On optimal partitions, individual values and cooperative games: Does a wiser agent always produce a higher value?

Abstract

We consider an optimal partition of resources (e.g. consumers) between several agents, given utility functions (“wisdoms”) for the agents and their capacities. This problem is a variant of optimal transport (Monge–Kantorovich) between two measure spaces where one of the measures is discrete (capacities) and the costs of transport are the wisdoms of the agents. We concentrate on the individual value for each agent under optimal partition and show that, counter-intuitively, this value may decrease if the agent’s wisdom is increased. Sufficient and necessary conditions for the monotonicity with respect to the wisdom functions of the individual values will be given, independently of the other agents. The sharpness of these conditions is also discussed. Motivated by the above we define a cooperative game based on optimal partition and investigate conditions for stability of the grand coalition.

Appendix

We prove that the function $\vec {m}\mapsto \Sigma _{\vec {\psi }}(\vec {m})$ as defined in (10) is strictly concave on the simplex (51). To prove this we recall some basic elements form convexity theory (see, e.g. [31, 32]):

• (i) If F is a convex function on $\mathbb R^N$ (say), then the sub gradient $\partial F$ at point $p\in \mathbb R^N$ is defined as follows: $\vec {q}\in \partial F(\vec {p})$ if and only if

  $$\begin{aligned} F(\vec {p}{\prime })-F(\vec {p})\ge \vec {q}\cdot (\vec {p}{\prime }-\vec {p}) \quad \forall \vec {p}{\prime }\in \mathbb R^N. \end{aligned}$$

• (ii) The Legendre transform of F:

  $$\begin{aligned} F^{*}(\vec {q}):= \sup _{\vec {p}\in \mathbb R^N} \vec {p}\cdot \vec {q}-F(\vec {p}), \end{aligned}$$

  and $Dom(F^{*})\subset \mathbb R^N$ is the set on which $F^{*}<\infty$.

• (iii) The function $F^{*}$ is convex (and closed), but $Dom(F^{*})$ can be a proper subset of $\mathbb R^N$ (or even an empty set).

• (iv) The subgradient of a convex function is non-empty (and convex) at any point in the proper domain of this function (i.e. at any point in which the function takes a value in $\mathbb R$).

• (v) Young’s inequality

  $$\begin{aligned} F(\vec {p})+F^{*}(\vec {q})\ge \vec {p}\cdot \vec {q}\end{aligned}$$

  holds for any pair of points $(\vec {p}, \vec {q})\in \mathbb R^N\times \mathbb R^N$. The equality holds iff $\vec {q}\in \partial F(\vec {p})$, iff $\vec {p}\in \partial F^{*}(\vec {q})$.

• (vi) The Legendre transform is involuting, i.e $F^{**}=F$ if F is convex and closed.

Returning to our case, let $F(\vec {p}):=\Xi ^0_{\vec {\psi }}(\vec {p})$. It is a convex function on $\mathbb R^N$ by Lemma 5.1. Moreover, its partial derivatives exists at any point in $\mathbb R^N$, which implies that its sub-gradient is s singleton. Recalling (9) with $\Xi _{\vec {\psi }}$ we get that

$$\begin{aligned} F^{*}(\vec {m})= -\Sigma _{\vec {\psi }}(-\vec {m}) \end{aligned}$$

takes finite values only on the simplex (51). So, we only have to prove the existence of a unique minimizer (10) on the set $S_N\cap \{\vec {m}\le \vec {M}\}$, which is a convex set as well. We prove below that $F^{*}$ is strictly convex on the simples. This implies the strict concavity of $\Sigma _{\vec {\psi }}$ on the same simplex.

Assume $F^{*}$ is not strictly convex. It means there exists $\vec {m}_1\not = \vec {m}_2\in S_N$ for which

$$\begin{aligned} F^{*}\left( \frac{\vec {m}_1+\vec {m}_2}{2}\right) =\frac{F^{*}(\vec {m}_1)+F^{*}(\vec {m}_2)}{2}. \end{aligned}$$

(80)

Let $\vec {m}:=\vec {m}_1/2+\vec {m}_2/2$, and $\vec {p}\in \partial F^{*}(\vec {m})$. Then, by (iv, v)

$$\begin{aligned} 0=F^{*}(\vec {m})+ F^{**}(\vec {p})-\vec {p}\cdot \vec {m}=F^{*}(\vec {m}) +F(\vec {p})-\vec {p}\cdot \vec {m}. \end{aligned}$$

(81)

By (80, 81):

$$\begin{aligned} \frac{1}{2}\left( F^{*}(\vec {m}_1)+F(\vec {p})-\vec {p}\cdot \vec {m}_1 \right) + \frac{1}{2}\left( F^{*}(\vec {m}_2)+F(\vec {p})-\vec {p}\cdot \vec {m}_2 \right) =0 \end{aligned}$$

while (v) also guarantees

$$\begin{aligned} F^{*}(\vec {m}_i)+F(\vec {p})-\vec {p}\cdot \vec {m}_i\ge 0, \quad i=1,2. \end{aligned}$$

It follows

$$\begin{aligned} F^{*}(\vec {m}_i)-F(\vec {p})+\vec {p}\cdot \vec {m}_i= 0, \quad i=1,2, \end{aligned}$$

so, by (v) again, $\{\vec {m}_1, \vec {m}_2\}\in \partial F(\vec {p})$, and we get a contradiction since $\partial F(\vec {p})$ is a singleton.