Question

Soapy Inc. and Suddies Inc., the only soappowder producers, collude and agree to share the market equally. If neither firm cheats, each makes \(\$ 1\) million. If one firm cheats, it makes \(\$ 1.5\) million, while the complier incurs a loss of \(\$ 0.5\) million. If both cheat, they break even. Neither firm can monitor the other's actions. a. What are the strategies in this game? Construct the payoff matrix for this game. b. If the game is played only once what is the equilibrium? Is it a dominant- strategy equilibrium? Explain.

Short answer

Both firms will cheat, resulting in a payoff of (0, 0). Cheating is the dominant strategy for both firms.

Step-by-step solution

Identify the players and strategies

The players in this game are Soapy Inc. and Suddies Inc. Each firm has two strategies: to 'Collude' or to 'Cheat'.

Determine the payoffs

If both firms collude, each makes \(1 million. If one cheats and the other colludes, the cheater makes \)1.5 million, and the colluder incurs a loss of \(0.5 million. If both cheat, they both break even and make \)0.

Construct the payoff matrix

Construct the payoff matrix for Soapy Inc. and Suddies Inc.:| | Suddies Inc. Colludes | Suddies Inc. Cheats ||---------------|-----------------------|---------------------|| Soapy Colludes | (1, 1) | (-0.5, 1.5) || Soapy Cheats | (1.5, -0.5) | (0, 0) |The first value in each cell represents Soapy Inc.'s payoff, and the second value represents Suddies Inc.'s payoff.

Determine the equilibrium

Analyze the payoff matrix to determine the equilibrium. If Soapy Inc. colludes, Suddies Inc. maximizes its payoff by cheating (getting 1.5 rather than 1). If Soapy Inc. cheats, Suddies Inc. should also cheat to avoid a loss (getting 0 rather than -0.5). The same analysis applies to Soapy Inc. Therefore, both firms cheating (0, 0) is the Nash equilibrium because neither firm can improve their payoff by changing their strategy unilaterally.

Identify if it's a dominant-strategy equilibrium

A strategy is dominant if it yields a higher payoff regardless of the other firm's strategy. Here, cheating is a dominant strategy for both firms because it results in a higher or equal payoff compared to colluding, irrespective of the other firm's action.

Key concepts

Nash Equilibrium

In game theory, a Nash Equilibrium occurs when each player's strategy is optimal, considering the strategies of the other players. This means no player can gain by unilaterally changing their strategy.
In the context of Soapy Inc. and Suddies Inc., both firms reach a Nash Equilibrium by choosing to cheat. If Soapy Inc. colludes, Suddies Inc. maximizes its gain by cheating, earning \(1.5 million. If Soapy Inc. cheats, Suddies Inc. should also cheat to avoid a loss, yielding \)0.
Thus, both cheating (0, 0) is the Nash Equilibrium. Neither firm can improve by switching strategies alone.

Dominant Strategy

A Dominant Strategy is one that provides a higher payoff for a player, no matter what the other players do. It's the best response in every situation.
For Soapy Inc. and Suddies Inc., cheating is a dominant strategy. If Soapy Inc. colludes, Suddies Inc. earns more by cheating (\(1.5 million versus \)1 million). If Soapy Inc. cheats, Suddies Inc. avoids a loss by also cheating (earning \(0 versus losing \)0.5 million).
The analysis is the same for Soapy Inc. Hence, cheating is the dominant strategy for both firms.

Payoff Matrix

The Payoff Matrix is a fundamental tool in game theory. It illustrates the payoffs for each player depending on the strategic choices made by all involved parties.
Let's break down the Payoff Matrix for Soapy Inc. and Suddies Inc. It looks like this:

• If both collude, they each make \(1 million.
• If one cheats and the other colludes, the cheater gets \)1.5 million and the colluder loses \(0.5 million.
• If both cheat, they break even, making \)0.

Here's a simplified version of the Matrix:

Soapy Inc. Colludes vs. Suddies Inc. Colludes: (1, 1)
Soapy Inc. Colludes vs. Suddies Inc. Cheats: (-0.5, 1.5)
Soapy Inc. Cheats vs. Suddies Inc. Colludes: (1.5, -0.5)
Soapy Inc. Cheats vs. Suddies Inc. Cheats: (0, 0)
The first number in each pair represents Soapy Inc.'s payoff, while the second is Suddies Inc.'s payoff.

Collusion

Collusion occurs when firms secretly agree to work together rather than compete, usually to increase profits. However, it can be unstable due to the temptation to cheat.
For Soapy Inc. and Suddies Inc., collusion would mean both firms agreeing to share the market equally, each earning \(1 million. However, without monitoring, the risk of defection is high.
If one cheats while the other colludes, the cheater gains \)1.5 million while the faithful firm loses $0.5 million. This results in a reach for mutual cheating as the more stable strategy, leading to the Nash Equilibrium where both firms cheat and make nothing. Collusion here is fragile, as the incentive to cheat disrupts the agreement.