|Jul2-12, 05:11 PM||#1|
Two-person sequential game, game theory
Here is my solution. Is it right? Thank you!
1. The problem statement, all variables and given/known data
An investor, whom we will refer to as player I, is considering bidding for a company called Fortune F. Currently, the shares of F are valued at $100. If I can take over the company it is known that F would be worth $110 per share under the new management. The investor I can make a bid of b (> 100) dollars per share in order to take over the company by buying 50% of the total shares of F.
The payoffs of the game are to be understood as follows. If the investor makes a bid of b dollars per share, then either fortune F can accept the bid and sell the shares to the investor at b dollars per share or reject the bid by buying back its shares at b dollars per share. If it accepts the bid, fortune F makes a profit of b − 100 dollars per share. The investor, however, makes 110 − b dollars per share after the value of the shares increase to $110 per share under the new management. If fortune F rejects the bid, then it has to buy back the shares at b dollars per share and loses b − 100 dollars per share. The investor in this case gets zero.
What should be the smallest and highest values of the bid b that a rational investor would make to the company Fortune?
3. The attempt at a solution
To answer this question, we need to make sure that both the investor and the company are at least indifferent when the bid is made. For sellers, the difference between the bid and the initial stock price must be equal to at least 0, which is b-100 ≥0 → b≥100. For the investor, 110-b≥0→b≤110. Thus, 100≤ b≤110.
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