomes,” let Δ(Ω) be the set of probability distributions on Ω, and let θ: S→Δ(Ω) be a function. We study the conjecture that for any utility in a generic set of n‐tuples of utilities on Ω there are finitely many distributions on Ω induced by the Nash equilibria of the game given by the induced utilities on S. We give a counterexample refuting the conjecture for n≥3. Several special cases of the conjecture follow from well known theorems, and we provide some generalizations of these results.     

Leverage and Default in Binomial Economies: A Complete Characterization

Our paper provides a complete characterization of leverage and default in binomial economies with financial assets serving as collateral. Our Binomial No‐Default Theorem states that any equilibrium is equivalent (in real allocations and prices) to another equilibrium in which there is no default. Thus actual default is irrelevant, though the potential for default drives the equilibrium and limits borrowing. This result is valid with arbitrary preferences and endowments, contingent or noncontingent promises, many assets and consumption goods, production, and multiple periods. We also show that only no‐default equilibria would be selected if there were the slightest cost of using collateral or handling default. Our Binomial Leverage Theorem shows that equilibrium Loan to Value (LTV) for noncontingent debt contracts is the ratio of the worst‐case return of the asset to the riskless gross rate of interest. In binomial economies, leverage is determined by down risk and not by volatility.     

Competition for a Majority

We define the class of two‐player zero‐sum games with payoffs having mild discontinuities, which in applications typically stem from how ties are resolved. For such games, we establish sufficient conditions for existence of a value of the game, maximin and minimax strategies for the players, and a Nash equilibrium. If all discontinuities favor one player, then a value exists and that player has a maximin strategy. A property called payoff approachability implies existence of an equilibrium, and that the resulting value is invariant: games with the same payoffs at points of continuity have the same value and ɛ‐equilibria. For voting games in which two candidates propose policies and a candidate wins election if a weighted majority of voters prefer his proposed policy, we provide tie‐breaking rules and assumptions about voters' preferences sufficient to imply payoff approachability. These assumptions are satisfied by generic preferences if the dimension of the space of policies exceeds the number of voters; or with no dimensional restriction, if the electorate is sufficiently large. Each Colonel Blotto game is a special case in which each candidate allocates a resource among several constituencies and a candidate gets votes from those allocated more than his opponent offers; in this case, for simple‐majority rule we prove existence of an equilibrium with zero probability of ties.     

Cursed Equilibrium

There is evidence that people do not fully take into account how other people's actions depend on these other people's information. This paper defines and applies a new equilibrium concept in games with private information, cursed equilibrium, which assumes that each player correctly predicts the distribution of other players' actions, but underestimates the degree to which these actions are correlated with other players' information. We apply the concept to common‐values auctions, where cursed equilibrium captures the widely observed phenomenon of the winner's curse, and to bilateral trade, where cursedness predicts trade in adverse‐selections settings for which conventional analysis predicts no trade. We also apply cursed equilibrium to voting and signalling models. We test a single‐parameter variant of our model that embeds Bayesian Nash equilibrium as a special case and find that parameter values that correspond to cursedness fit a broad range of experimental datasets better than the parameter value that corresponds to Bayesian Nash equilibrium.     

Split‐Award Procurement Auctions—Can Bayesian Equilibrium Strategies Predict Human Bidding Behavior in Multi‐Object Auctions?

We analyze if and when symmetric Bayes Nash equilibrium predictions can explain human bidding behavior in multi‐object auctions. We focus on two sealed‐bid split‐award auctions with ex ante split decisions as they can be regularly found in procurement practice. These auction formats are straightforward multi‐object extensions of the first‐price sealed‐bid auction. We derive the risk‐neutral symmetric Bayes Nash equilibrium strategies and find that, although the two auction mechanisms yield the same expected costs to the buyer, other aspects of the two models, including the equilibrium bidding strategies, differ significantly. The strategic considerations in these auction formats are more involved than in single‐lot first‐price sealed‐bid auctions, and it is questionable whether expected utility maximization can explain human bidding behavior in such multi‐object auctions. Therefore, we analyzed the predictive accuracy of our equilibrium strategies in the laboratory. In human subject experiments we found underbidding, which is in line with earlier experiments on single‐lot first‐price sealed‐bid auctions. To control for regret, we organize experiments against computerized bidders, who play the equilibrium strategy. In computerized experiments where bid functions are only used in a single auction, we found significant underbidding on low‐cost draws. In experiments where the bid function is reused in 100 auctions, we could also control effectively for risk aversion, and there is no significant difference of the average bidding behavior and the risk‐neutral Bayes Nash equilibrium bid function. The results suggest that strategic complexity does not serve as an explanation for underbidding in split‐award procurement auctions, but risk aversion does have a significant impact.     

Repeated Games Where the Payoffs and Monitoring Structure Are Unknown

This paper studies repeated games with imperfect public monitoring where the players are uncertain both about the payoff functions and about the relationship between the distribution of signals and the actions played. We introduce the concept of perfect public ex post equilibrium (PPXE), and show that it can be characterized with an extension of the techniques used to study perfect public equilibria. We develop identifiability conditions that are sufficient for a folk theorem; these conditions imply that there are PPXE in which the payoffs are approximately the same as if the monitoring structure and payoff functions were known. Finally, we define perfect type‐contingently public ex post equilibria (PTXE), which allows players to condition their actions on their initial private information, and we provide its linear programming characterization.     

Non‐Manipulable House Allocation With Rent Control

In many real‐life house allocation problems, rents are bounded from above by price ceilings imposed by a government or a local administration. This is known as rent control. Because some price equilibria may be disqualified given such restrictions, this paper proposes an alternative equilibrium concept, called rationing price equilibrium, tailored to capture the specific features of housing markets with rent control. An allocation rule that always selects a rationing price equilibrium is defined, and it is demonstrated to be constrained efficient and (group) non‐manipulable for “almost all” preference profiles. In its bounding cases, the rule reduces to a number of well‐known mechanisms from the matching literature. In this sense, the housing market with rent control investigated in this paper integrates several of the predominant matching models into a more general framework.     

Identification and Estimation of a Discrete Game of Complete Information

We discuss the identification and estimation of discrete games of complete information. Following Bresnahan and Reiss (1990, 1991), a discrete game is a generalization of a standard discrete choice model where utility depends on the actions of other players. Using recent algorithms to compute all of the Nash equilibria to a game, we propose simulation‐based estimators for static, discrete games. We demonstrate that the model is identified under weak functional form assumptions using exclusion restrictions and an identification at infinity approach. Monte Carlo evidence demonstrates that the estimator can perform well in moderately sized samples. As an application, we study entry decisions by construction contractors to bid on highway projects in California. We find that an equilibrium is more likely to be observed if it maximizes joint profits, has a higher Nash product, uses mixed strategies, and is not Pareto dominated by another equilibrium.     

Self‐Control and the Theory of Consumption

To study the behavior of agents who are susceptible to temptation in infinite horizon consumption problems under uncertainty, we define and characterize dynamic self‐control (DSC) preferences. DSC preferences are recursive and separable. In economies with DSC agents, equilibria exist but may be inefficient; in such equilibria, steady state consumption is independent of initial endowments and increases in self‐control. Increasing the preference for commitment while keeping self‐control constant increases the equity premium. Removing nonbinding constraints changes equilibrium allocations and prices. Debt contracts can be sustained even if the only feasible punishment for default is the termination of the contract.     

Reputation in Continuous‐Time Games

We study reputation dynamics in continuous‐time games in which a large player (e.g., government) faces a population of small players (e.g., households) and the large player's actions are imperfectly observable. The major part of our analysis examines the case in which public signals about the large player's actions are distorted by a Brownian motion and the large player is either a normal type, who plays strategically, or a behavioral type, who is committed to playing a stationary strategy. We obtain a clean characterization of sequential equilibria using ordinary differential equations and identify general conditions for the sequential equilibrium to be unique and Markovian in the small players' posterior belief. We find that a rich equilibrium dynamics arises when the small players assign positive prior probability to the behavioral type. By contrast, when it is common knowledge that the large player is the normal type, every public equilibrium of the continuous‐time game is payoff‐equivalent to one in which a static Nash equilibrium is played after every history. Finally, we examine variations of the model with Poisson signals and multiple behavioral types.     

Approximate versus Exact Equilibria in Dynamic Economies

This paper develops theoretical foundations for an error analysis of approximate equilibria in dynamic stochastic general equilibrium models with heterogeneous agents and incomplete financial markets. While there are several algorithms that compute prices and allocations for which agents' first‐order conditions are approximately satisfied (“approximate equilibria”), there are few results on how to interpret the errors in these candidate solutions and how to relate the computed allocations and prices to exact equilibrium allocations and prices. We give a simple example to illustrate that approximate equilibria might be very far from exact equilibria. We then interpret approximate equilibria as equilibria for close‐by economies; that is, for economies with close‐by individual endowments and preferences. We present an error analysis for two models that are commonly used in applications, an overlapping generations (OLG) model with stochastic production and an asset pricing model with infinitely lived agents. We provide sufficient conditions that ensure that approximate equilibria are close to exact equilibria of close‐by economies. Numerical examples illustrate the analysis.     

Modeling Strategic Behavior in The Competitive Newsvendor Problem: An Experimental Investigation

This study experimentally investigates ordering behavior in the competitive newsvendor problem. We consider a duopoly market setting with two identical newsvendors selling the same perishable goods in a common market. Our experimental results show that average observed orders systematically deviate from the Nash equilibrium, and exhibit a similar pull‐to‐center pattern as in the classic non‐competitive newsvendor experiments: average orders fall below the Nash equilibrium in the high‐margin condition, and above the Nash equilibrium in the low‐margin condition. More importantly, the observed orders in the duopoly market are significantly higher than that in the non‐competitive newsvendor market, even in situations where standard inventory models predict no difference. We explain the ordering behavior using a strategic experience‐weighted attraction (EWA) model, which captures players' propensities for strategic thinking in game settings. Our empirical analysis suggests that the strategic EWA model generates more accurate predictions of future ordering behavior than an existing linear adaptive model without concerning strategic thinking. Further analysis shows that individuals are heterogeneous in their propensities to be a strategic player. Our research indicates the importance of modeling strategic behavior when analyzing behavioral decisions in competitive (game) environments.