Abstract
This paper presents an axiomatization of a value for games with coalition structures which is an alternative to the Owen Value. The motor of this new axiomatization is a consistency axiom based on an associated game, which is not a reduced game. The new value of an n-player unanimity game is the compound average of the new values of all the (n-1)-player unanimity games. The new value of a unanimity game allocates to bigger coalitions a larger share of the total wealth. Note that the Owen value allocates to all the coalitions the same share independently of their size.
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References
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Appendix
Appendix
In this appendix we offer a comparison of the Owen value and of the value developed in this paper for unanimity games with coalition structures of at most four players.
\({\left\langle {N,{\user1{\mathcal{B}}}} \right\rangle }\) | \(\psi {\left( {N,{\user1{\mathcal{B}}},u_{N} } \right)}\) | Owen Value |
{1} | 1 | 1 |
{{1}, {2}} | \(\frac{1}{2},\frac{1}{2}\) | \(\frac{1}{2},\frac{1}{2}\) |
{{1, 2}} | \(\frac{1}{2},\frac{1}{2}\) | \(\frac{1}{2},\frac{1}{2}\) |
{{1}, {2}, {3}} | \(\frac{1}{3},\frac{1}{3},\frac{1}{3}\) | \(\frac{1}{3},\frac{1}{3},\frac{1}{3}\) |
{{1}, {2, 3}} | \(\frac{2}{7},\frac{5}{{14}},\frac{5}{{14}}\) | \(\frac{1}{2},\frac{1}{4},\frac{1}{4}\) |
{{1, 2, 3}} | \(\frac{1}{3},\frac{1}{3},\frac{1}{3}\) | \(\frac{1}{3},\frac{1}{3},\frac{1}{3}\) |
{{1}, {2}, {3}, {4}} | \(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\) | \(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\) |
{{1}, {2}, {3, 4}} | \(\frac{{22}}{{98}},\frac{{22}}{{98}},\frac{{27}}{{98}},\frac{{27}}{{98}}\) | \(\frac{1}{3},\frac{1}{3},\frac{1}{6},\frac{1}{6}\) |
{{1}, {2, 3, 4}} | \(\frac{4}{{21}},\frac{{17}}{{63}},\frac{{17}}{{63}},\frac{{17}}{{63}}\) | \(\frac{1}{2},\frac{1}{6},\frac{1}{6},\frac{1}{6}\) |
{{1, 2}, {3, 4}} | \(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\) | \(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\) |
{{1, 2, 3, 4}} | \(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\) | \(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\) |
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Hamiache, G. A value for games with coalition structures. Soc Choice Welfare 26, 93–105 (2006). https://doi.org/10.1007/s00355-005-0044-4
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DOI: https://doi.org/10.1007/s00355-005-0044-4