## Abstract

We explore the consequences of weakening the notion of incentive compatibility from strategy-proofness to ordinal Bayesian incentive compatibility (OBIC) in the random assignment model. If the common prior of the agents is the *uniform prior*, then a large class of random mechanisms are OBIC with respect to this prior—this includes the probabilistic serial mechanism. We then introduce a robust version of OBIC: a mechanism is *locally robust OBIC* if it is OBIC with respect *all* independent and identical priors in some neighborhood of a given independent and identical prior. We show that every locally robust OBIC mechanism satisfying a mild property called *elementary monotonicity* is strategy-proof. This leads to a strengthening of the impossibility result in Bogomolnaia and Moulin (J Econ Theory 100:295–328, 2001): if there are at least four agents, there is no locally robust OBIC and ordinally efficient mechanism satisfying equal treatment of equals.

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## Notes

Pycia and Ünver (2017) characterize the set of deterministic, strategy-proof, Pareto efficient, and non-bossy mechanisms in this model. This includes generalizations of the top-trading-cycle mechanism.

For instance, Bogomolnaia and Moulin (2001) show that the (uniform) random priority mechanism is ex-post efficient, strategy-proof and satisfies equal treatment of equals, but fails ordinal efficiency (which is stronger than ex-post efficiency).

Neutrality is a standard axiom in social choice theory which requires that objects are treated symmetrically. Elementary monotonicity is a monotonicity requirement of a mechanism. We define it formally in Sect. 4.

Katta and Sethuraman (2006) extend the simultaneous eating algorithm to allow for ties in preferences.

With three agent, the random priority mechanism satisfies these properties.

All our results extend even if the number of objects is not the same as the number of agents. We assume this only to compare our results with the random assignment literature, where this assumption is common.

Whenever we say an assignment, we mean a random assignment from now on.

The restriction to not consider cardinal mechanisms is arguably arbitrary. It is usually done to simplify the process of elicitation. Such restriction is also consistent with the literature on random assignment models. The set of incentive compatible mechanisms expand if we consider cardinal mechanisms (Miralles 2012; Abebe et al. 2020).

For three objects and three agents, we can show that the probabilistic serial mechanism cannot be OBIC with respect to a generic prior.

Note that we are not characterizing the set of \(\mu \) for which (3) has a solution. Our claim is only about the measure of the set of solutions.

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We are grateful to an anonymous referee, Sven Seuken, Timo Mennle, Arunava Sen, Dipjyoti Majumdar, Souvik Roy, and Wonki Cho for their comments.

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Dasgupta, S., Mishra, D. Ordinal Bayesian incentive compatibility in random assignment model.
*Rev Econ Design* **26**, 651–664 (2022). https://doi.org/10.1007/s10058-022-00289-4

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DOI: https://doi.org/10.1007/s10058-022-00289-4