Minimum Discount Factor for Grim-Trigger Strategy in a Repeated Duopoly Game

To find the minimum discount factor for a grim-trigger strategy to be a subgame perfect Nash equilibrium, we first need to define the grim-trigger strategy.

In a grim-trigger strategy, each firm plays the Cournot equilibrium strategy (i.e. produces q_i = (5 - q_j - c)/2) as long as the other firm does the same. However, if at any point in the game, the other firm deviates from the Cournot equilibrium strategy, then the deviating firm will be punished by the other firm playing the Nash equilibrium strategy of producing nothing (i.e. q_i = 0). This punishment will continue indefinitely until the deviating firm returns to playing the Cournot equilibrium strategy.

Now, let's consider the case where one firm deviates from the Cournot equilibrium strategy by producing more than its share of the market. If Firm 1 deviates by producing q_1 > (5 - q_2 - c)/2, then Firm 2 will respond by producing nothing (q_2 = 0) and will continue to do so indefinitely until Firm 1 returns to playing the Cournot equilibrium strategy. Similarly, if Firm 2 deviates by producing q_2 > (5 - q_1 - c)/2, then Firm 1 will respond by producing nothing (q_1 = 0) and will continue to do so indefinitely until Firm 2 returns to playing the Cournot equilibrium strategy.

Now, let's consider the case where both firms deviate from the Cournot equilibrium strategy. If both firms produce more than their share of the market (i.e. q_1 + q_2 > 5 - c), then both firms will receive negative payoffs and will continue to do so indefinitely until one or both firms return to playing the Cournot equilibrium strategy.

Given this grim-trigger strategy, we can now calculate the minimum discount factor for it to be a subgame perfect Nash equilibrium. This is the discount factor at which both firms are willing to continue playing the grim-trigger strategy indefinitely, rather than deviating and receiving the punishment.

Let V denote the discounted present value of a player's payoff. If a player deviates from the Cournot equilibrium strategy, then their discounted present value of payoff is:

V_deviate = -1 + δV

where -1 is the immediate punishment for deviating and δV is the discounted present value of future payoffs.

If a player continues to play the Cournot equilibrium strategy, then their discounted present value of payoff is:

V_Cournot = (5 - q_i - c)q_i + δV

where (5 - q_i - c)q_i is the payoff from the Cournot equilibrium strategy and δV is the discounted present value of future payoffs.

For the grim-trigger strategy to be a subgame perfect Nash equilibrium, both players must prefer to continue playing the Cournot equilibrium strategy rather than deviating. This means that:

V_Cournot > V_deviate

Simplifying this inequality, we get:

(5 - q_i - c)q_i + δV > -1 + δ(5 - q_i - c)q_i + δ^2V

Expanding this inequality, we get:

5q_iδ > c(1 - δ)q_i^2 + δ^2V - δV - 1

Solving for q_i, we get:

q_i < (5 - c - √(25 - 10c))/(2(1 - δ))

This is the maximum amount of the market share that a player can produce while still playing the Cournot equilibrium strategy. If a player produces more than this amount, then the other player will deviate and punish them.

Since both players have the same cost function, they will produce the same amount of the market share. Therefore, the maximum market share that either player can produce while still playing the Cournot equilibrium strategy is:

q < (5 - 2c - √(25 - 10c))/(4(1 - δ))

For the grim-trigger strategy to be a subgame perfect Nash equilibrium, both players must be willing to continue playing the Cournot equilibrium strategy even if the other player deviates. This means that:

δV_Cournot > -1 + δV_deviate

Simplifying this inequality, we get:

(5 - 2q - 2c)q + δV > -1 + δ(-1 + δV)

Expanding this inequality, we get:

5qδ > c(1 - δ)q^2 + δ^2V - δV - 1

Solving for q, we get:

q < (5 - 2c - √(25 - 10c))/(4(1 - δ))

This is the maximum market share that either player can produce while still playing the Cournot equilibrium strategy and being willing to continue playing even if the other player deviates.

Therefore, the minimum discount factor for the grim-trigger strategy to be a subgame perfect Nash equilibrium is the discount factor at which these two inequalities are equal:

q < (5 - 2c - √(25 - 10c))/(4(1 - δ))

q < (5 - c - √(25 - 10c))/(4(1 - δ))

Solving for δ, we get:

δ > (3 - √5)/5 ≈ 0.381966

Therefore, the minimum discount factor for the grim-trigger strategy to be a subgame perfect Nash equilibrium is δ = (3 - √5)/5 ≈ 0.381966.