Minimal Envy and Popular Matchings

We study ex-post fairness in the object allocation problem where objects are valuable and commonly owned. A matching is fair from individual perspective if it has only inevitable envy towards agents who received most preferred objects -- minimal envy matching. A matching is fair from social perspective if it is supported by majority against any other matching -- popular matching. Surprisingly, the two perspectives give the same outcome: when a popular matching exists it is equivalent to a minimal envy matching. We show the equivalence between global and local popularity: a matching is popular if and only if there does not exist a group of size up to 3 agents that decides to exchange their objects by majority, keeping the remaining matching fixed. We algorithmically show that an arbitrary matching is path-connected to a popular matching where along the path groups of up to 3 agents exchange their objects by majority. A market where random groups exchange objects by majority converges to a popular matching given such matching exists. When popular matching might not exist we define most popular matching as a matching that is popular among the largest subset of agents. We show that each minimal envy matching is a most popular matching and propose a polynomial-time algorithm to find them

Fair and consistent prize allocation in competitions

Given the final ranking of a competition, how should the total prize endowment be allocated among the competitors? We study consistent prize allocation rules satisfying elementary solidarity and fairness principles. In particular, we axiomatically characterize two families of rules satisfying anonymity, order preservation, and endowment monotonicity, which all fall between the Equal Division rule and the Winner-Takes-All rule. Specific characterizations of rules and subfamilies are directly obtained.Comment: 34 page

Fair and consistent prize allocation in competitions

Given the ranking of competitors, how should the prize endowment be allocated? This paper introduces and axiomatically studies the prize allocation problem. We focus on consistent prize allocation rules satisfying elementary solidarity and fairness principles. In particular, we derive several families of rules satisfying anonymity, order preservation, and endowment monotonicity, which all fall between the equal division rule and the winner-takes-all rule. Our results may help organizers to select the most suitable prize allocation rule for rank-order competitions

Fair and consistent prize allocation in competitions

Given the ranking of competitors, how should the prize endowment be allocated? This paper introduces and axiomatically studies the prize allocation problem. We focus on consistent prize allocation rules satisfying elementary solidarity and fairness principles. In particular, we derive several families of rules satisfying anonymity, order preservation, and endowment monotonicity, which all fall between the equal division rule and the winner-takes-all rule. Our results may help organizers to select the most suitable prize allocation rule for rank-order competitions