Abstract
In this paper, we give a new axiomatization of the weighted Shapley value. We investigate the asymmetric property of the value by focusing on the invariance of payoff after the change in the worths of singleton coalitions. We show that if the worths change by the same amount, then the Shapley value is invariant. On the other hand, if the worths change with multiplying by a positive weight, then the weighted Shapley value with the positive weight is invariant. Based on the invariance, we formulate a new axiom, \(\omega \)-Weak Addition Invariance. We prove that the weighted Shapley value is the unique solution function which satisfies \(\omega \)-Weak Addition Invariance and Dummy Player Property. In the proof, we introduce a new basis of the set of all games. The basis has two properties. First, when we express a game by a linear combination of the basis, coefficients coincide with the weighted Shapley value. Second, the basis induces the null space of the weighted Shapley value. By generalizing the new axiomatization, we also axiomatize the family of weighted Shapley values.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We can check that for any \(\omega \in \mathbb {R}^n_{++}\) and \(S\subset N\), \(|S|\ge 2\), player \(j\in N\backslash S\) is a null player in \(\chi _S^\omega \).
If \(S=N\), Condition \(3\) is redundant.
We explain the reason. Let us focus on the right-hand side of the equation in the statement. Then, it will be clear that \(R\) gains a non-zero value only if a game \(u_T^\omega \) such that \(T\subseteq S, T\subseteq R\) is chosen in the second summation. In this case, \(T\subseteq R\cap S\). So, if \(|R\cap S|=0\), \(R\) gains \(0\). This point of view also explains the first equality in the transformation below. For any player \(i\in R\cap S\), \(\omega _i\) is added when a coalition \(T\subseteq R\cap S\) such that \(i\in T\) is chosen. Such a coalition \(T\) is determined by choosing \(k\), \(0\le k \le r-1\), players from \((R\cap S)\backslash \{i\}\).
The weighted Shapley value with any positive weight can be expressed as a random order value. Given an arbitrary positive weight, we can derive a probability distribution over \(\mathbf {R}(N)\) with which the random order value coincides with the weighted Shapley value. For the way to calculate the probability, see, for example, Chun (1991).
Note that the games \(\chi _S^{1\omega }\) and \(\chi _S^\omega \) represent the same game.
The game \(u_{\{i\}}\) can be obtained by multiplying \(1/\omega _i\) to game \(\chi ^{\omega }_{\{i\}}\). After the scalar multiplication, linear independence is preserved.
For any player \(j\in N\), \(j\ne i\), we have \(\phi ^\omega _i(u_{\{j\}})=0\) since \(i\) is a null player. In addition, for any coalition \(S\subseteq N\), \(|S|\ge 2\), \(\phi ^\omega _i(\chi _S^{\omega })=0\) from Lemma 2.
We allow the possibility that the coordinate of \(\omega \) is not positive.
In a precise manner, the left-hand side is written as \(\psi (\{i,j\}, \chi _{\{i,j\}}^{(\omega _i, \omega _j)})\). For simplicity, we write \(\omega \) instead of \((\omega _i, \omega _j)\).
The difference between Strategic Invariance and Weak Strategic Invariance is that, in the case of Strategic Invariance, we assume linearity for positive scalar multiplication.
We use the same notations given by Kalai and Samet (1987).
The relationship between positive weight and weight system is discussed by Monderer et al. (1992).
References
Béal S, Rémila E, Solal P (2012) Axioms of invariance for TU-games. MPRA Paper No. 41530. http://mpra.ub.uni-muenchen.de/id/eprint/41530 Accessed 11 October 2013
Chun Y (1991) On the symmetric and weighted shapley values. Int J Game Theory 20:183–190
Derks JJM, Haller HH (1999) Null players out? Linear values for games with variable supports. Int Game Theory Rev 1:301–314
Harsanyi JC (1959) A bargaining model for cooperative n-person games. In: Tucker AW, Luce RD (eds) Contributions to the theory of games IV. Princeton University Press, Princeton, pp 325–355
Hart S, Mas-Colell A (1989) Potential, value and consistency. Econometrica 57:589–614
Kamijo Y, Kongo T (2010) Axiomatization of the Shapley value using the balanced cycle contributions property. Int J Game Theory 39:563–571
Kamijo Y, Kongo T (2012) Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value and the Banzhaf value. Eur J Oper Res 216:638–646
Kalai E, Samet D (1987) On weighted Shapley values. Int J Game Theory 16:205–222
Monderer D, Samet D, Shapley LS (1992) Weighted values and the core. Int J Game Theory 21(1):27–39
Pérez-Castrillo D, Wettstein D (2001) Bidding for the surplus: a non-cooperative approach to the Shapley value. J Econ Theory 100(2):274–294
Shapley LS (1953) A value for n-person games. In: Roth AE (ed) The Shapley value. Cambridge University Press, Cambridge, pp 41–48
Sobolev AI (1973) The functional equations that give the payoffs of the players in an n-person games. In: Vilkas E (ed) Advances in Game Theory, Izdat., “Mintis” Vilnius (in Russian), pp 151–153
Sobolev AI (1975) The characterization of optimality principles in cooperative games by functional equations. In: Vorobiev NN (ed) Mathematical methods in social sciences (in Russian), 6, Vilnius, pp 95–151
van den Brink R (2007) Null or nullifying players: the difference between the Shapley value and equal division solutions. J Econ Theory 136(1):767–775
Yokote K (2013) Strong addition invariance and axiomatization of the weighted Shapley value. Waseda Economics Working Paper Series No.13-001, Waseda University. http://www.waseda-pse.jp/file/keiken/32-13-001 Accessed 25 February 2014
Yokote K, Funaki Y, Kamijo Y (2013a) Linear basis approach to the Shapley value. Institute of Research in Contemporary Political and Economic Affairs Working Paper No.E1303, Waseda University. http://www.waseda-pse.jp/ircpea/jp/publish/working-paper-e-series/ Accessed 25 February 2014
Yokote K, Funaki Y, Kamijo K (2013b) Relationship between the Shapley value and other solution concepts. Institute of Research in Contemporary Political and Economic Affairs Working Paper No.E1304, Waseda University. http://www.waseda-pse.jp/ircpea/jp/publish/working-paper-e-series/ Accessed 25 February 2014
Young HP (1985) Monotonic solutions of cooperative games. Int J Game Theory 14(2):65–72
Author information
Authors and Affiliations
Corresponding author
Additional information
The author thanks Yukihiko Funaki and Yoshio Kamijo for their helpful comments and cooperation.
Rights and permissions
About this article
Cite this article
Yokote, K. Weak addition invariance and axiomatization of the weighted Shapley value. Int J Game Theory 44, 275–293 (2015). https://doi.org/10.1007/s00182-014-0429-7
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00182-014-0429-7