Pyramidal values

Abstract

We propose and analyze a new type of values for cooperative TU-games, 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 pyramidal-type 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.

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:

$$ \displaystyle \phi_i (N,v) = \sum_{\substack{S\subseteq N\\ i \notin S}} \frac{s!(n-s-1)!}{n!} \bigl( v(S\cup\{ i\}) - v(S) \bigr), \quad i=1,\dots,n, $$

(19)

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 2-person TU games into n-person cases. It is based on a two-sided 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:

$$r(S_k^{\pi})= \begin{cases} v(N), & \text{ if $k=n$,} \\ v(S_k^{\pi})+\frac{1}{2} (r(S_{k+1}^{\pi}) - v(S_k^{\pi}) - v(\{ \pi^{-1} (k+1)\}) ), & \text{ if $k\in\{1,\dots,n-1\}$.} \end{cases} $$

$r(S_{k}^{\pi})$ 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:

$$sr_{\pi^{-1}(k)}= \begin{cases} v(\{ \pi^{-1} (k)\}) + \frac{1}{2} (r(S_{k}^{\pi}) - v(S_{k-1}^{\pi}) - v(\{ \pi^{-1} (k)\}) ), & \text{ if $k\in\{ 2,\dots,n\}$,} \\ r(S_1^{\pi}) , & \text{ if $k=1$.} \end{cases} $$

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.,

$$\varPsi(v)=\frac{1}{n!} \sum_{\pi\in\varPi(N)} sr^{\pi} (v). $$

Definition 8

(Ju, Borm and Ruys 2007)

For every (N,v)∈G _n and α∈[0,1], the α-consensus value Ψ ^α(v) is defined as the average, over the set of all permutation Π(N), of the individual α-remainder vectors, i.e.,

$$\varPsi^{\alpha}(v)=\frac{1}{n!} \sum_{\pi\in\varPi(N)} (sr^{\pi})^{\alpha} (v). $$

Here, the α-remainder $r^{\alpha}(S_{k}^{\pi})$ and the individual α-remainder vector (sr ^π)^α(v) are defined as follows:

$$r^{\alpha}(S_k^{\pi})= \begin{cases} v(N), & \text{ if $k=n$,} \\ v(S_k^{\pi})+ (1-\alpha) (r^{\alpha}(S_{k+1}^{\pi}) - v(S_k^{\pi }) - v(\{ \pi^{-1} (k+1)\}) ), & \text{ if $k\in\{1,\dots,n-1\}$.} \end{cases} $$

and

$$(sr_{\pi^{-1}(k)})^{\alpha}= \begin{cases} v(\{ \pi^{-1} (k)\}) + \alpha (r^{\alpha}(S_{k}^{\pi }) - v(S_{k-1}^{\pi}) - v(\{ \pi^{-1} (k)\}) ), & \text{ if $k\in \{2,\dots,n\}$,} \\ r^{\alpha}(S_1^{\pi}) , & \text{ if $k=1$.} \end{cases} $$

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 α-dummy 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 α-consensus value Ψ ^α is the unique one-point solution concept on G _n that satisfies efficiency, symmetry, the α-dummy property and additivity.

• (b) The α-consensus value Ψ ^α is the unique function that satisfies efficiency, symmetry, the α-dummy 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 α-consensus value Ψ ^α is the unique function that satisfies efficiency and the α-equal welfare loss property over the class of TU games.

The Egalitarian Shapley values (Joosten 1996) make the trade-off 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 α-egalitarian Shapley value φ ^α(v) is given by

$$\varphi^{\alpha}(v)=\alpha\phi(v) + (1-\alpha) ED(v), $$

where ED(v) is the equal division value which distributes the worth v(N) equally among all players: $ED(v)=(\frac{v(N)}{n},\dots,\frac{v(N)}{n})$.