Problem Set #2
ISS 328 Summer, 2015
Michigan State University L. Martin
1. Let's play three yard football (The games are shown on Thursday afternoon between 4:45 and 5 on the SASN -Short Attention Span Network). The home team starts with the ball on the 1 yard line. The transition probabilities are given in the table
Outcome Probability
Lose 1 yard 0.2
Lose 2 yards 0.2
No gain 0.2
Gain 1 yard 0.2
Gain 2 yards 0.2
The first team to score wins. So, for example, if the home team gains 2 yards, it wins; if it loses 1 or 2 yards it loses. If it gains zero or 1 yard, the ball is turned over to the other team at the 1 yard line.
a. Is this game recursive? If so, which state recurs?
Yes, it is recursive. The ball will be turned over to the other team at the 1 yard line recurs whenever the first team gains zero or 1 yard.
b. Compute the following probabilities.
Home team wins on first possession _____0.2_____
Home team loses on first possession _____0.4_____
Ball is turned over to visitors _____0.4____
c. Draw the tree. You should have three lines emanating from the first node (home team wins, home team loses, ball is turned over. Also, if there is a state that recurs, do not go beyond that state.
Gain 2 yard (0.2) Win (0.2)
Gain 1 yard (0.2)
Start Turned over(0.4)
No gain (0.2)
Lose 1 yard (0.2)
Lose (0.4)
Lose 2 yard (0.2)
d. Write an equation for the value of having the ball at the 1 yard line. (Hint: let V be the value of having the ball on the 1 yard line. Then -V is the value of your opponent having the ball on the 1 yard line.) The value of winning is +1 (game) and the value of losing is -1 (game)
e. Solve the equation in part b. Value = ________________
f. What is the probability that the home team will win the game?
Probability = ____0.36__________
2. Overtime in three-yard football. The scored is tied at the end of regulation. The overtime rules are that the home team gets the ball on its one yard line with a chance to score. If it scores, the game is over. If it does not score, the visitors get a chance from their one yard line. If they score, the game is over. If neither team scores, the game is a tie. (We cannot have sudden death because of the short time slot.) As in problem 1, we are computing value from the perspective of the home team. The value of the home team winning is +1 (game), the value of a tie is 0 and the value of the home team losing is -1 (game).
a. Is the overtime recursive? (Hint: Is there a limit to the number of possessions or can the game continue indefinitely?)
b. The probabilities are the same as in problem one. Draw the tree.
c. What is the value of the game in the state where the visitors have possession (that is, after the home team has failed to score)?
d. What is the value of the game in the state where the home team has possession (that is, at the start of overtime)?
e. Compute the probability that the home team wins ___________
visiting team wins ___________
game is tied ___________
3. More on three yard football. The following table gives the strategies and payoffs for the first play.
Visitors
6 on the 7 on the 8 on the 9 on the
line line line line
fade -2 -1 1 -1
slant 1 0 2 1
run up 2 -1 2 2
the middle
run wide 1 -2 -1 -1
a. Write out the best response for the home team and the visitors.
Home team Visitors
6 on line ______ fade _______
7 on line ______ slant _______
8 on line ______ run middle _______
9 on line ______ run wide _______
b. What are the Nash equilibrium strategies?
Home _______________
Visitors _______________
c. Will the home team win the game on the first play?
Yes No
4. Marquise Brown, the power forward for Enormous State has two post moves: the jump hook with the left hand and the newly learned drop step. His defender must anticipate one move or the other. The probability that Marquise makes the shot is given in the table.
Defender
Anticipate Anticipate
Jump hook Drop step
Jump hook 0.3 0.9
Marquise
Drop step 0.6 0.2
a. Is there a Nash equilibrium in pure strategies? Explain briefly.
b. Determine the optimal mixing probability for Marquise.
c. What is the probability that he makes the shot.