Two-person sequential game, game theory

In summary, the smallest value of the bid b that a rational investor would make to the company Fortune is 100 and the highest value of the bid b that a rational investor would make to the company Fortune is 110. This ensures that both the investor and the company are at least indifferent when the bid is made.
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Here is my solution. Is it right? Thank you!

Homework Statement



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?

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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  • #2
Therefore, the smallest value of the bid b that a rational investor would make to the company Fortune is 100 and the highest value of the bid b that a rational investor would make to the company Fortune is 110.
 

1. What is a two-person sequential game?

A two-person sequential game is a type of game in game theory where two players take turns making decisions. Each player's decision is based on the previous player's decision, and the outcomes of the game are affected by both players' choices.

2. What is game theory?

Game theory is a branch of mathematics that studies decision-making in situations where multiple parties are involved and their decisions impact each other. It is commonly used in economics, political science, and other social sciences to analyze strategic interactions between individuals or organizations.

3. What is the difference between simultaneous and sequential games?

In simultaneous games, both players make their decisions at the same time without knowledge of the other player's decision. In sequential games, players take turns making decisions, and the outcome of the game is influenced by the previous player's decision.

4. How is a two-person sequential game represented?

A two-person sequential game is typically represented using a game tree, which shows the sequence of decisions and the possible outcomes at each decision point. The players' strategies and payoffs are also included in the game tree.

5. What are some real-world applications of two-person sequential games?

Two-person sequential games have been used to model various real-world scenarios, such as negotiations, military conflicts, and business strategies. Examples include the Prisoner's Dilemma, the Ultimatum Game, and the Centipede Game.

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