Temple University
Department of Economics

Economics 92
Principles of Economics, Honors

Homework 8
Extensive Form Games with Imperfect Information

Name

A. Credibility is a problem in many business circumstances: an employee promises to work hard in exchange for a higher wage, a borrower promises to operate wisely in order to protect the lender's capital, the insured promises not to take undue risks, etc.  In this problem we consider the game of Centipede.  There are two players Mille Pede and Chris Sallis.  There is a dollar on the table between them. Mille moves first.  She can either pick up the dollar and keep it for herself or she can leave it on the table.  If she leaves it on the table then it is automatically quadrupled and play passes to Chris.  Upon her turn Chris is confronted with snatching up the \$4 for herself, or splitting it with Chris as a reward for Chris passing up the opportunity to be greedy at the start. The game tree is shown as:

1. If Chris finds herself with a move at node C, what will she do?

2. Mille can look ahead and see what move Chris will make at node C.  With this in mind, what action should Mille take at node A?

3. Before the start of the game is it possible for Chris to make a credible commitment to 'share'? Yes  No

Consider a slightly modified version of the same game that Mille and Chris just played. There is an important change.  If at node B Chris says that she will share then she will be greeted with derision from her firends and so her payoff is -1.  For Mille a strategy consists of deciding what move to make should she find herself at node A.  Hence Mille has two strategies form which to choose.  For Chris a strategy consists of a pair of moves: the move that she would make should she find herself at B and the move that she would make should she find herself at node C. Hence, Chris has four strategies: <grab, grab>,  <grab, share>,  <share, grab> and <share, share>.

4. How many subgames are there in this new version of Centipede?

5. How many candidate solutions are there for the game given that Mille has two strategies from which to choose and Chris has four from which to choose?

6. Is the strategic profile {Wait, <share, share>} a subgame perfect equilibrium? Yes  No

7. Is the strategic profile {Wait, <grab, grab>} a subgame perfect equilibrium? Yes   No

8. Is the strategic profile {Wait, <share, grab>} a subgame perfect equilibrium? Yes  No

9. Is the strategic profile {Wait, <grab, share>} a subgame perfect equilibrium? Yes  No

10. Is the strategic profile {grab, <grab, grab>} a subgame perfect equilibrium? Yes  No

11. Is the strategic profile {grab, <grab, share>} a subgame perfect equilibrium? Yes  No

12. Is the strategic profile {grab, <share, grab>} a subgame perfect equilibrium? Yes  No

13. Is the strategic profile {grab, <share, share>} a subgame perfect equlibrium? Yes  No

14. What is the backward induction solution to the modified version of the Centipede game? (Pick one)

{Wait, <share, share>}

{Wait, <grab, grab>}

{Wait, <share, grab>}

{Wait, <grab, share>}

{grab, <grab, grab>}

{grab, <grab, share>}

{grab, <share, grab>}

{grab, <share, share>}

B. Entry and subsequent duopoly is a common model in economics because it teaches a number of valuable lessons.  In this model rivals compete on price.  Output is produced with zero cost, so revenue and profit are the same. Julie Yard must first decide whether or not she will enter the classical music recording business.  Once she has decided whether or not she will enter, Julie Yard and Strad E. Various, the incumbent firm, set their product price simultaneously.  If the two firms post the same price then they share the market equally, otherwise the lower priced firm gets all of the sales. The market demand curve is Q = 4-P. The game tree is as follows:

15. How many subgames are there?

16. Suppose Strad finds himself in the subgame beginning at node A.  What action should he take?

17. On a separate piece of paper write out the normal form of the subgame that begins with node B.
How many rows are there?
How many columns are there?

18. How many Nash equilibria can you find in the normal form game that you just wrote down in question 17?

19. Fill in the blanks for the actions chosen by Strad and Julie that correspond to the Nash equilibria in the subgame of question 17.

 Equilibrium # Julie Strad 1 2 3

20. Julie plays in how many proper subgames?

21. Strad plays in how many proper subgames?

22. A strategy for Strad consists of two parts: an action when he finds himself playing the game at node A and an action to be taken if he finds himself playing the game at node B.  A strategy for Julie also consists of two parts: The action she will take at the root and the action she will take should she find herself playing in the game at node B. Enter the complete strategies to be played by Julie and Strad that will lead to the Nash equilibria that you found in questions 17 - 19.

 Julie Strad At root At B At A At B 1 {< > < >} 2 {< > < >} 3 {< > < >}

23. From among the paths you entered in 22.,  enter the complete strategic profile(s) for Strad and Julie that results in a subgame perfect equilibrium:

 Julie Strad At root At B At A At B Sub-game perfect equilibrium {< > < >} {< > < >}

24. Which of the following is also subgame perfect equlibrium?

{<Out, 0>, <2, 0 >}

{<Out, 1>, <2, 1>}