Fair Allocation of Indivisible Items: Consequences of Correlated Preferences

Fair allocation of indivisible items among multiple individuals is a fundamental problem in collective choice. We consider the problem of allocating two of four indivisible items to each of two players, where the only known preference information is each player's strict ranking of the items. How is the rank correlation of preferences, as measured by Kendall Tau, related to properties that facilitate fair allocation such as the availability of envy-free, Pareto-optimal, maximin, and max BordaSum allocations? We also examine the relationship between the ranked correlation and features of Fallback Bargaining such as the depth of agreement and the probability of a (two-way) tie. We further categorize the players into two types, risk-averse and risk-acceptant, and analyze how player type affects various fair division properties. Our results suggest that increasing similarity of preferences tends to increase the number of Pareto-optimal and maximin allocations but to decrease the number of envy-free allocations. Higher rank correlation also makes Fallback Bargaining less compelling. Understanding how similarity of preference rankings influences the trade-offs among allocation properties gives new insight into the difficulties of fair allocation of indivisible items.