Abstract
The main focus of this note is on the interval Shapley value for interval games introduced by Alparslan Gök et al. (Cent. Eur. J. Oper. Res. 18:131–140, 2010). In the framework of interval games, we introduce the potential approach and prove that the interval Shapley value can be formulated as the vector of marginal contributions of a potential function.
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Notes
We use 〈N,v〉 and (N,v) to denote a TU game and an interval game, respectively.
A TU-game 〈N,v〉 is monotonic if v(S)≤v(T) for all S,T∈2N with S⊆T. We call an interval game (N,w) size monotonic, if its length TU game 〈N,|w|〉 is monotonic, where \(|w|(S)=\overline{w}(S)-\underline{w}(S)\) for all S⊆N. We denote by SMIGN the class of size monotonic interval games with player set N. Since 〈N,|w|〉 is monotonic for any (N,w) in KIGN, this implies KIGN⊆SMIGN.
The following example illustrates that the interval core does not satisfy the translation invariance property. That is, there exist an interval game (N,w) and an interval payoff vector a∈I(ℝ)N such that C(N,w+a)≠C(N,w)+a, where (w+a)(S)=w(S)+∑ i∈S a i for all S∈2N∖{∅}, and C(N,w)+a={b+a | b∈C(N,w)}.
Let N={1,2} and w({1})=[0,0], w({2})=[−1,0], w({1,2})=[0,0], and a=([1,3],[1,1]). Then (w+a)({1})=[1,3], (w+a)({2})=[0,1], (w+a)({1,2})=[2,4]. Let \(I=([\underline{I}_{1}, \overline{I}_{1}],[\underline{I}_{2}, \overline {I}_{2}]) \in C(w)\). By the definition of C(w), \(\underline{I}_{1} \geq0, \overline{I}_{1} \geq0, \underline{I}_{2} \geq -1, \overline{I}_{2}\geq0\), \(\underline{I}_{1} + \underline{I}_{2}=0\), and \(\overline{I}_{1} + \overline{I}_{2}=0\). Since \(\underline{I}_{1} \leq\overline{I}_{1}\), this forced \(\underline {I}_{1} =\overline{I}_{1}=0\). So, C(N,w)={([0,0],[0,0])}. Hence C(N,w)+a={a}={([1,3],[1,1])}. On the other hand, it is also easy to see that x=([2,3],[0,1])∈C(N,w+a). But x∉C(N,w)+a. Hence C(N,w+a)≠C(N,w)+a.
References
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Alparslan Gök SZ, Branzei R, Tijs S (2008b) Cooperative interval games arising from airport situations with interval data. Preprint no 107, Institute of Applied Mathematics, METU and Tilburg University, Center for Economic Research, The Netherlands, CentER DP 57
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Hwang, YA., Yang, WY. A note on potential approach under interval games. TOP 22, 571–577 (2014). https://doi.org/10.1007/s11750-012-0271-7
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DOI: https://doi.org/10.1007/s11750-012-0271-7