Scenario 1 (length: as needed)
Suppose two entities are considering collusion â€“ to make things â€˜legalâ€™, consider a situation similar to OPEC, except with only two countries: Saudi Arabia and Indonesia. The two countries have negotiated an agreement to restrict their production of petroleum. If both countries follow the agreement, the market price of petroleum will be high and both countries will make $100 million per year. If one country reneges and produces more petroleum than dictated in the agreement, then the market price will decrease. However, the increased production will offset the lower price for the country that reneges so that country will make $120 million per year, while the country who adhered to the agreement will make $75 million. If both countries renege on the agreement then the market price will drop further and both countries will make $80 million per year. The game is illustrated in the table below, with Indonesiaâ€™s payoff listed first and Saudi Arabiaâ€™s payoff listed second in every pair:
- Find the Nash Equilibria of this game.
- Suppose the game was repeated indefinitely. Explain how if both countries follow a trigger strategy (page 177 of your text) in which they adhere in the first period and continue to adhere to the agreement as long as the other country has always adhered but will renege otherwise leads to a long-term collusive arrangement. Hint: consider one country following the trigger strategy and determine what happens to the other countryâ€™s payoff if it decides to deviate from the strategy â€“ to play renege. What are the payoffs in that period and in all future periods?
Scenario 2 (length: as needed)
Consider the employee-employer relationship â€“ 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 payoffs of the â€œmonitoring gameâ€ are given below:
For the employer, this assumes that the worker receives 100 in wages, produces 200 worth of goods if the employee works and monitoring costs 20. From the employeeâ€™s point of view, the net benefit to the employee from working and getting paid is 100. If the worker can shirk and get paid the worker is better off, however the employee is fired if the worker shirks and the employer monitors and thus is worse off.
- Show that there are no pure strategy Nash equilibria in this game.
- What is the mixed strategy Nash equilibria? In other words, what is the probability that the employer will monitor? What is the probability that the employee will shirk? See the lecture for details on how to calculate the probabilities.
- Briefly interpret the Nash equilibria in words.
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 96 guests to arrive 5% of the time, 97 guests 10% of the time, 98 guests 20% of the time, 99 guests 30% of the time, 100 guests 25% of the time and 101 guests 10% of the time. The variable costs per occupied room and overbooking costs are the same as in the lecture.
- Calculate the expected revenue, expected variable costs and expected costs from overbooking.
- Using marginal analysis, should the hotel raise its price? Explain your answer.