Sustainability of Collusion in a finite Bertrand competition with N > 2 firms In a infinitely repeated Bertrand competition, collusion is sustainable if, and only, if, the following inequality is satisfied,
$\frac{\pi}{N(1-\delta)}\geq\pi$
Where $\pi$ is equilibrium profits, and $0 &lt; \delta &lt; 1$ is the discount factor. If the game is finite, I make the following claim:
Collusion is not sustainable, as each firm has an incentive to deviate from collusion in the last stage of the game. However, how would I go ahead and prove this claim?
My initial attempt, was to setup the following inequality,
$\frac{1-\delta^T}{1-\delta}$ $\frac{\pi}{N}$+$\pi_{\varepsilon}$ $\delta^{T-1}$ $\geq$ $\pi$,
where $\pi_{\varepsilon} &lt; \pi$, and is given to the deviating firm. Is this a valid approach, or should I change my approach?
You're already on the right track. Just need to reorganize your argument a little bit.
Hint: Prove by contradiction. Suppose that collusion is sustainable in an equilibrium. Show that the collusive strategy (i.e. choosing the monopoly price in every stage) is not optimal given that everybody else is doing the same. Then conclude that collusion cannot be an equilibrium, hence a contradiction.