On $\alpha$-constant-sum games

Abstract

Given any $\alpha \in [0,1]$, an $\alpha$-constant-sum game (abbreviated as $\alpha$-CS game) on a finite set of players, N, is a function that assigns a real number to any coalition $S\subseteq N$, such that the sum of the worth of the coalition S and the worth of its complementary coalition $N\backslash S$ is $\alpha$ times the worth of the grand coalition. This class contains the constant-sum games of Khmelnitskaya (Int J Game Theory 32:223–227, 2003) (for $\alpha = 1$) and games of threats of (Kohlberg and Neyman, Games Econ Behav 108:139–145, 2018) (for $\alpha = 0$) as special cases. An $\alpha$-CS game may not be a classical TU cooperative game as it may fail to satisfy the condition that the worth of the empty set is 0, except when $\alpha =1$. In this paper, we (i) extend the $\alpha$-quasi-Shapley value giving the Shapley value for constant-sum games and quasi-Shapley-value for threat games to any class of $\alpha$-CS games, (ii) extend the axiomatizations of Khmelnitskaya (2003) and Kohlberg and Neyman (2018) to any class of $\alpha$-CS games, and (iii) introduce a new efficiency axiom which, together with other classical axioms, characterizes a solution that is defined by exactly the Shapley value formula for any class of $\alpha$-CS games.

Appendix: The Proof of Theorem 2

Consider any $\alpha \in [0,1]$. Efficiency and symmetry follow from Theorem 1, while it is straightforward to see that the $\alpha$-quasi-Shapley value satisfies marginality on ${\mathcal {C}}_\alpha ^N$.

To prove uniqueness, suppose that $\phi :{\mathcal {C}}_\alpha ^N\rightarrow {\mathbb {R}}^N$ is a value on ${\mathcal {C}}_\alpha ^N$ that satisfies efficiency, symmetry and marginality.

According to Proposition 1, for any $\alpha$-CS game $\langle N,\mu \rangle \in {\mathcal {C}}_\alpha ^N$, there exist numbers $a_T \in {\mathbb {R}}$, $T \subseteq N$, $T\ne \emptyset$, such that

$$\begin{aligned} \mu =\sum _{T \subseteq N} a_T u^\alpha _T. \end{aligned}$$

(10)

.

Notice that in any game $\langle N,u_T^\alpha \rangle$, $\emptyset \ne T \subseteq N$, all players $i\in T$ are symmetric players, and every player $i\notin T$ is a null player, i.e., all of his marginal contributions are equal to zero.

Let the index I of a game $\langle N,\mu \rangle \in {\mathcal {C}}_\alpha ^N$ be the minimum number of terms under the summation in expression (10), i.e.,

$$\begin{aligned} \mu =\sum _{k=1}^I a_{T_k} u^\alpha _{T_k}, \end{aligned}$$

where all $a_{T_k}\ne 0$. Similar as the proof of uniqueness in Young (1985), we proceed the remaining part of the proof by induction on this index I (instead of the number of nonzero Harsanyi dividends).

If $I=0$, then $\langle N,\mu \rangle$ is a null game given by $\mu (S)=0$ for all $S \subseteq N$. Uniqueness of $\phi (N,\mu )$ then follows directly from efficiency and symmetry of $\phi$.

Assume now that $\phi (N,v)$ is uniquely determined whenever the index of $\langle N,\mu \rangle \in {\mathcal {C}}_\alpha ^N$ is at most I, and let $\langle N,\mu \rangle \in {\mathcal {C}}_\alpha ^N$ have index $I+1$ with expression

$$\begin{aligned} \mu =\sum _{k=1}^{I+1} a_{T_k} u^\alpha _{T_k}, \quad \text{ all } a_{T_k}\ne 0. \end{aligned}$$

Let $T=\cap _{k=1}^{I+1}T_k$. For all $i,j\in T$, symmetry implies that $\phi _i(N,\mu )=\phi _j(N,\mu )$. Hence, combined with the requirement of efficiency, it is sufficient to prove that $\phi _i(N,\mu )$ is uniquely determined when $i\notin T$. Define a game

$$\begin{aligned} \mu ^{(i)}=\sum _{k:i\in T_k}a_{T_k} u^\alpha _{T_k}. \end{aligned}$$

Obviously, the index of $\langle N,\mu ^{(i)}\rangle$ is at most I and, therefore, by the induction hypothesis, $\phi _i(N,\mu ^{(i)})$ is uniquely determined. Since i’s marginal contributions in the games $\langle N,\mu \rangle$ and $\langle N,\mu ^{(i)}\rangle$ coincide, by marginality of $\phi$, $\phi _i(N,\mu )=\phi _i(N,\mu ^{(i)})$ which is uniquely determined by the induction hypothesis.

Thus, we have shown that there can be only one value that satisfies efficiency, symmetry and marginality. Since the $\alpha$-quasi-Shapley value satisfies these axioms, it must be that $\phi =SH^{\alpha }$.