The uniform distributions puzzle.

This note deepens a problem proposed and discussed by Kadane and O'Hagan (JASA, 1995). Kadane and O'Hagan discuss the existence of a uniform probability on the set of natural numbers (they provide a su_cient and necessary condition for the existence of such a uniform probability). I question the practical relevance of their solution. I show that a purely _nitely additive measure on the set of natural numbers cannot be constructed (its existence needs some form of the Axiom of Choice, the prototype of a nonconstructive axiom).

Ordering infinite utility streams comes at the cost of a non-Ramsey set.

The existence of a Paretian and finitely anonymous ordering in the set of infnite utility streams implies the existence of a non-Ramsey set (a nonconstructive object whose existence requires the axiom of choice). Therefore, each Paretian and finitely anonymous quasi-ordering either is incomplete or does not have an explicit description. Hence, the possibility results of Svensson (1980) and of Bossert, Sprumont, and Suzumura (2006) do require the axiom of choice.Intergenerational justice; Pareto; Multi-period social choice; Axiom of choice; Constructivism;

The uniform distributions puzzle

Ordering infinite utility streams: maximal anonymity

Purely finitely additive measures are non-constructible objects

The existence of a purely finitely additive measure cannot be proved in Zermelo-Frankel set theory if the use of the Axiom of Choice is disallowed.Finitely additive probabilities; Charges; Axiom of choice; Constructivism.

Ordering infinite utility streams comes at the cost of a non-Ramsey set

The existence of a Paretian and finitely anonymous ordering in the set of infinite utility streams implies the existence of a non-Ramsey set (a nonconstructive object whose existence requires the axiom of choice). Therefore, each Paretian and finitely anonymous quasi-ordering either is incomplete or does not have an explicit description. Hence, the possibility results of Svensson (1980) and of Bossert, Sprumont, and Suzumura (2006) do require the axiom of choice.Intergenerational justice; Pareto; Multi-period social choice; Axiom of choice; Constructivism.

Lorenz comparisons of nine rules for the adjudication of conflicting claims.

Consider the following nine rules for adjudicating conflicting claims: the proportional, constrained equal awards, constrained equal losses, Talmud, Piniles’, constrained egalitarian, adjusted proportional, random arrival, and minimal overlap rules. For each pair of rules in this list, we examine whether or not the two rules are Lorenz comparable. We allow the comparison to depend upon whether the amount to divide is larger or smaller than the half-sum of claims. In addition, we provide Lorenz-based characterizations of the constrained equal awards, constrained equal losses, Talmud, Piniles’, constrained egalitarian, and minimal overlap rules.Rules; Claims problem; Bankruptcy; Constrained equal awards rule; Constrained equal losses rule; Talmud rule, Piniles’ rule; Random arrival rule; Minimal overlap rule;

The minimal dominant set is a non-empty core-extension.

A set of outcomes for a TU-game in characteristic function form is dominant if it is, with respect to an outsider-independent dominance relation, accessible (or admissible)and closed. This outsider-independent dominance relation is restrictive in the sense that a deviating coalition cannot determine the payoffs of those coalitions that are not involved in the deviation. The minimal (for inclusion) dominant set is non-empty and for a game with a non-empty coalition structure core, the minimal dominant set returns this core.

Value without absolute convergence

We address how the value of risky options should be assessed in the case where the sum of the probability-weighted payoffs is not absolutely convergent and thus dependent on the order in which the terms are summed (e.g., as in the Pasadena Paradox). We develop and partially defend a proposal according to which options should be evaluated on the basis of agreement among admissible (e.g., convex and quasi-symmetric) covering sequences of the constituents of value (i.e., probabilities and payoffs).

The coalition structure core is accessible.

We prove the existence of an upper bound for the number of blockings required to get from one imputation to another provided that accessibility holds. The bound depends only on the number of players in the TU game considered. For the class of games with non-empty cores this means that the core can be reached via a bounded sequence of blockings. Primitive recursive algorithms are provided to locate accessibility paths.Structure;

Lorenz comparisons of nine rules for the adjudication of conflicting claims

Consider the following nine rules for adjudicating conflicting claims: the proportional, constrained equal awards, constrained equal losses, Talmud, Piniles’, constrained egalitarian, adjusted proportional, random arrival, and minimal overlap rules. For each pair of rules in this list, we examine whether or not the two rules are Lorenz comparable. We allow the comparison to depend upon whether the amount to divide is larger or smaller than the half-sum of claims. In addition, we provide Lorenz-based characterizations of the constrained equal awards, constrained equal losses, Talmud, Piniles’, constrained egalitarian, and minimal overlap rules.Claims problem, Bankruptcy, Taxation, Lorenz dominance, Progressivity, Proportional rule, Constrained equal awards rule, Constrained equal losses rule, Talmud rule, Piniles’ rule, Constrained egalitarian rule, Adjusted proportional rule, Random arrival rule, Minimal overlap rule