1. Consider the following normal-form game:

U M N

D

L       C      R

(a)  [1pt] Calculate the rationalizable set R.

Answer∶   First, note that C is dominated by R. Second, note that D dominates M. hird, note that σ1  = (0．8﹐ 0﹐ 0．2) dominates N. Finally, in the game restricted to {U ﹐ D} × {L﹐ R} every strategy of every player is sometimes a best response (to simple beliefs). herefore, our iterated removal procedure terminates and we obtain R = {U ﹐ D} × {L﹐ R}.

(b)  [0.5pt] Find all Nash equilibria.

Answer∶  he game has two Nash equilibria: (U ﹐ L) and (D﹐ R).

(c)  [1pt] Find all mixed-strategy Nash equilibria.

Answer∶  We know that mNE are supported on R and so it’s sufcient to consider mixed strategies σ1 = (g﹐ 0﹐ 0﹐ 1 − g) and σ2 = (p﹐ 0﹐ 1 − p), denoted simply as g and p in calculations, respectively. Player 1 is indiferent when

11(U ﹐ p) = 5p + 3(1 − p) = 2p + 8(1 − p) = 11(D﹐ p) = 3p = 5(1 − p) = p =

Player 2 is indiferent when

12(g﹐ L) = 10g + 4(1 − g) = 4 = 12(g﹐ R) = g = 0

hus, we have best responses

Ⅰ(Ⅰ)0﹐         p :                    )Ⅰ

Ⅰ                                      Ⅰ 1﹐         g 5 0

Now we solve for all g and p such that g ∈ BR1(p) and p ∈ BR2(g).

Suppose g 5 0, then p ∈ BR2(g) = {1} implies p = 1 and then g ∈ BR1(1) = {1} implies g = 1. his corresponds to mNE ((1﹐ 0﹐ 0﹐ 0)﹐ (1﹐ 0﹐ 0)) or simply (U ﹐ L).

Suppose g = 0, then p ∈ BR2(g) = [0﹐ 1] does not restrict p at all. But in order for g = 0 ∈ BR1(p) we must have p *  . herefore, we obtain a continuum of such mNE parametrized by p: {((0﹐ 0﹐ 0﹐ 1)﹐ (p﹐ 0﹐ 1 − p)) ∶ p ∈ [0﹐ ]}.

It’s easy to verify graphically that there is indeed a continuum of mNE in this game by ploting the graphs of BR1 and BR2 and noticing that they intersect at (g = 1﹐ p = 1) and also over a line segment on the p axis.

2. Consider the Cournot oligopoly game in which frm i = 1﹐ … ﹐ n is choosing a quantity gi . he market inverse demand is given

by P = A − bO, where O = 〉i gi is the total supply and each frm’s average cost is c : A. In other words, each frm’s proft is given by 1i(g1 ﹐ … ﹐ gn) = (A − c − b〉j(n)=1 gj)gi .

(a)  [1pt] Find the best response function BRi(g−i) of frm i.

Answer∶ Consider frm i. he frst order condition is for its utility maximization problem with respect to gi is

a1i                         n

agi                      j=1

To solve it with respect to gi, group all terms with gi in the let-hand side and all other terms in the right-hand side (also swapping the sides):

2bgi = A − c − b 〉 gj

j二i

A − c − b 〉j二i gj

gi = BRi(g−i) =            2b           ．

(b)  [1pt] Find the symmetric Nash equilibrium.

Answer∶  Seting all gj = x, we get the equation:

x = BRi(x﹐ … ﹐ x) = A − c − b(n − 1)x

2bx = A − c − b(n − 1)x

(n + 1)bx = A − c

x = 

Hence, the symmetric Nash equilibrium is = (﹐ … ﹐ b (.

3. Consider a game in which simultaneously player 1 selects x ∈ s1 = [0﹐ 6] and player 2 selects 夕 ∈ s2 = [0﹐ 6]. he payofs are as follows:

11(x﹐ 夕) = 夕(1) − x2      12(x﹐ 夕) = x(1)夕2 − 夕2

(a)  [1pt] Calculate each player’s best response BRi(n) as a function of the opposing player’s pure strategy.

Hint: When you are maximizing a quadratic function with a negative quadratic coefcient over an interval, it is sufcient to use the frst order condition as long as its solution is in the inverval.

Answer∶  To fnd BR1(夕), we need to maximize 11(x﹐ 夕) with respect to x ∈ [0﹐ 6]. he frst order necessary condition is:

a11(x﹐ 夕)         16  

=             − 2x = 0．

ax          夕 + 2

Solving for x, we obtain x = 夕 . herefore, BR1(夕) = 夕 and, by symmetry, BR2(x) =  .     (b)  [1pt] Find the set B1 of all best responses of player 1 to simple beliefs about player 2’s strategy.

Answer∶ he range of BR1 over [0﹐ 6] is [BR1(6)﹐ BR1 (0)] = [1﹐ 4].

(c)  [1pt] Find and report the Nash equilibrium of this game.

Hint: 7e roots of ax2 + bx + c = 0 are given by −b ± 2(厂)

Answer∶  Any NE (x∗ ﹐ 夕∗ ) must satisfy x∗ = BR1(夕∗ )﹐ 夕∗ = BR2(x∗ ). Plugging in the second equation into the frst, we 

obtain

x∗ = BR1(夕∗ ) = BR1(BR2(x∗ )) = 夕∗  =

s (x∗ )2 + 2x∗ − 8 = 0

s x∗ =                          = −1 ± 3

2

Since x must be in [0﹐ 6], x∗ = 2. hen 夕∗ = BR2(2) = 2. he unique NE of this game is (2﹐ 2).

(d)  [1pt] Find and report the rationalizable set R. You may rely on a graphical solution to illustrate the iterated dominance procedure.

Hint: Your graph may be very stylized and does not have to be precise. It should aim to reaect the increasing/decreasing nature of the curves and where they intersect each other and the axes. It’s a good idea to calculate a couple of points (e.g. endpoints) to plot them.

Answer∶   he fgure shows1  a few rounds of removal of dominated strategies and we can see that the limit—the rationalizable set R —is a singleton. Since R must contain the Nash equilibrium, it follows that R = {2} × {2}．

0                                              6

6

0                                              6

0                                              6

6

0                                              6

0                                              6

6

0                                              6

4. here are two players. hey take stones from the pile of 6 stones. Player 1 can take only 2 or 3 stones. Player 2 can take only

2 or 4 stones. Players take turns, observe all previous moves, and player 1 moves frst. A player loses if she cannot make a legal move, while another player is declared to be a winner. Let the payof of winning equal to 1 and the payof of losing equal to 0.

(a)  [1pt] Represent the game in extensive form. (Note: Depict only legal moves).

1A sketch of a more rigorous argument is as follows. First, note that this is a game with convex strategy sets and monotone best responses. It is then straight- forward to show that in our defnitions of Bi﹐ UDi﹐ Ri  it is without loss of generality to restrict atention to pure strategies and pure beliefs.  herefore, when performing iterated removal of dominated strategies, we can remove strategies that are never best responses to simple beliefs. It is a bit more tricky to formally show convergence of Rk to a singleton.