Collusion and number of firms How would you answer the following question?

You work for a CEO of a large firm. He says to you, "In my experience collusion is less likely to be sustained as the number of firms in the market increases. Demonstrate this using a model of Bertrand Competition."
Iet's say we have n identical firms and an infinite horizon of time.

The n firms sustaining the collusion, will find optimal to fix the same price $p_m$ where $p_m$ is the price of the monopoly level and we define $\frac{\Pi^m}{n}$  as the profits each firm is obtaining by sustaining the collusion in each moment t.

Now, of course each firm can betray the others by fixing a price lower than $p_m$, namely $p_m-ε$, where ε is small, and by doing so, the firm will capture the entire demand because in this market the firms are doing the Bertrand competition. In other words, the firm by betraying the others, will get almost π_m at the time T=t. We will also assume that in all t > T no firms will make any profits, because they will punish the firm, by fixing the price in Bertrand competition.

The firm will defect if:

$π_m/n + δπ_m/n + δ^2π_m/n.... &lt; π_m+0+0....$

Where δ is the discount factor.

This can rewritten as:

$(\frac{π_m}{n})(\frac{1}{(1-δ)}) &lt; π_m$

We can now see that if n, the number of firms, increases then the profits by sustaining the collusion will decrease, so the above inequality will be more likely to be true. This means that a firm has less incentives to sustain a collusion when there are too many participants, because the profits will be divided among too many firms and the punishment will be seen as less heavy.