The impact of endowment heterogeneity and origin on contributions in best-shot public good games

Economists and psychologists have long argued the origin of wealth influences individual behavior. In a previous study (Cherry et al., 2005), we found the origin of endowment did not significantly affect behavior in linear public good games with summation contribution technology. In such games, however, both Nash behavior (everybody gives nothing) and social optimal behavior (everybody gives the entire endowment) call for symmetric levels of contributions. Results from this new study indicate that the origin of wealth might matter in more asymmetric situations, such as in a best-shot public good game with heterogeneous groups. Electronic Supplementary Material Supplementary material is available in the online version of this article at . JEL Classification C72, C92, H41     

Weak links, good shots and other public good games: Building on BBV

We suggest an alternative way of analyzing the canonical Bergstrom-Blume-Varian model of non-cooperative voluntary contributions to a public good that avoids the proliferation of dimensions as the number of players is increased. We exploit this approach to analyze models in which the aggregate level of public good is determined as a more general social composition function of individual gifts — specifically, as a CES form — rather than as an unweighted sum. We also analyze Hirshleifer's weakest-link and best-shot models. In each case, we characterize the set of equilibria, in some cases establishing existence of a unique equilibrium as well as briefly pointing out some interesting comparative static properties. We also study the weakest-link and best-shot limits of the CES composition function and show how the former can be used for equilibrium selection and the latter to establish that equilibria of some better-shot games are identical to those of the much simpler best-shot game.     

Voluntary contributions when the public good is not necessarily normal

We argue that there are interesting examples of privately provided public goods that do not satisfy the assumption of strict normality, and reconsider voluntary-contribution games in a more general framework. We show that, in general, (1) equalizing transfers between individuals with identical tastes can increase total supply of the public good, and (2) more of the public good can be supplied if agents move sequentially rather than simultaneously. These results are in sharp contrast to earlier conclusions derived in the literature under the assumption of strict normality.     

Renegotiation in repeated games

In repeated games, subgame-perfect equilibria involving threats of punishment may be implausible if punishing one player hurts the other(s). If players can renegotiate after a defection, such a punishment may not be carried out. We explore a solution concept that recognizes this fact, and show that in many games the prospect of renegotiation strictly limits the cooperative outcomes that can be sustained. We characterize those outcomes in general, and in the prisoner's dilemma, Cournot and Bertrand duopolies, and an advertising game in particular.     

Mindreading and endogenous beliefs in games

We argue that a Bayesian explanation of strategic choices in games requires introducing a psychological theory of belief formation. We highlight that beliefs in epistemic game theory are derived from the actual choice of the players, and cannot therefore explain why Bayesian rational players should play the strategy they actually chose. We introduce the players’ capacity of mindreading in a game theoretical framework with the simulation theory, and characterise the beliefs that Bayes rational players could endogenously form in games. We show in particular that those beliefs need not be ratifiable, and therefore that rational players can form action-dependent beliefs.     

On eliciting beliefs in strategic games

Several recent studies in experimental economics have tried to measure beliefs of subjects engaged in strategic games with other subjects. Using data from one such study we conduct an experiment where our experienced subjects observe early rounds of strategy choices from that study and are given monetary incentives to report forecasts of choices in later rounds. We elicit beliefs using three different scoring rules: linear, logarithmic, and quadratic. We compare forecasts across the scoring rules and compare the forecasts of our trained observers to forecasts of the actual players in the original experiment. We find significant differences across scoring rules. The improper linear scoring rule produces forecasts closer to 0 and 1 than the proper rules, and these forecasts are poorly calibrated. The two proper scoring rules induce significantly different distributions of forecasts. We find that forecasts by observers under both proper scoring rules are significantly different from the forecasts of the actual players, in terms of accuracy, calibration, and the distribution of forecasts. We also find evidence for belief convergence among the observers.     

Learning aspiration in repeated games

We study infinitely repeated symmetric 2×2 games played by bounded rational agents who follow a simple rule of thumb: each agent continues to play the same action if the current payoff exceeds the average of the past payoffs, and switches to the other action with a positive probability otherwise. By applying the stochastic approximation technique, we characterize the asymptotic outcomes for all 2×2 games. In the prisoners’ dilemma game, for example, the players cooperate in the limit if and only if the gain from defecting against cooperation is “modest.”     

Testing threats in repeated games

Under most game-theoretic solution concepts, equilibrium beliefs are justified by off-equilibrium events. I propose an equilibrium concept for infinitely repeated games, called “Nash Equilibrium with Tests” (NEWT), according to which players can only justify their equilibrium beliefs with events that take place on the equilibrium path itself. In NEWT, players test every threat that rationalizes a future non-myopic action that they take. The tests are an integral part of equilibrium behavior. Characterization of equilibrium outcomes departs from the classical “folk theorems”. The concept provides new insights into the impact of self-enforcement norms, such as reciprocity, on long-run cooperation.     

Monitoring institutions in indefinitely repeated games

Does monitoring past conduct facilitate intertemporal cooperation? We designed an experiment characterized by strategic uncertainty and multiple equilibria where coordinating on the efficient outcome is a challenge. Participants, interacting anonymously in a group, could pay a cost either to obtain information about their counterparts, or to create a freely available public record of individual conduct. Both monitoring institutions were actively employed. However, groups were unable to attain higher levels of cooperation compared to a treatment without monitoring. Information about past conduct alone thus appears to be ineffective in overcoming coordination challenges.     

Communication in repeated monetary policy games

A central bank and the public are engaged in an infinitely repeated monetary policy game with communication. For reasonable discount factors, there exists an equilibrium in which the central bank fully reveals its private information. The fully revealing equilibrium is superior to the uninformative equilibrium. The welfare gain of transparency increases with the slope of the Phillips curve, the natural rate of unemployment, and the degree of heterogeneity in the population. Transparency results in lower inflation but a higher variability of inflation.     

Adaptation and complexity in repeated games

The paper presents a learning model for two-player infinitely repeated games. In an inference step players construct minimally complex inferences of strategies based on observed play, and in an adaptation step players choose minimally complex best responses to an inference. When players randomly select an inference from a probability distribution with full support the set of steady states is a subset of the set of Nash equilibria in which only stage game Nash equilibria are played. When players make ‘cautious’ inferences the set of steady states is the subset of self-confirming equilibria with Nash outcome paths. When players use different inference rules, the set of steady states can lie between the previous two cases.     

Dismissals and quits in repeated games

This paper studies repeated games, where player 1 can decide to let the opponent continue or replace her by a new player. We also allow for the possibility of player 2 quitting the game. When only layoffs can occur, a folk theorem for finite horizons obtains due to the threat that termination of the relationship imposes on player 2. However, quits limit this result to those cases in which the outside option for player 2 is small (lower than some Nash equilibrium payoff of the stage game).     

Communication in repeated games with costly monitoring

We study repeated games with discounting where perfect monitoring is possible, but costly. It is shown that if players can make public announcements, then every payoff vector which is an interior point in the set of feasible and individually rational payoffs can be implemented in a sequential equilibrium of the repeated game when the discount factor is high enough. Thus, efficiency can be approximated even when the cost of monitoring is high, provided that the discount factor is high enough.     

On the complexity of repeated principal agent games

Summary We examine an infinitely repeated principal agent game without discounting (Radner [1985]), in which the agent may engage in multiple projects. We focus on linear strategies that summarize each history into a linear function of public outcomes, and select an action according to a single threshold rule. We claim that linear strategies significantly simplify the computation needed to make strategic decisions following each history. Despite the simplicity of linear strategies, we can virtually recover the folk theorem. For any individually rational payoff vector in the interior of the set of feasible expected payoff vectors, there exists a pair of linear strategies that form a Nash equilibrium supporting the target payoff. The equilibrium strategies and the equilibrium payoff vectors form a globally stable solution (Smale [1980]).I would like to thank Andrew Atkeson and the anonymous referee for helpful comments. John Curran and Hao Li provided excellent research assistance. Financial support from the National Science Foundation, the Alfred P. Sloan Foundation and the Division of Social Sciences at the University of Chicago is gratefully acknowledged.     

Estimating the causal effect of fertility on economic wellbeing: data requirements, identifying assumptions and estimation methods

This article aims to answer to what extent fertility has a causal effect on households’ economic wellbeing—an issue that has received considerable interest in development studies and policy analysis. However, only recently has this literature begun to give importance to adequate modelling for estimation of causal effects. We discuss several strategies for causal inference, stressing that their validity must be judged on the assumptions we can plausibly formulate in a given application, which in turn depends on the richness of available data. We contrast methods relying on the unconfoundedness assumption, which include regressions and propensity score matching, with instrumental variable methods. This discussion has a general importance, representing a set of guidelines that are useful for choosing an appropriate strategy of analysis. The discussion is valid for both cross-sectional or panel data.     

Coordination failure in repeated games with private monitoring

Players coordinate continuation play in repeated games with public monitoring. We investigate the robustness of such equilibrium behavior with respect to ex-ante small private-monitoring perturbations. We show that with full support of public signals, no perfect public equilibrium is robust if it induces a “regular” 2×2$2\times 2$ coordination game in the continuation play. This regularity condition is violated in all belief-free equilibria. Indeed, with an individual full rank condition, every interior belief-free equilibrium is robust. We also analyze block belief-free equilibria and point out that the notion of robustness is sensitive to whether we allow for uninterpretable signals.     

On the complexity of repeated principal agent games?

Summary. We examine an infinitely repeated principal agent game without discounting (Radner [1985] ), in which the agent may engage in multiple projects. We focus on “linear” strategies that summarize each history into a linear function of public outcomes, and select an action according to a single threshold rule. We claim that linear strategies significantly simplify the computation needed to make strategic decisions following each history. Despite the simplicity of linear strategies, we can virtually recover the folk theorem. For any individually rational payoff vector in the interior of the set of feasible expected payoff vectors, there exists a pair of linear strategies that form a Nash equilibrium supporting the target payoff. The equilibrium strategies and the equilibrium payoff vectors form a globally stable solution (Smale [1980] ).     

Communication in repeated network games with imperfect monitoring

I consider repeated games with private monitoring played on a network. Each player has a set of neighbors with whom he interacts: a player's payoff depends on his own and his neighbors' actions only. Monitoring is private and imperfect: each player observes his stage payoff but not the actions of his neighbors. Players can communicate costlessly at each stage: communication can be public, private or a mixture of both. Payoffs are assumed to be sensitive to unilateral deviations. First, for any network, a folk theorem holds if some Joint Pairwise Identifiability condition regarding payoff functions is satisfied. Second, a necessary and sufficient condition on the network topology for a folk theorem to hold for all payoff functions is that no two players have the same set of neighbors not counting each other.     

Learning from private information in noisy repeated games

We study the perfect type-contingently public ex-post equilibrium (PTXE) of repeated games where players observe imperfect public signals of the actions played, and both the payoff functions and the map from actions to signal distributions depend on an unknown state. The PTXE payoffs when players are patient are determined by the solutions to a family of linear programming problems. Using this characterization, we develop conditions under which play can be as if the players have learned the state. We provide a sufficient condition for the folk theorem, and a characterization of the PTXE payoffs in games with a known monitoring structure.