Problem
An individual hires an agent to sell an object for him. The object's owner does not know the actual value of the object, but he knows that it is worth nothing with probability 1/2, and worth $10 with probability 1/2. The agent will sell the object, report its sale price to the owner, and pay the owner part or all of the proceeds from the sale. Specifically
• If the agent reports that the sale price was $10, he pays R dollars to the owner.
• If the agent reports that the object was worth nothing and the owner accepts this report, the agent pays nothing to the owner. However, the owner can also choose to reject the report. If he does so, he must pay $2 to learn the true sale price of the object from an independent source. If he finds that the agent lied about the sale price, the owner receives the entire sale price ($10) while the agent pays a fine of $50 to the government
Assume that the both the owner and the agent act so as to maximize their own expected profits from the sale of the object. Let y be the probability that the agent, having actually sold the object for $10, claims it to have been worth nothing. Let x be the probability that the owner, having been told that the object was worth nothing, rejects the report. A Nash equilibrium in this situation is a pair (x∗, y∗) such that
i) If the agent lies with probability y*, the owner's expected profits are maximized when he rejects with probability x∗.
ii) If the owner rejects with probability x*, the agent's expected profits are maximized when he lies with probability y∗.
Find the Nash equilibrium for any R between 0 and 10. Find the expected profits of the owner and the agent for any R. What value of R (where 0 ≤ R ≤ 10) maximizes the owner's expected profits? What are the owner's and the agent's expected profits under this value of R? Does the sum of the owner's and agent's expected profits equal the true expected value of the object?
The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.