Econ 431: David Reiley
Due: Wednesday, 28 February 2007
Problem Set #4
The material in this problem set comes primarily
from Chapters 7 and 8, plus the related handout on
- (10 points) Play at least 50 rounds of rock-paper-scissors against the WWW Roshambot. Save your results, as shown on the View History
page, which will look something like this.
Hand in your results, along with one or two sentences to explain what ideas
or reasoning you used when choosing your strategies.
- (20 points) DS, question 7.3.
- (20 points) DS, question 7.5.
- (20 points) DS, question 7.7.
- (20 points) DS, question 7.9. To verify the equilibrium, compute the expected
payoffs for each of the pure strategies against the opponent's proposed equilibrium
mixed strategy. Show that the equilibrium mixture for each player includes
only those pure strategies with the highest payoffs. I recommend retaining
only three significant digits, or else you may confuse yourself with rounding
errors. (See the final extra-credit question below if you are interested in
computing this equilibrium from scratch.)
- (20 points) DS, question 7.10. For part (a), please just verify the entries
in row 2. You can assume the rest of the entries are correct.
- (20 points) DS, question 8.2. Hint for part (b): Consider doing a graph
like in Figure 7.9. Take the player with only two pure strategies, note that
you can summarize her mixed strategy with just a single variable, and plot
that variable on the horizontal axis.
- (20 points) DS, question 8.3.
- (20 points) DS, question 8.6. Note that in part (b), you are asked to compare
quantities between the old version of Chicken and the new version of Chicken.
- (20 points) DS, question 8.7. This is a hard problem, so here are some hints.
To find the mixed-strategy equilibrium, I recommend that you first eliminate
weakly dominated strategies for each player. Then I recommend considering
how each of the three players would mix over the two remaining strategies.
Since you have three players, the variables p and q won't
be enough. I suggest adding the variable r to represent the mixing
probability for the page player. You will need to show indifference between
the two strategies for the row player, for the column player, and for the
page player. To do this, you will need to make use of the multiplication rule
for probabilities. (For example, you will need to show that the page player
is indifferent between his two strategies. So you need to be able to answer
questions like: if the page player plays $15, what is his expected payoff?
It is the average of the four different payoff numbers he could get when he
plays $15, and the probability weightings of each of these four numbers depend
on the row and column strategies p and q.) Once you do all
this, you will have three equations in three unknowns, which you can then
solve for the final answer.
- (20 points) DS, question 8.8.
- (20 points) You might want to refer to the handout on simplified football when answering this question.
Consider the following simplified version of baseball. The pitcher can throw
either a fastball or a curveball, the batter can either swing at the pitch
or take (not swing). These choices are simultaneous for each pitch. On the
first pitch, if the batter swings at a curveball or takes a fastball, he strikes
out and gets 0. If the batter swings at a fastball, he has probability 0.75
of hitting a home run and getting 1, and probability 0.25 of hitting a fly
ball and getting 0. If the batter takes a curveball, there is a second pitch.
On the second pitch, the first three combinations (swing at a curveball, take
a fastball, and swing at a fastball) work as before; if the batter takes a
curveball second pitch, he walks and gets 0.25. (For fans of real baseball,
note that this is very similar to the game that occurs between a pitcher and
a batter starting at a count of two balls and two strikes. one strike and
you're out, two balls and you walk.)
This is a zero-sum game; the batter tries to maximize his expected score (probability-weighted
average payoff), and the pitcher tries to minimize the batter?s expected score.
Note that this is a sequential-move game (the two pitches) containing a simultaneous-move
game in each pitch.
- Write down the extensive-form game tree for this game.
- Solve this game using backward induction: construct a
table of payoffs for the second pitch and use these to determine the table
of payoffs for the first pitch. Show that on the first pitch, the batter
should take with probability 0.8.
- What is the pitcher's strategy in the subgame perfect equilibrium?
- What is thebatter's expected
score in this equilibrium?
- Explain intuitively why the batter's probability of swinging is so small.
- (Extra credit, 20 points) Return to the game described in DS, question 8.3,
and consider the discussion in Appendix 2 to Chapter 7. Now suppose that each
player is risk averse, represented by a square-root utility function, and
suppose that each player knows the other player's utility function. Then each
player wishes to maximize her expected utility rather than her expected
- (10 points) Find the new mixed-strategy equilibrium, along with the expected
utility to each player in equilibrium.
- (10 points) Explain intuitively how and why the equilibrium with risk-averse
players differs from the equilibrium with risk-neutral players differs
from the one you found in the original problem with risk-neutral players.
- (Extra credit, 15 points) Return to DS, question 7.9. You
may wish to use the Gambit software to compute the answers to these questions,
or you can try to compute the answers by hand (which is harder, because of the
number of possibilities to consider). When using Gambit, make sure to use
"float" rather than "rational" output when running the solver. Also, you
should check Gambit's answers as you did in the original version of the
What is the new equilibrium if we make each of the following
changes to the original payoff matrix?
- Change the payoff for (HC, C) to 0.10, making it
slightly more likely for the shooter to score on a high-center shot, but
leaving all other payoffs the same as in the original problem. Compare to
the original equilibrium.
- Change the payoff for (HL, L) to 0.70, making it more
likely for the shooter to score on a high-left shot, but leaving all other
payoffs the same as in the original problem. Compare to the original
- Change the payoff for (LC, C) to 0.50, making it more likely for the shooter
to score on a low-center shot, but leaving all other payoffs the same
as in the original problem. Compare to the original equilibrium.
- (Extra credit, 20 points) DS, question 8.9.