Two firms can produce either low (L), medium (M) or high (H) quantity. The payoff matrix is given by:
- What is the outcome of this game if firms play only once?
- Suppose the game is played for infinite periods and $\delta = 1/2$. Is it possible for firms to collude and choose $(L, L)$? Is it possible for firms to collude and choose $(M, M)$?
From the picture we can see that if the firms play once then the outcome is $(H,H)$. However I don't really know how to solve the second part and would really appreciate some help. What I do know in repeated infinite games that may be useful for this problem is the following.
Lets take the case of deviating from $(H,H)$ to $(M,M)$ for firm 1.
Equilibrium payoff = $3+3\delta+3\delta^2...=\frac{3}{1-\delta}$
Deviation payoff (I think that if firm 1 deviated and played $M$ then firm 2 would play L since they have higher payoff $(10)$ and so we would have)= $5+2\delta+2\delta^2... = 5+\frac{2\delta}{(1-\delta) } $
Where $\delta$ is the interest rate after the first round. I feel that this portion is irrelevant, however, since we are talking about collusion and should consider both firms, instead of just firm 1's strategy.