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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.

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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 $V(\{1,2,3\})-V(\{2,3\}) = 2 > V(\{1,2\})-V(\{2\}) = 1$.

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  • $\begingroup$ Yes, the intuition that LPG is not a convex game is clear, otherwise this game could be quite simple to understand. You got a typo in the concave case where the player name {2} instead of {1}. Thank you for the example. $\endgroup$
    – philipvn
    Commented Mar 6, 2016 at 13:13

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