Game Theory Assignment

This assignment gives you practice with mechanism design. You may do it individually or in groups of two students. In the latter case, submit one file with both names and ID numbers. Real life background One factor that complicates insurance policy design is the degree of control consumers have over their risks. For example, some automobile insurance risks are largely outside the control of consumers: e.g., drivers cannot control the weather or the design of their cars. Other risks are at least partly within consumers control: e.g., drivers can decide whether to say home during snowstorms and choose whether to exceed the speed limit. Consequently, insurance policies are sometimes designed to encourage consumers to reduce their risks. Hypothetical story An insurance company plans to sell automobiles policies that will cover the cost of repairing collision damage. The company plans to charge drivers $600 per year for this policy. but has not yet decided how large the deductible should be. Collision damage typically costs $5,000 to repair. The insurer knows that drivers can influence their collision risk, because each driver can choose to drive cautiously, normally, or aggressively. Driving cautiously means each driver has a 10% chance of incurring damage each year; driving normally gives a 20% chance; and driving aggressively gives a 35% chance. Driving normally gives drivers no extra pleasure or annoyance. Driving aggressively gives drivers enjoyment worth $100 per year. By contrast, driving cautiously reduces their enjoyment by an amount equivalent to a cost of $200 per year. Instead of buying insurance from the company, drivers can buy coverage from a government program. This coverage costs $1050 per year and has no deductible. Drivers will buy policies from the insurance company if their total cost is at least as low: the cost includes the premium, the expected cost of the deductible amount (if any) if a collision occurs, and the enjoyment benefit or annoyance cost (if any). Your task is to advise the insurance company regarding the best deductible to choose. Questions Treat this situation as a sequential-move game. The insurance company moves first by specifying the deductible. Drivers move second by choosing whether to purchase a policy from the insurer and how to drive their cars.

Question 1 below is a simple warm-up exercise; you can solve it using participation conditions or simple logic. Questions 2 and 3 require you to model, analyze, and interpret the situation using decision variables, participation conditions, and incentive compatibility conditions.

1. Suppose the insurance company sells its policy without a deductible.

i. Would drivers purchase the policy; if so, how would they drive?

ii. What would be the insurers annual profit or loss per customer: i.e., the premium they receive minus the expected cost of damage?

iii. What would be each customers annual cost: i.e., the premium they pay plus the pleasure or annoyance factor?

2. Suppose instead the insurance company wants to sell its policy with a deductible that encourages customers to drive cautiously. It also wants to maximize its profit.

i. What should the insurer choose for its deductible?

ii. Would drivers purchase the policy; if so, how would they drive?

iii. What would be the insurers annual profit or loss per customer: i.e., the premium they receive minus their share of the expected damage cost?

iv. What would be each customers annual cost: i.e., the premium they pay, plus their share of expected damage costs, plus the pleasure or annoyance factor?

3. Suppose instead the insurance company sells its policy with a deductible that encourages customers to drive normally. It also wants to maximize its profit.

i. What should the insurer choose for its deductible?

ii. Would drivers purchase the policy; if so, how would they drive?

iii. What would be the insurers annual profit or loss per customer: i.e., the premium they receive minus their share of the expected damage cost?

iv. What would be each customers annual cost: i.e., the premium they pay, plus their share of expected damage costs, plus the pleasure or annoyance factor?