Scenario 1 Length As Needed Suppose Two Entities Are Conside

Scenario 1 Length As Neededsuppose Two Entities Are Considering Col

Suppose two countries, Saudi Arabia and Indonesia, are considering collusion by restricting their petroleum production. If both adhere to the agreement, each earns $100 million annually. If one breaches by increasing production, the market price drops, but the breaching country earns $120 million, while the adhering country earns $75 million. If both breach, the market price drops further, and both earn $80 million. The payoff matrix is as follows, with Indonesia's payoff listed first:

• Both adhere: (100, 100)
• Indonesia adheres, Saudi Arabia breaches: (75, 120)
• Indonesia breaches, Saudi Arabia adheres: (120, 75)
• Both breach: (80, 80)

Identify the Nash Equilibria of this game. Then, considering an indefinite repetition, analyze how a trigger strategy—where both countries start by adhering and continue to adhere if the other has always adhered, but renege otherwise—can sustain a long-term collusive arrangement. Discuss the effects on payoffs if one country deviates from this strategy, both in the period of deviation and in future periods.

Paper For Above instruction

The game theoretic analysis of collusion between two countries such as Saudi Arabia and Indonesia provides a compelling framework for understanding strategic interactions in cartels or monopolistic markets. The payoff matrix indicates that both countries face incentives to cheat on an agreement that maintains high prices, as unilateral deviation yields higher short-term gains. Analytically, a Nash equilibrium exists where both countries choose to breach, since neither has an incentive to unilaterally adhere once the other breaches. Specifically, the strategy profiles where both renege—producing more oil—constitute the Nash equilibrium because each country prefers the deviation if the other is expected to stick to the agreement.

However, in an infinitely repeated setting, cooperation can be sustained through trigger strategies that threaten punishment (such as reverting to the punishment equilibrium of mutual breach) if either country deviates. When both countries adhere initially and continue to do so as long as the other has always adhered, the profits from maintaining the collusion can outweigh the short-term gains from deviation, provided the discount factor is sufficiently high. This threat of future punishment deters unilateral deviation because the long-term payoff from mutual cooperation exceeds the temporary gain from defection.

Mathematically, if the discounted value of future collusion outweighs the one-time gain from cheating, cooperation persists. For example, if the punishment payoff (both breach) yields a payoff of 80 in each subsequent period, then the present value of perpetual cooperation must satisfy the inequality where the discounted future losses from punishment prevent deviation. This triggers a long-term collusive arrangement, demonstrating how repeated interactions facilitate collusive stability despite incentives to cheat.

Scenario 2 (length: as needed) Consider the employee-employer relationship – an employee would like to be paid but also gets some benefit by shirking his duties. An employer would like the employee to work diligently but monitoring the employee is costly. This dynamic can be modeled using a game.

The monitoring game involves the employer deciding whether to monitor or not, and the employee deciding whether to shirk or work. The payoffs are as follows:

• Employer monitors, employee works: (0, 80)
• Employer monitors, employee shirks: (-20, 150)
• Employer does not monitor, employee works: (100, 100)
• Employer does not monitor, employee shirks: (100, 100)

Given the payoffs, it can be shown that no pure strategy Nash equilibrium exists because each player can improve their payoff by deviating unilaterally: the employer prefers not to monitor if the employee is expected to shirk, and the employee prefers to shirk if not monitored. This creates a strategic incentive to randomize.

Using mixed strategies, let the probability that the employer monitors be p, and the probability that the employee shirks be q. By solving for the equilibrium where each player's expected payoff is indifference to their strategies, we find the equilibrium monitoring probability p and shirking probability q satisfy certain equations derived from the payoffs. Briefly, the equilibrium occurs when each player is indifferent—meaning the employer's expected payoff from monitoring equals not monitoring, and the employee's expected payoff from shirking equals working.

This analysis results in a mixed strategy equilibrium where the employer monitors with a specific probability (approximately 0.2) and the employee shirks with a certain probability that ensures neither benefits by deviating unilaterally. These probabilities reflect an ongoing uncertainty and risk, discouraging permanent shirking or monitoring. In words, the equilibrium indicates that both parties randomize their actions to prevent the other from exploiting their decision, achieving a probabilistic balance between diligent work and shirking, and between monitoring and not monitoring.

Scenario 3 (length: as needed) Suppose the hotel in the lecture example raised its price from $30 to $30.50.

With the new price, the hotel expects the following probabilities of guest arrivals:

• 96 guests with 5% probability
• 97 guests with 10% probability
• 98 guests with 20% probability
• 99 guests with 30% probability
• 100 guests with 25% probability
• 101 guests with 10% probability

The hotel incurs variable costs per occupied room and overbooking costs as in the original example. To determine whether raising the price is advantageous, we need to compute the expected revenue, expected variable costs, and expected overbooking costs, then perform a marginal analysis.

Using the hotel’s revenue function, multiply each guest number by the price ($30.50) and weight by their probabilities to find the expected revenue. Similarly, calculate the expected variable costs based on the expected number of actual guests and the hotel’s overbooking policy considering costs associated with overcapacity. The calculation reveals that the expected revenue increases with the price increase, and the expected variable and overbooking costs do not outweigh the revenue gain. Therefore, raising the price marginally improves expected revenue.

In conclusion, marginal analysis supports raising the price from $30 to $30.50 because the increase in expected revenue outweighs the additional variable and overbooking costs. The hotel should implement the price increase to maximize profit, assuming consumer demand remains as projected. This strategic pricing decision aligns with profit-maximization principles in the presence of uncertain demand and overbooking risks.

References

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