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The Average Tree permission value for games with a permission tree

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Abstract

In the literature, various models of games with restricted cooperation can be found. In those models, instead of allowing for all subsets of the set of players to form, it is assumed that the set of feasible coalitions is a subset of the power set of the set of players. In this paper, we consider such sets of feasible coalitions that follow from a permission structure on the set of players, in which players need permission to cooperate with other players. We assume the permission structure to be an oriented tree. This means that there is one player at the top of the permission structure, and for every other player, there is a unique directed path from the top player to this player. We introduce a new solution for these games based on the idea of the Average Tree value for cycle-free communication graph games. We provide two axiomatizations for this new value and compare it with the conjunctive permission value.

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Notes

  1. For a survey we refer to Bilbao (2000).

  2. Other models of games with a hierarchy on the set of players are, for example, Faigle and Kern (1992) and Li and Li (2011) or the more general model of Derks and Peters (1993).

  3. When \((N,L)\) is connected, then \(K^{ij}_h,\,h =i,j\), are the two cones in \(L\) that result from deleting \(\{i,j\}\).

  4. A collection of feasible coalitions \(\mathcal{A} \subset 2^N\) is an antimatroid if, besides being union closed, it contains the empty set and it satisfies accessibility meaning that \(S \in \mathcal{A}\) implies that there is a player \(i \in S\) such that \(S {\setminus } \{i\} \in \mathcal{A}\), see Dilworth (1940) and Edelman and Jamison (1985).

  5. A TU game \((N,v)\) is monotone if \(v(S) \le v(T)\) whenever \(S \subset T\).

  6. A game \((N,v)\) is superadditive if \(v(S) + v(T) \le v(S \cup T)\) for all \(S, T \subset N\) with \(S \cap T = \emptyset \).

  7. Notice that also the Myerson value, being the Shapley value of the Myerson restricted game \(v^L\), need not to be in the core of \(v^L\), even when \(v\) is superadditive and \((N,L)\) is cycle-free.

  8. It weakens the null player property, which states that a null player earns zero payoff.

  9. In voting games necessary players are usually called veto players.

  10. Note that the last equality also follows because all players in \( \widehat{S}_D(j) \cup \{j\} \) are null players in \((N,v_D)\).

  11. Otherwise, collusion neutrality would follow immediately from van den Brink (2012a).

  12. The necessary player property states that for monotone games, every necessary player earns at least as much as any other player. This implies equal payoffs for the necessary players in monotone games. Efficiency and the necessary player property imply that all players earn zero in the null game \( v_{0} \) on \(D\) (given by \(v_0(S)=0\) for all \(S \subset N\)). With linearity it follows that necessary player symmetry is satisfied for all games since \(\varphi (N,u^T,D)+\varphi (N,-u^T,D)=\varphi (N,v_0,D)\) and \(u^T\) is monotone.

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Authors and Affiliations

  1. Department of Econometrics, Tinbergen Institute, VU University, De Boelelaan 1105, 1081 HV , Amsterdam, The Netherlands

    René van den Brink & Gerard van der Laan

  2. Department of Economics, Maastricht University, P.O. Box 616, 6200 MD , Maastricht, The Netherlands

    P. Jean-Jacques Herings

  3. CentER, Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000 LE , Tilburg, The Netherlands

    A. J. J. Talman

Authors
  1. René van den Brink
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  2. P. Jean-Jacques Herings
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  3. Gerard van der Laan
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  4. A. J. J. Talman
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Correspondence to P. Jean-Jacques Herings.

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We thank three anonymous referees for their valuable comments.

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van den Brink, R., Herings, P.JJ., van der Laan, G. et al. The Average Tree permission value for games with a permission tree. Econ Theory 58, 99–123 (2015). https://doi.org/10.1007/s00199-013-0796-5

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