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Fair reallocation in economies with single-peaked preferences

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Abstract

We consider the problem of fairly reallocating the individual endowments of a perfectly divisible good among agents with single-peaked preferences. As agents may have different individual endowments in this problem, the standard concept of envy-freeness cannot be applied directly. Thus, we propose a new concept of fairness, which we call envy-freeness for similarities. It requires that each agent whose best amount is strictly greater (less) than his individual endowment should not envy another agent whose best amount and assignment are strictly greater (less) than her individual endowment. We then construct a rule satisfying envy-freeness for similarities and some other desirable properties. In doing so, we propose a new extension of the well-known uniform rule for the reallocation problem, which we call the gross uniform reallocation rule. Moreover, we show that the gross uniform reallocation rule is the only rule that satisfies efficiency, individual rationality, strategy-proofness, and envy-freeness for similarities.

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Notes

  1. This “reallocation problem,” first analyzed by Klaus et al. (1997,1998a,1998b), is a natural extension of the problem of fairly allocating a social endowment of a perfectly divisible good among agents with single-peaked preferences (Sprumont 1991). For comprehensive surveys of the literature on private good economies in which agents have single-peaked preferences, see Klaus (1998) and Thomson (2014).

  2. This example is due to Klaus et al. (1997) and Klaus (1998).

  3. The history of this principle goes back to Aristotle’s formal principle of justice. See p. 15 of Rabinowicz (1979).

  4. The terms demander and supplier are proposed by Klaus et al. (1997, 1998a, b and 1998).

  5. The notion of envy-freeness in terms of changes in allocations is first formulated by Schmeidler and Vind (1972) in the context of pure exchange economies. Some studies incorporate individual endowments into the notion of envy-freeness in different contexts. For example, Yılmaz (2010) also mixes the notion of individual rationality with the notion of envy-freeness in the probabilistic assignment model. Note that there is no logical relationship between Yılmaz’s notion and our notion.

  6. Klaus et al. (1997, 1998a, b and 1998) just call it the “uniform reallocation rule.”

  7. In some situations, this rich agent’s envy could be justified. For example, if a “rich agent” got more initial endowments with a lot of hard work while a “poor agent” was lazy, a preferential treatment for the latter in a reallocation may cause the former’s envy. In other words, agents’ envy can occur also depending on how rightfully they got their initial endowments. We thank an anonymous referee for suggesting this interpretation.

  8. The uniform rule is introduced by Benassy (1982). Sprumont (1991) is the first to provide an axiomatic characterization of the uniform rule in the social endowment setting. Subsequently, there have been many studies on characterizing this rule in the same setting. See, for example, Ching (1994), Thomson (1994a, b, 1995, 1997), Chun (2006), Mizobuchi and Serizawa (2006), Serizawa (2006), Sakai and Wakayama (2015), and Wakayama (2017). See also Thomson (2014) for a survey of the characterizations of the uniform rule. Sönmez (1994) and Barberà et al. (1997) describe an algorithm that computes the allocation of the uniform rule.

  9. Sprumont (1991) proves this characterization result in the social endowment setting with single-peaked preferences, but this proof can be applied to our setting.

  10. As mentioned above, in our setting, it is well known that envy-freeness is incompatible with individual rationality.

  11. See Klaus et al. (1997, 1998a, b, 1998), Klaus (2001), Moreno (2002), and Bonifacio (2015).

  12. Note that this rule is a member of the family of “sequential rules” introduced by Barberà et al. (1997). A similar functional form appears in the social endowment setting under constraints. See Bergantiños et al. (2015).

  13. The notion of “strategy-proofness for same tops” states that an agent cannot gain if he truthfully reports his peak. This notion is formally defined as follows: For each \(R \in \mathscr {R}^n\), each \(i \in N\), and each \(R'_i \in \mathscr {R}\), if \(p(R_i) = p(R'_i)\), then \(f_i(R) \mathrel {R_i} f_i(R'_i, R_{-i})\).

  14. The notion of “peak-only” states that the amount assigned to agents depends only on their peaks.

  15. The notion of “group strategy-proofness” is formally defined as follows: For each \(R \in \mathscr {R}^n\), each \(C\subset N\), and each \(R'_{C} \in \mathscr {R}^{|C|}\), if there exists \(i\in C\) such that \(f_i(R'_{C}, R_{-C}) \mathrel {P_i} f_i(R)\), then for some \(j\in C\), \(f_j(R) \mathrel {P_j} f_j(R'_{C}, R_{-C})\).

  16. This fact can be seen using the following Serizawa’s (2006) result (Serizawa (2006) does not explicitly state this result, but the proof of his Proposition 7 actually shows this): strategy-proofness, efficiency, and “non-bossiness” (for each \(R \in \mathscr {R}^n\), each \(i \in N\), and each \(R'_{i} \in \mathscr {R}\), \(f_{i}(R) = f_{i}(R'_{i}, R_{-i})\) implies \(f(R) = f(R'_{i}, R_{-i})\)) imply group strategy-proofness. It is easy to check that the gross uniform reallocation rule is non-bossy. Then, the desired conclusion immediately follows from the Serizawa’s result.

  17. The notation \(I_1\) denotes the indifference relation associated with \(R_1\).

  18. This rule is inspired by Klaus (2010).

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Authors and Affiliations

  1. Faculty of Economics, Osaka University of Economics, 2-2-8, Osumi, Higashiyodogawa-ku, Osaka, 533-8533, Japan

    Kazuhiko Hashimoto

  2. Faculty of Economics, Ryukoku University, 67 Tsukamoto-cho, Fukakusa, Fushimi-ku, Kyoto, 612-8577, Japan

    Takuma Wakayama

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  1. Kazuhiko Hashimoto
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  2. Takuma Wakayama
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We are grateful to an associate editor and two anonymous referees of this journal for helpful comments. This work was supported by JSPS KAKENHI Grant Numbers JP26780117, JP15H03328, and JP16K03567.

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Hashimoto, K., Wakayama, T. Fair reallocation in economies with single-peaked preferences. Int J Game Theory 50, 773–785 (2021). https://doi.org/10.1007/s00182-021-00767-z

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