# CMPS272, Winter 2007, Section 01: Assignment 3: Game Dynamics

This assignment is due Tuesday January 30.

With the members of your group, talk over and write up solutions to the following.

### For Dan Friedman's Lectures

• Watson Ch 22 problems 2, 3; Ch 26 prob 2; and Ch 28 prob 3.8
• Dan is running behind on lectures; you can defer the Ch 26, 28 problems until next time!
• HD dynamics in 1 dimension.
• Write down a 2x2 matrix for a symmetric HD game for C=3 and V=2. (e.g., the HH payoff is (2-3) /2=-1/2.)
• Write down the dynamically equivalent normalized matrix with zeros on the main diagonal.
• Find the D(p) function (i.e., the payoff advantage for H when the probability of encountering H is p) for the HD matrix and confirm that it is the same as for the normalized matrix.
• Find the solution to D(p)=0. How does it change when you vary the values C and D?
• Draw the phase portrait for sign-preserving dynamics for your favorite values of C and D. Is the solution to D(p)=0 an ESS? A NE? An EE?
• HD dynamics in 2 dimensions.
• Write down the 2x2 bimatrix for an asymmetric HD game where C=3 and V=2 for both row and column players. (Comment: having the matrices the same (or more specifically A=B^T) is necessary for a symmetric game, but it is not sufficient. The key is that in the asymmetric game, there are two populations with fractions p and q respectively adopting the first pure strategy.With asymmetry, p and q can differ.)
• Find the D(p) and D(q) functions (for row and column players respectively).
• In (p,q) space, find the point D(p)=D(q)=0.
• Draw the phase portrait for sign-preserving dynamics.
• Interpret the points on the diagonal p=q as corresponding to the symmetric case. Use that interpretation to explain why the D(p)=D(q)=0 point is unstable (a saddle) here but is stable in the symmetric case.
• Show how the point D(p)=D(q)=0 changes when you vary the values C and D separately for row players and for column players.
• For your favorite values of C and V, find all Nash Equilibria (NE), all Evolutionary Equilibria (EE). Are any of them ESS?

### Manfred Warmuth's Lectures

• Let P be a payoff matrix for a zero-sum game.
• An entry P(i,j) is called a saddle point''
• if it is larger than any other entry in row i
• and smaller than any other entry in column j.
• 1. Show that a payoff matrix can have at most one saddle point.
• 2. Show that if P(i,j) is a saddle point of P, then P(i,j) is the minimax value of the game.
• You can use all the facts shown in the lecture notes and the minimax thm.

### For Barry Sinervo's Lectures

• Briefly read through the Bomze 1983, posted on this site. Go through and list the payoff matrix (write out the payoffs along with the number of the game) for those games with an internal equilibrium (which may or may not exhibit cyclical behavior). Come up with some scheme for classification of the dynamics in this subset of the 3-player game dynamics. Bomze provides a technical description of classification. However you do not need this technical classification in your answer, I am interested in how you might classify them (i.e., by the form of the payoffs or by the behavior of the dynamic are two options for classification, or a novel classification that you might describe).
• Bomze_83.pdf: Bomze's (1983) classification of three-player dynamics

I found that the above link for the Bomze article didn't work, so I attached the article again below. -Lesley

Turn in one copy per group, with all members' names noted, at the beginning of class Tuesday 1/30.