Abstract
A new value concept, called degree value, is proposed by employing the degree game induced by an original game for hypergraph communication situations (including graph communication situations). We provide an axiomatic characterization of the degree value for arbitrary hypergraph communication situations by applying component efficiency and balanced conference contributions, which is a natural extension of balanced link contributions introduced in Slikker (Int J Game Theory 33:505–514, 2005) for graph communication situations. By comparing the degree value with the position value and the Myerson value, it is verified that the degree value is a new allocation rule that differs from both the Myerson value and the position value, and the degree value highlights the important role of the degree of a player in hypergraph communication situations. Particularly, in a uniform hypergraph communication situation, where every conference contains the same number of players, we show that the degree value coincides with the position value.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Throughout this paper, we speak of graph when we refer to an undirected graph.
If \((N,v)\in {\mathcal {G}}^N\) is not zero-normalized in \((N,v,H)\in {\mathscr {H}}^N\), then the degree value is defined by \({\mathcal {D}}_i(N,v,H)=v(\{i\})+\sum _{l\in I(H)_i}Sh_l(I(H),w^0)\), for all \(i\in N\), where \(w^0(S)=(v^0)^N(H[S])\), for any \(S\subseteq I(H)\), and \(v^0(Q)=v(Q)-\sum _{i\in Q}v(\{i\})\), for any \(Q\subseteq N\).
References
Algaba E, Bilbao JM, Borm P, López JJ (2000) The position value for union stable systems. Math Methods Oper Res 52:221–236
Algaba E, Bilbao JM, Borm P, López JJ (2001) The Myerson value for union stable systems. Math Methods Oper Res 54(3):359–371
Borm P, Owen G, Tijs S (1992) On the position value for communication situations. SIAM J Discret Math 5:305–320
Casajus A (2007) The position value is the Myerson value, in a sense. Int J Game Theory 36:47–55
Kongo T (2010) Difference between the position value and the Myerson value is due to the existence of coalition structures. Int J Game Theory 39:669–675
Meessen R (1988) Communication games, Master’s thesis, Department of Mathematics. University of Nijmegen, the Netherlands (in Dutch)
Myerson RB (1977) Graphs and cooperation in games. Math Oper Res 2:225–229
Myerson RB (1980) Conference structures and fair allocation rules. Int J Game Theory 9:169–182
Shan E, Zhang G, Dong Y (2016) Component-wise proportional solutions for communication graph games. Math Soc Sci 81:22–28
Shan E, Zhang G (2016) Characterizations of the position value for hypergraph communication situations. arXiv:1606.00324
Shapley LS (1953) A value for \(n\)-person games. In: Kuhn H, Tucker AW (eds) Contributions to the Theory of Games II. Princeton University Press, Princeton, pp 307–317
Slikker M (2005) A characterization of the position value. Int J Game Theory 33:505–514
Slikker M, van den Nouweland A (2001) Social and economic networks in cooperative game theory. Kluwer, Norwell
van den Nouweland A, Borm P, Tijs S (1992) Allocation rules for hypergraph communication situations. Int J Game Theory 20:255–268
Acknowledgements
We thank an associate editor and two referees for valuable suggestions and comments. The authors also want to thank Dolf Talman and Anna Khmelnitskaya for many interesting and inspiring discussions. The research was supported in part by the National Nature Science Foundation of China (nos. 11571222, 11471210). The research of Guang Zhang was supported by the China Scholarship Council (CSC).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shan, E., Zhang, G. & Shan, X. The degree value for games with communication structure. Int J Game Theory 47, 857–871 (2018). https://doi.org/10.1007/s00182-018-0631-0
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00182-018-0631-0