# CMPS272, Winter 2007, Section 01: Assignment 2: Game Solutions

### For Dan Friedman's lectures:

With the members of your group, talk over and write up solutions to the following problems in Watson. Ch 2 probs 3, 5, 6. For each, write out the normal form and find all NE. Check whether each NE is also an ESS. Ch 5 prob 3; Ch 6 prob 4; Ch 7 probs 3 and 5. Ch 9 probs 1, 4, 5, and 9. Is (any) NE in problem 4 an ESS?

Turn in one copy per group, with all members' names noted, at the beginning of class Thurs 1/18.

### For Barry Sinervo's lectures:

Each enrolled student should complete this individually, and email the file to Barry Sinervo.

#### Problem 2a) Construct an excel spreadsheet of the hawk-dove game in which:

H is the frequency of hawks D is the frequency of doves (e.g., 1-H = D) V is the value of the resource C is the cost of fighting

The best way to make the spreadsheet is to construct a variable V in one cell near the top =4 (where the cells value is 4) and another cell for C =5 (where the cost of fighting is 5)

Then construct a payoff matrix (2X2) of player 1 and player 2 based on the value of these two cells (parameters)

Finally layout columns of row 1 for H, D, HXH (encounter frequencies of hawks), HXD, DXH, DXD, Wbar (average fitness), WH (fitness of hawks, e.g., from the point of view of player 1), WD (fitness of doves, e.g., from the point of view of player 1),

WH and WD are given by equations given in class (see class notes) and WH and WD are based on H, D, and the cells of the payoff matrix.

Player 2: Hawk Dove Player 1 Hawk (V-C)/2 V Dove 0 V/2

Better Formatting on Wiki

 Player 2: Hawk Dove Player 1 Hawk (V-C)/2 V Dove 0 V/2

Note this matrix can be placed in the excell spread sheet

If the HH payoff (e.g., Hawk on Hawk) is in cell C8 (for example) then you can refer to the cell in the rows described below (e.g., H,D,HXH,...,) as $C$8 which will maintain its position in a "fill down" operation. The same referencing should be used for all cells from the payoff matrix. But reference all other parameters normally (e.g., parameters H,D,HXH,..., Wbar, WD, WH).

To derive H at time t+1 [e.g., H(t+1)] use the formula: H(t+1) = H(t)*WH/Wbar

H(t+1) is the value immediately under H(t) in the rows of the spreadsheet. Fill in a value for D(t+1) by the same logic.

Then highlight the values (formulas) in row 1 for: HXH, HXD, DXH, DXD, WH, WD and perform a fill down from row 1 to row 2

Finally select row 2 (and only row 2) and carry out a fill down to 100 rows.

Voila: you have a simulation.

If you have any problems setting this up please email me (Barry Sinervo) and meet me sometime or during my office hours Tues Thurs. Any time between 2:30-3:30 pm.

#### Answer the following:

2b. Does the population oscillate? Why or why not?

(We will solve analytically for the equilibria in class Tuesday).