Abstract
We suggest a value for finite coalitional games with transferable utility that are enriched by non-negative weights for the players. In contrast to other weighted values, players stand for types of agents and weights are intended to represent the population sizes of these types. Therefore, weights do not only affect individual payoffs but also the joint payoff. Two principles guide the behavior of this value. Scarcity: the generation of worth is restricted by the scarcest type. Competition: only scarce types are rewarded. We find that the types’ payoffs for this value coincide with the payoffs assigned by the Mertens value to their type populations in an associated infinite game.
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Notes
Owen’s (1972) multi-linear extension is not in line with our interpretation of weights. The domain of this extension is the standard cube on the player set, where the players’ weights represent their (independent) probabilities of “cooperation”.
Nowak and Radzik (1995) use strong monotonicity in their characterizations of the weighted Shapley values.
Definitions and notation in this section closely follow [Neyman (2002), Sections 3 and 8].
One easily checks that strong monotonicity in the game can be replaced by a weaker requirement in Theorem 1, marginality in the game: For all \(v,w\in \mathbb {V},\) \(i\in N,\) and \(s\in \mathbb {R}_{+}^{N}\) such that \(v\left( S\cup \left( i\right) \right) -v\left( S\right) =w\left( S\cup \left( i\right) \right) -w\left( S\right) \) for all \(S\subseteq N{\setminus } \left\{ i\right\} ,\) we have \(\varphi _{i}\left( v,s\right) =\varphi _{i}\left( w,s\right) .\)
For \(v\in \mathbb {V}\) and \(s\in \mathbb {R}_{+}^{N}\), let \(v^{s}\in \mathbb {V}\) be given by \(v^{s}\left( T\right) =\bar{v}\left( s_{T}\right) \) for all \(T\subseteq N,\) where \(s_{T}\in \mathbb {R}_{+}^{N}\) is given by \(\left( s_{T}\right) _{i}=s_{i}\) for all \(i\in T\) and \(\left( s_{T}\right) _{i}=0\) for all \(i\in N{\setminus } T.\) A referee suggested the value \(\varphi ^{\left( 5\right) }\) given by \(\varphi ^{\left( 5\right) }\left( v,s\right) =\mathrm {Sh}\left( v^{s}\right) \) for all \(v\in \mathbb {V}\) and \(s\in \mathbb {R}_{+}^{N}\). One easily checks that \(\varphi ^{\left( 5\right) }=\varphi ^{\left( 3\right) } \).
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Acknowledgments
We are grateful to Frank Huettner, Michael Kramm, and Philippe Solal as well as two anonymous rerefees for valuable comments on this paper. Financial support by the Deutsche Forschungsgemeinschaft for André Casajus (Grant CA 266/4-1) is gratefully acknowledged.
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Casajus, A., Wiese, H. Scarcity, competition, and value. Int J Game Theory 46, 295–310 (2017). https://doi.org/10.1007/s00182-016-0536-8
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DOI: https://doi.org/10.1007/s00182-016-0536-8