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 compare 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 incompatible. 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.
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Notes
We also consider the possibility that agents receive more than one object. They thus have preferences over “sets of” objects.
Several interesting rules and their extension had been proposed and studied in the literature: the “serial rule” (Bogomolnaia and Moulin 2001; Katta and Sethuraman 2006; Athanassoglou and Sethuraman 2011; Kojima 2009; Yilmaz 2009, 2010; Heo 2014), the “random priority rule” (Abdulkadiroğlu and Sönmez 1998; Kojima 2009), the “uniform rule” (Chambers 2004), and the “priority rule” (Svensson 1994, 1999).
Under this relation, one probabilistic assignment stochastically dominates another one if and only if the former yields at least as much expected utility as the latter for any von-Neumann-Morgenstern utility representation consistent with the ordinal preferences (Bogomolnaia and Moulin 2001; Aziz et al. 2013).
See Thomson (2011) for various fairness notions proposed in the literature of resource allocation problems.
Preference is additive if there is a function that assigns a real number to each object, and the rankings over sets of objects are compared by adding these numbers.
We use the abbreviation “sd” in other axioms as well. The terminology is suggested by Thomson (2008).
This requirement is referred to as “ordinal efficiency” in Bogomolnaia and Moulin (2001).
One could require that (under the same hypothesis), it is not the case that (1) for each agent in the group, her probabilistic assignment when lying stochastically dominates her probabilistic assignment when telling the truth, or the two assignments are the same, and (2) there is at least one agent in the group that her probabilistic assignment when lying stochastically dominates her probabilistic assignment when telling the truth. This requirement is stronger than the one we consider here. Since our result (Theorem 3) is negative, it also holds under this stronger requirement.
A probabilistic assignment for an agent i is “comparable” (with respect to stochastic dominance) with another probabilistic assignment if either one assignment first-order stochastically dominates the other under agent i’s preference, or the entries in the two assignments are the same. Given an axiom (or a result involving that axiom), if it requires that for each problem and each agent i, (1) a probabilistic assignment for an agent i (given by a rule) is comparable with at least one other probabilistic assignment and (2) the former assignment is at least as desirable as the latter assignment for agent i, then we say that it “requires comparability of probabilistic assignments” (except for invariance properties). As noted in the next footnote, in fact, sd-strategy-proofness and sd-envy-freeness require that an assignment to be comparable with all other relevant assignments.
Note that sd-strategy-proofness requires that an agent i’s probabilistic assignment under truth-telling should be comparable with an assignment under any of agent \({\varvec{i}}\)’s report . Similarly, sd-envy-freeness requires that an agent’s probabilistic assignment should be comparable with each other agent’s assignment.
\(O^{\prime }\,R_i\,O^{\prime \prime }\) means that \(O^{\prime }\) is at least as desirable as \(O^{\prime \prime }\) for agent i.
\(R_{-i}\equiv R_{N\backslash \{i\}}\), i.e., the restriction of R to \(N\backslash \{i\}\).
\(R_{-S}\equiv R_{N\backslash \{S\}}\), i.e., the restriction of R to \(N\backslash \{S\}\).
When there are at least four agents, one can extend the proof of Bogomolnaia and Moulin (2001) [Theorem 2] to show that if agents receive more than one objects, the three axioms are incompatible. Thus our theorem is distinguished from theirs for the cases of two and three agents.
Recall that preferences have additive representations and preferences over O are strict. For each \(i\in N\), if \(v_{i}(a)>v_{i}(b)>v_{i}(c)>\cdots \), then we write
$$\begin{aligned} R_i:&\quad a,b,c,\ldots \\ \end{aligned}$$The serial rule is referred to as the “probabilistic serial mechanism” in Bogomolnaia and Moulin (2001) and Kojima (2009). Under the serial rule, each object is considered as an infinitely divisible good whose supply is 1. Agents “consume” the most favored available object at an equal speed until the supplies of all objects (q|N|) are exhausted. When the supply of a most preferred object is exhausted, agents consume their next most preferred object that is not exhausted, and so on. The fraction of object consumed by an agent is the probability of the agent receiving that object. If instead each agent starts consuming the most preferred q objects, then such a rule violates sd-efficiency (Che and Kojima 2010).
The random priority rule is referred to as the “random serial dictatorship” in Abdulkadiroğlu and Sönmez (1998) and “random priority mechanism” in Kojima (2009). Under the random priority rule, we take an order on the set of agents and let each agent choose her q most preferred objects among the remaining ones according to the order. Then, we consider all possible orders on the set of agents and place equal probabilities on the allocations obtained for such orders. If instead each agent only selects one object when her turn comes (and move to the second round if there are still remaining objects, and so on), then such a rule violates sd-strategy-proofness.
As for the random priority rule, if instead each agent only selects one object when her turn comes, then such a rule violates sd-strategy-proofness.
The serial rule (Bogomolnaia and Moulin 2001) satisfies these properties.
Kojima (2009) [Example 2, p.138] shows that the serial rule is not weakly sd-strategy-proof.
The difficulty of constructing such a rule comes from the fact that we do not have complete understanding of the characteristics of rules that satisfy sd-efficiency and weak sd-strategy-proofness. The priority rule associated with \(\prec \) is one of such rules, but it is not anonymous. If we give “priority” to some objects, i.e., assigning probabilities to those objects first, then we may not end up with an allocation that is sd-efficient. To make a rule that is not neutral but anonymous and sd-efficient, one can think of changing “consuming” speeds for some particular objects, based on the idea of the serial rule, but such a rule violates weak sd-strategy-proofness. To construct a rule that is not neutral but anonymous and weakly sd-strategy-proof, one may think of letting each agent consume her most preferred q objects, based on the idea of the serial rule (Che and Kojima 2010) (such a rule is weakly sd-strategy-proof, see Aziz, 2015), and change the consuming speeds for some objects, but such a rule violates sd-efficiency.
Note that by (3), \(\varphi _{1a}(R_1,R_2,R'_3,R'_4)=\varphi _{2a}(R_1,R_2,R'_3,R'_4)\le \frac{1}{2}\).
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Acknowledgements
We thank the editor, an associate editor, and an anonymous referee for their suggestions and comments. Kasajima is grateful to William Thomson for his guidance and many helpful comments. He also thanks Atila Abdulkadiroğlu, Paulo Barelli, John Duggan, Eun Jeong Heo, Bettina Klaus, Fuhito Kojima, and the seminar participants at Waseda University for their comments. All errors are ours.
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Two independent papers, Aziz (2016) and Kasajima (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.
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Aziz, H., Kasajima, Y. Impossibilities for probabilistic assignment. Soc Choice Welf 49, 255–275 (2017). https://doi.org/10.1007/s00355-017-1059-3
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DOI: https://doi.org/10.1007/s00355-017-1059-3