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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? She should grab.

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? She should garb.

3. Before the start of the game is it possible for Chris to make a credible commitment to 'share'? Yes  No No it is not possible for her to make a credible commitment to share since 'grab' is her dominant strategy.

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? There are three.

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? There are a total of eight.  To see this you can wirte down the strategic form of the game.

6. Is the strategic profile {Wait, <share, share>} a subgame perfect equilibrium? Yes  No No. The flaw in the plan is seen in either of the subgames at B or C. 

7. Is the strategic profile {Wait, <grab, grab>} a subgame perfect equilibrium? Yes   No No. The flaw in the plan is seen at the choice of Wait at the root.

8. Is the strategic profile {Wait, <share, grab>} a subgame perfect equilibrium? Yes  No No. The flaw is in the choice of 'share' if Chris finds herself at B.

There are five more strategic plans that might be used.  Of the eight complete contingent plans, only one is a subgame perfect equilibrium.  The one subgame perfect equilibrium is also the backward induction solution to the game.  In an extensive form game of perfect information with distinct payoffs there is only one backward induction solution and it is the only subgame perfect equlibrium. 

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:

9. How many subgames are there? Three

10. Suppose Strad finds himself in the subgame beginning at node A.  What action should he take? 2 is the price he should charge.

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

12. How many Nash equilibria can you find in the normal form game that you just wrote down in question 11? There are two.  One when they both charge a price of $0, and the other is when they both charge a price of $1.

13. Fill in the blanks for the actions chosen by Strad and Julie that correspond to the Nash equilibria in the subgame of question 11.

14. Julie plays in how many subgames? Two 

15. Strad plays in how many subgames? He plays in two proper subgames at nodes A and B, and he plays in the whole game which is itself a subgame.

16. 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 11 - 13.

17. Enter the complete strategic profile for Strad and Julie that results in a subgame perfect equilibrium: Both strategies in answer 16 are subgame perfect.  However, the strategic plan {<Enter, 1>, <2, 1>} is the backward induction solution to the game.  The subgame at B is dominance solvable using IEDS.  Recognizing this Julie will Enter and charge a price of $1.

There is a third strategic plan that is subgame perfect: {<Out, 0>, <2, 0>}. However, {<Out, 1>, <2, 1>} is not.  To see why this is so, we can write out a reduced version of the normal form of the game.

You can see that <Out,1> is not a best response to <2,1>, therefore {<Out, 1>, <2, 1>} is not a subgame perfect equlibrium.