Two firms are in the chocolate market. Each can choose to go for the high end of the market (high quality) or the low end (low quality). Resulting profits are given by the following payoff matrix:

a. What outcomes, if any, are Nash equilibria?

b. If the managers of both firms are conservative and each follows a maximin strategy, what will be the outcome?

c. What is the cooperative outcome?

d. Which firm benefits most from the cooperative outcome? How much would that firm need to offer the other to persuade it to collude?

• When Firm 1 chooses to go for the low end of the market, Firm 2 will get a higher profit by choosing the high end of the market (600 > -30).
• Further, if Firm 1 chooses to go for the high end of the market, Firm 2 will get a higher profit by choosing the low end of the market (800 > 50).

Thus, Firm 2 does not have a dominant strategy that exists regardless of Firm 1’s decision.

• Similarly, when Firm 2 chooses to go for the low end of the market, Firm 1 will get a higher profit by choosing the high end of the market (100 > -20).
• If Firm 2 chooses to go for the high end of the market, Firm 1 will get a higher profit by choosing the low end of the market (900 > 50).

Thus, even Firm 1 does not have a dominant strategy regardless of Firm 2’s decision.

As neither of the firms has a dominant strategy, a pure-strategy Nash equilibrium is absent.

However, mixed-strategy Nash equilibria exist at two points in the market:

• When Firm 1 chooses to Low, and Firm 2 chooses High.
• When Firm 1 chooses High, and Firm 2 chooses Low.

Neither of the firms has an incentive to deviate from these points as they can earn comparatively higher payoffs by adopting these strategies.

Thus, there are two mixed-strategy Nash equilibria at the points (Low, High) and (High, Low).

If both firms are conservative and each follows a maximin strategy, the payoff will be as follows:

• The lowest payoff that Firm 1 will receive on choosing Low is -20, and the lowest payoff on choosing High is 50. To minimize losses, Firm 1 will choose High.
• Similarly, the lowest payoff that Firm 2 receives on choosing Low is -30, and the lowest payoff on choosing High is 50. Firm 2 will also choose High to minimize losses.

Thus, using a conservative maximin strategy, both firms choose High and obtain profits worth 50 each.

If the firms opt for a cooperative outcome, they will decide to maximize the total profit earned by both firms; that is, they will be concerned with joint payoff maximization.

The total payoff earned when Firm 1 opts for Low, and Firm 2 opts for High is: \({\rm{900 + 600 = 1500}}\).

The total payoff earned when Firm 1 opts for High, and Firm 2 opts for Low is: \({\rm{100 + 800 = 900}}\).

Thus, the cooperative outcome will be for Firm 1 to choose Low and Firm 2 to choose High as the joint payoff will be maximum for that outcome.

When the firms opt for the cooperative outcome, Firm 1 benefits the most from it.

Had Firm 1 chosen High and Firm 2 chosen Low, Firm 1 would have gained 100, whereas Firm 2 would have gained 800.

However, when the firms choose the cooperative outcome, Firm 1 gains 900 (900>100), and Firm 2 gains 600 (600<800).

Thus, Firm 1 benefits from the cooperative outcome, whereas Firm 2 is at a comparative disadvantage.

As Firm 1 benefits from the cooperative outcome, it must offer Firm 2 some of its yields to persuade it to collude. Firm 2 loses profit worth 200\(\left( {{\rm{800 - 600 = 200}}} \right)\).

Thus, Firm 1 must atleast provide 200 units of its profit to Firm 2.

Firm 1 gains profit worth 800 \(\left( {{\rm{900 - 100 = 800}}} \right)\).

Thus, Firm 2 might not be happy with just 200 additional yields and will try to obtain the maximum possible gains of Firm 1 to enter into the collaboration.