Monotonicity implies linearity: characterizations of convex combinations of solutions to cooperative games

Abstract

The purpose of this study is to provide a comprehensive characterization of linear solutions to cooperative games by using monotonicity. A monotonicity axiom states an increase in certain parameters of a game as a hypothesis and states an increase in a player’s payoff as a conclusion. We focus on various parameters of a game and introduce new axioms. Combined with previous results, we prove that efficiency, symmetry and a monotonicity axiom characterize (i) four linear solutions in the literature, namely, the Shapley value, the equal division value, the CIS value and the ENSC value, and (ii) a class of solutions obtained by taking a convex combination of the above solutions. Our methodological contribution is to provide a new linear algebraic approach for characterizing solutions by monotonicity. Using a new basis of the linear space of TU games, we identify a class of games in which a solution that satisfies monotonicity is linear. Our approach provides some intuition for why monotonicity implies linearity.

Appendix

The purpose of this appendix is to introduce a solution $\varphi$ for $n=3$ that satisfies efficiency, symmetry, $cont+sur$-monotonicity, but is not a convex combination of linear solutions. Since $cont+sur$-monotonicity is stronger than $cont+gr+sur$-monotonicity, this solution is a counterexample to Theorems 1 and 2 for $n=3$.

Let $N=\{1,2,3\}$. We first focus on the unanimity games. For $\lambda \in {\mathbb {R}}$ and $T\subseteq N$, $T\ne \emptyset$, we define $\varphi (\lambda u_T)$ as follows:

$$\begin{aligned} \varphi (\lambda u_T)={\left\{ \begin{array}{ll} ED(\lambda u_T) &{} \text { if } \lambda <0 \text { and } |T|=2, \\ Sh(\lambda u_T) &{} \text { otherwise. } \end{array}\right. } \end{aligned}$$

Since $\{u_T\}_{\emptyset \ne T\subseteq N}$ is a basis of $\varGamma$, we can uniquely express $v\in \varGamma$ as a linear combination of $\{u_T\}_{\emptyset \ne T\subseteq N}$. Let $\{d^v_T\}_{\emptyset \ne T\subseteq N}$ denote the coefficients in the linear combination.¹⁰ We define $\varphi$ for a general game $v\in \varGamma$ as follows:

$$\begin{aligned} \varphi (v)=\sum _{T\subseteq N: T\ne \emptyset }\varphi (d^v_T u_T). \end{aligned}$$

Let $v,w\in \varGamma$ be games such that $\varDelta _1v(S) \ge \varDelta _1w(S)$ for all $S\subseteq N\backslash 1$ and $v(N)-\sum _{i\in N} v(i)\ge w(N)-\sum _{i\in N}w(i)$. Our goal is to prove that

$$\begin{aligned} \varphi _1(v)-\varphi _1(w)= & {} \sum _{T\in 2^N\backslash \emptyset } \varphi _1(d^v_T u_T)-\sum _{T\in 2^N\backslash \emptyset }\varphi _1(d^w_T u_T) \\= & {} \sum _{T\in 2^N\backslash \{\emptyset , \{2\}, \{3\}\}} \{\varphi _1(d^v_T u_T)-\varphi _1(d^w_T u_T)\}\ge 0. \end{aligned}$$

For each $T\subseteq N$, $T\ne \emptyset$, let $\delta _T=d^v_T-d^w_T$. We provide a claim that immediately follows from the definition of $\varphi$.

Claim 18

The following two statements hold : 

• (i) Let $i\in N\backslash \{1\}$. If $\delta _{1i}\ge 0,$ then $\varphi _1(d^v_{1i}u_{1i})-\varphi _1(d^w_{1i}u_{1i})\ge \frac{\delta _{1i}}{3}$. If $\delta _{1i}\le 0,$ then $\varphi _1(d^v_{1i}u_{1i})-\varphi _1(d^w_{1i}u_{1i})\ge \frac{\delta _{1i}}{2}$.

• (ii) If $\delta _{23}\ge 0,$ then $\varphi _1(d^v_{23}u_{23})-\varphi _1(d^w_{23}u_{23})\ge 0$. If $\delta _{23}\le 0,$ then $\varphi _1(d^v_{23}u_{23})-\varphi _1(d^w_{23}u_{23})\ge \frac{\delta _{23}}{3}$.

In what follows, Claim 18 is abbreviated as C18.

If $\delta _1<0$, then $v(1)<w(1)$, which is a contradiction with $\varDelta _1 v(S)\ge \varDelta _1 w(S)$ for all $S\subseteq N\backslash 1$. Thus, $\delta _1\ge 0$. Depending on the signs of $\delta _{12}$, $\delta _{13}$, $\delta _{23}$ and $\delta _{123}$, we consider 16 cases.

If $\delta _T\ge 0$ for all $T\subseteq N$, $|T|\ge 2$, then $\varphi _1(v)\ge \varphi _1(w)$ immediately holds. If $\delta _T<0$ for all $T\subseteq N$, $|T|\ge 2$, we obtain a contradiction with $v(N)-\sum _{i\in N} v(i)\ge w(N)-\sum _{i\in N} w(i)$. In what follows, we consider other 14 cases.

• Case 1: ${\varvec{\delta }}_{\mathbf{12}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}\ge \mathbf{0}$.

  In order that $v(12)-v(2)\ge w(12)-w(2)$, we must have $\delta _1\ge -\delta _{12}$. Then,

  $$\begin{aligned} \sum _{T\in 2^N\backslash \{\emptyset , \{2\}, \{3\}\}}\bigl \{\varphi _1(d^v_T u_T)-\varphi _1(d^w_T u_T)\bigr \}\mathop {\ge }\limits ^{\text {C18}} \delta _1+\frac{\delta _{12}}{2} \ge \delta _1+\delta _{12} \ge 0. \end{aligned}$$

• Case 2: ${\varvec{\delta }}_{\mathbf{12}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}\ge \mathbf{0}$.

  This case can be proved in the same way as Case 1.

• Case 3: ${\varvec{\delta }}_{\mathbf{12}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}<0$, ${\varvec{\delta }}_{\mathbf{123}}\ge \mathbf{0}$.

  In order that $v(N)-\sum _{i\in N} v(i)\ge w(N)-\sum _{i\in N}w(i)$, we must have $\delta _{12}+\delta _{13} +\delta _{123}\ge -\delta _{23}$. Then,

  $$\begin{aligned} \sum _{T\in 2^N\backslash \{\emptyset , \{2\}, \{3\}\}}\bigl \{\varphi _1(d^v_T u_T)-\varphi _1(d^w_T u_T)\bigr \} \mathop {\ge }\limits ^{\text {C18}} \frac{\delta _{12}+\delta _{13}+\delta _{123}}{3}+\frac{\delta _{23}}{3} \ge 0. \end{aligned}$$

• Case 4: ${\varvec{\delta }}_{\mathbf{12}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}<\mathbf{0}$.

  In order that $v(N)-v(23)\ge w(N)-w(23)$, we must have $\delta _1+\delta _{12}+\delta _{13}\ge -\delta _{123}$. Then,

  

• Case 5: ${\varvec{\delta }}_{\mathbf{12}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}\ge \mathbf{0}$.

  In order that $v(12)-v(2)\ge w(12)-w(2)$ and $v(13)-v(3)\ge w(13)-w(3)$, we must have $\delta _1\ge -\min \{\delta _{12}, \delta _{13}\}$. Then,

  

• Case 6: ${\varvec{\delta }}_{\mathbf{12}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}\ge \mathbf{0}$.

  In order that $v(12)-v(2)\ge w(12)-w(2)$, we must have $\delta _1\ge -\delta _{12}$. In order that $v(N)-\sum _{i\in N} v(i)\ge w(N)-\sum _{i\in N} w(i)$, we must have $\delta _{13} +\delta _{123}\ge -\delta _{12}-\delta _{23}\ge -\delta _{23}$. Then,

  

• Case 7: ${\varvec{\delta }}_{\mathbf{12}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}< \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}\ge \mathbf{0}$.

  This case can be proved in the same way as Case 6.

• Case 8: ${\varvec{\delta }}_{\mathbf{12}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}<\mathbf{0}$.

  In order that $v(12)-v(2)\ge w(12)-w(2)$ and $v(N)-v(23)\ge w(N)-w(23)$, we must have $\delta _1\ge -\delta _{12}$ and $\delta _{1}+\delta _{13}\ge -\delta _{12}-\delta _{123}\ge -\delta _{123}$. Then,

  

• Case 9: ${\varvec{\delta }}_{\mathbf{12}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}< \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}<\mathbf{0}$.

  This case can be proved in the same way as Case 8.

• Case 10: ${\varvec{\delta }}_{\mathbf{12}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}<\mathbf{0}$.

  In order that $v(N)-\sum _{i\in N} v(i)\ge w(N)-\sum _{i\in N} w(i)$, we must have $\delta _{12} +\delta _{13}\ge -\delta _{23}-\delta _{123}$. Then,

  

• Case 11: ${\varvec{\delta }}_{\mathbf{12}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}<\mathbf{0}$.

  In order that $v(12)-v(2)\ge w(12)-w(2)$, we must have $\delta _1\ge -\delta _{12}$. In order that $v(N)-\sum _{i\in N} v(i)\ge w(N)-\sum _{i\in N} w(i)$, we must have $\delta _{13}\ge -\delta _{12} -\delta _{23}-\delta _{123}\ge -\delta _{23}-\delta _{123}$. Then,

  

• Case 12: ${\varvec{\delta }}_{\mathbf{12}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}<\mathbf{0}$.

  This case can be proved in the same way as Case 11.

• Case 13: ${\varvec{\delta }}_{\mathbf{12}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}\ge \mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}<\mathbf{0}$.

  In order that $v(12)-v(2)\ge w(12)-w(2)$ and $v(13)-v(3)\ge w(13)-w(3)$, we must have $\delta _1\ge -\min \{\delta _{12}, \delta _{13}\}$. In order that $v(N)-v(23)\ge w(N)-w(23)$, we must have $\delta _1\ge -\delta _{12}-\delta _{13}-\delta _{123}$. Then,

  

• Case 14: ${\varvec{\delta }}_{\mathbf{12}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{13}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{23}}<\mathbf{0}$, ${\varvec{\delta }}_{\mathbf{123}}\ge \mathbf{0}$.

  In order that $v(12)-v(2)\ge w(12)-w(2)$ and $v(13)-v(3)\ge w(13)-w(3)$, we must have $\delta _1\ge -\min \{\delta _{12}, \delta _{13}\}$. In order that $v(N)-\sum _{i\in N} v(i)\ge w(N)-\sum _{i\in N} w(i)$, we must have $\delta _{123}\ge -\delta _{12} -\delta _{13}-\delta _{23}\ge -\delta _{23}$. Then,