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On the Core and Shapley Value for Regular Polynomial Games

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Abstract

Considering some classes of polynomial cooperative games, we describe the integral representation of the Shapley values and the support functions of their cores. Also, we analyze the relationship between the Shapley values and the polar forms of homogeneous polynomial games. The found formula for the support function of the core of a convex game is applied for the dual description of the Harsanyi sets of finite cooperative games. The main peculiarity of the proposed approach to the study of optimal solutions of game theory is a systematic use of the extensions of polynomial set functions to the corresponding measures on symmetric powers of the initial measure spaces.

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Notes

  1. The synonym Shapley vector is also used in the literature (see, for example, [2]).

  2. In what follows, when the indication of the feasibility of the domain of definition of the functions of the class \( {\mathcal{K}}\subseteq{\mathcal{V}} \) under consideration is relevant, we use the more detailed notation \( {\mathcal{K}}(B) \).

  3. The polar form of a homogeneous polynomial functional \( L \) of order \( n \) on a vector space \( X \) is the function \( L^{*}:X^{n}\rightarrow{𝕉} \) defined by the formula \( L^{*}(x_{1},\dots,x_{n}):=\frac{1}{n!}\sum\nolimits_{\omega\subseteq\{1,\dots,n\}}(-1)^{n-|\omega|}L(\sum\nolimits_{i\in\omega}x_{i}) \) (see, for example, [16, 18]).

  4. The homogeneity, polynomiality, and continuity of the functional \( P_{v} \) for a homogeneous game \( v \) were established in [5] (see also [17]).

  5. Recall (see [2]) that a finite game \( v\in V(Q) \) is called convex if \( v(e\cup e^{\prime})+v(e\cap e^{\prime})\geq v(e)+v(e^{\prime}) \) for all \( e,e^{\prime}\subseteq Q \).

  6. As usual, for \( x=(x_{i})_{i\in Q}\in{𝕉}^{Q} \) and \( e\subseteq Q \), we put \( x(e):=\sum\nolimits_{i\in e}x_{i} \).

  7. Recall [6, 21] that a game \( v \) is called almost positive if all its dividends \( v_{e} \) corresponding nontrivial coalitions are nonnegative: \( |e|\geq 2\Rightarrow v_{e}\geq 0 \).

  8. A player \( i\in e \) is called \( \pi \)-first (\( \pi \)-last) in \( e \) if all remaining player in \( e \) succeed \( i \) (precede \( i \)).

  9. A function \( \beta \) is called supermodular if \( \beta(e_{1}\cup e_{2})+\beta(e_{1}\cap e_{2})\geq\beta(e_{1})+\beta(e_{2}) \) for all \( e_{1},e_{2}\in{\mathcal{L}} \) (when there is always equality instead of inequality, \( \beta \) is called additive).

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Funding

The work was supported by the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences (Grant no. 0314–2019–0018) and the Russian Foundation for Basic Research (Grant no. 19–10–00910).

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  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    V. A. Vasil’ev

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  1. V. A. Vasil’ev
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Correspondence to V. A. Vasil’ev.

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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 1, pp. 77–94. https://doi.org/10.33048/smzh.2022.63.105

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Vasil’ev, V.A. On the Core and Shapley Value for Regular Polynomial Games. Sib Math J 63, 65–78 (2022). https://doi.org/10.1134/S0037446622010050

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