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An equilibrium analysis of the probabilistic serial mechanism

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Abstract

Two prominent mechanisms in the random assignment problem are the random priority (RP) and the probabilistic serial (PS). When agents are truthful, the outcomes obtained under PS have superior efficiency and fairness properties, but unlike RP, PS is vulnerable to strategizing. We study incentives of agents under PS. We find that when agents strategize, in equilibrium an outcome may be obtained under PS which is not efficient or fair and which is worse in some respects than the RP outcome. The results of our equilibrium analysis of PS call for caution when implementing it in “small” assignment problems.

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Notes

  1. See Roth and Sotomayor (1990) for an early survey of the literature on assignment problems allowing for monetary transfers.

  2. RP is called “random serial dictatorship” and has been shown to be equivalent to “core from random endowments” by Abdulkadiroğlu and Sönmez (1998).

  3. In this paper we use the sd- terminology (sd- standing for “stochastic dominance”), adapted from Thomson (2012). What is referred to as “ordinally efficient, envy-free, weakly envy-free, strategy-proof, and weakly strategy-proof” in BM’s paper are referred to in our paper as, respectively, “sd-efficient, sd-envy-free, weakly sd-envy-free, sd-strategy-proof, and weakly sd-strategy-proof.”

  4. For studies focusing exclusively on sd-efficiency, see McLennan (2002), Abdulkadiroğlu and Sönmez (2003) and Manea (2008).

  5. For other related studies, see Crès and Moulin (2001), Bogomolnaia and Moulin (2002), Kesten (2009), Budish et al. (2013) and Kesten and Ünver (2011).

  6. Budish et al. (2013) also pursue an equilibrium analysis for interpreting the outcomes of the celebrated “Boston mechanism” in the context of school choice.

  7. See, for example, Ehlers and Massó (2007) and the references therein.

  8. See Heo and Manjunath (2012) for a study of implementation in weak sd-NE and sd-NE of social choice correspondences in a randomized setting.

  9. Even though in the present context the central planner does not utilize monetary payments, agents themselves may resort to bribing one another in monetary terms when faced with a “bossy” mechanism.

  10. Note that \(r\) \(S\!D(\succ _{i}^{T})\, \overline{r}\) implies that \(r(s,\succ _{i}^{T})>\overline{r}(s,\succ _{i}^{T})\) for some \(s\in \{1,\ldots ,k-1\}\).

  11. Note that \(R\, S\!D(\succ ^{T})\, \overline{R}\) implies that \(R_{i}\,SD(\succ _{i}^{T})\, \overline{R}_{i}\) for some \(i\in I\).

  12. To put it in another way, a random assignment is sd-envy-free (weakly sd-envy-free) if no agent weakly sd-envies (sd-envies) another.

  13. Another open question is whether or not the truthful profile constitutes an sd-NE when an sd-NE exists. Notice that Theorem 1 says nothing about this; the truthful profile is indeed an sd-NE, however, in the cases that we considered where an sd-NE exists (under the “at most one desirable object” assumption and in Example 2).

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Acknowledgments

We would like to thank Utku Ünver, Morimutsu Kurino, and two anonymous referees for useful discussions and comments.

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Authors and Affiliations

  1. Özyeğin University, Istanbul, Turkey

    Özgün Ekici

  2. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Onur Kesten

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  1. Özgün Ekici
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Ekici, Ö., Kesten, O. An equilibrium analysis of the probabilistic serial mechanism. Int J Game Theory 45, 655–674 (2016). https://doi.org/10.1007/s00182-015-0475-9

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