Probabilistic Assignment: A Two-fold Axiomatic Approach

We study the problem of allocating objects by means of probabilistic mechanisms. Each agent has strict preferences over objects and ex post receives exactly one object. A standard approach in the literature is to extend agents' preferences over objects to preferences over lotteries de ned on those objects, using the rst-order stochastic dominance criterion, or the sd-extension. In a departure from this practice, we work with a general mapping, called an extension, from preferences over objects to preferences over lotteries. Our objective is to connect the extension operator theory in Cho (2012) with probabilistic assignment while maintaining an axiomatic perspective on each. Our main results concern axioms associated with the sd-, dl-, and ul-extensions. First, sd-e ciency, dl-e ciency, and ul-e ciency are equivalent (axioms are pre xed by the extensions they are associated with). Also, for each e ∈ {sd, dl, ul}, e-adjacent strategy-proofness is equivalent to e-strategy-proofness. For each e ∈ {sd, dl, ul}, a rule is e-strategy-proof i the welfare of each agent weakly decreases as he reports preference relations that are further and further away from the truth. Moreover, sd-strategy-proofness is equivalent to dl-strategy-proofness and ul-strategy-proofness. Finally, we propose a family of rules that generalizes the serial rule. Journal of Economic Literature Classi cation Numbers: C70, C78, D61, D63