ECON 3100 Winter 2009, Assignment 1

The assignment must be done by 3 p.m. February, 2nd. You should put your assignment in the special box on the secretary desk in the Department of Economics. Good luck!

1. (10%) Consider the following normal-form game

| | | |Player |Two | |
| | |A |B |C |D |
|Player |A |2,0 |2,1 |1,4 |2,3 |
|One |B |2,5 |0,-1 |4,2 |1,1 |
| |C |5,2 |3,0 |5,5 |4,1 |
| |D |7,4 |2,2 |-30,3 |3,2 |

(a) (2%) Name all strongly dominated strategies for both players?
(b) (2%) Name all weakly dominated strategies for both players?
(c) (3%) Name all of the equilibria using the method of eliminating strictly dominated strategies
(d) (3%) Name all the Nash equilibria (pure strategies)

2. (20%) Two companies, Sharp and Sony, are trying to decide on a standard for their new televisions. The two choices are 140 lines per inch (140 lpi) and 210 lines per inch. Sharp has lower costs than Sony, and will drive it out of the market if they compete head to head, using the same standard. Each company chooses which standard to use secretly, and after setting its new products to use the standard a company cannot change its mind.
Sony 's value if it is driven out of the market is 0. Sharp 's value if it is a monopoly is 100. If both firms survive in the market, then Sharp 's value will be 50 and Sony 's will be 30.
(a) (2%) Represent this game as a normal-form game.
(b) (4%) Identify all Nash Equilibria in pure strategies.
(c) (5%) Sony is thinking of randomizing between 140 lpi and 210 lpi, picking 140 lpi with probability R. It has discovered that there is a spy in its headquarters who will be able to tell Sharp the value of R, but not which standard is actually selected unless one of the nonrandom extremes of R=0 or R=1 is chosen. Should Sony randomize anyway, or give up and choose either R=0 or R=1? Explain.
d) (8%) Find equilibrium in mixed strategies.

3. (18%) Suppose that Bombardier had organized its sale to Air Canada or WestJet as a first-best price sealed-bid auction. The Value of wining for Air Canada is 80 and that for WestJet is 40. Possible bids are 20, 40, 60 or 80. Alternatively, each firm can decide not to participate in the auction. If bids are equal, Air Canada wins. The utility from wining is the Value of winning minus the bid.
(a) (6%) Represent this game as a normal-form game
(b) (6%) Find all Nash equilibria (in pure strategies)
(c) (6%) compare these results with the results we learned in class about first-best price auctions. Is it different or similar? If different, explain why.

4. (12%) There are two identical firms in a market, each with constant per- unit production costs of $20 per unit. The market demand is given by the equation P = 80 - 4Q, where P is the price per unit, in dollars, and Q is the number of units sold in the market. Determine the Cournot equilibrium quantities the firms will sell and the resulting market price.

5. (18%) Three firms produce an identical product, but at different costs: it costs Firm A $20 for every unit it produces, it costs Firm B $40 per unit, and it costs Firm C $80 per unit. The market demand is given by the equation P = 300 - Q, where P is the unit price, in dollars, and Q is the total number of units sold. (Since the firms' output is identical, they all sell at the same price.) These are the only three firms in this market. Determine the Cournot equilibrium production levels of the three firms and the resulting market price.

6. ECON3100 How hard is it? (22%)
Anton is facing a difficult choice. His class size will be 0, 15 or 30 students depending on how difficult he makes ECON3100. The numbers of Students will rise or fall in this dramatic fashion because the male students (15 in total) have committed not to take any classes that the female students (15 in total) do not take. The reverse however is not true. That is, the female students may decide to take a class the male students are free to choose whether to take or not to take it.
Because students of the same sex have agreed beforehand to coordinate their decisions there are only three players in the game: Anton, Female Students and Male Students. Furthermore because all students will attend the first class, and because of the pact made by the Male students noted in the previous paragraph the order of play is:
1. Anton
2. Female Students
3. Male Students
Anton’s decision will be to choose the level of course difficulty from one of the following three options: high, medium, or low.
The Female Students’ decision is to take the course or quit.
The Male Students may face a similar decision.
Anton prefers to have a course with a high level of difficulty. If he chooses this outcome and gets 30 students, his payoff for this outcome is 100, If only 15 his payoff is 50. If the course has medium difficulty and all 30 students take the class his payoff will be 45, and if only 15 students his payoff will be 20. Finally, a full course (30 Students) that is easy will give Anton a payoff of 25, and if the class has only 15 students his payoff will be 5. If no students take the class the payoff for Anton will be 30, since he’ll have more time for research.
Payoffs for Female students are as follows: If they take a hard class they will get 10, a medium class 25 and an easy class 5. They don’t care if male students take the class or not. The Female students’ payoff for not taking the class is 15.
Male Students receive a payoff of 20 for a hard class, 25 for a medium and the same for an easy class. Their payoff for not taking the class is 15. (a) Write the game tree for this game (11%) (b) What is the outcome of this game? Explain (11%)