Temporal Resolution of Uncertainty and Dynamic Choice Theory

We consider dynamic choice behavior under conditions of uncertainty, with emphasis on the timing of the resolution of uncertainty. Choice behavior in which an individual distinguishes between lotteries based on the times at which their uncertainty resolves is axiomatized and represented, thus the result is choice behavior which cannot be represented by a single cardinal utility function on the vector of payoffs. Both descriptive and normative treatments of the problem are given and are shown to be equivalent. Various specializations are provided, including an extension of "separable" utility and representation by a single cardinal utility function. CONSIDER THE FOLLOWING idealization of a dynamic choice problem with uncertainty. At each in a finite, discrete sequence of times t = 0, 1, . . ., T, an individual must choose an action d,. His choice is constrained by what we temporarily call the state at time t, xt. Then some random event takes place, determining an immediate payoff zt to the individual and the next state xt+l. The probability distribution of the pair (zt, xt+l) is determined by the action dt. The standard approach in analyzing this problem, which we will call the payoff vector approach, assumes that the individual's choice behavior is representable as follows: He has a von Neumann-Morgenstern utility function U defined on the vector of payoffs (z0, z1, . . ., ZT). Each strategy (which is a contingent plan for choosing actions given states) induces a probability distribution on the vector of payoffs. So the individual's choice of action is that specified by any optimal strategy, any strategy which maximizes the expectation of utility among all strategies (assuming sufficient conditions so that an optimal strategy exists). This paper presents an axiomatic treatment of the dynamic choice problem which is more general than the payoff vector approach, but which still permits tractable analysis. The fundamental difference between our treatment and the payoff vector approach lies in our treatment of the temporal resolution of uncertainty: In our models, uncertainty is "dated" by the time of its resolution, and the individual regards uncertainties resolving at different times as being different. For example, consider a situation in which a fair coin is to be flipped. If it comes up heads, the payoff vector will be (zo, z1) = (5, 10); if it is tails, the vector will be (5, 0). Because z0 = 5 in either case, the coin flip can take place at either time 0 or time 1. It will not matter when the flip occurs to someone who has cardinal utility on the vector of payoffs. But an individual can obey our axioms and prefer either one to the other. One justification for our approach is the well known "timeless-temporal" or "joint time-risk" feature of some models (usually models which are not "complete"). For example, preferences on income streams which are induced from primitive preferences on consumption streams in general depend upon when the