In cooperative game theory, the linear production game (LPG) is defined by letting the characteristic function have the form of a linear programming problem.


Does anyone know if the LPG is a convex game or not? If not, could you give a counterexample?
Thank you.



Is the linear production game a convex game?
I don't think it is. Consider the following LP


$$\max x$$
$$s.t.$$
$$x \leq \sum\_{s\in S} b\_s^1$$
$$x \leq \sum\_{s\in S} b\_s^2$$


Now consider $S=\{1,2,3\}$ with $b\_1=(1,0)$, $b\_2=(0,1)$, $b\_3=(0,1)$.
Then $V(\{1\}) = 0$, $V(\{1,2\}) = 1$, $V(\{1,2,3\}) = 1$, $V(\{1,3\}) = 1$,
then $V(\{1,2,3\})-V(\{1,2\}) = 0 < V(\{1,3\})-V(\{1\}) = 1$,
so it is not convex.


It is also not concave since if $b\_1=(2,0)$, $b\_2=(0,1)$ and $b\_3=(0,1)$,
then $V(\{1,2,3\})-V(\{2,3\}) = 2 > V(\{1,2\})-V(\{2\}) = 1$.


