The Division Problem of Chances
Rasoul Ramezanian
TL;DR
The Division Problem of Chances explores fair division of recurring allocations when agents value lotteries over outcomes rather than single items. It introduces Uniform Rules for Dividing Chances (URC) as a two-phase mechanism that exhausts excess-demand objects via the uniform rule and then completes feasibility through a coordinated Phase 2, proving URC achieves strategy-proofness, efficiency, replacement monotonicity, in-betweenness, anonymity, and envy-freeness. A central result shows that any mechanism satisfying these properties is welfare-equivalent to URC, i.e., yields the same closeness to agents’ ideal lotteries, up to welfare considerations. The paper further contrasts URC with Serial Dictatorship (SDC) and Proportional Division of Chances (PDC) to illustrate the logical landscape of axioms, and discusses extensions to two-sided markets and Markov-chain preferences, highlighting both theoretical and practical implications for fair division in repeated matching contexts.
Abstract
In frequently repeated matching scenarios, individuals may require diversification in their choices. Therefore, when faced with a set of potential outcomes, each individual may have an ideal lottery over outcomes that represents their preferred option. This suggests that, as people seek variety, their favorite choice is not a particular outcome, but rather a lottery over them as their peak for their preferences. We explore matching problems in situations where agents' preferences are represented by ideal lotteries. Our focus lies in addressing the challenge of dividing chances in matching, where agents express their preferences over a set of objects through ideal lotteries that reflect their single-peaked preferences. We discuss properties such as strategy proofness, replacement monotonicity, (Pareto) efficiency, in-betweenness, non-bossiness, envy-freeness, and anonymity in the context of dividing chances, and propose a class of mechanisms called URC mechanisms that satisfy these properties. Subsequently, we prove that if a mechanism for dividing chances is strategy proof, (Pareto) efficient, replacement monotonic, in-between, non-bossy, and anonymous (or envy free), then it is equivalent in terms of welfare to a URC mechanism.