Cartels and the Natural Incentive to Defect

def. Cartels are when firms collude to act as a single monopolistic firm.

However, cartels cooperating is not a stable set of strategies, because it has a tendency to unravel, as it is a Prisoner’s Dillemma situation.

We can analyze this for two firms in the following two ways:

  1. Using the profit for the hypothetical monopoly firm (graph a)

    1. Two firms each produce , and thus together produce at .
    2. If I secretely produces one more unit of good: …both will lose red area together [=half each] …I will gain the profit of selling the additional units [blue area]

    → Thus I am incentivized to defect and produce more.

  2. Using residual demand for one of the firms

    1. Assume the other firm is diligently cooperating to produce their .
    2. I will face a residual demand curve [=] which is shifted left
    3. my best option is to produce at which is , bigger than the promised

    → Thus I am incentivized to defect and produce more

Prisoner’s Dillemma situation:※


Aside on Prisoner’s Dilemma


def. Prisoner’s Dillemma (PD) is any game where the following payoff structure holds… …where :

Finitely Iterated PD

In a finitely iterated PD, the Subgame Perfect Nash Equilibrium is to defect every game.


Solving from the last game:

  1. Last Game: Knowing there will be no more games [= same as one-shot game], your dominant strategy is defection.
  2. 2nd to last: Knowing that next game both will defect, your dominant strategy is defection.
  3. The whole game unravels into a defection.

Infinitely Iterated PD

Game design with:

  • Infinitely many games
  • probability that one will meet []
  • discount parameter []

In this case there are multiple Nash Equilibirum Strategies

  1. ALL C is a BR to ALL C → (ALL C, ALL C) is an NE
  2. TRIGGER is a BR to TRIGGER → (TRIGGER, TRIGGER) is an NE Trigger Strategy:= strategy where the opponent’s current action triggers my future behavior. e.g. “I will coop. until opp. defects, and defect always after that.”
  3. TFT is a BR to TFT → (TFT, TFT) is an NE

These strategies are subgame perfect. Let’s prove that for (TRIGGER, TRIGGER):

  1. If we’ve always cooperated before, this subgame is same as original game → (TRIGGER, TRIGGER) is an NE
  2. If there was non-cooperation before, this subgame has NE: → (ALL D, ALL D) is an NE

→ All subgames have NE (TRIGGER, TRIGGER).