Question

Sometimes oligopolies in the same industry are very different in size. Suppose we have a duopoly where one firm (Firm A) is large and the other firm (Firm B) is small, as the prisoner's dilemma box in Table 10.4 shows. $$\begin{array}{l|l|l}\hline & \begin{array}{l}\text { Firm B colludes with Firm } \\\\\text { A }\end{array} & \begin{array}{l}\text { Firm B cheats by selling more } \\\\\text { output }\end{array} \\\\\hline \text { Firm A colludes with Firm B } & \begin{array}{l}\text { A gets } \$ 1,000, \text { B gets } \\\\\$ 100\end{array} & \text { A gets \$800, B gets \$200 } \\\\\hline \begin{array}{l}\text { Firm A cheats by selling more } \\ \text { output }\end{array} & \begin{array}{l}\text { A gets \$1,050, B gets } \\\\\$ 50\end{array} & \text { A gets \$500, B gets \$20 } \\\\\hline\end{array}$$ Assuming that both firms know the payoffs, what is the likely outcome in this case?

Short answer

The likely outcome in this case is that both firms cheat, with Firm A earning $500 and Firm B earning $20. This is because both firms have a dominant strategy to cheat, as they each earn more when they cheat regardless of the other firm's decision.

Step-by-step solution

Analyze the Payoff Matrix

The first step is to examine the payoff matrix and understand the incentives of each firm. The payoff matrix can be represented as: $$ \begin{array}{l|l|l}\hline & \begin{array}{l}\text { Firm B colludes with Firm A }\end{array} & \begin{array}{l}\text { Firm B cheats by selling more output }\end{array} \\\\\hline \text { Firm A colludes with Firm B } & \begin{array}{l}\text { A gets } \$ 1,000, \text { B gets } \\\\\$ 100\end{array} & \text { A gets \$800, B gets \$200 } \\\\\hline \begin{array}{l}\text { Firm A cheats by selling more } \\\ \text { output }\end{array} & \begin{array}{l}\text { A gets \$1,050, B gets } \\\\\$ 50\end{array} & \text { A gets \$500, B gets \$20 } \\\\\hline\end{array} $$

Identify the Dominant Strategy for Firm A

Now that we have analyzed the payoff matrix, we shall find the dominant strategy for Firm A. When Firm B colludes: - If Firm A colludes, it earns $1000. - If Firm A cheats, it earns $1050. When Firm B cheats: - If Firm A colludes, it earns $800. - If Firm A cheats, it earns $500. Comparing the payoffs, we see that Firm A has a dominant strategy to cheat, as it always earns more (\(1050 > \)1000\( and \)500 > \(800\)) when it cheats, regardless of Firm B's decision.

Identify the Dominant Strategy for Firm B

Next, we shall find the dominant strategy for Firm B. When Firm A colludes: - If Firm B colludes, it earns $100. - If Firm B cheats, it earns $200. When Firm A cheats: - If Firm B colludes, it earns $50. - If Firm B cheats, it earns $20. Comparing the payoffs, we see that Firm B also has a dominant strategy to cheat, as it always earns more (\(200 > \)100\( and \)20 > \(50\)) when it cheats, regardless of Firm A's decision.

Determine the Likely Outcome

Since both firms have a dominant strategy to cheat, the likely outcome will be one in which both firms cheat. In this scenario, the payoff would be: - Firm A gets $500. - Firm B gets $20. Hence, the likely outcome for this duopoly is that both firms cheat, leading to Firm A earning \(500 and Firm B earning \)20.

Key concepts

Duopoly

Imagine two dominant firms, let's call them Firm A and Firm B, controlling the majority of the market share in their industry. This situation is known as a duopoly, which is a special case of an oligopoly where only two firms have significant influence over a market.

Normally, these firms face a high degree of interdependence because each firm's decisions affect the other's profits. If one firm lowers its prices, for example, the other might have to follow to remain competitive, potentially leading to a price war. Conversely, they could also implicitly or explicitly agree to fix prices, which may be illegal. In the duopoly from our exercise, the firms face the classic choice of either colluding to maximize joint profits or cheating for potential short-term gains.

Prisoner's Dilemma

The prisoner's dilemma is a standard example of a game analyzed in game theory that demonstrates why two rational individuals might not cooperate, even if it appears that it is in their best interest to do so.

Applying the prisoner's dilemma to Firm A and Firm B in our exercise, each firm acts as a 'prisoner' facing two options: to collude or to cheat. Collusion would lead to both firms earning decent profits (Firm A: \(1,000, Firm B: \)100), while mutual cheating leads to both earning significantly less (Firm A: \(500, Firm B: \)20). The dilemma occurs because each firm has an incentive to cheat to attempt to gain more profit, leading to a worse outcome for both if the other firm also cheats.

Dominant Strategy

In game theory, a dominant strategy is the best action a player can take regardless of what the opponent does. For both Firm A and Firm B, cheating is their dominant strategy since it yields a higher payoff no matter the decision of the other. When deciding between collusion and cheating, Firm A and Firm B compare their payoffs in different scenarios. For Firm A, cheating results in a higher payoff than collusion, whether Firm B decides to cheat (\(500 > \)800) or collude (\(1,050 > \)1,000). Similarly, Firm B gains more by cheating, irrespective of Firm A's actions (\(200 > \)100, \(20 > \)50). Hence, the exercise concludes that the likely outcome is both firms cheating, with payoffs of \(500 and \)20, respectively.

Understanding these concepts is fundamental in analyzing market behaviors between competitive firms and is pivotal for strategic decision-making in various economic and business models.