The effect of endogenous timing on coordination under asymmetric information: An experimental study

This paper investigates the role of endogenous timing of decisions on coordination under asymmetric information. In the equilibrium of a global coordination game, where players choose the timing of their decision, a player who has sufficiently high beliefs about the state of the economy undertakes an investment without delay. This decision (potentially) triggers an investment by the other player whose beliefs would have led to inaction otherwise. Endogenous timing has two distinct effects on coordination: a learning effect (early decisions reveal information) and a complementarity effect (early decisions eliminate strategic uncertainty for late movers). The experiments that we conduct to test these theoretical results show that the learning effect of timing has more impact on the subjects' behavior than the complementarity effect. We also observe that subjects' welfare improves significantly under endogenous timing.     

Disclosure of information in matching markets with non-transferable utility

We present a model of two-sided matching where utility is non-transferable and information about individualsʼ skills is private, utilities are strictly increasing in the partnerʼs skill and satisfy increasing differences. Skills can be either revealed or kept hidden, but while agents on one side have verifiable skills, agents on the other side have skills that are unverifiable unless certified, and certification is costly. Agents who have revealed their skill enter a standard matching market, while others are matched randomly. We find that in equilibrium only agents with skills above a cutoff reveal, and then they match assortatively. We show that an equilibrium always exists, and we discuss multiplicity. Increasing differences play an important role to shape equilibria, and we remark that this is unusual in matching models with non-transferable utility. We close the paper with some comparative statics exercises where we show the existence of non-trivial externalities and welfare implications.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

Learning to play games in extensive form by valuation

Game theoretic models of learning which are based on the strategic form of the game cannot explain learning in games with large extensive form. We study learning in such games by using valuation of moves. A valuation for a player is a numeric assessment of her moves that purports to reflect their desirability. We consider a myopic player, who chooses moves with the highest valuation. Each time the game is played, the player revises her valuation by assigning the payoff obtained in the play to each of the moves she has made. We show for a repeated win-lose game that if the player has a winning strategy in the stage game, there is almost surely a time after which she always wins. When a player has more than two payoffs, a more elaborate learning procedure is required. We consider one that associates with each move the average payoff in the rounds in which this move was made. When all players adopt this learning procedure, with some perturbations, then, with probability 1 there is a time after which strategies that are close to subgame perfect equilibrium are played. A single player who adopts this procedure can guarantee only her individually rational payoff.     

Interactive beliefs,epistemic independence and strong rationalizability

We use a universal, extensive form interactive beliefs system to provide an epistemic characterization of a weak and a strong notion of rationalizability with independent beliefs. The weak solution concept is equivalent to backward induction in generic perfect information games where no player moves more than once in any play. The strong solution concept is related to explicability (Reny,22) and is outcome-equivalent to backward induction in generic games of perfect information.     

A simple model of homophily in social networks

Biases in meeting opportunities have been recently shown to play a key role for the emergence of homophily in social networks (see Currarini et al., 2009). The aim of this paper is to provide a simple microfoundation of these biases in a model where the size and type-composition of the meeting pools are shaped by agents׳ socialization decisions. In particular, agents either inbreed (direct search only to similar types) or outbreed (direct search to population at large). When outbreeding is costly, this is shown to induce stark equilibrium behavior of a threshold type: agents “inbreed” (i.e. mostly meet their own type) if, and only if, their group is above certain size. We show that this threshold equilibrium generates patterns of in-group and cross-group ties that are consistent with empirical evidence of homophily in two paradigmatic instances: high school friendships and interethnic marriages.     

Measuring agents’ reaction to private and public information in games with strategic complementarities

In games with strategic complementarities, public information about the state of the world has a larger impact on equilibrium actions than private information of the same precision, because public signals are more informative about the likely behavior of others. We present an experiment in which agents’ optimal actions are a weighted average of the fundamental state and their expectations of other agents’ actions. We measure the responses to public and private signals. We find that, on average, subjects put a larger weight on the public signal. In line with theoretical predictions, as the relative weight of the coordination component in a player’s utility increases, players put more weight on the public signal when making their choices. However, the weight is smaller than in equilibrium, which indicates that subjects underestimate the information contained in public signals about other players’ beliefs.     

Conservative belief and rationality

Playersʼ beliefs may be incompatible, in the sense that player i can assign probability 1 to an event E to which player j assigns probability 0. One way to block incompatibility is to assume a common prior. We consider here a different approach: we require playersʼ beliefs to be conservative, in the sense that all players must ascribe the actual world positive probability. We show that common conservative belief of rationality (CCBR) characterizes strategies in the support of a subjective correlated equilibrium where all playersʼ beliefs have common support. We also define a notion of strong rationalizability, and show that it is characterized by CCBR.     

Preference shocks from aggregation: time series data evidence

Abstract.  We investigate the impact of preference shocks on the aggregate dynamics of the U.S. economy in the context of a neoclassical growth model derived from aggregation. The aggregation result we use is as follows: if markets are complete and if agents have identical preferences of the addilog type, then the heterogeneous‐agent economy where agents are subject to idiosyncratic productivity shocks behaves as if there was a representative consumer who faces shocks to preferences and technology. We estimate the parameters in the aggregation‐based model from the aggregate time‐series data and compute the numerical solution. We find that the preference shocks play an important role in the aggregate labour‐market fluctuations. JEL classification: C73, D90, E21     

Beliefs and Public Good Provision with Anonymous Contributors

We analyze a static game of public good contributions where finitely many anonymous players have heterogeneous preferences about the public good and heterogeneous beliefs about the distribution of preferences. In the unique symmetric equilibrium, the only individuals who make positive contributions are those who most value the public good and who are also the most pessimistic; that is, according to their beliefs, the proportion of players who most like the public good is smaller than it would be according to any other possible belief. We predict whether the aggregate contribution is larger or smaller than it would be in an analogous game with complete information and heterogeneous preferences, by comparing the beliefs of contributors with the true distribution of preferences. A trade‐off between preferences and beliefs arises if there is no individual who simultaneously has the highest preference type and the most pessimistic belief. In this case, there is a symmetric equilibrium, and multiple symmetric equilibria occur only if there are more than two preference types.     

Learning,Matching, and Aggregation

Fictitious play and stimulus–response/reinforcement learning are examined in the context of a large population where agents are repeatedly randomly matched. We show that the aggregation of this learning behavior can be qualitatively different from learning at the level of the individual. This aggregate dynamic belongs to the same class of simply defined dynamic as do several formulations of evolutionary dynamics. We obtain sufficient conditions for convergence and divergence which are valid for the whole class of dynamics. These results are therefore robust to most specifications of adaptive behavior.Journal of Economic LiteratureClassification Numbers: C72, D83.     

A spectrum of equilibration processes in game theory

In game theory, four dynamic processes converging towards an equilibrium are distinguished and ordered by way of agents' decreasing cognitive capacities. In the eductive process, each player has enough information to simulate perfectly the others' behavior and gets immediately to the equilibrium. In epistemic learning, each player updates his beliefs about others' future strategies, with regard to their sequentially observed actions. In behavioral learning, each player modifies his own strategies according to the observed payoffs obtained from his past actions. In the evolutionary process, each agent has a fixed strategy and reproduces in proportion to the utilities obtained through stochastic interactions. All along the spectrum, longer term dynamics makes up for weaker rationality, and physical relations substitute for mental interactions. Convergence, if any, is towards an always stronger equilibrium notion and selection of an equilibrium state becomes more sensitive to context and history. The processes can be mixed if associated to different periods, agents or mechanisms and deepened if obtained by formal reasoning principles.     

Evolutionary stability of discrimination under observability

We study the evolution of preferences under perfect and almost perfect observability in symmetric 2-player games. We demonstrate that if nature can choose from a sufficiently general preference space, which includes preferences over outcomes that may depend on the opponent's preference-type, then, in most games, only discriminating preferences (treating different types of opponents differently in the same situation) can be evolutionary stable and some discriminating types are stable in a very strong sense in all games. We use these discriminating types to show that any symmetric outcome which gives players more than their minmax value in material payoffs (fitness) can be seen as equilibrium play of a player population with such strongly stable preferences.     

INCOMPLETE INFORMATION GAMES WITH MULTIPLE PRIORS*

We present a model of incomplete information games, where each player is endowed with a set of priors. Upon arrival of private information, it is assumed that each player “updates” his set of priors to a set of posterior beliefs, and then evaluates his actions by the most pessimistic posterior beliefs. So each player's preferences may exhibit aversion to ambiguity or uncertainty. We define a couple of equilibrium concepts, establish existence results for them, and demonstrate by examples how players’ views on uncertainty about the environment affect the strategic outcomes.     

Bargaining over a Common Good with Private Information*

Consider two agents who make sequential claims on a common good, receiving their respective claims only if these are compatible. We let the first mover be privately informed about the size of the good. Conventional theory predicts multiple equilibria, and the intuitive criterion predicts that the first player claims the entire good. Our experimental results reject the intuitive criterion. Private information reduces the claim and the profit of the first mover. However, we cannot reject that the subjects play according to a perfect Bayesian equilibrium (PBE). Most subjects play according to a PBE, and almost all successful coordination occurs at the even split.     

Learning in society

In an individual experimentation problem a decision maker learns only from his own experience. It is well known that an optimal experimentation strategy for such problems sometimes results in the best alternative being dropped altogether, which is the so-called “Rothschild effect.” Many experimentation problems of interest, however, involve learning from both individual experience and the experience of others. This paper shows that learning in society can overcome the Rothschild effect. We consider an economy with a continuum of infinitely lived players in which each player faces a multi-armed bandit and in each period a player observes the action choice of another randomly chosen player. We show that social conformity always happens in the long run, and we use this fact to derive a condition on the distribution of prior beliefs that implies that the fraction of players who choose the best alternative always converges to one in the long run.     

Evolutionary stability of discrimination under observability

We study the evolution of preferences under perfect and almost perfect observability in symmetric 2-player games. We demonstrate that if nature can choose from a sufficiently general preference space, which includes preferences over outcomes that may depend on the opponent's preference-type, then, in most games, only discriminating preferences (treating different types of opponents differently in the same situation) can be evolutionary stable and some discriminating types are stable in a very strong sense in all games. We use these discriminating types to show that any symmetric outcome which gives players more than their minmax value in material payoffs (fitness) can be seen as equilibrium play of a player population with such strongly stable preferences.     

Religion and the reception of marginalism in Britain

Did their religious beliefs play any role in the interest British economists took in marginal utility theory in the final third of the Nineteenth Century? I contrast the beliefs of the leading economists—Mill and his followers—who were unsympathetic to marginalism with the beliefs of those economists who embraced it. Mill’s followers were themselves sharply divided on key methodological questions, and I ask whether or not this division, too, may owe something to differing religious beliefs. I conclude, provisionally, that religion may indeed have played a significant role in determining the methodological predisposition of British economists between 1860 and 1900.     

What to maximize if you must

The assumption that decision makers choose actions to maximize their preferences is a central tenet in economics, often justified formally or informally by appealing to evolutionary arguments. In contrast, we show that in almost every game and for almost every family of distortions of a player's actual payoffs, some degree of this distortion is beneficial to the player, and will not be driven out by any evolutionary process involving payoff-monotonic selection dynamics. Consequently, under any such selection dynamics the population will not converge to payoff-maximizing behavior. We also show that payoff-maximizing behavior need not prevail when preferences are imperfectly observed.