Abstract
This work deals with the weighted excesses of players in cooperative games which are obtained by summing up all the weighted excesses of all coalitions to which they belong. We first show that the resulting payoff vector is the corresponding least square value by lexicographically minimizing the individual weighted excesses of players over the preimputation set, and thus give an alternative characterization of the least square values. Second, we show that these results give rise to lower and upper bounds for the core payoff vectors and, using these bounds, we show that the least square values can be seen as the center of a polyhedron defined by these bounds. This provides a second new characterization of the least square values. Third, we show that the individually rational least square value is the solution that lexicographically minimizes the individual weighted excesses of players over the imputation set.
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Notes
A cost game is defined similar to a (profit) game, except that the interpretation of the worth of a coalition is the total cost that the coalition of players has to face jointly. Some solutions need to be redefined accordingly, for example the (anti-)core of a cost game is the set of efficient payoff vectors such that no coalition pays more than its own cost.
This follows from substituting \(a^p_i(v)\) in \(lo^p_i(v)+up^p_i(v)= \frac{a^p_i(v)-\beta v(N)}{\alpha } + \frac{\alpha v(N)-\sum _{S\subseteq N\setminus \{i\}}p_sv(S)}{\alpha } = \frac{\sum _{S \subseteq N \setminus \{i\}} p_sv(S) + (\alpha -\beta ) v(N)-\sum _{S \subseteq N, S \ni i} p_sv(S)}{\alpha } = \frac{\sum _{S\subseteq N\setminus \{i\}}(p_{s+1}v(S\cup \{i\})-p_{s}v(S))+(\alpha -\beta )v(N)}{\alpha }\).
This follows from substituting \(a^p_i(v)\) in \(lo^p_i(v)-up^p_i(v)= \frac{a^p_i(v)-\beta v(N)}{\alpha } - \frac{\alpha v(N)-\sum _{S\subseteq N\setminus \{i\}}p_sv(S)}{\alpha } = \frac{\sum _{S \subseteq N \setminus \{i\}} p_sv(S) - (\alpha +\beta ) v(N)+\sum _{S \subseteq N, S \ni i} p_sv(S)}{\alpha } = \frac{\sum _{S\subseteq N}p_sv(S)-(\alpha +\beta )v(N)}{\alpha }\).
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Acknowledgements
The research has been supported by the National Natural Science Foundation of China (Grant Nos. 72071158) and the China Scholarship Council (Grant No. 201906290164). We thank two anonymous reviewers for comments on an earlier draft of this paper.
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Zhang, X., Brink, R.v.d., Estévez-Fernández, A. et al. Individual weighted excess and least square values. Math Meth Oper Res 95, 281–296 (2022). https://doi.org/10.1007/s00186-022-00781-1
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DOI: https://doi.org/10.1007/s00186-022-00781-1