Picking Sequences and Monotonicity in Weighted Fair Division

We study the problem of fairly allocating indivisible items to agents with different entitlements, which captures, for example, the distribution of ministries among political parties in a coalition government. Our focus is on picking sequences derived from common apportionment methods, including five traditional divisor methods and the quota method. We paint a complete picture of these methods in relation to known envy-freeness and proportionality relaxations for indivisible items as well as monotonicity properties with respect to the resource, population, and weights. In addition, we provide characterizations of picking sequences satisfying each of the fairness notions, and show that the well-studied maximum Nash welfare solution fails resource- and population-monotonicity even in the unweighted setting. Our results serve as an argument in favor of using picking sequences in weighted fair division problems.

Population Monotonicity in Public Good Economies with Single Dipped Preferences

<p>We study public good economies with variable population. We consider the problem of locating a single public good along a segment when agents have single dipped preferences. We analyze population monotonicity along with the standard properties Pareto efficiency, continuity and no-veto power. We show that there is no rule satisfying these properties together.</p>

The Role of Replication-Invariance: Two Answers Concerning the Problem of Fair Division When Preferences Are Single-Peaked

We consider the problem of allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. A rule that has played a central role in the previous analysis of the problem is the so-called uniform rule. Thomson (1995b) proved that the uniform rule is the only rule satisfying Pareto optimality, no-envy, one-sided population-monotonicity, and replication-invariance. Replacing one-sided population-monotonicity by one-sided replacement-domination yields another characterization of the uniform rule (Thomson, 1997a). Until now, the independence of replication-invariance from the other properties in these characterizations was an open problem. In this note we prove this independence by means of a single example.

THE DUTTA-RAY SOLUTION ON THE CLASS OF CONVEX GAMES: A GENERALIZATION AND MONOTONICITY PROPERTIES

This paper considers generalized Lorenz-maximal solutions in the core of a convex TU-game and demonstrates that such solutions satisfy coalitional monotonicity and population monotonicity.