Name Key

A. Market Niche is important to any firm since it confirms a certain degree of market power and pricing discretion.  This is a problem about entry and market niche; and it is a first problem in mixed strategy.

There is a new market in intelligent human-like robots, but there is room for only one firm.  Miss Anne Thrope and Andy Royd own firms that could manufacture the new robots.  If only one firm enters then it earns profits of $100. If they both enter then each of them loses $50.  If neither of them enters then no one earns a profit.  The payoffs can be summarized as

1. Does Miss Anne Thrope have a dominant strategy? No. The payoff inequality reverses as you read across the rows.

2. Does Andy Royd have a dominant strategy? No. The game is symmetric, so the payoff inequalities change as you read down the columns.

3. Using "best response", how many pure strategy equilibria are there? There are two: {Enter, Out} and {Out, Enter}

4. With what probability should Andy Royd play "Enter?"

Andy wants to play Enter with a probability p such that Thrope is indifferent between playing Enter and Out. Hence he should choose p so that -50p+100(1-p) = 0p+0(1-p) = 0.  Solving for p we get p=2/3

5. With what probability should Miss Anne Thrope play "Stay Out?"

The game is symmetric so we know she will play Enter with probability 2/3.  Therefore she plays Stay Out with probability 1/3.

6. What is the expected payoff to Andy Royd from playing the mixed strategy?

-50(4/9) +100(2/9) = 0

7. What percent of the time will too many firms enter the robot market? 4/9

8. In the sense of maximizing total payoff to all players, is the mixed strategy solution that you have found an efficient outcome? Yes  No  No. Either of the asymmetric pure strategy equilibria would result in total payoff of 100.  When both players use a mixed strategy the total payoff is zero, on average.

9. In the sense that on average each player does as well as the other, is the mixed strategy solution fair? Yes  No

Yes.  In expectation they each get a payoff of 0.

10. Is there a way to reconcile fairness and efficiency? Yes  No No.  That is, the answer is no unless we allow them to collude, play an asymmetric solution and split the payoff.

B. Video System Coordination (VHS v. Beta) was a problem when VCR's were first introduced.  A similar problem emerged with DVD and high density TV so it is worth looking closely at coordination games and mixed strategies.  Sony and Panasonic can choose from either of two formats, VHS or Beta.  If they use the same format then they are better off.  If they use different formats then no one is better off.  They payoffs can be summarized as

11. There are two pure strategy equilibria in this game.  What are they?

12. Are the pure strategy solutions fair in the sense of equal payoffs? Yes  No Yes

13. Are the pure strategy solutions efficient? Yes  No Yes

14. With what probability should Sony play Beta in a mixed strategy? 1/2  The reasoning is the same as that in the previous problem.

15. What is Sony's expected payoff in the mixed strategy solution? 1/2

16. Is the mixed strategy solution to the game fair in the sense of both players getting the same payoff? Yes  No Yes

17. Is the mixed strategy solution to the game efficient in the sense of maximizing the total payoff for all players? Yes  No No. With the mixed strategy each player earns 1/2 in expectation, for a total of 1.  If the could coordinate on either Beta or VHS then the combined payoff would be 2.

C. Sales are foremost in our minds now that the Christmas shopping season is behind us.  With expansion of retail outlets and the poor results of the season Anna Conda, the VP for Sales and Marketing, must decide on a new strategy: Periodic sales or 'everyday low prices.'  When Conda's firm has a sale they pick up new informed buyers who were waiting for a price drop, as well as winning customers from their rival.  If both firms have a sale the Anna is worse off.  If Anna holds her price when Turnitif has a sale then she will pick up some customers who want her trademark.  Anna Conda's research assistant has determined that payoffs in their retail market are

18. Does Conda have a dominant strategy? Yes   No No

19. Does Turnitif have a dominant strategy? Yes   No No

20. Is there an equilibrium in pure strategies? Yes   No No

21. If Anna Conda starts a new campaign in which she promises sale prices all of the time what will be her profits? 90

22. Is there a mixed strategy equilibrium? If your answer is yes then fill in the blanks:

Anna wants to choose q so that Al is indifferent between Normal and Sale.  That is,

 175q+100(1-q)=150q+120(1-q)

25q-20+20q=0

q=4/9

Turnitif wants to choose p so that Anna is indifferent between Normal and Sale. That is,

100p+150(1-p) = 200p + 90(1-p)

-100p +60 - 60p = 0

p = 3/8