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Similarities in axiomatizations: equal surplus division value and first-price auctions

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Abstract

We compare axiomatizations between a value for cooperative games with transferable utilities (TU games), and a rule for auctions. The equal surplus division value on the set of zero-monotonic TU games is characterized by the following: individual rationality, Pareto efficiency, and equal effect of players’ nullification on others. Meanwhile, first-price auctions, on the general preference domain, are characterized by individual rationality, envy-freeness, and weak equal effect of buyers’ nullification on others. Here, envy-freeness implies Pareto efficiency in the model of auctions. Given the agents’ general preferences in the auction model, the characteristic of a weak equal effect of buyers’ nullification on others weakens the requirement of equal effect of players’ nullification on others. Although the two models are different, the corresponding axioms in both models require conditions corresponding to each other. In particular, individual rationality requires voluntary participation of agents, Pareto efficiency (or its stronger axiom of envy-freeness in the model of auctions) requires outcomes with no waste, and (weak) equal effect of players’ (buyers’) nullification on others requires equal treatment of agents when an agent is nullified. Therefore, in terms of axiomatizations, we can similarly interpret the equal surplus division value and first-price auctions.

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Notes

  1. Buyers who do not obtain the item in an auction may still end up paying. A typical example is the payment of participation fee for auctions.

  2. See also Graham and Marshall (1987) for incentive aspects in their model.

  3. van den Brink (2007) also studies the same situation and characterizes the convex combination between the equal surplus division and equal division values. Further, Branzei et al. (2009) studies the relationships between the complete information auction games and peer group games.

  4. Yokote and Funaki (2017) consider only a case with greater than or equal to six players.

  5. An excellent survey on this property is given by Thomson (2016).

  6. The first-price auctions and the second-price auctions have been recently compared from a different viewpoint from axiomatizations (see Shimoji 2017).

  7. Peleg and Sudhölter (2003) call this condition weak superadditivity.

  8. Ferrières (2017) introduces an equivalent axiom to \(\hbox {EENO}_G\) in TU games.

  9. This weaker variation of \(\hbox {EENO}_G\) is introduced by Kongo (2019) to axiomatize the set of weighted surplus division values. Following Casajus (2017), who weaken the balanced contributions property of Myerson (1980) in this way, corresponding generalizations of existing axioms in TU games are well-investigated. For example, Casajus and Yokote (2017) weakens differential marginality in Casajus (2011), and Casajus (2018a) engages in symmetry. Furthermore, further generalizations of such axioms are discussed in Casajus (2018b) and Casajus (2019).

  10. From (P1), there exists \(WP(\succsim _i) \in {\mathbb {R}}\) such that \((0,0) \sim _i (1,WP(\succsim _i))\). Suppose that \(WP(\succsim _i)<0\). From (P2), there exists \(m_i \in (WP(\succsim _i), 0)\) such that \((1,WP(\succsim _i)) \succ _i (1, m_i) \). From (P3), \((1,0) \succsim _i (0,0)\). Therefore, we obtain \((1,0) \succ _i (1,m_i)\). But, this contradicts (P2) because \(0>m_i\).

  11. One possible way to apply this axiom to TU games is that \(\varphi _i(v)=\varphi _j(v)\) for any \(i,j \in N\) and any \(v \in V(N)\), which has a condition representing strict egalitarianism. This condition does not imply \(\hbox {PE}_G\) and, together with \(\hbox {PE}_G\), this condition clearly implies the equal division value \(\varphi ^e\).

  12. This rule is the same as that in Adachi and Kongo (2013).

  13. In Sakai (2008), this property is formalized as an axiom called non-imposition. Clearly, non-imposition is a weaker requirement than \(\hbox {IR}_A\).

  14. This lemma is the same as Lemma 1 in Adachi and Kongo (2013).

  15. The other case of \(a_i(\succsim '_j,\succsim _{-j})=0\) and \(a_j(\succsim '_j,\succsim _{-j})=1\) can be shown by replacing i and j.

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Acknowledgements

The author is grateful to the editor and an anonymous referee for their comments that assisted in improving this manuscript. This study was funded by JSPS KAKENHI (grant number 19K01569).

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  1. Faculty of Economics, Fukuoka University, 8-19-1 Nanakuma, Jonan-ku, Fukuoka, 814-0180, Japan

    Takumi Kongo

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Kongo, T. Similarities in axiomatizations: equal surplus division value and first-price auctions. Rev Econ Design 24, 199–213 (2020). https://doi.org/10.1007/s10058-020-00233-4

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