1. The following matrix depicts payoffs to two department stores, Sears and Ames, when they adopt different combinations of appeals towards different income classes of customers.
(a) (4 points) Does either company have any strongly or weakly dominated strategies?
ANSWER. Ames: High is strongly dominated by either Medium or Low.
Sears: High is strongly dominated by Medium.
(b) (4 points) Does either company have any strongly or weakly dominant strategies?
ANSWER. Ames: High is dominated, so it can't be dominant. Medium
is the best response to Sears's Low and Low is the best response to
Sears's High, so no strategy is dominant for Ames.
Sears: High is dominated, so it cannot be dominant. Medium is the
best response to Medium and Low is the best response to High, so no
strategy is dominant for Sears.
A strategy being "dominant" does not just mean "good in some
circumstances". It means that that it is a best response to any
strategy the other player might use (though it is only weakly
dominant if in some circumstances there is another strategy that does
just as well).
(c) (4 points) What are the Nash equilibria? (consider only non-random strategies)
ANSWER. We can rule out the strictly dominated strategies from being part of Nash equilibria. That leaves Low and Medium for each player. (Low, Medium) is the unique Nash equilibrium.
Sears
High-income Medium-income Low-income
High-income 0,0 1,1 2,3
Ames
Medium-income 4,2 2,3 4,2
Low-income 5,1 3,2 3,0
2. Two companies, Acme and Basic, 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. Acme has lower costs than Basic,
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. Basic's value if it is driven out of the market is 0. Acme's value if it is a monopoly is 100. If both firms survive in the market, then Acme's value will be 50 and Basic's will be 30.
a. (4 pts.) Represent this game in a 2x2 matrix.
ANSWER:
Basic
140 lpi 210 lpi
140 lpi 100,0 50,30
Acme
210 lpi 50,30 100,0
b. (4 pts.) Identify all Nash Equilibria in nonrandom strategies.
ANSWER. There are no Nash equilibria, except in mixed strategies, in which each player randomizes.
c. (4 pts.) Basic 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 Acme 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 Basic randomize anyway, or give up and choose either R=0 or R=1? Explain.
ANSWER. Basic should indeed randomize between 140 lpi and 210 lpi. You will learn later in the course how to compute the probability Basic should use for 140 lpi. Here, simply note that if Basic chooses any probability R greater than 50 percent, Acme will choose 140 lpi and drive Basic out with probability of more than 50 percent. If Basic chooses any probability R less than 50 percent, Acme will choose 210 lpi and drive Basic out with probability of more than 50 percent. If R is exactly 50 percent, then Basic survives with probability 50 percent, which is the best it can do.
3. Vice-presidents Smith and Jones are the two contenders to be
the next president of Indiana Steel when the current president
retires in five years. They know that if both stay with Indiana
Steel, they will be in a tournament with each other and after both
working very hard and flattering the board of directors, they will
each have a 50 percent chance of becoming president. Smith suggested that in keeping with the Indiana tradition of basketball being the game of life, they should short-circuit the succession process by using a foul shooting contest. The two would take turns trying to throw a basketball into the hoop from the foul line, and whoever succeeded first would stay with Indiana Steel, while the loser would quit and find a job elsewhere. Smith says that since both of them know that he, Smith, succeeds in 50 percent of his shots from the foul line whereas Jones only succeeds 30 percent of the time, he is willing to let Jones shoot first. Also, the rules will be that each player only gets one shot. If they both miss, they will give up and go back to working hard.
While each vice-president would like the other to quit, each prefers that they both stay to himself having to leave. Also, there is another option both of them are considering; namely, to miss on purpose by rolling the ball along the floor.
a. (8 points) Draw a game tree that expresses the game described above, being sure to label the payoffs, the actions on each branch, and the probabilities of all moves by Nature. You may assume that a player's from being the only one to stay at the company is 2; his payoff to leaving is -2; and his payoff to staying when the other player stays also is 0.
ANSWER. At the first node, Jones chooses SHOOT or ROLL. If Jones chooses SHOOT, then with probability .3 he succeeds, and Smith resigns, for payoffs of (2, -2) for the two players. If Jones fails, or if he chooses ROLL, Smith chooses SHOOT or ROLL. If Smith chooses ROLL, then the payoffs are (0,0). If Smith chooses SHOOT, then with probability . 5 he succeeds and the payoffs are (-2,2). If he fails, which has probability .5, then the payoffs are (0,0), since neither resigns.
b. (4 points) What is Smith's optimal strategy? What is Jones's optimal strategy?
ANSWER. Work back from the end. If Jones has chosen SHOOT and failed, Smith will choose SHOOT too, since he has a choice between a sure 0 and a 50 percent 0, 50 percent 2. Smith's expected payoff from that is .5(-2)= -1. So if Jones chooses SHOOT, he will get +2 with probability .3 and -1 with probability .7, an expected value of -0.1.
If Jones chooses ROLL, then Smith will SHOOT, since he has a choice between a sure 0 and a 50 percent 0, 50 percent 2. This will give Jones an expected payoff of .5(0) + .5(2) = -1.
Jones should therefore SHOOT, and Smith should SHOOT too, whatever Jones does.
Does that depend on the particular payoffs I chose? No--- only on their rankings. More generally, if Jones has chosen SHOOT and failed, Smith will choose SHOOT too, since he has a choice between a sure W and a 50 percent W, 50 percent something bigger. Jones's expected payoff from that is .5(X)+ .5 (X-Y). So if Jones chooses SHOOT, he will get some large payoff Z>X with probability .3, and .5(X)+ .5 (X- Y) with probability .7, an expected value of .3Z + .7X -.35Y.
If Jones chooses ROLL, then Smith will SHOOT, since he has a choice between a sure W and a 50 percent W, 50 percent something bigger. This will give Jones an expected payoff of .5(X) + .5(X-Y), or X - .5Y. That is less than .3Z + .7X -.35Y, so Jones should SHOOT.
Notes on some wrong answers.
1. Rolling the ball on the floor is a move in the game tree that
needs to be considered.
2. It is wrong to say that a player's optimal strategy is to "succeed
in putting the ball through the basket". That is not a choice the
player can make-- it is up to chance. All he can decide is whether to
shoot the ball or roll it on the ground.
3. The rules say that whichever player succeeds first
wins. Hence, if Jones succeeds, the contest is over, and the payoffs
are 2 for him and -2 for Smith.
4. I gave partial credit for a wrong answer in a later part of the
question if it followed logically from a mistake in an earlier
part.
5. This is a one-shot game, so if Jones rolls the ball, Smith will
not respond by rolling also-- Smith will try to shoot a basket.
6. Rolling the ball is not the same as forfeiting the contest. A
player loses only if the other player shoots a basket first, and the
other player might miss or roll.
Note also that since the probability Jones wins the contest is .3 and the probability that Smith wins is (1-.3)(.5)= .35, the contest gives Smith an advantage. Jones's expected payoff from participating in the contest is .3(2) + .35 (-2) + .35 (0) = -.1, which is worse than not participating. Smith's expected payoff is .3(-2) + .35 (2) + .35 (0) = .1. It was not necessary to analyze the participation decision, however, to get full credit for the question-- doing just the contest subgame was enough. If you did, however, then I gave credit for an answer that explained that Jones would not participate.
A variant on the problem would be to set the payoff to losing the contest and having to leave at -1 instead of -2. That makes sense if leaving at least allows the player to take it easy and not work so hard. With that change, the game becomes positive-sum, and both Smith and Jones would gain (in expected value) from participating in the contest.
c. (4 points) When both players use their optimal strategies what is the probability that Jones stays with the company?
ANSWER. Jones stays if he wins his first shot (.3), or if he misses and Smith misses too (.7)(.5), which gives him probability .65 of staying.