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Assortative Matching with Externalities and Farsighted Agents

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Abstract

We consider a one-to-one assortative matching problem in which matched pairs compete for a prize. With externalities, the standard solution concept—pairwise stable matching—may not exist. In this paper, we consider farsighted agents and analyze the largest consistent set (LCS) of Chwe (J Econ Theory 63:299-325, 1994). Despite the assortative structure of the problem, LCS tend to be large with the standard effectiveness functions: LCS can be the set of all matchings, including an empty matching with no matched pair. By modifying the effectiveness function motivated by Knuth (Marriages stables. Les Presses de l’Universite de Montreal, Montreal, 1976), LCS becomes a singleton of the positive assortative matching. Our results suggest that the choice of the effectiveness function can significantly impact the solution for a matching problem with externalities.

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Notes

  1. Thus, there is a free-rider problem in this pairs competition. For example, if a low-ability male is paired with a high-ability female, then a low-ability male agent may not make much effort, free-riding on his partner.

  2. An effectiveness function assigns a resulting matching to every combination of matching and a pair deviating from it [37]. The effectiveness function in Roth and Vande Vate [38] are commonly used (see Diamantoudi and Xue [12], Diamantoudi et al. [11], and Kojima and Unver [26]).

  3. Diamantoudi and Xue [12] consider LCS in a class of characteristic function games (thus with no externalities across coalitions): a hedonic game is an NTU game in which there is a single payoff vector for each coalition [3, 7]. Diamantoudi and Xue [12] show that if a hedonic game satisfies a top-coalition property introduced by Banerjee et al. [3], which includes Becker’s positive assortative matching problem (1973) as a special case, then LCS is equivalent to a singleton set of the core, which is the positive assortative matching in a two-sided, one-to-one matching context.

  4. In the companion paper, Imamura et al. [22] show that if we use the Knuth effectiveness function via swapping, then there is a unique (myopically) pairwise stable matching (via swapping) which is the assortative matching.

  5. More generally, there is a large literature of theory of coalition formation with externalities, starting from Hart and Kurz [19]. For surveys from various aspects, see Bloch [5], Ray [33] and Ray and Vohra [34].

  6. Dutta and Vartiainen [14] introduced history dependence to the rational expectations farsighted stability in Dutta and Vohra [13] to ensure nonemptiness of solutions for all finite problems.

  7. See Jackson [7] and Leo et al. [29] as well.

  8. Herings et al. [20, 21] show that the farsighted stable set is robust to allowing deviations from coalitions of arbitrary size instead of only considering pairwise or individual deviations when the effectiveness correspondence satisfies coalitional sovereignty.

  9. This is a group contest game with endogenous group formation. Group formation in contests is first analyzed by Bloch et al. [6]. Here, we consider a specific problem in which groups need to be pairs in two-sided matching setup. See Imamura et al. [22] for details.

  10. This CES aggregator function becomes a linear function (perfect substitutes) when \(\sigma =1\), and becomes a Cobb-Douglas function when \(\sigma =0\) in the limit.

  11. Please see Imamura et al. [22] for detailed derivations.

  12. The results derived from this example continue to hold for a sufficiently small \(\epsilon >0\).

  13. Our definition of a consistent set is slightly different from Chwe’s ([10] original definition, though it is easy to show that the resulting largest consistent sets are the same in either definition.

  14. Recently, in the standard marriage problems without externalities, Herings et al. [20, 21] have shown that the farsighted stable set is robust to allowing deviations from coalitions of arbitrary size instead of only considering pairwise or individual deviations.

  15. Dutta and Vohra [13] is closely related to Konishi and Ray [28], especially when coalitional moves are restricted to deterministic ones, having absorbing states.

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Acknowledgements

We are most grateful to two anonymous referees of the journal for their helpful comments and suggestions. We also thank Keisuke Bando, Andrew Copland, Chen-Yu Pan, Dimitar Simeonov, Utku Ünver, and Bumin Yenmez for their comments.

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Authors and Affiliations

  1. The University of Tokyo Market Design Center, Tokyo, Japan

    Kenzo Imamura

  2. Department of Economics, Boston College, Chestnut Hill, MA, USA

    Hideo Konishi

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  1. Kenzo Imamura
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  2. Hideo Konishi
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Correspondence to Hideo Konishi.

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This article is part of the topical collection “Group Formation and Farsightedness” edited by Francis Bloch, Ana Mauleon and Vincent Vannetelbosch.

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Imamura, K., Konishi, H. Assortative Matching with Externalities and Farsighted Agents. Dyn Games Appl 13, 497–509 (2023). https://doi.org/10.1007/s13235-022-00462-y

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