Equal support from others for unproductive players: efficient and linear values that satisfy the equal treatment and weak null player out properties for cooperative games

In cooperative games with transferable utilities defined on a variable set of players, we characterize the family of values satisfying efficiency, linearity, the equal treatment property, and the weak null player out property. The last property weakens the usual null player out property, and together with efficiency, it is interpreted as considering equal support from others for null players. Together with the fact that efficiency, linearity, and the equal treatment property characterize the Shapley value along with the usual null player out property, our result reveals how weakening the null player out property can expand the possibilities of solutions. The characterized family contains well-known values in the literature, such as the Shapley value, equal division value, equal surplus division value, and the egalitarian non-separable contributions value, etc. In addition, each value in the characterized family is determined by an infinite sequence of real numbers. Furthermore, the equal treatment property in our characterization can be replaced by the balanced contributions property for symmetric players. Comparing this result with the existing one also shows that how weakening the balanced contributions property more can expand the possibilities of solutions.

1. See Sect. 2.3 for definitions of the three basic axioms.

2. Yokote et al. (2021) consider the redistributions of payoffs to null players from a different perspective than this manuscript.

3. Moreover, by replacing the equal treatment property with a stronger axiom, they characterize the convex combination of the two values (i.e., egalitarian Shapley values).

4. We should probably call (N, v) a cooperative game with transferable utilities, however for the sake of simplicity, we call it just a game.

5. This fact is obtained by using Eqs. (17) and (18) in the proof of Theorem 1 in Sect. 4.2 and linearity of \(\varphi ^t\).

6. To be more precise, efficiency and null player out property together imply the null player axiom, which requires that any null player obtains zero payoffs in any game, and it is well known that this null player axiom characterizes the Shapley value together with the three basic axioms (on both \(\Gamma (N)\) and \(\Gamma \), Shapley 1953b).

7. The fact that the universe of players are infinite is important for Theorem 1 (and Theorem 3 below). For a discussion on the domain of games on a fixed player set and all their restricted games, see Theorem 2 below.

8. From the equation (3.5) in Choudhury et al. (2021), their generalized egalitarian Shapley values on \(\Gamma (N)\) can be a counterpart of the r-egalitarian Shapley values on \(\Gamma \) in Yokote et al. (2018). Thus, they are also different from the family of values characterized in our Theorem 1.

9. More precisely, Chameni (2012) and Radzik and Driessen (2013) use symmetry which is stronger than the equal treatment property. However, from Lemma 2 in Sect. 4.1 in this manuscript, symmetry is replaceable by the equal treatment property.

10. Yokote et al. (2019) call the weak null player out property “differential null player out”.

11. A counterexample is shown when \(n=2\) and 3; it is still an open question whether or not the same result is obtained when \(n=4\) or 5.

12. \({\bar{n}}\) represents the cardinality of the set \({\bar{N}}\).

13. Similarly, if we use weak monotonicity (van den Brink et al., 2013) instead of cont+gr+sur-monotonicity, the egaritarian Shapley value is characterized on \({\bar{\Gamma }}({\bar{N}})\) with \({\bar{n}} \ne 2\) and \({\bar{n}}<\infty \). This can be a parallel results of Casajus and Huettner (2014) on \(\Gamma (N)\) with \(n \ne 2\) and \(n<\infty \).

14. Because desirability implies the equal treatment property, for this axiom it is more accurate to say that strengthening an existing axiom rather than adding a new axiom.

15. This fact is parallel to the discussion by Radzik and Driessen (2013) that efficiency, linearity, and desirability together imply general acceptability on \(\Gamma (N)\).

16. Radzik and Driessen (2013) show that efficiency, linearity, desirability and well-known positivity (Kalai & Samet, 1987, also known as monotonicity in Radzik and Driessen 2013) together imply social acceptability on \(\Gamma (N)\). However, this does not imply that social acceptability can be replaced by the pair of desirability and positivity in (iii) of Theorem 3. A typical example of \(\alpha \) that satisfies the condition (iii) and the value obtained from it lacks positivity is \(\alpha =(1,0,0,\dots )\).

17. In the existing literature, the equal treatment property in this manuscript is often referred to as symmetry, and the symmetry in this manuscript is often referred to as anonymity.

18. More precisely, Theorem 2 in Malawski (2020) shows the fact by using additivity instead of linearity. Additivity is a special case of linearity with \(a=b=1\).

19. For each \(R \subseteq N {\setminus } ij\), \(v(R \cup i)\) appears in \(\lambda ^{(N,v)}_{T \cup i}\) for any \(T \subseteq N {\setminus } ij\) with \(T \supseteq R\). The coefficient of \(v(R \cup i)\) in \(\lambda ^{(N,v)}_{T \cup i}\) is \((-1)^{t-r}\). Furthermore, there are \(\left( {\begin{array}{c}n-2-r\\ t-r\end{array}}\right) \) coalitions that contain R and that have t elements. The same is true for \(v(R \cup j)\).

The author is grateful to Yukihiko Funaki, Marcin Malawski, and participants in the 17th European Meeting on Game Theory (SING17), 2022 Asian Meeting of the Econometric Society in East and South-East Asia (AMES2022), and the Summer Workshop on Game Theory and Experimental Economics 2022 for valuable comments on the previous version of this manuscript. This work was supported by JSPS KAKENHI Grant No. 19K01569.

Authors and Affiliations

1. Faculty of Economics, Fukuoka University, 8-19-1 Nanakuma, Jonan-ku, Fukuoka, 814-0180, Japan

   Takumi Kongo

Corresponding author

Correspondence to Takumi Kongo.

Conflict of interest

The author have no relevant financial or non-financial interests to disclose.

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