# 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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