Abstract
Multinomial values were previously introduced by one of the authors in reliability and extended later to all cooperative games. Here, we present for this subfamily of probabilistic values three new results, previously stated only for binomial semivalues in the literature. They concern the dimension of the subspace spanned by the multinomial values and two characterizations: one, individual, for each multinomial value; another, collective, for the whole subfamily they form. Finally, an application to simple games is provided.
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Notes
In particular, all binomial semivalues, but also the Shapley value, satisfy this property.
The reader might well like to see this argument illustrated for, say, \(n=4\). Let, e.g., \(\mathbf{p}=(p_1,p_2,0,0)\) with \(p_1,p_2>0\). For singletons, we have \(\phi _i[u_{\{j\}}]=1\) if \(i=j\) and 0 otherwise. For any other nonempty T, \(\phi _i[u_T]=0\) if \(i\notin T\). Coalition \(\{1,2\}\) is the only of type (a), and we get \(\phi _i[u_{\{1,2\}}]=p_j\) if \(\{i,j\}=\{1,2\}\). Next, for coalitions \(\{1,3\}\), \(\{1,4\}\), \(\{2,3\}\), \(\{2,4\}\), \(\{1,2,3\}\) and \(\{1,2,4\}\) we have \(\phi _i[u_T]=0\) if \(i=1,2\). Then, from \(\phi _1[u_{\{1,2,3\}}]+\phi _2[u_{\{1,2,3\}}]+\phi _3[u_{\{1,2,3\}}]=p_1p_2\) we find \(\phi _3[u_{\{1,2,3\}}]=p_1p_2\), and something analogous occurs for the remaining coalitions of the list. Finally, for \(\{3,4\}\), \(\{1,3,4\}\), \(\{2,3,4\}\) and N, we find \(\phi _i[u_T]=0\) for all \(i\in N\). Here, we are exclusively applying, as in the proof of Theorem 4.6, linearity (of course), positivity, the dummy player property, the \(\mathbf{p}\)-multinomial total power property, and the property of \(\mathbf{p}\)-weighted payoffs for unanimity games.
The dictatorial index \(\psi ^0\) satisfies (i) only, with \(p_0=1\) and \(\mu =0\). The marginal index \(\psi ^1\) satisfies (ii) only, with \(p_{n-1}=1\) and \(\mu '=0\). Any other binomial semivalue, with \(q\ne 0,1\), satisfies (i) and (ii) because \(\mu =\dfrac{1-q}{q}\ne 0\); thus, \(q=\dfrac{\mu }{1+\mu }\) and \(p_0=\dfrac{1}{(1+\mu )^{n-1}}\).
We use this term to emphasize that exceptionality corresponds to the next option, that of magnetic player.
Some points of the argument that follows are well illustrated by, e.g., the particular case \(n=4\).
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Acknowledgements
The authors wish to thank two anonymous reviewers for their helpful comments.
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This research project was partially supported by funds from the Spanish Ministry of Economy and Competitiveness (MINECO) and from the European Union (FEDER funds) under Grant MTM2015-66818-P (MINECO/FEDER).
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Carreras, F., Puente, M.A. A note on multinomial probabilistic values. TOP 26, 164–186 (2018). https://doi.org/10.1007/s11750-017-0464-1
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DOI: https://doi.org/10.1007/s11750-017-0464-1