Existance of concave balanced function

by Eval   Last Updated May 15, 2019 18:20 PM

We know that the core of a balanced game is non-empty. The convexity ensures balancedness. However, I was wondering if a concave function too satisfies balancedness condition.

The balancedness is defined according to Bondareva–Shapley theorem as follows: Let the pair $(N, v)$ be a cooperative game in characteristic function form, where $N$ is the set of players and where the value function $v:2^{N} \to \mathbb {R}$ is defined on $N$'s power set (the set of all subsets of $N$).

The core of $(N, v)$ is non-empty if and only if for every function $\alpha :2^{N \setminus \emptyset} \to [0,1]$, where

$ \forall i\in N:\sum _{S\in 2^{N}:\;i\in S}\alpha (S)=1$ the following condition holds:

$\sum_{S\in 2^{N\setminus \emptyset}}\alpha (S)v(S)\leq v(N) $



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