An effort to link cartel and Nash equilibrium

Question

In definition, Nash equilibrium is

where the optimal outcome of a game is one where no player has an incentive to deviate from their chosen strategy after considering an opponent's choice

Let's say about a cartel case where firms in a cartel collude together. So, for one firm, it has two choices: betray or not betray. If the firm betrayed, the firms will get the amnesty from the government regardless how other firms react. If the firms not betray, if other firms betray on the cartel, this firm will loose all. The best choice is not any firm will betray.

However, I do not know why it falled into a Nash equilibrium as stated in this link

On the other hand, the behavior of cartels can be also be considered a prisoner’s dilemma. All members of a cartel can collectively enrich themselves by restricting output to keep the price that each receives high enough to capture economic rents from consumers, but each cartel member individually has an incentive to cheat on the cartel and increase output to also capture rents away from the other cartel members.

I do not understand what does "each cartel member individually has an incentive to cheat on the cartel and increase output to also capture rents away from the other cartel members" mean, by which I can link this example to Nash equilibrium.

In short, my main question here is what is "increase output to also capture rents away from the other cartel members"

Answer 1

You're considering two distinct games here. In the game you're describing, collusion is illegal, and if the government finds out about it, there will be collective punishment. Hence, it is not in the interest of any firm to be a whistleblower. (But what if the government rewards whistle-blowing? Can they design rewards such that the game that the cartel is playing becomes a Prisoner's Dilemma?)

In the other game---the prisoner's dilemma---, collusion is not "illegal" per se; the question is whether a particular firm can benefit by deviating from the collective agreement. There are two basic economic models to think about this situation: price competition and competition in quantity. What you're citing is the latter, but the former is easier to think about. In a one-shot interaction, each firm will find it in her best interest to betray the other firms by selling below the cartel price to be able to serve the whole market; collusion cannot be sustained. (But as @Giskard points out: What if the firms interact over a long time horizon? Can the other firms punish the traitor so that she does not deviate from the collective agreements in the first place?)