Econ 431: David Reiley

Due: Wednesday, 28 February 2007

The material in this problem set comes primarily from Chapters 7 and 8, plus the related handout on simplified football.

- (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 money payoff.

- (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
problem above.

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 equilibrium.
- 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.