Trading probabilities along cycles

Abstract

Consider the problem of allocating indivisible objects when agents are endowed with fractional amounts and rules can assign lotteries. We study a natural generalization (to the probabilistic domain) of Gale’s Top Trading Cycles. The latter features an algorithm wherein agents trade objects along a cycle—in our new family of rules, agents now trade probabilities of objects along a cycle. We ask if the attractive properties, namely efficiency, individual rationality, and strategy-proofness extended in the stochastic dominance sense, carry over to the Trading-Probabilities-Along-Cycles (TPAC) rules. All of these rules are sd-efficient. We characterize separately the subclass of TPAC rules satisfying the sd-endowment lower bound and sd-strategy-proofness. Regarding fairness, we follow in spirit to the no-envy in net trade condition of Schmeidler and Vind (1972), where the set of allocations satisfying the property essentially coincides with the set of competitive equilibria, and augment the notion appropriately for our environment. We further generalize the TPAC family while extending results on sd-efficiency and the sd-endowment lower bound, and provide sufficient conditions on parameters for the rules to arbitrarily closely satisfy the sd-no-envy in net trade.