Abstract
We propose and analyze a new type of values for cooperative TUgames, which we call pyramidal values. Assuming that the grand coalition is sequentially formed, and all orderings are equally likely, we define a pyramidal value to be any expected payoff in which the entrant player receives a salary, and the rest of his marginal contribution to the just formed coalition is distributed among the incumbent players. We relate the pyramidaltype sharing scheme we propose with other sharing schemes, and we also obtain some known values by means of this kind of pyramidal procedures. In particular, we show that the Shapley value can be obtained by means of an interesting pyramidal procedure that distributes nonzero dividends among the incumbents. As a result, we obtain an alternative formulation of the Shapley value based on a measure of complementarity between two players. Finally, we introduce the family of proportional pyramidal values, in which an incumbent receives a dividend in proportion to his initial investment, measured by means of his marginal contribution.
This is a preview of subscription content, access via your institution.
Notes
 1.
In the sequel, for convenience, we will write singleton {i} just as i.
 2.
Note that these dividends are not the same as the wellknown Harsanyi dividends, which are associated to coalitions, not only to agents.
 3.
Those negative shares can be interpreted as investments on human capital.
 4.
This concept was implicitly first considered by Owen (1972) who defined the covalue q _{ ij }(v) of i and j. The interaction index (11) differs from Owen’s covalue in the number of orders in which \(\Delta_{ij}^{2} (S)\) is considered. Owen’s covalue takes into account only those orders in which i, S players, and j arrive in the first places.
References
Béal, S., Rémila, E., & Solal, P. (2012). Compensations in the Shapley value and the compensation solutions for graph games. International Journal of Game Theory, 41, 157–178.
Casajus, A., & Huettner, F. (2013). Null players, solidarity, and the egalitarian Shapley values. Journal of Mathematical Economics, 49, 58–61.
Castro, J., Gomez, D., & Tejada, J. (2009). Polynomial calculation of the Shapley value based on sampling. Computers and Operations Research, 36, 1726–1730.
Derks, J. J. M., & Haller, H. H. (1999). Null players out? Linear values for games with variable supports. International Game Theory Review, 1, 301–314.
Grabisch, M., & Roubens, M. (1999). An axiomatic approach to the concept of interaction among players in cooperative games. International Journal of Game Theory, 28, 547–565.
Ichiisi, T. (1981). Supermodularity: applications to convex games and to the greedy algorithm for LP. Journal of Economic Theory, 25, 283–286.
Joosten, R. (1996). Dynamics, equilibria and values. Dissertation, Maastricht University.
Ju, Y., Borm, P., & Ruys, P. (2007). The consensus value: a new solution concept for cooperative games. Social Choice and Welfare, 28, 685–703.
Kalai, E., & Samet, D. (1987). On weighted Shapley values. International Journal of Game Theory, 16, 205–222.
Malawski, M. (2013). “Procedural” values for cooperative games. International Journal of Game Theory, 42, 305–324.
Maschler, M., & Peleg, B. (1966). A characterization, existence proof and dimension bounds for the kernel of a game. Pacific Journal of Mathematics, 18, 289–328.
Myerson, R. B. (1977). Graphs and cooperation in games. Mathematics of Operations Research, 2, 225–229.
Owen, G. (1972). Multilinear extensions of games. Management Sciences, 18, 64–79.
Segal, I. (2003). Collusion, exclusion and inclusion in randomorder bargaining. Review of Economic Studies, 70, 439–460.
Shapley, L. S. (1953). A value for nperson games. In Contributions to the theory of games II, pp. 307–317.
van den Brink, R., Funaki, Y., & Ju, Y. (2013). Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values. Social Choice and Welfare, 40, 693–714.
Weber, R. J. (1988). Probabilistic values for games. In A. Roth (Ed.), The Shapley value: Essays in honor of Lloyd S. Shapley (pp. 101–119). Cambridge: Cambridge University Press.
Young, H. P. (1985). Monotonic solutions of cooperative games. International Journal of Game Theory, 14, 65–72.
Acknowledgements
We would like to warmly thank the referees for their careful reports, that definitely improved the quality of our paper and made it much more readable and interesting.
Author information
Affiliations
Corresponding author
Additional information
This research has been supported by I+D+i research project MTM201127892 from the Government of Spain.
Appendix
Appendix
In this appendix we collect the formal definitions of all the known values analyzed in Sect. 3, as well as the characterization results we have used.
Theorem 1
(Shapley 1953)
There exists a unique value satisfying the efficiency, symmetry, dummy, and additivity axioms. It is the Shapley value, which is defined for every (N,v)∈G _{ n } as follows:
where s=S denotes the cardinality of coalition S⊆N.
The Consensus value (Ju et al. 2007) is aimed to generalize the standard solution for 2person TU games into nperson cases. It is based on a twosided negotiation process that can be understood as a standardized remainder rule described by the following vectors. The reader is referred to Ju et al. (2007) for a detailed exposition of this rule.
Definition 6
(Ju, Borm and Ruys 2007)
Let (N,v)∈G _{ n }, and π∈Π(N) be a given permutation. Define \(S_{k}^{\pi}=\{ \pi^{1}(1),\dots, \pi^{1}(k)\}\subseteq N\) and \(S_{0}^{\pi }=\emptyset\). Then, the standardized remainder for coalition \(S_{k}^{\pi }\), \(r(S_{k}^{\pi})\), is recursively defined as follows:
\(r(S_{k}^{\pi})\) is the value left for \(S_{k}^{\pi}\) after allocating surpluses to earlier leavers \(N\setminus S_{k}^{\pi}\). Then, the standardized remainder vector, sr ^{π}(v), which corresponds to the situation where the players leave the game one by one in the order (π ^{−1}(n),…,π ^{−1}(1)), is defined recursively by:
Definition 7
(Ju, Borm and Ruys 2007)
For every (N,v)∈G _{ n }, the consensus value Ψ(v) is defined as the average, over the set of all permutation Π(N), of the individual standardized remainder vectors, i.e.,
Definition 8
(Ju, Borm and Ruys 2007)
For every (N,v)∈G _{ n } and α∈[0,1], the αconsensus value Ψ ^{α}(v) is defined as the average, over the set of all permutation Π(N), of the individual αremainder vectors, i.e.,
Here, the αremainder \(r^{\alpha}(S_{k}^{\pi})\) and the individual αremainder vector (sr ^{π})^{α}(v) are defined as follows:
and
The authors introduce the following property in order to characterize the family of consensus values. The next theorem corresponds to Theorem 5 in Ju, Borm and Ruys (2007). We make use of (a) characterization.
Definition 9
(Ju, Borm and Ruys 2007)
A value \(\varphi: G_{n}\rightarrow\mathbb{R}^{n}\) verifies the αdummy property if \(\varphi_{i}(v)=\alpha v(i) + (1\alpha) ( v(i) + \frac{v(N)\sum_{j\in N} v(j)}{n} )\), for all (N,v)∈G _{ n }, and every dummy player i∈N with respect to v.
Theorem 2
(Ju, Borm and Ruys 2007)

(a)
The αconsensus value Ψ ^{α} is the unique onepoint solution concept on G _{ n } that satisfies efficiency, symmetry, the αdummy property and additivity.

(b)
The αconsensus value Ψ ^{α} is the unique function that satisfies efficiency, symmetry, the αdummy property and the transfer property over the class of TU games.

(c)
For any v∈G _{ n }, it holds that
$$\varPsi^{\alpha}(v)=\alpha\alpha\phi(v) + (1\alpha) E(v), $$where E(v) is the equal surplus solution of v, i.e., \(E_{i}(v)=v(i)+\frac{v(N)\sum_{j\in N} v(j)}{n}\).

(d)
The αconsensus value Ψ ^{α} is the unique function that satisfies efficiency and the αequal welfare loss property over the class of TU games.
The Egalitarian Shapley values (Joosten 1996) make the tradeoff between marginalism and egalitarianism by means of convex combinations of the Shapley value and the equal division solution.
Definition 10
(Joosten 1996)
For every (N,v)∈G _{ n } and α∈[0,1], the αegalitarian Shapley value φ ^{α}(v) is given by
where ED(v) is the equal division value which distributes the worth v(N) equally among all players: \(ED(v)=(\frac{v(N)}{n},\dots,\frac{v(N)}{n})\).
Rights and permissions
About this article
Cite this article
Flores, R., Molina, E. & Tejada, J. Pyramidal values. Ann Oper Res 217, 233–252 (2014). https://doi.org/10.1007/s104790131509y
Published:
Issue Date:
Keywords
 Game theory
 TU games
 Pyramidal values
 Procedural values
 Shapley value
 Covalues
 Consensus values
 Egalitarian Shapley values