Soc Choice Welf (2017) 49:255–275
Impossibilities for probabilistic assignment
· Yoichi Kasajima
Received: 12 November 2016 / Accepted: 15 May 2017 / Published online: 29 May 2017
© Springer-Verlag Berlin Heidelberg 2017
Abstract We consider the problem of assigning objects probabilistically among a
group of agents who may have multi-unit demands. Each agent has linear preferences
over the (set of) objects. The most commonly used extension of preferences to com-
pare probabilistic assignments is by means of stochastic dominance, which leads to
corresponding notions of envy-freeness, efﬁciency, and strategy-proofness. We show
that equal treatment of equals, efﬁciency, and strategy-proofness are incompatible.
Moreover, anonymity, neutrality, efﬁciency, and weak strategy-proofness are incom-
patible. If we strengthen weak strategy-proofness to weak group strategy-proofness,
then when agents have single-unit demands, anonymity, neutrality, efﬁciency, and
weak group strategy-proofness become incompatible.
Two independent papers, Aziz (2016)andKasajima (2011), were merged into the current paper. Kasajima
(2011) was based on a chapter of his Ph.D. thesis at the University of Rochester. Aziz is supported by a
Julius Career Award. Kasajima acknowledges support from the JSPS KAKENHI Grant Numbers
22830102 and 16K03561.
Data61, CSIRO and UNSW, Sydney, Australia
School of Social Sciences, Waseda University, 1-6-1 Nishi-Waseda,
Shinjuku-ku, Tokyo 1698050, Japan