Abstract
By focusing on players’ relative contributions, we study some properties for values in positive cooperative games with transferable utilities. The well-known properties of symmetry (also known as “equal treatment of equals”) and marginality are based on players’ marginal contributions to coalitions. Both Myerson’s balanced contributions property and its generalization of the balanced cycle contributions property (Kamijo and Kongo Int J of Game Theory 39:563–571, 2010; BCC) are based on players’ marginal contributions to other players. We define relative versions of marginality and BCC by replacing marginal contributions with relative contributions, and examine efficient values satisfying each of the two properties. On the class of positive games, a relative variation of marginality is incompatible with efficiency, and together with efficiency and the invariance property with respect to the payoffs of players under a player deletion, a relative variation of BCC characterizes the proportional value and egalitarian value in a unified manner.
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Notes
Note that a value on \(\mathbf {G}\) is defined as \(f: \mathbf {G} \rightarrow \mathbb {R}^n\).
An order \((i_1,i_2,\ldots ,i_{n})\) is a sequence of \(n\) players such that each player appears only once.
Note that the quotient is well-defined by the definition of a value on \(\mathbf {G}_+\).
In Ortmann (2000), this property is called “preserving ratios.”
Note that if we consider the class of one-person games, this does not hold because any value satisfies relativity on the class of one-person games.
In the definitions of proportional and quasi-proportional players on \(\mathbf {G}\), we (implicitly or explicitly) require a condition \(v(k)=0\) (see Kamijo and Kongo 2012). This condition is irrelevant on \(\mathbf {G}_+\), and thus, we omit it.
Tijs and Driessen (1986) also examine ID. They call it the “dummy out property.”
Note that when \(Q(S,v)=v(S)\), a \(Q\)-player is a null player that is a dummy player with its singleton coalition worth being zero; there is no null player in games on \(\mathbf {G}_+\).
Theorem 3 seems to contradict Lemma 1 (ii) in Kamijo and Kongo (2012). However, the domains are different. On \(\mathbf {G}\), the contradiction mentioned in the proof of Theorem 3 does not matter when \(v(i)=v(j)=v(\{i,j\})=0\).
Casajus (2011a) proves that on any convex cone \(\mathcal {C} \subseteq \mathbf {G}\), differential marginality is equivalent to fairness introduced by Brink (2001). Differential marginality also characterizes the Banzhaf value (Banzhaf 1965; Casajus 2011b) and the Owen value (Owen 1977; Casajus 2010).
The null player property requires that a null player obtains zero value. Note that in the class of positive game, there are no null players. However, most characterizations using the null player property hold when we replace null players with dummy players. Therefore, to compare the results on the classes of all games and positive games, we sometimes replace null players with dummy players.
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Acknowledgments
The authors are grateful to an anonymous referee for valuable comments and suggestions. This study was partly supported by JSPS KAKENHI Grant Number 24730174 (Kongo).
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Kamijo, Y., Kongo, T. Properties based on relative contributions for cooperative games with transferable utilities. Theory Decis 78, 77–87 (2015). https://doi.org/10.1007/s11238-013-9402-3
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DOI: https://doi.org/10.1007/s11238-013-9402-3