Axioms for an individual's preferences over time, taken from the present perspective, usually assume that the individual will live, or expects to live, throughout a given horizon span. This paper offers an axiomatization that explicitly recognizes the uncertainty of an individual's lifetime. It divides a horizon span into n periods and assumes that if death is not immediate then it will occur at the end of one of the periods. The theory is based on an unconditional preference relation over potential future consumption streams that accounts for uncertain lifetime, along with a conditional preference order that is based on the hypothesis that death will occur at the end of period i. There is a conditional order for each i from 1 to n. The utility representation involves an order-preserving utility function for each of the n conditional orders such that one potential consumption stream is unconditionally preferred to another if, and only if, the sum of the conditional utilities for the first stream exceeds the sum of the conditional utilities for the second. It is argued that the theory seems fairly reasonable only if probability of survival does not depend significantly on past consumption.

[A one-way expected utility representation has the expected utility of one probability measure greater than the expected utility of another probability measure whenever the first is preferred to the second. It requires preferences to be acyclic but not necessarily transitive, and does not require indifference to be transitive. Preference axioms which are sufficient for one-way expected utility for sets of simple probability measures have been presented before (see [8]). This paper uses additional axioms to extend the one-way representation to sets of discrete and more general probability measures.]

An axiomatization is presented for a Savage-type conditional subjective expected utility model. The axioms consist of extensions of the Herstein-Milnor [11] axioms for measurable utility, a generalization of an averaging condition in Bolker [4], and several structural conditions. The structural conditions are examined in some detail, and examples are given to show what happens to the numerical model when they do not hold. The numerical model expresses the utility of an act (or mixed act), given an event, as a weighted sum of the utilities of the act given events that partition the initial event, the weights being personal probabilities for the partition events conditioned on the initial event. The theory is compared to Savage's theory [18] and to a version of the theory of Luce and Krantz [14] for conditional expected utility. 1. DECISION UNDER UNCERTAINTY THE PREDICAMENT BETWEEN mathematical tractability and situational reality that is characteristic of mathematical models in the behavioral sciences is epitomized in the axiomatizations of subjective expected utility models. These axiomatizations include structural conditions that facilitate the derivation of the desired numerical representations for preference. Unfortunately, actual situations of decision making under uncertainty often fail to exhibit the structural properties that occur in the axiomatizations. Thus there is real concern about the applicability of such models to realistic decision situations. As might be expected, decision theorists have attempted to alleviate this predicament by weakening the structural conditions while maintaining the ability to derive the desired model from the axioms. An early move in this direction was made by Suppes [21] in his alternative to Savage's axiomatization [18]. The more recent axiomatizations of Bolker [3 and 4], based on Jeffrey's decision model [12], and of Pfanzagl [15 and 16] and Luce and Krantz [14], continue this line of research. The present paper is a further effort in this direction. To understand its approach we shall first review briefly some other theories. The formulation of the paper is set in the context of Savage's states-of-the-world approach to decision under uncertainty, and I shall therefore focus the discussion within this context. We suppose that the decision maker is to select an alternative, or act, from a set of acts. The consequence of his decision will depend not only on the selected act but also on which state in a set of exclusive and exhaustive states of the world obtains. The state that obtains is not known beforehand by the decision maker and does not depend on the selected act.3

A summation social choice function is a social choice function whose choice sets are determinable from maximum sums of utilities that preserve individual preference. Assuming the set of alternatives is finite and individual preferences are irreflexive and transitive, a unanimity-type condition is shown to be necessary and sufficient for a social choice function to be a summation social choice function. The effects of conditions of voter independence, anonymity, and neutrality are noted.

[A general definition of majority decision in terms of a hierarchy of voting councils has been given by Murakami [3, 4]. The present article establishes a set of necessary and sufficient conditions for Murakami's majority decision or representative system in terms of properties of a group decision function for two alternatives. One corollary of the general theorem is Murakami's conjecture, which says that if a group decision function is dual, strongly monotonic, and nondictatorial, then it is a representative system.]

[The results of A. K. Sen and P. K. Pattanaik on sufficient conditions for the transitivity of simple majorities are extended to the case where it is not assumed that the individual's indifference relations are transitive.]