Self-tuning experience weighted attraction learning in games

Self-tuning experience weighted attraction (EWA) is a one-parameter theory of learning in games. It addresses a criticism that an earlier model (EWA) has too many parameters, by fixing some parameters at plausible values and replacing others with functions of experience so that they no longer need to be estimated. Consequently, it is econometrically simpler than the popular weighted fictitious play and reinforcement learning models. The functions of experience which replace free parameters “self-tune” over time, adjusting in a way that selects a sensible learning rule to capture subjects’ choice dynamics. For instance, the self-tuning EWA model can turn from a weighted fictitious play into an averaging reinforcement learning as subjects equilibrate and learn to ignore inferior foregone payoffs. The theory was tested on seven different games, and compared to the earlier parametric EWA model and a one-parameter stochastic equilibrium theory (QRE). Self-tuning EWA does as well as EWA in predicting behavior in new games, even though it has fewer parameters, and fits reliably better than the QRE equilibrium benchmark.     

English auctions with resale: An experimental study

I design and test a simple English auction and two English auctions with resale, but with different informational backgrounds. All three treatments theoretically have the same equilibrium. I find, however, that the possibility of resale alters behavior significantly. In the two treatments with resale, subjects deviated from both the Nash prediction and the common results about bidding behavior in English auctions. Subjects tend to overbid, when they are certain they can reap the whole surplus in the resale market. I employ different models like QRE and levels of reasoning and conclude that overbidding can be explained as a rational response to the noisy environment in markets with human participants, that is, as rational decision making when anticipating others to make errors. When the outcome of the resale market is not certain, there is significant signaling behavior and auction prices tend to be lower than the Nash prediction.     

On the beliefs off the path: Equilibrium refinement due to quantal response and level-k

The extensive form game we study has multiple perfect equilibria, but it has a unique limiting logit equilibrium (QRE) and a unique level-k prediction as k approaches infinity. The convergence paths of QRE and level-k are different, but they converge to the same limit point. We analyze whether subjects adapt beliefs when gaining experience, and if so whether they take the QRE or the level-k learning path. We estimate transitions between level-k and QRE belief rules using Markov-switching rule learning models. The analysis reveals that subjects take the level-k learning path and that they advance gradually, switching from level 1 to 2, from level 2 to equilibrium, and reverting to level 1 after observing opponents deviating from equilibrium. The steady state therefore contains a mixture of behavioral rules: levels 0, 1, 2, and equilibrium with weights of 2.9%, 16.6%, 37.9%, and 42.6%, respectively.     

Heterogeneous quantal response equilibrium and cognitive hierarchies

We explore an equilibrium model of games where behavior is given by logit response functions, but payoff responsiveness and beliefs about others' responsiveness are heterogeneous. We study two substantively different ways of extending quantal response equilibrium (QRE) to this setting: (1) Heterogeneus QRE, where players share identical correct beliefs about the distribution of payoff responsiveness; and (2) Truncated QRE, where players have downward looking beliefs, systematically underestimating others' responsiveness. We show that the cognitive hierarchy model is a special case of Truncated QRE. We conduct experiments designed to differentiate these approaches. We find significant evidence of payoff responsive stochastic choice, and of heterogeneity and downward looking beliefs in some games.     

Optimistic irrationality and overbidding in private value auctions

Bidding one’s value in a second-price, private-value auction is a weakly dominant solution (Vickrey in J Finance 16(1):8–37, 1961), but repeated experimental studies find more overbidding than underbidding. We propose a model of optimistically irrational bidders who understand that there are possible gains and losses associated with higher bids but who may overestimate the additional probability of winning and/or underestimate the potential losses when bidding above value. These bidders may fail to discover the dominant strategy—despite the fact that the dominant strategy only requires rationality from bidders—but respond in a common sense way to out-of-equilibrium outcomes. By varying the monetary consequences of losing money in experimental auctions we observe more overbidding when the cost to losing money is low, and less overbidding when the cost is high. Our findings lend themselves to models in which less than fully rational bidders respond systematically to out-of-equilibrium incentives, and we find that our model better fits the effects of our manipulations than most of the existing models we consider.     

Regular Quantal Response Equilibrium

The structural Quantal Response Equilibrium (QRE) generalizes the Nash equilibrium by augmenting payoffs with random elements that are not removed in some limit. This approach has been widely used both as a theoretical framework to study comparative statics of games and as an econometric framework to analyze experimental and field data. The framework of structural QRE is flexible: it can be applied to arbitrary finite games and incorporate very general error structures. Restrictions on the error structure are needed, however, to place testable restrictions on the data (Haile et al., 2004). This paper proposes a reduced-form approach, based on quantal response functions that replace the best-response functions underlying the Nash equilibrium. We define a regular QRE as a fixed point of quantal response functions that satisfies four axioms: continuity, interiority, responsiveness, and monotonicity. We show that these conditions are not vacuous and demonstrate with an example that they imply economically sensible restrictions on data consistent with laboratory observations. The reduced-form approach allows for a richer set of regular quantal response functions, which has proven useful for estimation purposes. JEL Classification: D62, C73     

Qualitatively Rational Expectations and Adjustment in the Specific-Factors Model

This paper considers adjustment in a dynamic specific-factors model with endogenous capital stocks. Investment is analyzed under the assumption that expectations are rational with respect to qualitative aspects of the adjustment process (qualitatively rational expectations, QRE). QRE leave considerable scope for systematic errors in expectations formation. Adjustment under rational expectations is similar to QRE adjustment; however, only in the former case is the speed of adjustment optimal. Overshooting of capital stocks is possible and may be optimal. Comparative-static analysis shows an asymmetry between inflows of labor and capital: only capital inflows may cause a Rybczynski effect.     

TESTING THE QUANTAL RESPONSE HYPOTHESIS

We develop a nonparametric test for consistency of player behavior with the quantal response equilibrium (QRE). The test exploits a characterization of the equilibrium choice probabilities in any structural QRE as the gradient of a convex function; thereby, QRE‐consistent choices satisfy the cyclic monotonicity inequalities. Our testing procedure utilizes recent econometric results for moment inequality models. We assess our test using lab experimental data from a series of generalized matching pennies games. We reject the QRE hypothesis in the pooled data but cannot reject individual‐level quantal response behavior for over half of the subjects.     

Overbidding in fixed rate tenders—an empirical assessment of alternative explanations

This paper investigates various theories explaining banks’ overbidding in the fixed rate tenders of the European Central Bank (ECB). Using auction data from both the Bundesbank and the ECB, we show that none of the theories can on its own explain the observed overbidding. This implies that the proposed new rules by the ECB, aimed at neutralizing interest rate expectations, would not eliminate overbidding if the rationing rule in the fixed rate tenders remains unchanged.     

Sequential equilibrium in monotone games: A theory-based analysis of experimental data

A monotone game is an extensive-form game with complete information, simultaneous moves and an irreversibility structure on strategies. It captures a variety of situations in which players make partial commitments and allows us to characterize conditions under which equilibria result in socially desirable outcomes. However, since the game has many equilibrium outcomes, the theory lacks predictive power. To produce stronger predictions, one can restrict attention to the set of sequential equilibria, or Markov equilibria, or symmetric equilibria, or pure-strategy equilibria. This paper explores the relationship between equilibrium behavior in a class of monotone games, namely voluntary contribution games, and the behavior of human subjects in an experimental setting. Several key features of the symmetric Markov perfect equilibrium (SMPE) are consistent with the data. To judge how well the SMPE fits the data, we estimate a model of Quantal Response Equilibrium (QRE) [R. McKelvey, T. Palfrey, Quantal response equilibria for normal form games, Games Econ. Behav. 10 (1995) 6-38; R. McKelvey, T. Palfrey, Quantal response equilibria for extensive form games, Exp. Econ. 1 (1998) 9-41] and find that the decision rules of the QRE model are qualitatively very similar to the empirical choice probabilities.     

Limited Consumer Attention in International Trade

This paper introduces a model of limited consumer attention into an otherwise standard new trade theory model with love‐of‐variety preferences and heterogeneous firms. In this setting, we show that international integration needs not be welfare enhancing if the consumers' capacity to gather and process information is limited. Rather, it intensifies competition for scarce consumer attention, which causes mutual overbidding of producers in their advertising expenditures. The mutual overbidding renders advertising—which is informative in principle—wasteful and diverts purchases to imported goods at an inefficient scale. Wasteful advertising provides scope for policy intervention in the form of an advertising tax. However, if the tax instrument is not allowed to discriminate against foreign producers, it cannot eliminate inefficient diversion of consumer purchases to imports; hence it needs not be successful in securing gains from international integration in this framework.     

Bidding in private-value auctions with uncertain values

Auctions are often used to sell idiosyncratic goods difficult for potential bidders to value ex ante. Laboratory auctions with uncertainty over final values in this experiment resulted in 18% and 27% of bids above the expected value of the item in private-value first-price and English auctions, respectively. Risk-seeking preferences as measured on an individual decision task cannot explain overbidding and the first-price auction results suggest that risk aversion may not be a good explanation for bidding behavior observed with certain values. Several candidate explanations fail to explain overbidding, rather it appears to stem from some bidders who are prone to overbidding. Relative to first-price auctions, the size and frequency of overbids are significantly larger in English auctions, while more English auctions are won by overbidders. Differences between the formats appear to be driven by the dynamic nature of English auctions which is consistent with popular notions of “auction fever.”     

SUBJECTIVE PROBABILITIES IN GAMES: AN APPLICATION TO THE OVERBIDDING PUZZLE*

This article illustrates how the joint elicitation of subjective probabilities and preferences may help us understand behavior in games. We conduct an experiment to test whether biased probabilistic beliefs may explain overbidding in first‐price auctions. The experimental outcomes indicate that subjects underestimate their probability of winning the auction, and indeed overbid. When provided with feedback on the precision of their predictions, subjects learn to make better predictions, and to curb significantly overbidding. The structural estimation of different behavioral models suggests that biased probabilistic beliefs are a driving force behind overbidding, and that risk aversion plays a lesser role than previously believed.     

Quantal-response equilibrium models of the ultimatum bargaining game

This paper investigates the implications of quantal response equilibrium (QRE) models [McKelvey and Palfrey, 1995, Games Econ. Behav. 10, 6–38; 1998, Exper. Econ. 1, 9–41] in the ultimatum bargaining game. It is shown that, in a normal-form QRE (NQRE), each bargainer's decision depends critically on the anticipated behavior of the other, and there is a NQRE in which the proposer makes any offer between zero and equal split as a strict best response. The application of NQRE to the experimental data [Slonim and Roth, 1998, Econometrica 66, 569–596] suggests that the history dependence observed in the experiment is a result of the strategic interactions between bargainers.     

Endowment effects in contests

We experimentally study overbidding in contests and find that overbidding is significantly higher when subjects are given a large per-experiment endowment rather than when the endowment is given per-period. Risk-aversion and non-monetary utility of winning can partially explain our findings.     

Auctions with Anticipated Emotions: Overbidding,Underbidding, and Optimal Reserve Prices*

The experimental literature has documented that there is overbidding in second‐price auctions, regardless of the valuations of bidders. In contrast, in first‐price auctions, there tends to be overbidding for large valuations, but underbidding for small valuations. We show that the experimental evidence can be rationalized by a simple extension of the standard auction model, where bidders anticipate (constant) positive or negative emotions caused by the mere fact of winning or losing. Even if the emotional (dis‐)utilities are very small, the revenue‐maximizing reserve price might be significantly different from the standard model. Moreover, decreases with the number of bidders.     

The Repo Auctions of the European Central Bank and the Vanishing Quota Puzzle*

Until June 2000 the European Central Bank (ECB) used fixed rate tenders for its weekly repo auctions. A switch to variable rate tenders became necessary due to massive overbidding by banks. In this paper we introduce a stylized game among banks to investigate this overbidding phenomenon. Our results confirm the weakness of the fixed rate tender format and indicate that the ECB's liquidity management has significantly improved since the switch to the variable rate system. Yet recent episodes of rate cut expectations suggest that the ECB's practice of setting a minimum bid rate should be abandoned in favor of a more symmetric interest rate target.     

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.     

An experimental investigation of overdissipation in the all pay auction

Pervasive overbidding represents a well-documented feature of all-pay auctions. Aggregate bids exceed Nash predictions in laboratory experiments, and individuals often submit bids that guarantee negative profits. This paper examines three factors that may reduce pervasive overbidding: (a) repetition (experience), (b) reputation (strangers vs. partners), and (c) active participation. Reputation and repetition reduce aggregate overdissipation, but they eliminate it only in conjunction with active participation.     

Over- and Underbidding in Central Bank Open Market Operations Conducted as Fixed Rate Tender

Abstract. Open market operations play a key role in allocating central bank funds to the banking system and thereby in steering short‐term interest rates in line with the stance of monetary policy. Many central banks apply so‐called ‘fixed rate tender’ auctions in their open market operations. This paper presents, on the basis of a survey of central bank experience, a model of bidding in such tenders. In their conduct of fixed rate tenders, many central banks faced specifically an ‘under‐’ and an ‘overbidding’ problem. These phenomena are revisited in the light of the proposed model, and the more general question of the optimal tender procedure and allotment policy of central banks is addressed.