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.     

Preference Evolution,Two-Speed Dynamics,and Rapid Social Change

We present a dynamic analysis of the evolution of preferences in a strategic environment. In our model, each player's behavior depends on both the game's payoffs and his idiosyncratic biases, but only the game's payoffs determine his evolutionary success. Dynamics run at two speeds at once; while natural selection slowly reshapes the distribution of preferences, players quickly learn to behave as their preferences dictate. We establish the existence and uniqueness of the paired trajectories of society's preferences and behavior. While aggregate behavior adjusts smoothly in equilibration games, in coordination games aggregate behavior can jump discretely in an instant of evolutionary time. Journal of Economic Literature Classification Numbers: C72, C73.     

Endogenous Move Structure and Voluntary Provision of Public Goods: Theory and Experiment

In this paper, we examine voluntary contributions to a public good, embedding Varian's (1994) voluntary contribution game in extended games that allow players to choose the timing of their contributions. We show that predicted outcomes are sensitive to the structure of the extended game, and also to the extent to which players care about payoff inequalities. We then report a laboratory experiment based on these extended games. We find that behavior is similar in the two extended games: subjects avoid the detrimental move order of Varian's model, where a person with a high value of the public good commits to a low contribution, and instead players tend to delay contributions. These results suggest that commitment opportunities may be less damaging to public good provision than previously thought.     

The British gambler's fallacy

People facing choices under uncertainty, and gamblers in particular, are often subject to statistical fallacies. This paper explores the hypothesis that if lotto players were subject to the ‘gambler's fallacy’, predictable fluctuations in the number of jackpots would occur. Evidence, based on a Poisson regression model in which the number of winning bets is conditional on the history of draws, indicates that number selection in the UK is only marginally affected by the history of draws.     

The importance of higher-order beliefs to successful coordination

Beliefs about other players’ strategies are crucial in determining outcomes for coordination games. If players are to coordinate on an efficient equilibrium, they must believe that others will coordinate with them. In many settings there is uncertainty about beliefs as well as strategies. Do people consider these “higher-order” beliefs (beliefs about beliefs) when making coordination decisions? I design a modified stag hunt experiment that allows me to identify how these higher-order beliefs and uncertainty about higher-order beliefs matter for coordination. Players prefer to invest especially when they believe that others are “optimistic” that they will invest; but knowledge that others think them unlikely to invest does not cause players to behave differently than when they do not know what their partners think about them. Thus resolving uncertainty about beliefs can result in marked efficiency gains.     

An Experiment to Evaluate Bayesian Learning of Nash Equilibrium Play

We report on an experiment designed to evaluate the empirical implications of Jordan's model of Bayesian learning in games of incomplete information. A finite example is constructed in which the model generates unique predictions of subjects' choices in nearly all periods. When the “true” game defined by players' private information was one with a unique equilibrium in pure strategies, the experimental subjects' play converged to the equilibrium, as Jordan's theory predicts, even when the subjects had not attained complete information about one another. But when there were two pure strategy equilibria, the theory's predictions were not consistent with observed behavior. Journal of Economic Literature Classification numbers: D83, C72, C92.     

Heterogeneous network games: Conflicting preferences

In many economic situations, a player pursues coordination or anti-coordination with her neighbors on a network, but she also has intrinsic preferences among the available options. We here introduce a model which allows to analyze this issue by means of a simple framework in which players endowed with an idiosyncratic identity interact on a social network through strategic complements or substitutes. We classify the possible types of Nash equilibria under complete information, finding two thresholds for switching action that relate to the two-player setup of the games. This structure of equilibria is considerably reduced when turning to incomplete information, in a setup in which players only know the distribution of the number of neighbors of the network. For high degrees of heterogeneity in the population the equilibria is such that every player can choose her preferred action, whereas if one of the identities is in the minority frustration ensues.     

Random-player games

This paper introduces general games with incomplete information in which the number, as well as the types or identities, of the participating players are determined by chance and might not be known to the players when they make their choices of actions. In these games, the selection of the number and types of players is modeled as a finite point process on a suitable type space. Definitions of pure-strategy, mixed-strategy, and correlated equilibria in random-player games are given, extending the corresponding ones for finite games, Bayesian games, and games with population uncertainty, which may all be considered as special cases of random-player games.     

Robustness to strategic uncertainty

We introduce a criterion for robustness to strategic uncertainty in games with continuum strategy sets. We model a player's uncertainty about another player's strategy as an atomless probability distribution over that player's strategy set. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others. When payoff functions are continuous we show that our criterion is a refinement of Nash equilibrium and we also give sufficient conditions for existence of a robust strategy profile. In addition, we apply the criterion to Bertrand games with convex costs, a class of games with discontinuous payoff functions and a continuum of Nash equilibria. We show that it then selects a unique Nash equilibrium, in agreement with some recent experimental findings.     

On ‘Informationally Robust Equilibria’ for Bimatrix Games

Informationally robust equilibria (IRE) are introduced in Robson (Games Econ Behav 7: 233–245, 1994) as a refinement of Nash equilibria for strategic games. Such equilibria are limits of a sequence of (subgame perfect) Nash equilibria in perturbed games where with small probability information about the strategic behavior is revealed to other players (information leakage). Focusing on bimatrix games, we consider a type of informationally robust equilibria and derive a number of properties they form a non-empty and closed subset of the Nash equilibria. Moreover, IRE is a strict concept in the sense that the IRE are independent of the exact sequence of probabilities with which information is leaked. The set of IRE, like the set of Nash equilibria, is the finite union of polytopes. In potential games, there is an IRE in pure strategies. In zero-sum games, the set of IRE has a product structure and its elements can be computed efficiently by using linear programming. We also discuss extensions to games with infinite strategy spaces and more than two players. The authors would like to thank Marieke Quant for her helpful comments.     

Preferences and beliefs in a sequential social dilemma: a within-subjects analysis

In empirical analyses of games, preferences and beliefs are typically treated as independent. However, if beliefs and preferences interact, this may have implications for the interpretation of observed behavior. Our sequential social dilemma experiment allows us to separate different interaction channels. When subjects play both roles in such experiments, a positive correlation between first- and second-mover behavior is frequently reported. We find that the observed correlation primarily originates via an indirect channel, where second-mover decisions influence beliefs through a consensus effect, and the first-mover decision is a best response to these beliefs. Specifically, beliefs about second-mover cooperation are biased toward own second-mover behavior, and most subjects best respond to stated beliefs. However, we also find evidence for a direct, preference-based channel. When first movers know the true probability of second-mover cooperation, subjects' own second moves still have predictive power regarding their first moves.     

Dragon Slaying with Ambiguity: Theory and Experiments

This paper studies the impact of ambiguity in the best‐shot and weakest‐link models of public good provision. The models are first analyzed theoretically. Then, we conduct experiments to study how ambiguity affects behavior in these games. We test whether subjects' perception of ambiguity differs between a local opponent and a foreign one. We find that an ambiguity‐safe strategy is often chosen by subjects. This is compatible with the hypothesis that ambiguity aversion influences behavior in games. Subjects tend to choose contributions above (respectively, below) the Nash equilibrium in the best‐shot (respectively, weakest‐link) model.     

Economic policy uncertainty in the US and China and their impact on the global markets

The recent US-China trade conflict has caused substantial uncertainty in the global markets. What is the rationale of this conflict? Is the rising of Chinese economy imposing a realistic threat to the US-led post-war international system? Using economic policy uncertainty in each of these two key global economic players as a measure of policy stance, this paper builds a time series model following Diebold and Yilmaz (2014) to estimate the influence of both the US and China on several key international markets, namely, stock, credit, energy and commodity markets. We find that, although China has become more influential, the US's dominant position still holds in all the markets. The results suggest that concerns regarding China's competition with the US in shaping the global world order are more likely to be driven by political factors rather than economic motives.     

SEQUENTIAL TWO‐PLAYER GAMES WITH AMBIGUITY*

If players' beliefs are strictly nonadditive, the Dempster–Shafer updating rule can be used to define beliefs off the equilibrium path. We define an equilibrium concept in sequential two‐person games where players update their beliefs with the Dempster–Shafer updating rule. We show that in the limit as uncertainty tends to zero, our equilibrium approximates Bayesian Nash equilibrium. We argue that our equilibrium can be used to define a refinement of Bayesian Nash equilibrium by imposing context‐dependent constraints on beliefs under uncertainty.     

Behavior and Learning in the “Dirty Faces” Game

This paper examines the Dirty Faces problem as a Bayesian game. The equilibrium in the general form of the game requires the extreme assumption of common knowledge of rationality. However, for any finite number of players, the exact number of steps of iterated rationality necessary for the equilibrium to arise depends on the number of players of a particular type, allowing the game to be used to bound the number of steps satisfied by actual players. The game differs from other games used to study iterated rationality in that all players are better off when common knowledge of rationality is satisfied. While behavior in experiments is inconsistent with the game-theoretic prediction at the group level, individual level behavior shows a greater degree of consistency with theory and with previous results on iterated rationality. Finally, there is some evidence of learning in repeated play.     

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.     

Strategic games beyond expected utility

This paper argues that Nash equilibrium is a solution where all strategic uncertainty has been resolved and, therefore, inappropriate to model situations that involve ??ambiguity.?? Instead, to capture what players will do in the presence of some strategic uncertainty, takes a solution concept that is closed under best replies. It is shown that such a solution concept, fixed sets under the best reply correspondence, exists for a class of games significantly wider than those games for which generalizations of Nash equilibrium exist. In particular, this solution can do without the expected utility hypothesis.     

The One-Shot-Deviation Principle for Sequential Rationality

This article analyzes the fictitious play process originally proposed for strategic form games by Brown (1951) and Robinson (1951). We interpret the process as a model of preplay thinking performed by players before acting in a one-shot game. This model is one of bounded rationality. We discuss how fictitious play should then be defined for extensive form games and conclude that this is somewhat problematic. We therefore study two alternative definitions. For either of these, under a weak condition of initial uncertainty, a convergence point of a fictitious play sequence is a sequential equilibrium. For generic games of perfect information initial uncertainty also implies convergence of fictitious play.Journal of Economic LiteratureClassification Number: C72.     

Fictitious Play in Extensive Form Games

This article analyzes the fictitious play process originally proposed for strategic form games by Brown (1951) and Robinson (1951). We interpret the process as a model of preplay thinking performed by players before acting in a one-shot game. This model is one of bounded rationality. We discuss how fictitious play should then be defined for extensive form games and conclude that this is somewhat problematic. We therefore study two alternative definitions. For either of these, under a weak condition of initial uncertainty, a convergence point of a fictitious play sequence is a sequential equilibrium. For generic games of perfect information initial uncertainty also implies convergence of fictitious play.Journal of Economic LiteratureClassification Number: C72.     

Measuring Strategic Uncertainty in Coordination Games

This paper proposes a method to measure strategic uncertainty by eliciting certainty equivalents analogous to measuring risk attitudes in lotteries. We apply this method by conducting experiments on a class of one-shot coordination games with strategic complementarities and choices between simple lotteries and sure payoff alternatives, both framed in a similar way. Despite the multiplicity of equilibria in the coordination games, aggregate behaviour is fairly predictable. The pure or mixed Nash equilibria cannot describe subjects' behaviour. We present two global games with private information about monetary payoffs and about risk aversion. While previous literature treats the parameters of a global game as given, we estimate them and show that both models describe observed behaviour well. The global-game selection for vanishing noise of private signals offers a good recommendation for actual players, given the observed distribution of actions. We also deduce subjective beliefs and compare them with objective probabilities.