:We address the question, in decision theory, of how the value of risky options should be assessed when they have no finite standard expected value, that is, where the sum of the probability-weighted payoffs is infinite or not well defined. We endorse, combine and extend the proposal of Easwaran to evaluate options on the basis of their weak expected value, and the proposal of Colyvan to rank options on the basis of their relative expected value.

We show that it is not possible to extend the ranking of one-stage lotteries based on their weak-expectation to a reflexive and transitive relation on the collection of one- and two-stage lotteries that satisfies two basic axioms, the minimal value axiom and the reduction axiom. We propose an extension that satisfies only the first axiom. This ranking takes payoffs, their probabilities, and the tree structure into account.

We address the question of how finitely additive moral value theories (such as utilitarianism) should rank worlds when there are an infinite number of locations of value (people, times, etc.). In the finite case, finitely additive theories satisfy both Weak Pareto and a strong anonymity condition. In the infinite case, however, these two conditions are incompatible, and thus a question arises as to which of these two conditions should be rejected. In a recent contribution, Hamkins and Montero (2000) have argued (...) in favor of an anonymity-like isomorphism principle and against Weak Pareto. After casting doubt on their criticism of Weak Pareto, we show how it, in combination with certain other plausible principles, generates a plausible and fairly strong principle for the infinite case. We further show that where locations are the same in all worlds, but have no natural order, this principle turns out to be equivalent to a strengthening of a principle defended by Vallentyne and Kagan (1997), and also to a weakened version of the catching-up criterion developed by Atsumi (1965) and by von Weizsäcker (1965). Footnotes1 For valuable comments, we would like to thank Marc Fleurbaey, Bart Capéau, Joel Hamkins, Barbara Montero, Tim Mulgan, and two anonymous referees for this journal. (shrink)

The existence of a transitive, complete, and weakly independent relation on the full set of gambles implies the existence of a non-Ramsey set. Therefore, each transitive and weakly independent relation on the set of gambles either is incomplete or does not have an explicit description. Whatever tools decision theory makes available, there will always be decision problems where these tools fail us. In this sense, decision theory remains incomplete.

One result of this note is about the nonconstructivity of countably infinite lotteries: even if we impose very weak conditions on the assignment of probabilities to subsets of natural numbers we cannot prove the existence of such assignments constructively, i.e., without something such as the axiom of choice. This is a corollary to a more general theorem about large-small filters, a concept that extends the concept of free ultrafilters. The main theorem is that proving the existence of large-small filters requires (...) a nonconstructive axiom like AC. (shrink)

The note addresses the problem of how utilitarianism and other finitely additive theories of value should evaluate infinitely long utility streams. We use the axiomatic approach and show that finite anonymity does not apply in an infinite framework. A stronger anonymity demand (fixed step anonymity) is proposed and motivated. Finally, we construct an ordering criterion that combines fixed step anonymity and strong monotonicity.

One of the main results in topological social choice states the non-existence of a continuous, anonymous, and unanimous aggregation rule on spheres. This note provides a proof based upon simple methods such as integration.

Using post-war election results for the Belgian House of Representatives, the power relations among political parties are analysed by calculating power- and satisfaction indices. Also, a participation index has been constructed to calculate the probability that a party will join a government coalition.Since the election of 1981 the traditional parties join the same Banzhof power and participation probability. The other parties represent no power and participation valueat all.