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Extended Random Assignment Mechanisms on a Family of Good Sets

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Abstract

For the problem of allocating indivisible goods to agents, we recently generalized the probabilistic serial (PS) mechanism proposed by Bogomolnaia and Moulin (J Econ Theory 100(2):295–328, 2001). We generalized the constraints with the fixed quota on each good to a set of inequality constraints induced by a polytope called polymatroid (Fujishige et al. in ACM Trans Econ Comput 6(1):1–28, 2018. https://doi.org/10.1145/3175496; Math Program 178(1–2):485–501, 2019). The main contribution of this paper was to extend the previous mechanism to allow indifference among goods. We show that the extended PS mechanism is ordinally efficient and envy-free. We also characterize the mechanism by lexicographic optimization. Finally, a lottery, an integral decomposition mechanism is outlined.

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Data availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

Notes

  1. Ordinal efficiency and envy-freeness are defined later. An assignment mechanism is strategy-proof if an agent cannot obtain a better off allocation by falsifying the true preference.

  2. The cardinality of \(\varGamma\), \(|\varGamma |\), is a submodular function (Lemma 2.1 in [31]).

  3. It is called dichotomous preferences in papers [8, 15].

  4. The term was used by Katta and Sethurama [8].

  5. By Lemma 3, the composition \(\rho \circ \varGamma : 2^N \rightarrow {\mathbb {R}}\) is a submodular function. Denote \(\rho \circ \varGamma\) by \(\varGamma '\), the uniqueness of \({{\bar{X}}}^*\) in (18) then follows from the same discussion as the one given by Katta and Sethurama [8].

  6. We omit bar over notations and specify them by iteration steps.

  7. By our assumption of \(\sum _{i}d(i) \ge \rho (E)\), resource E exhausted before all demands by agents satisfied, hence \(E_p\) (\(p\ge 1\)) are used as the termination condition.

  8. It shuld be pointed out that Algorithm I is not exactly a special case because \(N_p \subset N\) ( \(p>1\)) does not occur in Algorithm II, it clearly is a simpler case.

  9. Here, \(\lambda \mathbf{d}\in {\mathbb {R}}^N_{\ge 0}\) can be viewed as a modular function on N given in Sect. 2.2.

  10. Here, “sd” stands for (first-order) stochastic dominance, employed in paper [1].

  11. Also as treated in papers [12, 26], for agent \(i \in N\), we can put good \(e\in E\) with \(P(i,e)=0\) to agent i’s least preferred one without changing the problem if necessary.

  12. This is a standard procedure for finding an expression of a given point x in a relative interior of a polytope P as a combination of extreme points of P.

  13. The existing of the minimal face has been shown by Lemma 7.1 of paper [12].

  14. Here \(\partial ^- \varphi\) is the direct sum of base polyhedron \(\mathrm{B}(\rho )\) induced by associated submodular equalities (49)\(\sim\)(52), refer to paper [12].

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Acknowledgements

We are deeply indebted to Prof. Satoru Fujishige for the foundation of this research and his generously sharing and advices. We are also grateful to anonymous reviewers for their valuable comments to improve the presentation of the present paper.

Funding

Y. Sano’s work was partially supported by JSPS KAKENHI Grant Number 19K03598. And P. Zhan’s work was partially supported by JSPS KAKENHI Grant Number 20K04970.

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Authors and Affiliations

  1. Division of Information Engineering, Faculty of Engineering, Information and Systems, University of Tsukuba, Ibaraki, 305-8573, Japan

    Yoshio Sano

  2. Department of Communication and Business, Edogawa University, Nagareyama, Chiba, 270-0198, Japan

    Ping Zhan

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  1. Yoshio Sano
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Correspondence to Ping Zhan.

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Sano, Y., Zhan, P. Extended Random Assignment Mechanisms on a Family of Good Sets. Oper. Res. Forum 2, 52 (2021). https://doi.org/10.1007/s43069-021-00095-8

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