Random reduction consistency of the Weber set, the core and the anti-core

Abstract

In this paper we introduce a new consistency condition and provide characterizations for several solution concepts in TU cooperative game theory. Our new consistency condition, which we call the random reduction consistency, requires the consistency of payoff vectors assigned by a solution concept when one of the players is removed with some probability. We show that the random reduction consistency and other standard properties characterize the Weber set, the convex hull of the marginal contribution vectors. Another salient feature of random reduction consistency is that, by slightly changing its definition, we can characterize the core and the anti-core in a parallel manner. Our result enables us to compare the difference between the three solution concepts from the viewpoint of consistency.

Appendix: independence of properties

In this appendix we show independence of the set of properties we used to characterize the Weber set, the core and the anti-core.

1.1 Independence of NE, M-RRC, and M-CRRC

Set $\sigma (N,v) = \emptyset$ for all $(N,v) \in \Gamma$. We can easily see that $\sigma$ satisfies M-RRC and M-CRRC but fails to satisfy NE.

To see the independence of M-RRC, set $\sigma (N,v) = PI(N,v)$ for all $(N,v) \in \Gamma$. It is evident that $\sigma = PI$ satisfies ZNP and M-CRRC. Now consider a game $(N,v) \in \Gamma$ such that $N = \{ 1,2 \}$, $v(N) = 1$, $v( \{ 1 \} ) = 2$, and $v( \{ 2 \} ) = -2$. Then we have $PI(v^{mar}_{\{1\}}) = \{ 3 \}$ and $PI(v^{mar}_{\{2\}}) = \{ -1 \}$. If a preimputaion $x = (x_{1} ,x_{2})$ can be represented as the way prescribed in the Eq. (1), we have

$$\begin{aligned}&x_{1} = p_{1}v( \{ 1 \} ) + p_{2}y^{\{ 1 \}}_{1} = 2p_{1} + 3p_{2} \ge 2, \\&x_{2} = p_{1}y^{\{ 2 \}}_{2} + p_{2}v( \{ 2 \} ) = -p_{1} - 2p_{2} \le -1. \end{aligned}$$

Hence, for example $(1,0) \in PI(N,v)$ can have no such representation. This indicates that $\sigma = PI$ violates M-RRC.

In order to confirm the independence of M-CRRC, set $\sigma (N,v) = \{ Sh(N,v) \}$ for all $(N,v) \in \Gamma$, where Sh(N, v) is the Shapley value of (N, v). Clearly $\sigma$ satisfies NE. To see that $\sigma$ satisfies M-RRC, it is sufficient to follow the proof of Lemma 1, with setting $t_{\pi } = 1/|N|!$ in the Eq. (2) . Now consider a game $(N,v) \in \Gamma$ such that $N = \{ 1,2,3 \}$, $v(N) = 2$, $v(S) = 1$ for $|S| = 2$, and $v( \{ i \} ) = 0$ for $i = 1,2,3$. Then for all $i \in N$, $Sh(v^{mar}_{N\backslash \{i\}}) = (1,1)$. Set $(p_{1},p_{2},p_{3}) = (1,0,0) \in \Delta (N)$. Then if we set

$$\begin{aligned}&x_{1} = p_{1}v( \{ 1 \} ) + p_{2} Sh_{1}(v^{mar}_{N\backslash \{2\}}) + p_{3} Sh_{1}(v^{mar}_{N\backslash \{3\}}) = 0, \\&x_{2} = p_{1}Sh_{2}(v^{mar}_{N\backslash \{1\}}) + p_{2} v( \{ 2 \} ) + p_{3} Sh_{2}(v^{mar}_{N\backslash \{3\}}) = 1, \\&x_{3} = p_{1}Sh_{3}(v^{mar}_{N\backslash \{1\}}) + p_{2} Sh_{3}(v^{mar}_{N\backslash \{2\}}) + p_{3} v( \{ 3 \} ) = 1, \end{aligned}$$

we have $x = (x_{1},x_{2},x_{3}) \in PI(N,v)$. However $x \ne Sh(N,v) = (2/3,2/3,2/3)$, indicating that $\sigma$ dose not satisfy M-CRRC.

1.2 Independence of ZNP, IR, COV, CP-PRRC, and CP-CPRRC

Again, the empty solution $\sigma (N,v) = \emptyset$ satisfies all the properties except ZNP.

If we take the solution $\sigma (N,v) = PI(N,v)$, we can see that it satisfies all the properties except IR. Indeed the violation of IR is evident. It is also clear that PI satisfies ZNP, COV and CP-CPRRC. To see that PI possesses CP-PRRC, notice that for any $x \in PI(N, v)$, it is true that $x_{|N {\setminus } \{ i \}} \in PI(v^{com}_{N\backslash \{i\}, x}) \cap PI(v^{prj}_{N\backslash \{i\}, x})$ for all $i \in N$ where $x_{|N {\setminus } \{ i \}} = (x_{k})_{ k \in N {\setminus } \{ i \}}$, that is, the projection of x on ${\mathbb {R}}^{N {\setminus } \{ i \}}$.

To see the independence of COV, consider the solution $\sigma (N,v)$ defined as follows.

$$\begin{aligned} \sigma (N,v) = {\left\{ \begin{array}{ll} \{ 0 \} &{} \text {if }0 \in C(N,v), \\ \emptyset &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

It clearly satisfies ZNP and IR. It is also evident that this solution violates COV. Notice that if $0 \in C(N,v)$, we have $0 \in C(v^{com}_{N\backslash \{j\}, 0}) \cap C(v^{prj}_{N\backslash \{j\}, 0})$ for each $j \in N$. Therefore $\sigma (N,v) = \{ 0 \}$ implies $\sigma (v^{com}_{N\backslash \{j\}, x}) \cap \sigma (v^{prj}_{N\backslash \{j\}, x}) = \{ 0 \}$ for each $j \in N$. By this fact, we can see that $\sigma$ satisfies CP-PRRC and CP-CPRRC.

Next we will show the independence of CP-PRRC. Consider $\sigma (N,v) = IR(N,v) = \{ x \in PI(N,v) \; | \; x_{i} \ge v( \{ i \} ) , \; \forall i \in N \}$, the set of all individually rational preimputations. $\sigma = IR$ satisfies ZNP since $\{ 0 \} = IR(N,v)$ for any zero non-positive games. It clearly satisfies IR and COV. Notice that the proof of Lemma 4 shows that $\sigma = IR$ satisfies CP-CPRRC [see Eqs. (5, 6)]. Now consider a game (N, v) such that $N = \{ 1,2,3 \}$ and the coalition function is defined as follows;

$$\begin{aligned}&v(N) = 1, \; v( \{ 1,2 \} ) = 3, \; v( \{ 1,3 \} ) = v( \{ 2,3 \} ) = 0 \\&v( \{ 1 \} ) = 1, v( \{ 2 \} ) = 0, v( \{ 3 \} ) = -1. \end{aligned}$$

Then we have $IR(N,v) \ne \emptyset$, but $IR(v^{com}_{N\backslash \{1\}, x}) \cap IR(v^{prj}_{N\backslash \{1\}, x}) = \emptyset$ for all $x\in IR(N,v)$. Therefore any $x \in IR(N,v)$ cannot have the representation as the way prescribed in the Eq. (7). This shows the violation of CP-PRRC.

Finally, we will show the independence of CP-CPRRC. Consider the solution $\sigma$ defined as follows.

$$\begin{aligned} \sigma (N,v) = {\left\{ \begin{array}{ll} C(N,v) &{} \text {if }|N| \le 3, \\ \emptyset &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

It clearly satisfies the all the properties except CP-CPRRC. The violation of CP-CPRRC is also easily verified (consider, for example, any four-person game).