Escaping Nash Inflation

An ordinary differential equation (ODE) gives the mean dynamics that govern the convergence to self-confirming equilibria of self-referential systems under discounted least squares learning. Another ODE governs escape dynamics that recurrently propel away from a self-confirming equilibrium. In a model with a unique self-confirming equilibrium, the escape dynamics make the government discover too strong a version of the natural rate hypothesis. The escape route dynamics cause recurrent outcomes close to the Ramsey (commitment) inflation rate in a model with an adaptive government.
"If an unlikely event occurs, it is very likely to occur in the most likely way."
Michael Harrison     

Believers vs. deniers: Climate change and environmental policy polarization

This paper theoretically studies the consequences of partisanship with an application to environmental policy. We model an election between a right-wing and a left-wing candidate who strategically propose environmental policies to gain the support of an electorate divided based on their climate change beliefs and productive assets. While environmental regulations imply a trade-off between a more sustainable environment and higher incomes for all voters, climate change believers have a higher belief in human activity-induced climate change, which translates into greater expected environmental benefits from policy, and high-asset voters care relatively more about mitigating economic costs. Voters view the left-wing candidate as more effective in addressing environmental challenges, whereas her right-wing opponent is the better candidate to deliver relief from the economic burden of regulations. In equilibrium, there exists policy divergence and the right-wing candidate always proposes the more pro-industry policy. We find that higher asset inequality moves equilibrium policies in a pro-industry direction as long as high-asset voters are ideologically more homogeneous than low-asset ones. Equilibrium policies become further polarized with greater partisanship as those voters with the same climate change belief hold similar ideologies.     

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.     

Altruism and the Political Economy of Income Taxation

I develop a model in which altruistic agents vote over quadratic income tax schedules. Agents have heterogeneous preferences and productivities, and the model incorporates the incentive effects of taxation. The main result of the paper establishes the existence of a self-confirming majority rule equilibrium in which agents' labor supply decisions are optimal given their tax liabilities and the tax policy is a majority rule equilibrium given the labor supply decisions. In equilibrium the actions by all agents confirm their expectations, but such expectations may be incorrect out of equilibrium. The model generates majority rule voting equilibria that involve progressive taxation.     

Knightian games and robustness to ambiguity

This paper introduces a notion of robustness to ambiguous beliefs for Bayesian Nash equilibria. An equilibrium is robust if the corresponding strategies remain approximately optimal for a class of games with ambiguous beliefs that results from an appropriately defined perturbation of the belief structure of the original non-ambiguous belief game. The robustness definition is based on a novel definition of equilibrium for games with ambiguous beliefs that requires equilibrium strategies to be approximate best responses for all measures that define a player's belief. Conditions are derived under which robustness is characterized by a newly defined strategic continuity property, which can be verified without reference to perturbations and corresponding ambiguous belief games.     

On the Evolution of Pareto‐Optimal Behavior in Repeated Coordination Problems

I characterize the asymptotic behavior of a society facing a repeated‐common‐interest game. In this society, new individuals enter to replace their “parents” at random times. Each entrant has possibly different beliefs about others' behavior than his or her predecessor. A self‐confirming equilibrium (SCE) belief process describes an evolution of beliefs in this society consistent with a self‐confirming equilibrium of the repeated game. The main result shows that for any common‐interest game, the Pareto‐dominant equilibrium is a globally absorbing state of the behavioral dynamics when the SCE beliefs of new entrants satisfy certain independence and full‐support properties. This result does not involve either of the usual assumptions of myopia or large inertia common in evolutionary models, nor is this result possible if only Nash rather than self‐confirming equilibria are considered.     

Stated Beliefs and Play in Normal-Form Games

Using data on one-shot games, we investigate whether players' actions can be viewed as responses to underlying expectations about their opponent's behaviour. In our laboratory experiments, subjects play a set of 14 two-person 3×3 games and state beliefs about which actions they expect their opponents to play. The data sets from the two tasks are largely inconsistent. Rather, we find evidence that the subjects perceive the games differently when they (i) choose actions and (ii) state beliefs—their stated beliefs reveal deeper strategic thinking than their actions. On average, they fail to best respond to their own stated beliefs in almost half of the games. The inconsistency is confirmed by estimates of a unified statistical model that jointly uses the actions and the belief statements. There, we can control for decision noise and formulate a statistical test that rejects consistency. Effects of the belief elicitation procedure on subsequent actions are mostly insignificant.     

A model of noisy introspection

We present a theoretical model of noisy introspection designed to explain behavior in games played only once. The model determines layers of beliefs about others' beliefs about …, etc., but allows for surprises by relaxing the equilibrium requirement that belief distributions coincide with decision distributions. Noise is injected into iterated conjectures about others' decisions and beliefs, which causes the predictions to differ from those of deterministic models of iterated thinking, e.g., rationalizability. The paper contains a convergence proof that implies existence and uniqueness of the outcome of the iterated thought process. In addition, estimated introspection and noise parameters for data from 37 one-shot matrix games are reported. The accuracy of the model is compared with that of several alternatives.     

The Facts and the Values of the Lucas Critique

In his influential 1976 paper, ‘Econometric Policy Evaluation: A Critique,’ Robert E. Lucas, Jr. presented the policy non-invariance argument, also known as the Lucas critique (LC). Drawing on the work of Putnam and Walsh, this paper discusses how the LC, like all works of scientific inquiry, contains values entangled with scientific facts, and argues that the Lucas critique devalued and revalued the highest values in macroeconomic science, a process known as ‘transvaluation.’ Most importantly, the LC worked to operationalize a shift in values that undermined belief in economists’ ability and responsibility to make meaningful interventions in the economy. Employing the language and concepts of continental philosophy, this paper discusses the meaning and effect of the LC on the values embedded in contemporary macroeconomic science.     

Bayesian learning and convergence to Nash equilibria without common priors

Summary. Consider an infinitely repeated game where each player is characterized by a “type” which may be unknown to the other players in the game. Suppose further that each player's belief about others is independent of that player's type. Impose an absolute continuity condition on the ex ante beliefs of players (weaker than mutual absolute continuity). Then any limit point of beliefs of players about the future of the game conditional on the past lies in the set of Nash or Subjective equilibria. Our assumption does not require common priors so is weaker than Jordan (1991); however our conclusion is weaker, we obtain convergence to subjective and not necessarily Nash equilibria. Our model is a generalization of the Kalai and Lehrer (1993) model. Our assumption is weaker than theirs. However, our conclusion is also weaker, and shows that limit points of beliefs, and not actual play, are subjective equilibria. Received: March 3, 1995; revised version: February 17, 1997     

Subjectivism,indeterminacy and the structure of economic analysis

Subjectivists tend to reject the use of mechanistic laws in the explanation of human behaviour. There is a widespread belief, even in Austrian circles, that this rejection has destructive implications for all theory. The aim of this paper is to show that this belief is mistaken and that deterministic laws of behaviour need not play an essential role in economic theory. Behaviour can still be analysed and understood without using such laws. In addition, economics is not only about the explanation of behaviour, but also about the identification of which behaviour, by whom, is responsible for certain economic phenomena. The relevant methodology for these kinds of analyses is worked out and topical controversies in subjectivist economics, such as the role of the economics of knowledge, the possibility for dynamic theory and the relation between subjectivism and Keynes's method, are addressed.     

Does the Lucas critique apply during hyperinflation?: empirical evidence from four hyperinflationary episodes

Farmer (Review of Economic Studies, 58, 321–32, 1991) suggests that in a model in which there are multiple rational expectations equilibria agents may find it useful to coordinate their expectations in a unique rational expectations equilibrium which is supported by a self-fulfilling forecast rule having the property of being immune to the Lucas Critique. In this paper, Farmer's hypothesis is tested using data from hyperinflationary episodes. It is believed that those episodes are suitable for testing this hypothesis because an agent who lives in a hyperinflationary environment usually faces frequent changes in policy regime. The agent may thus choose a selffulfilling forecast rule which is immune to the Lucas Critique as a way of hedging against unanticipated policy regime switches. The empirical results show mixed evidence on Farmer's hypothesis during the hyperinflationary episodes studied.     

Self-referential thinking and equilibrium as states of mind in games: fMRI evidence

Sixteen subjects' brain activity were scanned using fMRI as they made choices, expressed beliefs, and expressed iterated 2nd-order beliefs (what they think others believe they will do) in eight games. Cingulate cortex and prefrontal areas (active in “theory of mind” and social reasoning) are differentially activated in making choices versus expressing beliefs. Forming self-referential 2nd-order beliefs about what others think you will do seems to be a mixture of processes used to make choices and form beliefs. In equilibrium, there is little difference in neural activity across choice and belief tasks; there is a purely neural definition of equilibrium as a “state of mind.” “Strategic IQ,” actual earnings from choices and accurate beliefs, is negatively correlated with activity in the insula, suggesting poor strategic thinkers are too self-focused, and is positively correlated with ventral striatal activity (suggesting that high IQ subjects are spending more mental energy predicting rewards).     

How Much Common Belief Is Necessary for a Convention?

We study Bayesian coordination games in which players choose actions conditional on the realization of their respective signals. Due to differential information, the players do not have common knowledge that a particular game is being played. However, they do have common beliefs with specified probabilities concerning their environment. In our framework, any equilibrium set of rules must be simple enough so that the actions of all players are common belief with probability 1 at every state. Common belief with probability close to 1 will not do.Journal of Economic LiteratureClassification Numbers: C72, D82.     

Consensus Consumer and Intertemporal Asset Pricing with Heterogeneous Beliefs

The aim of this paper is to analyse the impact of heterogeneous beliefs in an otherwise standard competitive complete market economy. The construction of a consensus probability belief, as well as a consensus consumer, is shown to be valid modulo an aggregation bias, which takes the form of a discount factor. In classical cases, the consensus probability belief is a risk tolerance weighted average of the individual beliefs, and the discount factor is proportional to beliefs dispersion. This discount factor makes the heterogeneous beliefs setting fundamentally different from the homogeneous beliefs setting, and it is consistent with the interpretation of beliefs heterogeneity as a source of risk.
We then use our construction to rewrite in a simple way the equilibrium characteristics (market price of risk, risk premium, risk-free rate) in a heterogeneous beliefs framework and to analyse the impact of beliefs heterogeneity. Finally, we show that it is possible to construct specific parametrizations of the heterogeneous beliefs model that lead to globally higher risk premia and lower risk-free rates.     

Cooperative equilibria in finite horizon noncooperative supergames

It is well known that cooperative outcomes can be supported by noncooperative equilibrium strategy combinations in games that are infinitely repeated; however, the example of the repeated prisoner's dilemma has led people to conclude that such cooperative outcomes cannot be sustained if the game is repeated finitely often. In this paper it is proved that the belief about finitely repeated games is false. Conditions are given that allow the existence of cooperative outcomes in finitely repeated games that are supported by subgame perfect noncooperative equilibrium strategies.     

Learning with perfect information

For extensive form games with perfect information, consider a learning process in which, at any iteration, each player unilaterally deviates to a best response to his current conjectures of others' strategies; and then updates his conjectures in accordance with the induced play of the game. We show that, for generic payoffs, the outcome of the game becomes stationary, and is consistent with Nash equilibrium. In general, if payoffs have ties or if players observe more of each others' strategies than is revealed by plays of the game, the same result holds provided a rationality constraint is imposed on unilateral deviations: no player changes his moves in subgames that he deems unreachable, unless he stands to improve his payoff there. Moreover, with this constraint, the sequence of strategies and conjectures also becomes stationary, and yields a self-confirming equilibrium.     

Crime and punishment in the “American Dream”

We observe that countries where belief in the “American dream”(i.e., effort pays) prevails also set harsher punishment for criminals. We know that beliefs are also correlated with several features of the economic system (taxation, social insurance, etc). Our objective is to study the joint determination of these three features (beliefs, punitiveness and economic system) in a way that replicates the observed empirical patterns. We present a model where beliefs determine the types of contracts that firms offer and whether workers exert effort. Some workers become criminals, depending on their luck in the labor market, the expected punishment, and an individual shock that we call “meanness”. It is this meanness level that a penal system based on “retribution” tries to detect when deciding the severity of the punishment. We find that when initial beliefs differ, two equilibria can emerge out of identical fundamentals. In the “American” (as opposed to the “French”) equilibrium, belief in the “American dream” is commonplace, workers exert effort, there are high powered contracts (and income is unequally distributed) and punishments are harsh. Economists who believe that deterrence (rather than retribution) shapes punishment can interpret the meanness parameter as pessimism about future economic opportunities and verify that two similar equilibria emerge.     

Random belief equilibrium in normal form games

In defining random belief equilibrium (RBE) in finite, normal form games we assume a player's beliefs about others' strategy choices are randomly drawn from a belief distribution that is dispersed around a central strategy profile, the focus. At an RBE: (1) Each chooses a best response relative to her beliefs. (2) Each player's expected choice coincides with the focus of the other players' belief distributions. RBE provides a statistical framework for estimation which we apply to data from three experimental games. We also characterize the limit-RBE as players' beliefs converge to certainty. When atoms in the belief distributions vanish in the limit, not all limit-RBE (called robust equilibria) are trembling hand perfect Nash equilibria and not all perfect equilibria are robust.     

Mutually acceptable courses of action

We offer a game-theoretic framework that enables the analysis of situations whereby rational individuals with different beliefs and views of the world agree to a shared course of action. We introduce a new solution concept: a mutually acceptable course of action, which can be viewed as an (incomplete) contract or a social norm that free rational individuals would be willing to follow for their own diverse reasons. We show that by varying the degree of completeness of the underlying course of action, our concept can be related to commonly used solutions, such as perfect equilibrium, perfect Bayesian equilibrium, (rationalizable) self-confirming equilibrium, and rationalizable outcomes. We are grateful to the editor and an anonymous referee for very useful and helpful comments and suggestions. Earlier versions of this paper were circulated under the title “Towering over Babel: Worlds Apart but Acting Together”. We thank participants at the World Congress of Game Theory Society (2004), the International Conference on Game Theory (2004), the European Meeting (2004), and the Econometric Society World Congress (2005). Financial support from SSHRC and NSERC of Canada, and the National Science Council of Taiwan are gratefully acknowledged.