Knowledge that Transforms

IN THIS PAPER we investigate the properties of the winning bid in a sealed bid tender auction where each player has private information. We find that it is possible for the winning bid to converge in probability to the true value of the object at auction, even though no bidder knows the true value. Necessary and sufficient conditions for this phenomenon are derived, extending and generalizing certain of Wilson's results [3]. We study an auction in which a seller offers to sell at the highest bid an item of unknown value V. The kth bidder receives a private signal Sk (for k = 1, 2,.. .) and submits a bid without knowledge of the other signals. A finitely additive probability measure P reflects the bidders' unanimous beliefs about V and the signals. Conditional on V, the signals are independent and identically distributed. The signals take their values in some space &'. With n bidders, a bidding strategy for k is a function Pnk: 9' -> R. k's strategy specifies that upon receiving the signal Sk, he shall bid Pnk(Sk).2 Thus the winning

AbstractIt is well known that the returns on various betting opportunities at a racetrack are determined by a competitive bidding of the bettors in a natural environment of their decision making. In this paper, two simple bets of unknown but identical winning probabilities are identified. An analysis of 1,089 observations shows the data are consistent with the hypothesis that both bets are identically priced, an implication of an efficient speculative market.

Finite horizon sequential decision problems with a temporal von NeumannMorgenstern criterion are analyzed. This criterion, as developed in [7], is a generalization of von Neumann-Morgenstern (expected) utility of the vector of rewards, wherein an individual's preferences concerning the timing of the resolution of uncertainty are taken into account. The preference theory underlying this criterion is reviewed and then extended in natural fashion to yield preferences for strategies in sequential decision problems. The main result is that value functions for sequential decision problems can be defined by a dynamic programming recursion using the functions which represent the original preferences, and these value functions represent the preferences defined on strategies. This permits citation of standard results from the dynamic programming literature, concerning the existence of (memoryless) strategies which are optimal with respect to the given preference relation.

THIS PAPER IS a preliminary investigation of dynamics under uncertainty. We attempt to develop a general approach to the continuous time stochastic processes that arise in dynamic economics from the maximizing behavior of agents. The analysis builds on recent results of Bismut [2, 3] concerning the characterization of the extrema of stochastic variational problems over a finite horizon and on our own investigations [6, 7, 20, 21] of the stability properties of the equations of dynamic economics.2 We consider a class of discounted infinite horizon maximum problems. While it is convenient to pose the basic economic problem as a stochastic control problem, to obtain the full benefit of Bismut's elegant characterization of a maximizing process it is convenient to transform this problem into an equivalent stochastic variational problem along the lines indicated by Rockafellar [27] in the deterministic case and generalized by Bismut [2] to the stochastic case. Within this framework we show that the idea of a competitive path introduced in the continuous time deterministic case in [21] generalizes in a natural way in the case of uncertainty to a competitive process. We show, under a concavity assumption on the basic integrand of the problem, that a competitive process which satisfies a transversality condition is optimal under a discounted catching up criterion (Section 2). In Section 3 we examine the sample path properties of a competitive process. If for almost every realization of a competitive process the associated dual price process generates a path of subgradients for the value function, we call the process McKenzie competitive, since it was McKenzie [22] who first recognized the importance of this property in the deterministic case. We show that two McKenzie competitive processes starting from distinct nonrandom initial conditions converge almost surely if the processes are bounded almost surely and if a certain curvature condition is satisfied by the Hamiltonian of the system. The earlier convergence result extensively studied in the deterministic case thus continues to hold in the stochastic case. The problem of finding sufficient conditions for the existence of a McKenzie competitive process remains an open problem. Section 4 examines the long-run behavior of the probability measure associated with a competitive process. We give conditions under which a McKenzie competitive process is a Markov process with an invariant probability measure and

[In this paper we consider the lognormal, gamma, beta, and Singh-Maddala functions as descriptive models for the distribution of family income for 1960 and 1969 through 1975. Least squares and two efficient estimation techniques are used to estimate the unknown parameters. Alternative functional forms for comparable estimation techniques can then be contrasted and estimation techniques for a given functional form can be compared. We note that estimates of population characteristics depend upon the functional form and estimation technique selected.]

2Bayesian paradigm. Following it, probability assessments and utility functions are defined independently. It seems reasonable to suggest that groups adopt a similar approach.3 That is, the group should deal separately with the two areas of potential disagreement. By some prescribed procedure, the group aggregates the individual probability assessments into a group probability assessment. Similarly, a group utility function is constructed by aggregating individual utility functions. The group probability assessment and utility function are multiplied in the usual manner to give expected utilities. The action offering the highest expected utility is chosen.4 Most real world decision processes, we recognize, make less than conscientious attempts to separate beliefs from values, either in debate or at time of decision.5