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Haavelmo (1944) proposed a probabilistic structure for econometric modeling, aiming to make econometrics useful for decision making. His fundamental contribution has become thoroughly embedded in econometric research, yet it could not answer all the deep issues that the author raised. Notably, Haavelmo struggled to formalize the implications for decision making of the fact that models can at most approximate actuality. In the same period, Wald (1939, 1945) initiated his own seminal development of statistical decision theory. Haavelmo favorably cited Wald, but econometrics did not embrace statistical decision theory. Instead, it focused on study of identification, estimation, and statistical inference. This paper proposes use of statistical decision theory to evaluate the performance of models in decision making. I consider the common practice of as‐if optimization: specification of a model, point estimation of its parameters, and use of the point estimate to make a decision that would be optimal if the estimate were accurate. A central theme is that one should evaluate as‐if optimization or any other model‐based decision rule by its performance across the state space, listing all states of nature that one believes feasible, not across the model space. I apply the theme to prediction and treatment choice. Statistical decision theory is conceptually simple, but application is often challenging. Advancing computation is the primary task to complete the foundations sketched by Haavelmo and Wald. [ABSTRACT FROM AUTHOR]

Measurements of ambiguity attitudes have so far focused on artificial events, where (subjective) beliefs can be derived from symmetry of events and can be then controlled for. For natural events as relevant in applications, such a symmetry and corresponding control are usually absent, precluding traditional measurement methods. This paper introduces two indexes of ambiguity attitudes, one for aversion and the other for insensitivity/perception, for which we can control for likelihood beliefs even if these are unknown. Hence, we can now measure ambiguity attitudes for natural events. Our indexes are valid under many ambiguity theories, do not require expected utility for risk, and are easy to elicit in practice. We use our indexes to investigate time pressure under ambiguity. People do not become more ambiguity averse under time pressure but become more insensitive (perceive more ambiguity). These findings are plausible and, hence, support the validity of our indexes. [ABSTRACT FROM AUTHOR]

This paper axiomatizes the robust control criterion of multiplier preferences introduced by Hansen and Sargent (2001). The axiomatization relates multiplier preferences to other classes of preferences studied in decision theory, in particular, the variational preferences recently introduced by Maccheroni, Marinacci, and Rustichini (2006a). This paper also establishes a link between the parameters of the multiplier criterion and the observable behavior of the agent. This link enables measurement of the parameters on the basis of observable choice data and provides a useful tool for applications. [ABSTRACT FROM AUTHOR]

This paper provides nonparametric identification results for a class of latent utility models with additively separable unobservable heterogeneity. These results apply to existing models of discrete choice, bundles, decisions under uncertainty, and matching. Under an independence assumption, such models admit a representative agent. As a result, we can identify how regressors alter the desirability of goods using only average demands. Moreover, average indirect utility ("welfare") is identified without needing to specify or identify the distribution of unobservable heterogeneity. [ABSTRACT FROM AUTHOR]

This paper begins by observing that any reflexive binary (preference) relation (over risky prospects) that satisfies the independence axiom admits a form of expected utility representation. We refer to this representation notion as the coalitional minmax expected utility representation. By adding the remaining properties of the expected utility theorem, namely, continuity, completeness, and transitivity, one by one, we find how this representation gets sharper and sharper, thereby deducing the versions of this classical theorem in which any combination of these properties is dropped from its statement. This approach also allows us to weaken transitivity in this theorem, rather than eliminate it entirely, say, to quasitransitivity or acyclicity. Apart from providing a unified dissection of the expected utility theorem, these results are relevant for the growing literature on boundedly rational choice in which revealed preference relations often lack the properties of completeness and/or transitivity (but often satisfy the independence axiom). They are also especially suitable for the (yet overlooked) case in which the decision‐maker is made up of distinct individuals and, consequently, transitivity is routinely violated. Finally, and perhaps more importantly, we show that our representation theorems allow us to answer many economic questions that are posed in terms of nontransitive/incomplete preferences, say, about the maximization of preferences, the existence of Nash equilibrium, the preference for portfolio diversification, and the possibility of the preference reversal phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, I construct players' prior beliefs and show that these prior beliefs lead the players to learn to play an approximate Nash equilibrium uniformly in any infinitely repeated slightly perturbed game with discounting and perfect monitoring. That is, given any ε > 0, there exists a (single) profile of players' prior beliefs that leads play to almost surely converge to an ε-Nash equilibrium uniformly for any (finite normal form) stage game with slight payoff perturbation and any discount factor less than 1. [ABSTRACT FROM AUTHOR]

The paper studies the representation and characterization of risks generated by a continuum of random variables. The Main Theorem is a characterization of a broad class of continuum processes in terms of the decomposition of risk into aggregate and idiosyncratic components, and in terms of the approximation of the continuum process by finite collections of random variables. This characterization is used to study decision making problems with anonymous and state-independent payoffs. An Extension Theorem shows that if such a payoff function is defined on simple processes, then it has a unique continuous extension to the class of processes characterized in this paper. This extension is formulated without reference to sample realizations and with minimal restrictions on the patterns of correlation between the random variables. As an application, the theory is used to develop a new model of large games which emphasizes the explicit description of the players' randomizations. This model is used to study the class of environments in which Schmeidler's (1973) representation of strategic uncertainty in large games is valid. [ABSTRACT FROM AUTHOR]

Suppose, for example, that for each I i i , we observe HT ht , for some parametric family of distributions HT ht . Assume independence between units I i i , and let HT ht be the maximum likelihood estimator based on HT ht . Asymptotic Approximations Gu and Koenker regard the normality assumption on HT ht as an approximation. [Extracted from the article]

This paper applies some general concepts in decision theory to a linear panel data model. A simple version of the model is an autoregression with a separate intercept for each unit in the cross section, with errors that are independent and identically distributed with a normal distribution. There is a parameter of interest γ and a nuisance parameter τ, a N x K matrix, where N is the cross-section sample size. The focus is on dealing with the incidental parameters problem created by a potentially high-dimension nuisance parameter. We adopt a "fixed-effects" approach that seeks to protect against any sequence of incidental parameters. We transform τ to (δ p, ω), where δ is a J x K matrix of coefficients from the least-squares projection of τ on a N x J matrix x of strictly exogenous variables, p is a K x K symmetric, positive semidefinite matrix obtained from the residual sums of squares and cross-products in the projection of τ on x, and ω is a (N - J) x K matrix whose columns are orthogonal and have unit length. The model is invariant under the actions of a group on the sample space and the parameter space, and we find a maximal invariant statistic. The distribution of the maximal invariant statistic does not depend upon ω. There is a unique invariant distribution for ω. We use this invariant distribution as a prior distribution to obtain an integrated likelihood function. It depends upon the observation only through the maximal invariant statistic. We use the maximal invariant statistic to construct a marginal likelihood function, so we can eliminate ω by integration with respect to the invariant prior distribution or by working with the marginal likelihood function. The two approaches coincide. Decision rules based on the invariant distribution for ω have a minimax property. Given a loss function that does not depend upon ω and given a prior distribution for (γ, δ p), we show how to minimize the average--with respect to the prior distribution for (γ, δ, p)--of the maximum risk, where the maximum is with respect to ω. There is a family of prior distributions for (δ, p) that leads to a simple closed form for the integrated likelihood function. This integrated likelihood function coincides with the likelihood function for a normal, correlated random-effects model. Under random sampling, the corresponding quasi maximum likelihood estimator is consistent for γ as N → ∞, with a standard limiting distribution. The limit results do not require normality or homoskedasticity (conditional on x) assumptions. [ABSTRACT FROM AUTHOR]

There are typically multiple equilibrium outcomes in the Crawford-Sobel (CS) model of strategic information transmission. This paper identifies a simple condition on equilibrium payoffs, called NITS (no incentive to separate), that selects among CS equilibria. Under a commonly used regularity condition, only the equilibrium with the maximal number of induced actions satisfies NITS. We discuss various justifications for NITS, including perturbed cheap-talk games with nonstrategic players or costly lying. We also apply NITS to other models of cheap talk, illustrating its potential beyond the CS framework. [ABSTRACT FROM AUTHOR]

This paper develops a nonparametric theory of preferences over one's own and others' monetary payoffs. We introduce "more altruistic than" (MAT), a partial ordering over such preferences, and interpret it with known parametric models. We also introduce and illustrate "more generous than" (MGT), a partial ordering over opportunity sets. Several recent studies focus on two-player extensive form games of complete information in which the first mover (FM) chooses a more or less generous opportunity set for the second mover (SM). Here reciprocity can be formalized as the assertion that an MGT choice by the FM will elicit MAT preferences in the SM. A further assertion is that the effect on preferences is stronger for acts of commission by FM than for acts of omission. We state and prove propositions on the observable consequences of these assertions. Finally, empirical support for the propositions is found in existing data from investment and dictator games, the carrot and stick game, and the Stackelberg duopoly game and in new data from Stackelberg mini-games. [ABSTRACT FROM AUTHOR]

This paper applies some general concepts in decision theory to a simple instrumental variables model. There are two endogenous variables linked by a single structural equation; k of the exogenous variables are excluded from this structural equation and provide the instrumental variables (IV). The reduced-form distribution of the endogenous variables conditional on the exogenous variables corresponds to independent draws from a bivariate normal distribution with linear regression functions and a known covariance matrix. A canonical form of the model has parameter vector (ρ, φ, ω), where φ is the parameter of interest and is normalized to be a point on the unit circle. The reduced-form coefficients on the instrumental variables are split into a scalar parameter ρ and a parameter vector ω, which is normalized to be a point on the (k - 1)-dimensional unit sphere; ρ measures the strength of the association between the endogenous variables and the instrumental variables, and ω is a measure of direction. A prior distribution is introduced for the IV model. The parameters φ, ρ, and ω are treated as independent random variables. The distribution for φ is uniform on the unit circle; the distribution for ω is uniform on the unit sphere with dimension k - 1. These choices arise from the solution of a minimax problem. The prior for ρ is left general. It turns out that given any positive value for ρ, the Bayes estimator of φ does not depend on ρ; it equals the maximum-likelihood estimator. This Bayes estimator has constant risk; because it minimizes average risk with respect to a proper prior, it is minimax. The same general concepts are applied to obtain confidence intervals. The prior distribution is used in two ways. The first way is to integrate out the nuisance parameter ω in the IV model. This gives an integrated likelihood function with two scalar parameters, φ and ρ. Inverting a likelihood ratio test, based on the integrated likelihood function, provides a confidence interval for φ. This lacks finite sample optimality, but invariance arguments show that the risk function depends only on ρ and not on φ or ω. The second approach to confidence sets aims for finite sample optimality by setting up a loss function that trades off coverage against the length of the interval. The automatic uniform priors are used for φ and ω, but a prior is also needed for the scalar ρ, and no guidance is offered on this choice. The Bayes rule is a highest posterior density set. Invariance arguments show that the risk function depends only on ρ and not on φ or ω. The optimality result combines average risk and maximum risk. The confidence set minimizes the average--with respect to the prior distribution for ρ--of the maximum risk, where the maximization is with respect to φ and ω. [ABSTRACT FROM AUTHOR]

An unresolved problem in Bayesian decision theory is how to value and price information. This paper resolves both problems assuming inexpensive information. Building on Large Deviation Theory, we produce a generically complete asymptotic order on samples of i.i.d. signals in finite-state, finite-action models. Computing the marginal value of an additional signal, we find it is eventually exponentially falling in quantity, and higher for lower quality signals. We provide a precise formula for the information demand, valid at low prices: asymptotically a constant times the log price, and falling in the signal quality for a given price. [ABSTRACT FROM AUTHOR]

The main contribution of this paper is the development and application of cryptographic techniques to the design of strategic communication mechanisms. One of the main assumptions in cryptography is the limitation of the computational power available to agents. We introduce the concept of limited computational complexity, and by borrowing results from cryptography, we construct a communication protocol to establish that every correlated equilibrium of a two-person game with rational payoffs can be achieved by means of computationally restricted unmediated communication. This result provides an example in game theory where limitations of computational abilities of players are helpful in solving implementation problems. More specifically, it is possible to construct mechanisms with the property that profitable deviations are too complicated to compute. [ABSTRACT FROM AUTHOR]

This paper offers a new approach to the study of economic problems usually modeled as games of incomplete information with discontinuous payoffs. Typically, the discontinuities arise from indeterminacies (ties) in the underlying problem. The point of view taken here is that the tie-breaking rules that resolve these indeterminacies should be viewed as part of the solution rather than part of the description of the model. A solution is therefore a tie-breaking rule together with strategies satisfying the usual best-response criterion. When information is incomplete, solutions need not exist; that is, there may be no tie-breaking rule that is compatible with the existence of strategy profiles satisfying the usual best-response criteria. It is shown that the introduction of incentive compatible communication (cheap talk) restores existence. [ABSTRACT FROM AUTHOR]

We propose a positive model of empirical science in which an analyst makes a report to an audience after observing some data. Agents in the audience may differ in their beliefs or objectives, and may therefore update or act differently following a given report. We contrast the proposed model with a classical model of statistics in which the report directly determines the payoff. We identify settings in which the predictions of the proposed model differ from those of the classical model, and seem to better match practice. [ABSTRACT FROM AUTHOR]

The paper discusses pure strategy dominance. The Bayesian approach to rationality in noncooperative normal-form games is gaining popularity. Its approach is based on the postulate that rational players should treat strategic and nonstrategic uncertainly similarly in standard decision theory. Using the approach, a player is called rational if he forms a subjective probability measure defined on the set of strategy combinations of other players. A result fundamental to the Bayesian approach to games is that a strategy maximizes expected utility for some subjective belief. No restrictions are imposed on the players beliefs hence, nothing is assumed about the players knowledge.

This paper provides two results that are useful in proving the existence of and characterizing separating equilibria in signaling games. These are games in which a privately informed agent takes an action which is observed by uninformed agents before they in turn choose actions. Depending on the equilibrium, the privately informed agent's action could reveal the private information, partially or completely. If the private information is completely revealed, the equilibrium is called separating. A key element in the analysis of separating equilibria is the examination of the implied incentive compatibility constraints (namely, that no type should have an incentive to take an action other than the prescribed equilibrium action). It is shown that these constraints together with one of two conditions imply differentiability of strategies when the set of possible types is an interval. The first condition, implied by sequentiality in many games, is an initial value condition on the informed agent's strategy. The second condition is a monotonicity condition which is similar to the familiar single crossing condition. The monotonicity condition turns out to be necessary and sufficient for there to be a strictly monotonic strategy satisfying the incentive compatibility constraints. As a direct consequence of these two results, the analysis of Milgrom and Roberts (1982) is considerably strengthened. [ABSTRACT FROM AUTHOR]

This paper studies the sequential equilibria of signaling games. It introduces a new solution concept, divine equilibrium, that refines the set of sequential equilibria by requiring that off-the-equilibrium-path beliefs satisfy an additional restriction. This restriction rules out implausible sequential equilibria in many examples. We show that divine equilibria exist by demonstrating that a sequential equilibrium that fails to be divine cannot be in a stable component. However, the stable component of signaling games is typically smaller than the set of divine equilibria. We demonstrate this fact through examples. We also present a characterization of the stable equilibria in generic signaling games. [ABSTRACT FROM AUTHOR]

This paper examines the nature of rational choice in strategic games. Although there are many reasons why an agent might select a Nash equilibrium strategy in a particular game, rationality alone does not require him to do so. A natural extension of widely accepted axioms for rational choice under uncertainty to strategic environments generates an alternative class of strategies, labelled "rationalizable." It is argued that no rationalizable strategy can be discarded on the basis of rationality alone, and that all rationally justifiable strategies are members of the rationalizable set. The properties of rationalizable strategies are studied, and refinements are considered. [ABSTRACT FROM AUTHOR]

This paper explores the fundamental problem of what can be inferred about the outcome of a noncooperative game, from the rationality of the players and from the information they possess. The answer is summarized in a solution concept called rationalizability. Strategy profiles that are rationalizable are not always Nash equilibria; conversely, the information in an extensive form game often allows certain "unreasonable" Nash equilibria to be excluded from the set of rationalizable profiles. A stronger form of rationalizability is appropriate if players are known to be not merely "rational" but also "cautious." [ABSTRACT FROM AUTHOR]

The main result of this paper is a generalization of the quasilinear mean of Nagumo, Kolmogorov, and de Finetti. We prove that the most general class of mean values, denoted by Mαϕ, satisfying Consistency with Certainty, Betweenness, Substitution- independence, Continuity, and Extension, is characterized by a continuous, nonvanishing weight function α and a continuous, strictly monotone value-like function ϕ. The quasilinear mean Mϕ, results whenever the weight function α is constant. Existence conditions and consistency conditions with first and higher degree stochastic dominance are derived and an extension of a well known inequality among quasilinear means, which is related to Pratt's condition for comparative risk aversion, is obtained. Under the interpretation of mean value as a certainty equivalent for a lottery, the Mαϕ mean gives rise to a generalization of the expected utility hypothesis which has testable implications, one of which is the resolution of the Allais "paradox." The Mαϕ mean can also be used to model the equally-distributed-equivalent or representative income corresponding to an income distribution. This generates a family of relative and absolute inequality measures and a related family of weighted utilitarian social welfare functions. [ABSTRACT FROM AUTHOR]

This paper presents an expected utility theory for state-dependent preferences. It proposes axioms that permit the joint derivation of subjective probabilities and utilities when the decision maker's preferences are not independent of the prevailing state of nature. In addition to the usual von Neumann-Morgenstern axioms, these axioms also include the requirement that the decision-maker's actual preferences are consistent with his preferences contingent on an hypothetical probability distribution over the states of nature. Two versions of the consistency axiom are introduced and their significance in the context of Bayesian decision theory is discussed. [ABSTRACT FROM AUTHOR]

A fundamental assumption in much of game theory and economics is that all the relevant information for determining the rational play of a game is contained in its structural description. Recent experimental studies of bargaining have demonstrated effects due to information not included in the classical models of games of complete information. The goal of the experiment reported here is to separate these effects into components that can be attributed to the possession of specific information by specific bargainers, and to assess the extent to which the observed behavior can be characterized as equilibrium behavior. The results of the experiment permit us to identify such component effects, in equilibrium, including effects that depend on whether certain information is common knowledge or not. The paper closes with some speculation on the causes of these effects. [ABSTRACT FROM AUTHOR]

The paper discusses a rather robust resolution of the prisoner's dilemma in the language of an arms race. A class of good strategies is given where the solution is Pareto optimal, stable, and a Nash equilibrium. It demonstrates how good strategies with optimal solutions might bifurcate into strategies with the worst solutions. The dynamics is introduced in a rather abstract setting for non-cooperative games. A way of taking the past into account, with limited memory, is axiomatized in the decision making.

The article develops an adaptive learning decision rule for multiperiod problems in which the objective function is quadratic and the econometric model consists of a system of linear difference equations. Certainty equivalence or linear decision rules were discussed that deals with uncertainty in the parameters by which the model is treated as independent, identically distributed random variables in each period which yields the sequential updating rules. The rule developed in the paper incorporates the effect of policy variables on the learning which is expected to occure throughout the remainder of the planning period. The decision rule yields to economic analysis in terms of the stock and value of information.

The preliminary test estimator lot predictor) is studied in the context of linear normal regression models. The estimator (or predictor) obtained after the preliminary test is known to be inadmissible with respect to squared error loss under certain assumptions. Nevertheless, it is still widely used in practical application of regression analysis, particularly in econometrics. The object of the present paper is to tabulate the optimal significance points in the preliminary test for practical use. The optimality is based on the minimax regret principle. It is shown that if, as is usual, we take the significance point equal to the 5 per cent point or 10 per cent point, the risk is extremely large for some parameter value. The optimal significance point of the preliminary t test decreases slightly as the degrees of freedom increase, but it is nearly constant; i.e., it lies between 1.370 to 1.380 if d.f. is greater than 6. [ABSTRACT FROM AUTHOR]

The von Neumann-Morgenstern theory of games does not yield determinate solutions (corresponding to unique payoff vectors) for two-person variable-sum games and for n-person games. The present paper outlines a general theory of rational behavior in game situations which does yield determinate solutions for all classes of games. The theory is based on two classes of rationality postulates: those defining rational behavior as such, and those defining rational expectations concerning the other players' behavior. It is argued that this new approach greatly increases the possibilities for the application of game theory in economics and the other social sciences. [ABSTRACT FROM AUTHOR]

Treatment for depression is complex, requiring decisions that may involve trade‐offs between exploiting treatments with the highest expected value and experimenting with treatments with higher possible payoffs. Using patient claims data, we show that among skilled doctors, using a broader portfolio of drugs predicts better patient outcomes, except in cases where doctors' decisions violate loose professional guidelines. We introduce a behavioral model of decision making guided by our empirical observations. The model's novel feature is that the trade‐off between exploitation and experimentation depends on the doctor's diagnostic skill. The model predicts that higher diagnostic skill leads to greater diversity in drug choice and better matching of drugs to patients even among doctors with the same initial beliefs regarding drug effectiveness. Consistent with the finding that guideline violations predict poorer patient outcomes, simulations of the model suggest that increasing the number of possible drug choices can lower performance. [ABSTRACT FROM AUTHOR]

We find that Epstein's (2010) Ellsberg-style thought experiments pose, contrary to his claims, no paradox or difficulty for the smooth ambiguity model of decision making under uncertainty developed by Klibanoff, Marinacci, and Mukerji (2005). Not only are the thought experiments naturally handled by the smooth ambiguity model, but our reanalysis shows that they highlight some of its strengths compared to models such as the maxmin expected utility model (Gilboa and Schmeidler (1989)). In particular, these examples pose no challenge to the model's foundations-interpretation of the model as affording a separation of ambiguity and ambiguity attitude or the potential for calibrating ambiguity attitude in the model. [ABSTRACT FROM AUTHOR]