An Equivalence Between Fair Division and Wagering Mechanisms

We draw a surprising and direct mathematical equivalence between the class of fair division mechanisms, designed to allocate divisible goods without money, and the class of weakly budget-balanced wagering mechanisms, designed to elicit probabilities. Although this correspondence between fair division and wagering has applications in both settings, we focus on its implications for the design of incentive-compatible fair division mechanisms. In particular, we show that applying the correspondence to competitive scoring rules, a prominent class of wagering mechanisms based on proper scoring rules, yields the first incentive-compatible fair division mechanism that is both fair (proportional and envy-free) and responsive to agent preferences. Moreover, for two agents, we show that competitive scoring rules characterize the whole class of nonwasteful and incentive-compatible fair division mechanisms, subject to mild technical conditions. As one of several consequences, this allows us to resolve an open question about the best possible approximation to optimal utilitarian welfare that can be achieved by any incentive-compatible mechanism. Finally, because the equivalence greatly expands the set of known incentive-compatible fair division mechanisms, we conclude with an evaluation of this entire set, comparing the mechanisms' axiomatic properties and examining their welfare performance in simulation.