The intermediate set and limiting superdifferential for coalitional games: between the core and the Weber set

Abstract

We introduce the intermediate set as an interpolating solution concept between the core and the Weber set of a coalitional game. The new solution is defined as the limiting superdifferential of the Lovász extension and thus it completes the hierarchy of variational objects used to represent the core (Fréchet superdifferential) and the Weber set (Clarke superdifferential). It is shown that the intermediate set is a non-convex solution containing the Pareto optimal payoff vectors that depend on some chain of coalitions and marginal coalitional contributions with respect to the chain. A detailed comparison between the intermediate set and other set-valued solutions is provided. We compute the exact form of intermediate set for all games and provide its simplified characterization for the simple games and the glove game.

Appendices

Appendix A: Superdifferentials

In this section we will define the selected concepts of variational (nonsmooth) analysis, mainly various superdifferentials which generalize the superdifferential of concave functions. Since these superdifferentials will be computed only for the Lovász extension, we will confine to defining superdifferentials only for piecewise affine functions. Even though the computation of these objects may be rather a challenging task, see e.g. Adam et al. (2016) and Henrion and Outrata (2008), the presented framework allows for a significant simplification. For the general approach based on upper semicontinuous functions, we refer the reader to Rockafellar and Wets (1998).

The standard monographs on variational analysis, Mordukhovich (2006), Rockafellar (1970) or Rockafellar and Wets (1998), follow the approach usual in convex analysis by dealing with subdifferentials instead of superdifferentials. However, most of the results can be easily transformed to the setting of superdifferentials, usually by reversing inequalities only.

Definition 4

Let $f:{\mathbb R}^n\rightarrow {\mathbb R}$ be a piecewise affine function and ${\bar{\mathbf {x}}}\in {\mathbb R}^n$. We say that $\mathbf {x}^*\in {\mathbb R}^n$ is a

• regular (Fréchet) supergradient of f at ${\bar{\mathbf {x}}}$ if there exists neighborhood $\mathcal {X}$ of ${\bar{\mathbf {x}}}$ such that for all $\mathbf {x}\in \mathcal {X}$ we have

  $$\begin{aligned} f(\mathbf {x}) - f({\bar{\mathbf {x}}}) \le \langle \mathbf {x}^*,\mathbf {x}-{\bar{\mathbf {x}}}\rangle ; \end{aligned}$$

• limiting (Mordukhovich) supergradient of f at ${\bar{\mathbf {x}}}$ if for every neighborhood $\mathcal {X}$ of ${\bar{\mathbf {x}}}$ there exists $\mathbf {x}\in \mathcal {X}$ such that $\mathbf {x}^*$ is a Fréchet supergradient of f at $\mathbf {x}$;

• convexified (Clarke) supergradient of f at ${\bar{\mathbf {x}}}$ if

  $$\begin{aligned} \mathbf {x}^*\in {\text {conv}}\{\mathbf {y}\in \mathbb {R}^n|\ \forall \,\text {neighborhood }\mathcal {X}\text { of }{\bar{\mathbf {x}}}\exists \,\mathbf {x}\in \mathcal {X}\cap D\text { with }\mathbf {y}= \nabla f(\mathbf {x})\}, \end{aligned}$$

  where

  $$\begin{aligned} D:= \{\mathbf {x}\in {\mathbb R}^n|\ f\text { is differentiable at }\mathbf {x}\}. \end{aligned}$$

The collection of all (regular, limiting, convexified) supergradients of f at ${\bar{\mathbf {x}}}$ is called (Fréchet, limiting, Clarke) superdifferential and it is denoted by ${\hat{\partial }}f({\bar{\mathbf {x}}})$, ${\partial }f({\bar{\mathbf {x}}})$ and ${\overline{\partial }}f({\bar{\mathbf {x}}})$, respectively.

Remark 6

The previous definition can be found e.g. in (Rockafellar and Wets 1998, Definition 8.3). Note that in the original definition term $o(||\mathbf {x}-{\bar{\mathbf {x}}}||)$ is added. Because we work with piecewise affine functions, this term is superfluous. If f is concave, then all the above superdifferentials coincide with the standard superdifferential for concave functions. $\square$

It is possible to show that

$$\begin{aligned} {\hat{\partial }}f({\bar{\mathbf {x}}}) \subseteq {\partial }f({\bar{\mathbf {x}}}) \subseteq {\overline{\partial }}f({\bar{\mathbf {x}}}), \quad {\bar{\mathbf {x}}}\in {\mathbb R}^n, \end{aligned}$$

where all the inequalities may be strict. According to Rockafellar and Wets (1998, Theorem 8.49) we have the following relation between the limiting and the Clarke superdifferential for every piecewise affine function f:

$$\begin{aligned} {\overline{\partial }}f({\bar{\mathbf {x}}}) = {\text {conv}}{\partial }f({\bar{\mathbf {x}}}). \end{aligned}$$

We will show the differences among the three discussed superdifferentials.

Fig. 3

Supergradients for a piecewise affine function f

Example 5

Let $f:{\mathbb R}\rightarrow {\mathbb R}$ be defined by

$$\begin{aligned} f(x) = {\left\{ \begin{array}{ll} x &{}\quad \text {if } x\in (-\infty ,0],\\ 0 &{}\quad \text {if } x\in [0,1],\\ x-1 &{}\quad \text {if } x\in [1,\infty ). \end{array}\right. } \end{aligned}$$

This function is depicted in Fig. 3. Consider points $\bar{x}=0$ and $\bar{y}=1$. The locally supporting hyperplanes from the definition of Fréchet superdifferential at $\bar{x}$ are depicted in the figure. Note that there are no affine majorants for f at $\bar{y}$, which means that the Fréchet superdifferential is empty at this point. Thus we obtain

$$\begin{aligned} {\hat{\partial }}f(\bar{x})&= [0,1], \quad {\hat{\partial }}f(\bar{y}) = \emptyset , \\ {\partial }f(\bar{x})&= [0,1], \quad {\partial }f(\bar{y}) = \{0,1\}, \\ {\overline{\partial }}f(\bar{x})&= [0,1], \quad {\overline{\partial }}f(\bar{y}) = [0,1]. \end{aligned}$$

$\square$

The superdifferential sum rule is employed frequently in this paper. The following proposition collects the results of Rockafellar and Wets (1998, Exercise 8.8, Corollary 10.9, Exercise 10.10).

Proposition 4

Let $f_1,f_2:{\mathbb R}^n\rightarrow {\mathbb R}$ be piecewise affine functions. Then

$$\begin{aligned} \partial (f_1+f_2)(\mathbf {x})\subseteq \partial f_1(\mathbf {x})+\partial f_2(\mathbf {x}), \quad \mathbf {x}\in \mathbb {R}^n. \end{aligned}$$

Moreover, if at least one of the functions is smooth around $\mathbf {x}$, we obtain equality in the previous relation.

Appendix B: Proof of Theorem 1

To prove Theorem 1, consider first a game $v\in \varGamma (N)$, fix ${\bar{\mathbf {x}}}\in {\mathbb R}^n$ and choose any $\pi \in \varPi ({\bar{\mathbf {x}}})$. Then there are necessarily unique integers

$$\begin{aligned} 0=L_0<L_1<\dots <L_k=n \end{aligned}$$

such that $L_i-L_{i-1}$ is the number of coordinates of ${\bar{\mathbf {x}}}$ which have the i-th greatest distinct value in the order given by $\pi$:

$$\begin{aligned} {\bar{\mathbf {x}}}_{\pi (1)} \!= \!\dots \!=\! {\bar{\mathbf {x}}}_{\pi (L_1)}>{\bar{\mathbf {x}}}_{\pi (L_1+1)} = \dots \!=\! {\bar{\mathbf {x}}}_{\pi (L_2)} \!>\! \dots > {\bar{\mathbf {x}}}_{\pi (L_{k-1}+1)} = \dots = {\bar{\mathbf {x}}}_{\pi (L_k)}. \end{aligned}$$

Define

$$\begin{aligned} C_i:=\{\pi (1),\dots ,\pi (L_i)\} \end{aligned}$$

and observe that $C_i$ is independent of the choice of $\pi \in \varPi ({\bar{\mathbf {x}}})$. Take any $\mathbf {x}$ sufficiently close to ${\bar{\mathbf {x}}}$ and select some $\rho \in \varPi (\mathbf {x})$. Then $\rho \in \varPi ({\bar{\mathbf {x}}})$ and

$$\begin{aligned} V_j^\rho (\mathbf {x})\subseteq & {} V_j^\rho ({\bar{\mathbf {x}}}),\quad j=1,\dots ,n,\\ V_{L_i}^\rho (\mathbf {x})= & {} V_{L_i}^\rho ({\bar{\mathbf {x}}})=C_i,\quad i=1,\dots ,k. \end{aligned}$$

This allows us to write $\hat{v}$ in a separable structure

$$\begin{aligned} \hat{v}(\mathbf {x}) = \sum _{i=1}^k\hat{v}_i(\mathbf {x}_{C_i\setminus C_{i-1}}), \end{aligned}$$

(20)

where $\mathbf {x}_{A}$ is the restriction of $\mathbf {x}$ to components A and $\hat{v}_i:{\mathbb R}^{|B_i|}\rightarrow {\mathbb R}$ is defined as

$$\begin{aligned} \hat{v}_i(\mathbf {y}) = \sum _{j=1}^{|B_k|}y_{\varphi (j)}\left[ v(C_{i-1}\cup V_j^\varphi (\mathbf {y})) - v(C_{i-1}\cup V_{j-1}^\varphi (\mathbf {y}))\right] , \end{aligned}$$

where $\varphi \in \varPi (\mathbf {y})$. We now fix a constant $c>0$, coalition $B\subseteq C_i\setminus C_{i-1}$ and denoting a to be the common value of ${\bar{\mathbf {x}}}$ on $C_i\setminus C_{i-1}$, we obtain

$$\begin{aligned}&\hat{v}_i(({\bar{\mathbf {x}}}+c\chi _B)_{C_i\setminus C_{i-1}}) = a\left[ (v(C_i) - v(C_{i-1}\cup B)\right] + (a+c)\\&\quad \left[ (v(C_{i-1}\cup B) - v(C_{i-1})\right] ,\hat{v}_i({\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}) = a\left[ (v(C_i) - v(C_{i-1})\right] , \end{aligned}$$

so that

$$\begin{aligned} \hat{v}_i({\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}+c\chi _B) - \hat{v}_i({\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}) = c\left[ (v(C_{i-1}\cup B) - v(C_{i-1})\right] . \end{aligned}$$

(21a)

When we choose $B=N$, we can move in the opposite direction as well, obtaining

$$\begin{aligned} \hat{v}_i({\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}-c\chi _N) - \hat{v}_i({\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}) = c\left[ (v(C_i) - v(C_{i-1})\right] . \end{aligned}$$

(21b)

Now we prove the following lemma.

Lemma 11

For any $i\in \{1,\dots ,k\}$ we have

$$\begin{aligned} {\hat{\partial }}\hat{v}_i({\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}) = \left\{ \mathbf {x}^*\left| \ \begin{array}{l} \mathbf {x}^*(C_i\setminus C_{i-1}) = v(C_i) - v(C_{i-1}),\\ \qquad \qquad \,\,\,\mathbf {x}^*(B) \ge v(C_{i-1}\cup B) - v(C_{i-1})\text { for all } B\subseteq C_i\setminus C_{i-1}\end{array} \right. \right\} . \end{aligned}$$

Proof

The definition of Fréchet superdifferential and the piecewise affinity of $\hat{v}_i$ give

$$\begin{aligned} {\hat{\partial }}\hat{v}_i({\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}) = \{\mathbf {x}^*|\ \hat{v}_i(\mathbf {y}) - \hat{v}_i({\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}})\le \langle \mathbf {x}^*,\mathbf {y}-{\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}\rangle \text { for all }\mathbf {y}\text { close to }{\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}\}. \end{aligned}$$

Consider now any $\mathbf {x}^*\in {\hat{\partial }}\hat{v}_i({\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}})$, any $B\subseteq C_i\setminus C_{i-1}$, and put $\mathbf {y}= {\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}+c \chi _B$, where $c>0$ is sufficiently small. By realizing that $\langle \mathbf {x}^*,\mathbf {y}-{\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}\rangle = c\mathbf {x}^*(B)$ and from relation (21a) it follows that

$$\begin{aligned} \mathbf {x}^*(B) \ge v(C_{i-1}\cup B) - v(C_{i-1}). \end{aligned}$$

Similarly from (21b) we obtain equality in the previous relation for $B=N$. This finishes the proof of the first inclusion.

Consider now any $\mathbf {x}^*$ from the right–hand side of the formula in Lemma 11 and fix any $\mathbf {y}$ from a sufficiently small neighborhood of ${\bar{\mathbf {x}}}_{C\setminus C_{i-1}}$. Defining

$$\begin{aligned} \begin{aligned} \mathbf {y}^0&:= {\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}-\chi _{\{1,\dots ,|C_i\setminus C_{i-1}|\}},\\ \mathbf {y}^j&:= {\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}+\chi _{\{\varphi (1)\dots \varphi (j)\}},\quad j=1,\dots ,|C_i\setminus C_{i-1}|, \end{aligned} \end{aligned}$$

we have

$$\begin{aligned} \mathbf {y}\in {\text {conv}}\left\{ \mathbf {y}^0, \mathbf {y}^1, \dots , \mathbf {y}^{|C_i\setminus C_{i-1}|}\right\} . \end{aligned}$$

From the assumption and from (21) we obtain that

$$\begin{aligned} \hat{v}_i(\mathbf {y}^j) - \hat{v}_i({\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}})\le \langle \mathbf {x}^*,\mathbf {y}^j -{\bar{\mathbf {x}}}_{C_i\setminus C_{i-1}}\rangle \end{aligned}$$

(22)

for all $j=0,\dots ,|C_i\setminus C_{i-1}|$. Since $\hat{v}_i$ is linear on very particular domains and since $\mathbf {y}$ lies in the convex hull of the above points, we obtain that formula (22) holds also for $\mathbf {y}$. This finishes the proof. $\square$

The decomposition (20) together with Lemma 11 imply that Theorem 1 holds true.