The authors discuss the unity between the two standard approaches to noncooperative solution concepts for games. The decision-theoretic approach starts from the assumption that the rationality of the players is common knowledge. This leads to the notion of correlated rationalizability. It is shown that correlated rationalizability is e quivalent to a posteriori equilibrium-a refinement of subjective corr elated equilibrium. Hence a decision-theoretic justification for the equilibrium approach to game theory is provided. An analogous equival ence result is proved between independent rationalizability, which is the appropriate concept if each player believes that the others act independently, and conditionally independent a posteriori equilibrium. Copyright 1987 by The Econometric Society.
In a multiple-regression model the residual variance is an unknown function of the explanatory variables, and estimated by nearest-neighbor nonparametric regression. The resulting weighted least-squares estimator of the regression coefficients is shown to be adaptive, in the sense of having the same asymptotic distribution, to first order, as estimators based on knowledge of the actual variance function or a finite parameterization of it. A similar result was established by R. J. Carrol l (1982) using kernel estimation and under substantially more restrictive conditions on the data generating process than ours. Extensions to various other models seem to be possible. Copyright 1987 by The Econometric Society.
The analogy principle proposes that population parameters be estimated by sample statistics which make known properties of the population hold as closely as possible in the sample. Applications of the analogy principle are ubiquitous. Nevertheless, estimation theory has not been studied from a consistent analog perspective. This paper makes a start.
Buyer's preferences over auctions depend on their measure of absolute risk aversion. If it is constant and they have independent private values, they are indifferent between a first-price auction (FPA), a second-price auction (SPA), and a first-price auction in which the number of bidders is revealed before bids are taken (FPA- R). If they have decreasing absolute risk aversion, they prefer the SPA to the FPA-R to the FPA. Their preference for the SPA is diminished to the extent that their values are affiliated. Affiliation also causes them to prefer the number of bidders to be revealed, whereas the seller then prefers to keep it secret. Copyright 1987 by The Econometric Society.
When a decision rule is implemented using a Bayesian incentive compatible mechanism in which the messages are publically obser vable, the players' information is augmented by their observation of each others' strategies. In this paper the authors study the set of Bayesian implementable decision rules which have the further property that the information c onveyed in the process of thier implementation does not invalidate the optimality of the players' strategies. Such rules are called posterior implementable. The authors concentrate on a two-person problem with two possible decisions and, for this problem, they obtain a complete characterization of the set of posterior implementable decision rules. Copyright 1987 by The Econometric Society.
This paper offers an interpretive comparison of the Arrow-Pratt and Ross characterizations of comparative risk aversion for expected utility maximizers. The tools used in this comparison are then applied to obtain a strengthening of the Ross cha racterization. This strengthened result is in turn extended to the ca se of general, smooth, nonexpected utility preferences over probabili ty distributions. Copyright 1987 by The Econometric Society.
Empirically estimated flexible functional forms frequently fail to satisfy the appropriate theoretical curvature conditions. Lau and Gallant and Golub have worked out methods for imposing the appropriate curvature conditions locally, but those local techniques frequently fail to yield satisfactory results. We develop two methods for imposing curvature conditions globally in the context of cost function estimation. The first method adopts Lau's technique to a generalization of a functional form first proposed by McFadden. Using this Generalized McFadden functional form, it turns out that imposing the appropriate curvature conditions at one data point imposes the conditions globally. The second method adopts a technique used by McFadden and Barnett, which is based on the fact that a non-negative sum of concave functions will be concave. Our various suggested techniques are illustrated using the U.S. Manufacturing data utilized by Berndt and Khaled
THIS PAPER DISCUSSES the problem of aggregating probability judgements and shows that Arrow-type paradoxes arise in this context, just as in the context of aggregating preferences. The need to aggregate probability judgements may arise in several different sorts of situation. Two examples which concern decision making under risk are: (i) when an individual, prior to making a decision, consults a number of experts who differ in their assessments of the probabilities of alternative states of nature; (ii) when the individuals constituting a society have to make a joint decision on the basis of identical utility functions but, again, differing assessments of the probabilities of alternative states of nature.2 We illustrate (i) above by means of a simple example: An individual consults two legal experts, who give him their probability judgements concerning the success or failure (there are just two possible outcomes) of proposed litigation. Scenario 1: The individual knows that litigation will succeed if judge J presides and barrister B defends, but will not succeed otherwise. From the experts' probability judgements the individual is able to infer that one expert has received a message that judge J will preside, and the other a message that barrister B will defend. (Either message, or both, may be false.) Given these messages, the individual updates in Bayesian fashion the prior probabilities he assigns to success and failure. Scenario 2: Success or failure depends on interpretation of the law. Pooling the information on which the experts' probability judgements are based is, we suppose, impracticable. (Perhaps they meet to discuss the case and cannot agree.) Aggregation of probability judgements in situations such as Scenario 1 (and according to axioms which we discuss in the next section) is clearly a wrong procedure which may lead to totally erroneous results. In situations such as Scenario 2 such aggregation may however have a useful role. Formulae for aggregating probability judgements have been discussed by, among others, Wagner (1982), Bordley (1982), Genest, Weerahandi, and Zidek (1984), and Fishburn and Rubinstein (1984). In this paper we generalize some results contained in the latter two papers, before going on to consider paradoxes. An early, Arrow-type impossibility result, based on rather strong assumptions, is due to Dalkey (1972). In Section 2 notation and axioms are introduced, and some results given. In Section 3 three propositions are stated and proved.