Econ 4020 Game Theory

Nash Equilibrium II1. Consider the two-person strategic form game in which N = {1, 2}, A1 = A2 = R, and

the utility functions are:

u1(a1, a2) = a1a22 − a21

u2(a1, a2) = 8a2 − a1a22.Find the Nash equilibria, or show that none exist.

Solution: Each player’s utility function is concave in his own choice, so we can findbest response functions by looking at first-order conditions. Differentiate player 1’sutility function with respect to a1 to see that his best response function is B1(a2) =a22/2. Suppose first that a2 > 0. Then the corresponding calculation for player 2 givesB2(a1) = 4/a1. In a Nash equilibrium, a∗1 = B1(a

∗2) and a∗2 = B2(a

∗1), and the solution is

clearly a∗a = a∗2 = 2.

Player 2 cannot play a2 = 0 in any Nash equilibrium. If a2 = 0, then player 1’s bestresponse is a1 = 0. If a1 = 0, then player 2’s utility is strictly increasing in a2, so shehas no best response; in particular, 0 is not a best response.

2. Consider the game described by the following tree:

I II II1 I2 U −4, 1

O1

1, 1

O2

−1, 0

D

3, 2

(a) What are player I’s strategies?

(b) What are player II’s strategies?

(c) What is the normal form of the game?

(d) Find all the Nash equilibria.

Solution:

(a) A1 = {I1U, I1D,O1U,O1D}.

(b) A2 = {I2, O2}.

(c)Player 2

Player 1I1U I1D O1U O1D

I2 −4, 1 3, 2 1, 1 1, 1O2 −1, 0 −1, 0 1, 1 1, 1

(d) The pure equilibria are O1x,O2 where x is either U or D. In any mixed equilibrium,player 2 chooses O2, player 1 chooses O1 and randomizes over U and D.

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3. Find all the (pure and mixed) Nash equilibria, pure and mixed, for the following games:

(a)L R

T 2,4 0,0B 1,6 3,7

Solution: T, L and B,R are both pure-strategy Nash equilibria. Prob{L} = 3/4,Prob{T} = 1/5 is a mixed-strategy equilibrium.

(b)L C R

T 1,7 1,5 3,4B 2,3 0,4 0,6

Solution: A 50:50 mix of L and R dominates C, so the game reduces to

L RT 1,7 3,4B 2,3 0,6

This game has no pure equilibria. It has a mixed equilibrium in which Prob{L} =3/4 and Prob{T} = 1/2.

(c)

L C RT 8,3 3,5 6,3M 3,3 5,5 4,8B 5,2 3,7 4,9

Solution: A mixture of T and M dominates B, so after one round of eliminationof dominated strategies we have

L C RT 8,3 3,5 6,3M 3,3 5,5 4,8

In this game L is dominated, and so the game further reduces to

C RT 3,5 6,3M 5,5 4,8

This game has only a mixed equilibrium, with Prob{C} = 1/2 and Prob{T} = 3/5.

4. Consider the following parametric set of normal form games:

L C RT x, x x, 0 x, 0M 0, x 2, 0 0, 2B 0, x 0, 2 2, 0

Compute the pure and mixed strategy Nash equilibria for this game, and note how theydepend upon x. In particular, what is the difference between x > 1 and x < 1.

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Solution:

x < 0: T and L are dominated. The reduced game has a single mixed equilibrium, withProb{C} = Prob{M} = 1/2.

0 ≤ x ≤ 1: T, L is a Nash equilibrium. Prob{C} = Prob{R} = Prob{M} = Prob{B} =1/2 is a mixed equilibrium. Also, Prob{L} = Prob{T} = 1 − x and Prob{C} =Prob{R} = Prob{M} = Prob{B} = x/2 is a mixed equilibrium. (Notice the firstand third equilibria are the same at x = 1.

1 < x ≤ 2: T, L is a Nash equilibrium. Prob{L} = Prob{T} = 1− x/2 and Prob{C} =Prob{M} = x/2 is a Nash equilbrium. Prob{L} = Prob{T} = 1 − x/2 andProb{R} = Prob{B} = x/2 is a Nash equilbrium. Notice that the second andthird equilibria collapse to the first at x = 2. Notice that there are four equilibriaat x = 1.

x > 2: T and L dominate all other strategies, and so T, L is the unique Nash equilibrium,and it is a dominant strategy equilibrium.

5. Consider the following parametric set of normal form games:

L RU 1,1 0,0D 0,0 λ, λ

For λ between −1 and 1, plot for each λ the set of all points p such that p is theprobability of U in a Nash equilibrium of the game. For example, at λ = 2 (out of therange of the computation I am asking you for, there are three Nash equilibria. In onep = 1. In another, p = 0. In the third, p = 2/3. So you would plot the points (2, 0),(2, 2/3) and (2, 1).

Solution:

-1 1

1

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6. Repeat the plotting exercise for the following parametric set of game:

L RU 1,1 0,0D 0,0 λ, 2

Solution:

-1 1

1

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