Cartels and Collusion

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 $\frac{1}{2}{x}^M$, and thus together produce $x^M$ at $p^M$.
   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 $\frac{1}{2}{x}^M$.
   2. I will face a residual demand curve [=$D^r$] which is shifted left $0.5x^M$
   3. my best option is to produce at $M{R}^r=MC$ which is $0.75x^M$, bigger than the promised $0.5x^M$

   → Thus I am incentivized to defect and produce more

Prisoner’s Dillemma situation:※

Aside on Prisoner’s Dilemma §

Quote

def. Prisoner’s Dillemma (PD) is any game where the following payoff structure holds… …where $T>R>P>S$:

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. …
4. The whole game unravels into a defection.

Infinitely Iterated PD §

Game design with:

• Infinitely many games
• probability that one will meet [$\gamma$]
• discount parameter [$PV=\frac{FV}{r}$]

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).

∴ (TRIGGER, TRIGGER) is a SPNE