Abstract
Albizuri and Aurrekoetxea (Soc Choice Welf 26:571–596, 2006a) and Albizuri et al. (Games Econ Behav 57:1–17, 2006b) defined values for games in which the players are organized into an a priori coalition configuration. In games with coalition configuration, we suppose that players organize themselves into coalitions that are not necessarily disjoint. A player can belong to more than one a priori coalition. In this paper we redefine coalition configuration values by using the concept of share function, as introduced by van der Laan and van den Brink (Theory Decis 53:61–86, 2002). A share function assigns to every player in a game its share in the worth to be distributed. We also define and characterize a general class of share function for games with coalition configuration which contains among other values those introduced by Albizuri et al.
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Notes
A relation is: reflexive if every element in a set is related to itself; symmetric if it holds for each element a and b that if a is related to b then b is related to a ; and transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c.
A value function is efficient if, for every TU-game, it exactly distributes the worth that can be obtained by the “grand coalition”.
The dummy axiom requires that a player who does not add to or detract from any coalition when he joins it earns a zero payoff. According to the additivity axiom, if we sum two TU-games then the payoff distributed in the sum game is equal to the sum of the payoff of separate games.
Owen (1977) introduced this game under the name of “quotient game”.
In this formulation, the external game does not appear clearly, but is useful to determine the payoff in the internal game.
\(i\) and \(j\) are symmetric in \(\left( N,v\right) \in {\mathcal {G}}\) if \(v(S\cup \left\{ i\right\} )=v(S\cup \left\{ j\right\} )\) for all \(S\subseteq N\backslash \left\{ i,j\right\} \).
See van den Brink and van der Laan (2005) for a discussion about null games and share functions.
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We are grateful to two anonymous referees for very valuable comments and suggestions.
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Andjiga, N.G., Courtin, S. Coalition configurations and share functions. Ann Oper Res 225, 3–25 (2015). https://doi.org/10.1007/s10479-014-1754-8
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DOI: https://doi.org/10.1007/s10479-014-1754-8