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.     

Let׳s talk it over: Coordination via preplay communication with level-k thinking

This paper reconsiders Farrell׳s (1987) and Rabin׳s (1994) analyses of coordination via preplay communication, focusing on Farrell׳s analysis of Battle of the Sexes. Replacing their equilibrium and rationalizability assumptions with a structural non-equilibrium model based on level-k thinking, I reevaluate Farrell and Rabin׳s assumptions on how players use language and their conclusions on the limits of communication in bringing about coordination. The analysis partly supports their assumptions about how players use language, but suggests that their “agreements” do not reflect a full meeting of the minds. A level-k analysis also yields very different conclusions about the effectiveness of communication.     

Level-k analysis of experimental centipede games

As one of the best-known examples of the paradox of backward induction, centipede games have prompted a host of studies with various approaches and explanations (McKelvey and Palfrey, 1992, Fey et al., 1996, Nagel and Tang, 1998, Rapoport et al., 2003, Palacios-Huerta and Volij, 2009). Focusing on initial plays observed in experiments, this paper attempts to offer another explanation based on thorough study of level-k models as applied to these games. Borrowing ideas from the cognitive hierarchy model (Camerer et al., 2004), the authors constructed a group of models based on levels of rationality, and also tested for various assumptions on the play of the most naïve player type in these models. It was found that level-k models generally perform better than the agent quantal response equilibrium (AQRE) model and its variant with altruistic player types for increasing-pie centipede games, while the AQRE model with altruistic player types performs better in constant-pie games.     

Privacy concerns,voluntary disclosure of information,and unraveling: An experiment

We study the voluntary revelation of private information in a labor-market experiment where workers can reveal their productivity at a cost. While rational revelation improves a worker׳s payoff, it imposes a negative externality on others and may trigger further revelation. Such unraveling can be observed frequently in our data although less often than predicted. Equilibrium play is more likely when subjects are predicted to conceal their productivity than when they should reveal. This tendency of under-revelation, especially of low-productivity workers, is consistent with the level-k model. A loaded frame where the private information concerns the workers׳ health status leads to less revelation than a neutral frame.     

Authority and communication in the laboratory

We report findings from experiments on two delegation–communication games. An uninformed principal chooses whether to fully delegate her decision-making authority to an informed agent or to retain the authority and communicate with the agent via cheap talk to obtain decision-relevant information. In the game in which the delegation outcome is payoff-dominated by both the truthful and the babbling communication outcomes, we find that principal-subjects almost always retain their authority and agent-subjects communicate truthfully. Significantly more choices of delegation than of communication are observed in another game in which the delegation outcome payoff-dominates the unique babbling communication outcome; yet there is a non-negligible fraction of principal-subjects who holds on to their authority and agent-subjects who transmits some information. A level-k analysis of the game indicates that a principal-subject “under-delegates” due to the belief that her less-than-fully-strategic opponent will provide information; such belief is in turn consistent with the actual play.     

Out of your mind: Eliciting individual reasoning in one shot games

We experimentally investigate the fundamental element of the level-k model of reasoning, the level-0 actions and beliefs. We use data from a novel experimental design that allows us to obtain incentivised written accounts of individuals' reasoning. In particular, these accounts allow to infer level-0 beliefs. Level-0 beliefs are not significantly different from 50, and almost 60% of higher level players start their reasoning from a level-0 belief of exactly 50. We also estimate that around one third of the participants play non-strategically. The non-strategic level-0 actions are not uniformly distributed.     

A unified approach to comparative statics puzzles in experiments

The paper shows that several game-theoretic solution concepts provide similar comparative statics predictions over a wide class of games. I start from the observation that, in many experiments, behavior is affected by parameter shifts that leave the Nash equilibrium unchanged. I explain the direction of change with a heuristic structural approach, using properties such as strategic complementarities and increasing differences. I show that the approach is consistent with general comparative statics results for (i) the Nash equilibrium of a game with perturbed payoff functions, (ii) the quantal response equilibrium, (iii) level-k reasoning. I also relate the structural approach to equilibrium selection concepts.     

The minimum approval mechanism implements the efficient public good allocation theoretically and experimentally

We propose the minimum approval mechanism (MAM) for a standard linear public good environment with two players. Players simultaneously and privately choose their contributions to the public good in the first stage. In the second stage, they simultaneously decide whether to approve the otherʼs choice. Both contribute what they choose in the first stage if both players approve; otherwise, both contribute the minimum of the two choices in the first stage. The MAM implements the Pareto-efficient allocation in multiple solution concepts including backward elimination of weakly dominated strategies (BEWDS), limit logit agent quantal response equilibrium, subgame perfect minimax regret equilibrium, level-k thinking, and diagonalization heuristics. Moreover, the MAM is unique under plausible conditions. Overall, contributions in the MAM experiment averaged 94.9%. Quantifying subjectsʼ responses to the questionnaire reveal heterogeneous reasoning processes and highlight the importance of developing mechanisms that implement the desired social choice outcome in multiple solution concepts.     

The rate of convergence to perfect competition of matching and bargaining mechanisms

We study the steady state of a market with incoming cohorts of buyers and sellers who are matched pairwise and bargain under private information. A friction parameter is τ, the length of the time period until the next meeting. We provide a necessary and sufficient condition for the convergence of mechanism outcomes to perfect competition at the linear rate in τ, which is shown to be the fastest possible among all bargaining mechanisms. The condition requires that buyers and sellers always retain some bargaining power. The bargaining mechanisms that satisfy this condition are called nonvanishing bargaining power (NBP) mechanisms. Simple random proposer take-it-or-leave-it protocols are NBP, while k-double auctions (k-DA) are not. We find that k-DAs have equilibria that converge to perfect competition at a linear rate, converge at a slower rate or even do not converge at all.     

Cognitive ability and strategic sophistication

In three experiments we examine the extent to which strategic sophistication (i.e., inductive reasoning, iterative dominance and level-k thinking) is determined by broader cognitive skills. In the first experiment we replicate previous results showing strong associations between cognitive ability and sophistication in a game of iterative dominance and show that similar results arise in a game requiring induction. In the second two experiments we extend the literature in new directions. In Experiment 2 we modify the games to better capture participantsʼ ability to reason inductively and predict the sophistication of others and, again, find strong associations between cognitive ability, measured using a common IQ test, and sophistication. In Experiment 3 we examine more closely the causal nature of the relationship between cognitive ability and sophistication. We use a standard tool from cognitive psychology to randomly shock the cognitive ability of participants and show that this significantly affects game performance.     

Games of fixed rank: a hierarchy of bimatrix games

We propose and investigate a hierarchy of bimatrix games (A, B), whose (entry-wise) sum of the pay-off matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank k-games strictly generalizes the class of zero-sum games, but is a very special case of general bimatrix games. We study both the expressive power and the algorithmic behavior of these games. Specifically, we show that even for k = 1 the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components. While the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank, we present polynomial time algorithms for finding an ε-approximation.     

North–South technology transfer in unionised multinationals

We study how incentives for North–South technology transfers in multinational enterprises are affected by labour market institutions. If workers are collectively organised, incentives for technology transfers are partly governed by firms' desire to curb trade union power. Higher union bargaining power in the North leads to more technology transfer along two different dimensions – skill upgrading of Southern workers and quality upgrading of products produced in the South – possibly to the extent that the utility of Northern workers decline. Policies to raise the wage levels of Southern workers might spur technology transfer if wages are initially very low, but have a dampening effect on North–South technology transfer once the Southern wage level has surpassed a certain threshold level. These conclusions are reached in a setting where a unionised multinational multiproduct firm produces vertically differentiated products in Northern and Southern subsidiaries.     

Coordination and Information in Critical Mass Games: An Experimental Study

We present experimental results on a repeated coordination game with Pareto-ranked equilibria in which a payoff from choosing an action is positive only if a critical mass of players choose that action. We design a baseline version of the game in which payoffs remain constant for values above the critical mass, and an increasing returns version in which payoffs keep increasing for values above the critical mass. We test the predictive power of security and payoff-dominance under different information treatments. Our results show that convergence to the payoff-dominant equilibrium is the modal limit outcome when players have full information about others' previous round choices, while this outcome never occurs in the remaining treatments. The paths of play in some groups reveal a tacit dynamic coordination by which groups converge to the efficient equilibrium in a step-like manner. Moreover, the frequency and speed of convergence to the payoff-dominant equilibrium are higher, ceteris paribus, when increasing returns are present. Finally, successful coordination seems to crucially depend on players' willingness to signal to others the choice of the action supporting the efficient equilibrium.     

Rates of information aggregation in common value auctions

We study the rates at which transaction prices aggregate information in common value auctions under the different information structures in Wilson (Rev. Econ. Stud. 44 (1977) 511) and Pesendorfer and Swinkels (Econometrica 65 (1997) 1247). We consider uniform-price auctions in which k identical objects of unknown value are auctioned to n bidders, where both n and k are allowed to diverge to infinity, and k/n converges to a number in [0,1). The Wilson assumptions lead to information aggregation at a rate proportional to , but the price aggregates information at a rate proportional to in the PS setting. We also consider English auctions, and investigate whether the extra information revealed in equilibrium improves convergence rates in these auctions.     

International sign predictability of stock returns: The role of the United States

We study the directional predictability of monthly excess stock market returns in the U.S. and ten other markets using univariate and bivariate binary response models. We introduce a new bivariate (two-equation) probit model that allows us to examine the benefits of predicting the signs of returns jointly, focusing on the predictive power originating from the U.S. to foreign markets. Our in-sample and out-of-sample forecasting results indicate superior predictive performance of the new model over competing univariate binary response models, and conventional predictive regressions, by statistical measures and market timing performance. This highlights the importance of predictive information from the U.S. to the other markets providing also practical improvement in investors' market timing decisions.     

Using eye-tracking to model attribute non-attendance in choice experiments

We investigate revealed attribute attendance in discrete choice experiments using eye-tracking. A simple theoretical framework is proposed in which choices are a function of visual attention. Consistent with the existing literature, the assumption that participants use all the available information to make their decisions does not hold. The results also illustrate that model fit and predictive power are greatly increased by using visual attendance measures as regressors. The use of eye-tracking technology has value for measuring revealed attention and to benchmark with existing choice models.     

Stochastic stability in a double auction

In a k-double auction, a buyer and a seller must simultaneously announce a bid and an ask price respectively. Exchange of the indivisible good takes place if and only if the bid is at least as high as the ask, the trading price being the bid price with probability k and the ask price with probability (1−k). We show that the stable equilibria of a complete information k-double approximate an asymmetric Nash bargaining solution with the seller's bargaining power decreasing in k.Note that ceteris paribus, the payoffs of the seller of the one-shot game increase in k. Nevertheless, as the stochastically stable equilibrium price decreases in k, choosing the seller's favorite price with a relatively higher probability in individual encounters makes him worse off in the long run.     

Observability and overcoming coordination failure in organizations: An experimental study

Motivated by problems of coordination failure in organizations, we examine how overcoming coordination failure and maintaining coordination depend on the ability of individuals to observe others’ choices. Subjects’ payoffs depend on coordinating at high effort levels in a weak-link game. Treatments vary along two dimensions. First, subjects either start with low financial incentives for coordination, which typically leads to coordination failure, and then are switched to higher incentives or start with high incentives, which usually yield effective coordination, and are switched to low incentives. Second, as the key treatment variable, subjects either observe the effort levels chosen by all individuals in their experimental group (full feedback) or observe only the minimum effort (limited feedback). We find three primary results: (1) When starting from coordination failure the use of full feedback improves subjects’ ability to overcome coordination failure, (2) When starting with good coordination the use of full feedback has no effect on subjects’ ability to avoid slipping into coordination failure, and (3) History-dependence, defined as dependence of current effort levels on past incentives, is strengthened by the use of full feedback. Electronic Supplementary Material Supplementary material is available in the online version of this article at . JEL Classification C92, D23, J31, L23, M52     

The gains and losses to predicting nominal income by disaggregating via the new classical aggregate supply and rational expectations hypotheses

In this paper we compaia the predictive power of two types of model of nominal income: one based on a simple single equation aggregate framework; the second disaggregated into price level and output components. The source of the decomposition of nominal income of the type of model that is considered here are the twin hypotheses of rational expectations and structural neutrality. The model chosen as being representative of this approach to macroeconomic model building and against which some single equation models are compared isBarro's [1978] model of the price level and output in the U.S.     

Testing theories of behavior for extensive-form two-player two-stage games

We examine choices in two-player extensive-form games that give subjects opportunities for individualistic as well as other-regarding behavior, and where each subject makes choices in a variety of games. Following an extensive search over models, where we estimate a single parameter vector for all the games rather than different parameter vectors for each game, we find that (1) the level-n model organizes the data well, (2) other-regarding behaviors in these games appear to consist primarily of egalitarian and utilitarian behaviors, and (3) there is no significant evidence for reciprocal behavior. Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.