1. Consider the chicken game represented by the following payoff table.

ROW

SLOW

SPEED

COLUMN

SLOW     SPEED

(a) Find and represent graphically the players’best response functions, and show that the game

has three Nash equilibra:

● one in which ROW plays SPEED and COLUMN plays SLOW (let’s call it NE1 ),

● one in which ROW plays SLOW and COLUMN plays SPEED (let’s call it NE2 ),

● one in which both players play SLOW with probability and SPEED with probability (let’s call it NE3 ).

2 points

Let’s introduce the following notation: PrROW (SLOW) = p and PrCOLUMN (SLOW) = q. We now can write the players’expected payoffs.

EuROW (SLOW |q . SLOW + (1 _ q) . SPEED)   =   2q

EuROW (SPEED |q . SLOW + (1 _ q) . SPEED)   =   4q + (_1) . (1 _ q) = 5q _ 1 EuCOLUMN (SLOW |p . SLOW + (1 _ p) . SPEED)   =   2p

EuCOLUMN (SPEED |p . SLOW + (1 _ p) . SPEED)   =   4p + (_1) . (1 _ p) = 5p _ 1

The players’best-response function are:

, 1

p(q)   =   , 0

． [0< 1]

, 1

q(p)   =   , 0

． [0< 1]

if q ←

if q ￥ < and

if q =

if p ←

if p ￥

The three Nash equilibria of the game are:

NE1     =   {SPEED < SLOW }<

NE2     =   {SLOW < SPEED }< and

NE3     =   { . SLOW + . SPEED < . SLOW + . SPEED } .

Graphically:

(b) Compute each player’s expected payoff in each of the three Nash equilibria of the game.

1 point

In NE1 , the payoffs are (4; 0).

In NE2 , the payoffs are (0; 4).

In NE3 , the payoffs are . . (2; 2) + . . (0; 4) + . . (4; 0) + . . (_1; _1) = = . [(2; 2) + 2 . (0; 4) + 2 . (4; 0) + 4 . (_1; _1)] = ( ; ) = ( ; ).

(c) Deduce that the game obtained by repeating the above chicken game twice has at least nine subgame-perfect equilibria.

3 points

Note that we have a subgame-perfect Nash equilibrium in the twice repeated chicken game  if, irrespectively from the history of the game, in each stage players choose actions that con- stitute a Nash equilibrium in the stage game.

In other words, we have a subgame-perfect Nash equilibrium in the twice repeated chicken  game if players play according to NEi in the first moment in time and play according to NEj in each and every subgame that starts in the second moment in time (note that there are 4     such subgames).

Note that both i and j can take on three different values, so we have already constructed 9 equilibria. We can construct more by allowing players to play according different NEi in  different subgames that start in the second moment in time.

2. Suppose that a class of 100 students is comparing two careers – lawyer or engineer. An engineer   gets a take-home pay of 100 thousand dollars per year, irrespective of the number of people who   choose this career. Lawyers make work for each other, so as the total number of lawyers increases, the income of each lawyer increases – up to a point. Ultimately, the competition between them      drives down the income each earns.

Specifically, if there are N lawyers, each will get 100N _ N2 thousand dollars a year. The annual cost of running a legal practice (office space, secretary, paralegals, access to online reference ser-   vices, and so forth) is 800 thousand dollars. Therefore, each lawyer takes home 100N _ N2  _ 800 thousand dollars a year when there are N of them.

For simplicity, let’s assume that the class size of 100 students is large enough, so that any one per- son makes only a very small difference on income levels.

(a) Draw a graph showing the take-home income of each lawyer and each engineer (on the verti- cal axis) as a function of the number of lawyers (on the horizontal axis).

1 point

(b) When career choices are made in an uncoordinated way, what are the possible Nash equi-

librium outcomes? Indicate whether the Nash equilibria that you have found are stable or unstable.

3 points

When N  =  0, lawyers earn _800 thousand dollars while engineer make 100 thousand.      There are no lawyers and no one would want to become one. This points constitutes a Nash- equilibrium outcome of the game (N1  = 0).

When N =  100, lawyers earn _800 thousand dollars while engineer make 100 thousand. There are no engineers, but a lawyers would have incentives to switch to engineering. This point does not constitute a Nash-equilibrium outcome of the game.

When the two professions earn the same income, we have other two Nash-equilibrium out- comes of the game. 100N _ N2 _ 800 = 100 二 N2  = 10 and N3  = 90.

In order to determine the stability of the equilibria, note that for N < 10 and for N ￥ 90   the expected payoff for engineers is larger than the expected payoff for lawyers. And that for 10 < N < 90, it is the other way around. These imply that from a situation with N < 10 or N ￥ 90, the number of lawyers would tend to decrease, while from a situation with 10 < N < 90, the number of lawyers would tend to increase. These results are illustrated with arrows along the horizontal axis. We then can conclude that N1 = 0 and N3 = 90 are stable equilibrium outcomes, but N2 = 10 is unstable.

(c) Now suppose that the whole class decides how many should become lawyers, aiming to          maximize the total take-home income of the whole class. What will be the number of lawyers?

● Find the mathematical expression that shows the take-home income of the whole class as a function of the number of lawyers.

T (N) = N . (100N _ N2 _ 800) + (100 _ N) . 100 = _N3 + 100N2 _ 900N + 10 000

● If you know calculus, find the function’s maximum point.

max T (N)   = _N3 + 100N2 _ 900N + 10 000

N

FoC :          _3N2 + 200N _ 900 = 0

soC :                  _6N + 200 < 0

N*  = (10 +^73) · 61.8 二 62

● If you do not know calculus, show that the take-home income of the whole class is larger when the number of lawyers is 60 (than in the stable equilibrium).

T (N)   =   _N3 + 100N2 _ 900N + 10 000

T (60)   =   100 000

T (0)   =   10 000

T (80)   =   66 000

2 points

3. In a simple version of the holdup problem, Alice has $3 million, which she is thinking of invest-

ing in Bob’s company. If she makes the investment, Bob can either work or slack. If he slacks, he  consumes Alice’s investment, and she gets nothing. If he works, Alice doubles her investment, and Bob nets $2 million.

(a) Represent the game in extensive form, and find its subgame-perfect Nash equilibrium.

2

The subgame-perfect Nash equilibrium of the game is {DON’T INVEST; SLACK }.

(b) Represent the game in strategic form, and find all its pure-strategy Nash equilibria.

2

points

points

BOB

WORK    SLACK

The game has only one Nash equilibrium in pure strategies. It is {DON’T INVEST; SLACK }.

4. Suppose that the following payoff table represents the ultimate chicken game between the two

nuclear superpowers during the Cold War.

WAIT

USA

ATTACK

USSR

ATTACK

Recall the following words from the Dr. Strangelove movie.

President Merkin Muffley:

“But, how is it possible for this thing to be triggered automatically, and at the same time impossible to untrigger?”

Strangelove:

Mr. President, it is not only possible, it is essential. That is the whole idea of this ma-    chine, you know. Deterrence is the art of producing in the mind of the enemy ... the fear to attack. And so, because of the automated and irrevocable decision making process    which rules out human meddling, the doomsday machine is terrifying. It’s simple to      understand. And completely credible, and convincing.

● Show how such a doomsday machine changes the above game in the following two hypothet- ical situations. Represent the game with a payoff table, explain players’ incentives and the    resulting equilibrium in each situation.

Situation (a) The USSR has a doomsday machine, but the USA does not have a doomsday machine.

2 points

In this situation an attack by the USA is going to be retaliated.

USSR

WAIT                        ATTACK

The players’ best responses are indicated with boxes around payoffs in the table above. Now WAIT is a dominant strategy for the USA. The Nash equilibrium of the game is:  {WAIT ; ATTACK }.

Situation (b) Both the USSR and the US have a doomsday machine.

2 points

In this situation any attack by any of the players is going to be retaliated by the other.

USA

WAIT

ATTACK

USSR

ATTACK