Gametheory Homework

Problem Set 3: Due in class on Tuesday July 21. Solutions to this homework will be posted

right after class hence no late submissions will be accepted. Test 3 on the content of this

homework will be given on July 28 at 9:00am sharp.

Problem 1 (2p)

(a) In the game from the previous problem set (“old lady crossing the street”) identify all pure strategy Nash equilibria.

(b) In another game from the previous homework (battle of the Bismarck Sea) identify all pure strategy Nash equilibria.

Problem 2 (2p)

Using only 0 and 1 as payoffs construct a 44 game with two pure strategy Nash

equilibria.

Problem 3 (6p)

To formulate the problem below I need to introduce the following definition: Consider a

vector of payoffs (x, y) where x and y are payoffs to Row and Column respectively; we

will say that (x*, y*) is “better” than (x, y), or equivalently that (x, y) is “worse” than

(x*, y*), if x* > x and y* > y. For example, the Prisoner’s Dilemma game in lecture

notes is an example of a game with one Nash equilibrium in pure strategies where the

vector of payoffs in this equilibrium is “worse” than some other vector of payoffs in the

game. Now, using only 0, 1, 2, 3 and 4 as payoffs:

(a) Construct a 33 game in which there is only one Nash equilibrium in pure strategies and the vector of payoffs in this equilibrium is “worse” than some other

vector of payoffs in the game.

(b) Construct a 33 game in which there are two Nash equilibria in pure strategies and the vector of payoffs in one Nash equilibrium is “worse” than the vector of

payoffs in the other Nash equilibrium.

(c) Construct a 33 game in which there are two Nash equilibria in pure strategies and the vector of payoffs in both is “worse” than some other vector of payoffs in

the game.

Problem 4 (4p)

Suppose that political ideologies can be represented on a 5 point discrete scale. The scale

identifies five clearly distinguishable ideologies on a spectrum from the far left (1) to the far

right (5). Assume that your electorate is evenly spread over the whole political spectrum

with 20% identifying with each of the five ideologies. The only two political parties that are

about to form (the country is a new democracy) can choose their ideologies anywhere on the

5 point scale. We assume that voters will vote for party that is closest to their ideology and

if both parties are in equal distance the voters will split evenly between them.

Model this situation as a game. Represent payoffs as percent of votes a party gets (the other

party gets the remaining percentage of votes).

(a) Solve the game through iterated dominance. (b) To get the solution in (a) do you need to assume cardinal payoffs or ordinal payoffs

would be sufficient? In other words, to arrive at the equilibrium in (a) do you

merely need to assume that parties prefer more votes over less or that they also have

to prefer 40% over 20% twice as strongly as 30% over 20%?

Page 2 of 3

Problem 5 (Dixit and Skeath, and Watson) (6p)

(1) Write out all (pure) strategies of each player in the three games below. Also, for each

game (2) mark in bold all branches that will be taken at each node, (3) circle the payoffs the

players will get in equilibrium, and (4) write a pair of strategies that constitute the rollback

equilibrium.

A

B

A

B

C

X

Y

X

Y

X

Y

1,2

1

2

2,8

1,3

1,3

8,2

8,1

U

B

D

A

B

A

3,8

1

2

X

Y

X

Y

P

Q 1

1

8,1

1,2

2,0

6,6

5,5

0,0

C

ONE EXTRA CREDIT PROBLEM

Problem 6 (2 extra credit points)

Assume the same situation as in Problem 4 above with a different distribution of voters:

16%, 16%, 26%, 26%, 16% in each ideology from 1 to 5 respectively. Model this situation

as a game. Again, represent payoffs as percent of votes a party gets.

Can you solve this game through iterated dominance? Does the model and the analysis

make sense for ordinal payoffs only, cardinal payoffs only, or both?

Assume now that the distribution of voters is 10%, 10%, 10%, 60%, 10% in each

ideology from 1 to 5. Can you solve this game through iterated dominance?

Any general observations from what you have concluded in all three cases (i.e., Problem

4 and the two cases here)? What, do you think, is the general theorem that applies to any

distribution of voters (I am only asking here for an informed speculation/guess, not a

rigorous proof)?

A 1

1,0

1 1

2 2

2 2,2

4,5

3,2

2,3

5,4

0,1

B

C

D

E

F

Q

R

M

N

O

P