The average tree value for hypergraph games

We consider transferable utility cooperative games (TU games) with limited cooperation introduced by a hypergraph communication structure, the so-called hypergraph games. A hypergraph communication structure is given by a collection of coalitions, the hyperlinks of the hypergraph, for which it is assumed that only coalitions that are hyperlinks or connected unions of hyperlinks are able to cooperate and realize their worth. We introduce the average tree value for hypergraph games, which assigns to each player the average of the player’s marginal contributions with respect to a particular collection of rooted spanning trees of the hypergraph, and study its properties. We show that the average tree value is stable on the subclass of superadditive cycle-free hypergraph games. We also provide axiomatizations of the average tree value on the subclasses of cycle-free hypergraph games, hypertree games, and cycle hypergraph games.

1. The position value for graph games is first defined in Meessen (1988) and later studied and axiomatized by Borm, Owen, and Tijs, cf. Borm et al. (1992).

2. A collection of coalitions \(\mathcal B\) on N is a building set on N if (i) for any \(S,Q\in \mathcal B\) such that \(S\cap Q\ne \emptyset \) it holds that \(S\cup Q\in \mathcal B\), and (ii) \(\{i\}\in \mathcal B\) for all \(i\in \mathcal B\), and therefore, it is also a union stable system, cf. Algaba et al. (2001).

3. This boils down to player i being a null player in the TU game \(v^H\). Such player is also known as superfluous player in, e.g., Slikker and van den Nouweland (2001).

Authors and Affiliations

1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

   Liying Kang

2. Faculty of Applied Mathematics, Saint-Petersburg State University, Universitetskii prospekt 35, Peterhof, Saint-Petersburg, Russia, 198504

   Anna Khmelnitskaya

3. School of Management, Shanghai University, Shanghai, 200444, People’s Republic of China

   Erfang Shan

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

   Dolf Talman

5. Business School, University of Shanghai for Science and Technology, Shanghai, 200093, People’s Republic of China

   Guang Zhang

Corresponding author

Correspondence to Dolf Talman.

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The research of Anna Khmelnitskaya was supported by RFBR (Russian Foundation for Basic Research) grant #18-01-00780. Her research was done partially during her stay at the University of Twente, whose hospitality is highly appreciated. The research of Guang Zhang was supported by NSFC (National Natural Science Foundation of China) Grant #71901145.

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