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Nullified equal loss property and equal division values

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Abstract

We provide characterizations of the equal division values and their convex mixtures, using a new axiom on a fixed player set based on player nullification which requires that if a player becomes null, then any two other players are equally affected. Two economic applications are also introduced concerning bargaining under risk and common-pool resource appropriation.

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Notes

  1. In that article, player i’s nullification is denoted \(v^{\mathbf {N}i}\).

  2. This game is denoted \(v_S\) in Neyman (1989).

  3. These games are called “almost null games” in Kongo (2016).

  4. This framework may seem analog to hyperplane games introduced in Maschler and Owen (1989) but with the main difference that \(W_S\) is shared here by the whole fixed player set N and not by the subset \(N\backslash S\), as in the framework of cooperative games without sidepayments.

  5. The following result 1 remains true if \(F_i\) is any strictly increasing continuous function of \(w_i\).

  6. Abusing notation, \(\pi \) also denotes the induced additive TU-game. In a more axiomatic way to handle the certainty equivalence, one may introduce here a replaceability axiom like in Smorodinsky (2005), in which only the disagreement point is allowed to be random (see axiom 7 and corollary 2). However, our approach sticks to an illustrative aim.

  7. More precisely, we assume that the players cannot make binding agreements on their individual efforts, which would instead gives rise to coalition-wise efforts, modelled by a game in partition function form.

  8. Although our approach is parallel to that of Sen, we need not model “social consciousness”, nor “social goodwill” in individual preoccupations.

  9. A game v contains a veto player \(h\in N\) if \(v(S)=0\) whenever \(h \notin S\).

  10. A nullifying player \(i \in N\) in a game v is such that \(v(S)=0\) whenever \(i \in S\).

  11. A dummy player \(i \in N\) in a game v is such that \(v(S)=v(S\backslash i)+v(i)\) whenever \(i \in S\).

  12. A proportional player (or per-capita null player) \(i \in N\) in a game v is such that \(\frac{v(S)}{s}=\frac{v(S\backslash i)}{s-1}\) whenever \(i \in S\).

  13. A dummifying player \(i \in N\) in a game v is such that \(v(S)=\sum _{j \in S}v(j)\) whenever \(i \in S\).

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Authors and Affiliations

  1. CRESE EA3190, Univ. Bourgogne Franche-Comté, 25000, Besançon, France

    Sylvain Ferrières

  2. Chair of Economics and Information Systems, HHL Leipzig Graduate School of Management, Jahnallee 59, 04109, Leipzig, Germany

    Sylvain Ferrières

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  1. Sylvain Ferrières
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Correspondence to Sylvain Ferrières.

Additional information

The author wishes to thank Hervé Moulin for helpful comments and most relevant references as well as William Thomson for fruitful discussions. This article has also benefited from many improving suggestions from two anonymous reviewers.

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Ferrières, S. Nullified equal loss property and equal division values. Theory Decis 83, 385–406 (2017). https://doi.org/10.1007/s11238-017-9604-1

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