Abstract
This paper analyzes Harsanyi power solutions for cooperative games in which partial cooperation is based on union stable systems. These structures contain as particular cases the widely studied communication graph games and permission structures, among others. In this context, we provide axiomatic characterizations of the Harsanyi power solutions which distribute the Harsanyi dividends proportional to weights determined by a power measure for union stable systems. Moreover, the well-known Myerson value is exactly the Harsanyi power solution for the equal power measure, and on a special subclass of union stable systems the position value coincides with the Harsanyi power solution obtained for the influence power measure.
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Notes
Recently, Faigle et al. (2010) have established union stable systems as the more general systems where it is possible to define a meaningful notion of supermodularity that generalizes Shapley’s original convexity concept for classical cooperative games.
In modern societies large international firms have an important impact on society, and therefore members of the Boards of Directors of such firms have a repercussion on society and it is interesting to measure, in some sense, this influence.
Although in the beginning of this section, we mentioned that we always take as player set N={1,…,n}, the definition of the conference game is the only occasion where we deviate from this. Note that the player set in a conference game is still derived from a structure on N.
Consider, for example the union stable system \(\mathcal{F}=\{\{1,2\},\{1,2,3\}\}\). Note that \((N,v,\mathcal{F})\) ∉USI N since {1,2,3} is a support and it can also be written as the union of the supports {1,2} and {1,2,3}. Further, consider the unanimity game v=u {1,2,3}. Then the influence measure on N={1,2,3} yields \(I(N,\mathcal{F})=(5/6,5/6,1/3)\), so \(\varphi^{I}(N,v,\mathcal {F})=(5/6,5/6,1/3)\). Deleting from \(\mathcal{F}\) the superfluous support {1,2}, we obtain the union stable system \(\mathcal{F}^{\prime}=\{\{1,2,3\}\}\). Now the influence measure is given by \(I(N,\mathcal{F}^{\prime })=(1/3,1/3,1/3)\), so \(\varphi^{I}(N,v,\mathcal{F}^{\prime})=(1/3,1/3,1/3)\), showing that the outcome changed although we only deleted a superfluous support. Also, on \((N,v,\mathcal{F})\) the position value is not equal to \(\varphi ^{I}(N,v,\mathcal{F})\) since \(\pi(N,v,\mathcal{F})=(1/3,1/3,1/3)\).
The proof can be obtained from the corresponding proof for the Myerson value in Algaba et al. (2012) by applying σ-point unanimity instead of point unanimity at every occasion where this is used.
The influence property used in Algaba et al. (2000) to axiomatize the position value, is obtained by taking σ=I, and moreover applying this to all support anonymous union stable structures which include the support unanimous union stable structures, i.e. it states that for each \(( N,v, \mathcal{F} ) \in \mathit{US}^{N}\) that is support anonymous, the payoffs to the players are proportional to their influence. This axiomatization still holds when the influence property is weakened by requiring it only for support unanimous union stable structures.
Although van den Brink et al. (2011b) is about games on union closed systems, the proof is the same.
A similar remark can be made about strengthening the σ-influence property using support anonymous union stable structures, i.e. triples \(( N,v,\mathcal{F} ) \in \mathit{US}^{N}\) such that there exists a function \(f: \{ 0,1,\ldots,\vert \mathcal{C}\vert \} \rightarrow\mathbb{R}\) with \(v^{\mathcal{C}} ( \mathcal{A} ) =f ( \vert \mathcal{A}\vert )\), for all \(\mathcal{A}\subseteq\mathcal{C}\). In Algaba et al. (2000) is shown that the position value satisfies support anonymity, which is used to characterize this solution on USI N.
References
Algaba, E., Bilbao, J. M., Borm, P., & López, J. J. (2000). The position value for union stable systems. Mathematical Methods of Operational Research, 52, 221–236.
Algaba, E., Bilbao, J. M., Borm, P., & López, J. J. (2001). The Myerson value for union stable structures. Mathematical Methods of Operational Research, 54, 359–371.
Algaba, E., Bilbao, J. M., van den Brink, R., & Jiménez-Losada, A. (2004a). Cooperative games on antimatroids. Discrete Mathematics, 282, 1–15.
Algaba, E., Bilbao, J. M., & López, J. J. (2004b). The position value in communication structures. Mathematical Methods of Operational Research, 59, 465–477.
Algaba, E., Bilbao, J. M., van den Brink, R., & López, J. J. (2012). The Myerson value and superfluous supports in union stable systems. Forthcoming in Journal of Optimization Theory and Applications. doi:10.1007/s10957-012-0077-7.
Bilbao, J. M. (2003). Cooperative games on augmenting systems. SIAM Journal on Discrete Mathematics, 17, 122–133.
Borm, P., Owen, G., & Tijs, S. H. (1992). On the position value for communication situations. SIAM Journal on Discrete Mathematics, 5, 305–320.
van den Brink, R. (1997). An axiomatization of the disjunctive permission value for games with a permission structure. International Journal of Game Theory, 26, 27–43.
van den Brink, R. (2011). On hierarchies and communication. Forthcoming in Social Choice and Welfare. doi:10.1007/s00355-011-0557-y.
van den Brink, R., van der Laan, G., & Pruzhansky, V. (2011a). Harsanyi power solutions for graph-restricted games. International Journal of Game Theory, 40, 87–110.
van den Brink, R., Katsev, I., & van der Laan, G. (2011b). Axiomatizations of two types of Shapley values for games on union closed systems. Economic Theory, 47, 175–188.
Conyon, M. J., & Muldoon, M. R. (2006). The small world of corporate boards. Journal of Business Finance & Accounting, 33, 1321–1343.
Derks, J., Haller, H., & Peters, H. (2000). The selectope for cooperative TU-games. International Journal of Game Theory, 29, 23–38.
Derks, J., van der Laan, G., & Vasil’ev, V. A. (2010). On the Harsanyi payoff vectors and Harsanyi imputations. Theory and Decision, 68, 301–310.
Gilles, R. P., Owen, G., & van den Brink, R. (1992). Games with permission structures: the conjunctive approach. International Journal of Game Theory, 20, 277–293.
Faigle, U., & Kern, W. (1992). The Shapley value for cooperative games under precedence constraints. International Journal of Game Theory, 21, 249–266.
Faigle, U., Grabish, M., & Heyne, M. (2010). Monge extensions of cooperation and communication structures. European Journal of Operational Research, 206, 104–110.
Harsanyi, J. C. (1959). A bargaining model for cooperative n-person games. In A. W. Tucker & R. D. Luce (Eds.), Contributions to the theory of games (Vol. IV, pp. 325–355). Princeton: Princeton University Press.
Meessen, R. (1988). Communication games. Master’s thesis, Dept of Mathematics, University of Nijmegen, The Netherlands (in Dutch).
Mizruchi, M. S., & Bunting, D. (1981). Influence in corporate networks: an examination of four measures. Administrative Science Quarterly, 26, 475–489.
Myerson, R. B. (1977). Graphs and cooperation in games. Mathematics of Operations Research, 2, 225–229.
Myerson, R. B. (1980). Conference structures and fair allocation rules. International Journal of Game Theory, 9, 169–182.
van den Nouweland, N. A. (1993). Games and graphs in economics situations. PhD dissertation, Tilburg University, The Netherlands.
van den Nouweland, A., Borm, P., & Tijs, S. H. (1992). Allocation rules for hypergraph communication situations. International Journal of Game Theory, 20, 255–268.
Owen, G. (1986). Values of graph-restricted games. SIAM Journal on Algebraic and Discrete Methods, 7, 210–220.
Potters, J. A. M., & Reijnierse, H. (1995). Γ-Component additive games. International Journal of Game Theory, 24, 49–56.
Shapley, L. S. (1953). A value for n-person games. In H. W. Kuhn & A. W. Tucker (Eds.), Contributions to the theory of games (Vol. II, pp. 307–317). Princeton: Princeton University Press.
Vasil’ev, V. A. (1982). Support function on the core of a convex game. Optimizacija, 21, 30–35 (in Russian).
Vasil’ev, V. A. (2003). Extreme points of the Weber polytope. Discretnyi Analiz i Issledovaniye Operatsyi, Series 1, 10, 17–55 (in Russian).
Acknowledgements
This research was finished while the first author was visiting Tinbergen Institute and VU University Amsterdam, under grant Ref. 24022011 of Seville University. Also, this visit was partially supported by Tinbergen Institute. Moreover, this work was presented in some conferences under financial support of the project ECO201017766.
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Algaba, E., Bilbao, J.M. & van den Brink, R. Harsanyi power solutions for games on union stable systems. Ann Oper Res 225, 27–44 (2015). https://doi.org/10.1007/s10479-012-1216-0
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DOI: https://doi.org/10.1007/s10479-012-1216-0