Economics: Game Theory, problem set

Economics: Game Theory, problem set.

Problem set now attached!

Fall

2012

GAME THEORY IN THE SOCIAL SCIENCES
Problem Set 2

(Due in Lecture Tuesday, September 25)

1. In lecture we studied how political parties or candidates choose their positions when all

they care about is winning.

This question explores what happens when political candidates

care not only about winning but also about the policies they espouse.

The U.S. Congress is up for grabs in the election this November. Bipartisanship has

again broken down, and the parties are very polarized. The more liberal faction of the

Democratic Party now dominates that party and a more conservation faction dominates the

Republican Party. As the election approaches, these parties are trying to stake out positions

that reflect their own policy preferences and will attract enough voters to win. To simplify

matters, suppose the parties have to choose a position along a left-right spectrum and can

adopt one of the following positions: Liberal (L), Liberal leaning centrist (LC), Middle of

the Road (M), Conservative Centrist, (CC), Conservative (C). These positions are

represented on the line below where the distance between any two neighboring positions is

the same.

Since the voter’s ideal points are evenly distributed along the political spectrum,

the party whose position is closer to the middle of the road (M) wins. If, for example, the

Republicans, R, choose a position of CC and the Democrats, D, announces, L, then R

would win because CC is closer to M. If the parties run on platforms that are equally

distant from M, each party is equally likely to win. Finally, each party chooses its

platform secretly.

As noted above, R and D care about policies as well as winning. D’s von

Neumann-Morgenstern payoffs are: 5 for winning with L; 4 for winning with LC; 3 for

winning with M; 2 for winning with CC; 1 for winning with C; -1 for losing with L; -2

for losing with LC; -3 for losing with M; -4 for losing with CC; and –5 for losing with C.

R’s von Neumann-Morgenstern payoffs are the opposite: 5 for winning with C;

for winning with CC; 3 for winning with M; 2 for winning with LC; 1 for winning with

L; -1 for losing with C; -2 for losing with CC; -3 for losing with M; -4 for losing with

LC; and –5 for losing with L.

(a) Suppose R adopts position C and D chooses LC. What are their payoffs?

(b) Suppose R adopts C and D chooses the more extreme position L. What are the

parities payoffs?

(d) Solve the game by iterated deletion of dominated strategies. Be sure to indicate

the order of deletion, what dominates what, and whether this is strict or weak

dominance.

(e) What is/are the pure strategy Nash equilibria of this game?

(f) Finally, suppose that there are three parties instead of two. Call the third party S

(for spoiler) and assume that S has the same preferences that D does. Consider

the three-player game in which each player selects his position secretly; each

voter votes for the party whose position is closest to her ideal point; and the party

that receives the most votes wins. If each candidate chooses M, is this a Nash

equilibrium of the game? Explain

why or why not.

2. In class and in the problem above, we have seen that even ideologically motivated parties

(i.e., parties that care about the policies they implement as well as being elected) will still

be drawn to the center of the political spectrum and run on policy platforms that the

median voter favors. This question examines what happens when the parties care about

policies and there is uncertainty about what policies the median voter prefers.

Two parties, Dem and Rep, simultaneously have to choose one of three party

platforms Left (L), Center (C), and Right (R) as illustrated in the figure.

If Dem wins with policy L, its payoff is 4+p where p measures how much it cares

about policy and not just winning. If Dem wins with policies C or R, its payoff is 4. And

Dem’s payoff to losing is 0.

If Rep wins with policy R, its payoff is 4+p where, again, p measures how much it

cares about policy and not just winning. If Rep wins with policies C or L, its payoff is 4.

And Rep’s payoff to losing is 0.

The party that runs on a policy closer to the median voter’s preference wins. If,

for example, the median voter prefers L and Dem and Rep run on platforms C and R,

respectively, then Dem wins (because C is closer to L) with payoffs 4 and 0 for Dem and

Rep. If the parties’ platforms are equally far from the median’s preferred policy, each

party wins with probability ½ and loses with probability ½.

L C R

However, the parties are unsure of the median voter’s preference. Polling data

indicates that there is a 25% chance that the median favors L, a 50% chance that the

median favors C, and a 25% chance that the median favors R.

The strategic form for this game is:

Rep

L C R

2 ,
2

2
p



3 ,
4



?, ?

Dem C

3, 1

2, 2

4
1 ,3

p


2, 2

1, 3

2
2 ,2

p


(a) Will an ideologically motivated party ever choose the extreme position favored by

the other party. More concretely, is it ever rational for Dem to choose R? Explain

why or why not.

In answering (b)- (d) assume that the premium to winning with one’s preferred policy is

p=6, so that Dem’s and Rep’s respective payoffs to winning with L and R are 4+p=10.

(b) Suppose Dem chooses L and Rep chooses R. The payoffs to these actions depend

on the median’s preferred policy. Suppose the median prefers L, what are Dem’s

and Rep’s payoffs? What are Dem’s and Rep’s expected payoffs if the median’s

preferred policy is C?

chance that the median favors L, a 50% chance that the median favors C, and a

25% chance that the median favors R. What is Dem’s expected payoff to the

strategies (L,R)? What is Rep’s expected payoff?

(d) What is/are the Nash equilibria of the game?

(e) If the parties do not care enough about policy (i.e., if p is small enough), they will

be drawn to the center. For what values of p is (C,C) the solution when the game

is solved by iterated deletion of strictly dominated strategies?

3. In international relations, states are sometimes assumed to be concerned about how well

they are doing relative to other states. This problem examines that issue. Recall the divide

the dollar game discussed in class. There are two players, 1 and 2, and each secretly decides

how much of the dollar to demand. If the sum of the demands is less than a dollar, each

player receives what it demanded. If the demands exceed a dollar, each receives nothing.

In class we also assumed that each player only cared about its monetary payoff and

showed that any division of the dollar was a Nash-equilibrium outcome.

(a) Verify that .25 for player 1 and .75 for 2 is a Nash-equilibrium outcome by

specifying a strategy for each player that produces this outcome and by showing

that neither player can benefit by deviating from its strategy given that the other

player follows its strategy.

Now assume that each player cares in part about how well it does relative to the other

player. Suppose in particular that if 1 receives a monetary payoff of m1 and 2 receives a

monetary payoff of m2, then 1’s utility to this outcome is u m m m1 1 2 1  ( ) . (Note that 1’s

utility increases as its monetary payoff, m1, increases and decreases as the difference

between 2’s payoff and its own increases. This last part formalizes the assumption that 1

cares about how well it does compared to 2.) 2’s utility is given by u m m

2 2

1 2

  ( ) .

(b) Is .25 for 1 and .75 for 2 a Nash equilibrium outcome. Be sure to justify your

answer.

the players care about their relative gains?

4. In this problem you will trace out a person’s von Neumann-Morgenstern utility function for

money. Let L(p) be the lottery that pays $1000 with probability p and zero with probability

1-p. To anchor the scale, take u( )0 0 and (1000) 10.u  We begin by telling our subject

that p .2 and asking how much money we would have to give her in order to make her

indifferent between that amount of money and the lottery when p .2 . She answers

dollars. We then repeat the question for different values of p. The table on the next pages

summarizes the results:

0.0

1.0

Certainty

Equivalent

$64

$216

$512

$1000

(a) Draw this person’s Von Neumann-Morgenstern’s utility function.

(b) Using the graph in (a), consider the lottery which pays $4 with probability ¼ and $10

with probability ¾.

What is the expected utility of this lottery?

What is the certainty equivalent of this lottery?

probability ¼, and $10 with probability 7/24. What is the utility of this lottery?

5. Many states, including California, now use lotteries as a way of raising money. Are people

who purchase lottery tickets risk acceptant, risk neutral, or risk averse. Be sure to explain

your answer.

6. Consider the inspection game below where the numbers in the cells are the player’s

monetary payoffs.

Inspector

Confess Not Confess

Comply

9,9

9,25

Inspectee

Not

Comply

0,4

16,0

Suppose the players are risk averse. In particular, each player’s utility for an amount of

money m is equal to the square root of that amount: u m m( )  . What is the mixed-

strategy Nash equilibrium of the game?

7. Following the events of September 11, the United States increased its airport security. The

inspection game below studies several aspects of the problem. In the game, a Challenger

has to decide whether or not to challenge security and Security has to decide whether or not

to inspect. (Assume further that all payoffs are Von Neumann-Morgenstern.)

The Challenger’s payoff to challenging and being inspected, t, depends on how good

security is. Suppose that an effort to challenge security will be detected with probability d.

If the challenge is detected, the Challenger’s payoff to being caught is –8. If the challenge

is not detected, the Challenger’s payoff is the same as it is if there is no inspection, i.e., the

Challenger’s payoff is a.

Challenger

Challenge Security

(c)

Not Challenge

Inspect (i) (d)

-5, t

0, 5

Security

Not Inspect

-15, a

5, 0

In parts (a)-(e), assume that the payoff to a successful challenge is 12, i.e., a=12.

(a) What is the Challenger’s payoff t to attempting to breech security if security is initially not

very good and the probability of detection is only ¼, i.e., if d=¼?

(b) What is/are the equilibria of the game given that the probability of detection is only ¼?

For (c)-(g), suppose security improves and the probability of being detected rises to ¾.

(d) Draw Security’s best-reply correspondence?

(e) What is the equilibrium probability that Security inspects? That the Challenger challenges?

(f) Suppose that the Challenger becomes more determined. In particular the payoff to a

successful challenge, a, increases from 12 to 20. What is the equilibrium probability that

Security inspects?

(g) Although the Challenger becomes more determined, the probability of a challenge is the

same in (e) where a=12 and in (f) where a=20. Why doesn’t this change in the

Challenger’s payoffs affect its strategy?

8. President Obama signed the Dodd-Frank Wall Street Reform and Consumer Protection Act

into law in July 2010. This act creates a Consumer Financial Protection Bureau which will

try to ensure that financial services companies follow standard practices in making loans. In

this problem, the Bureau must decide how to design a program to audit a financial services

company, F. There are two options. The first is a program that is expensive to set up but has

a low cost-per-audit. The second is less expensive to set up but has a high cost-per-audit.

In the game below, B has to decide whether or not to audit F. F has to decide whether

to comply with standard practices or to evade the new rules and hope it does not get caught.

If F complies, it gets a profit of 40. If it gets away with evading the new standards, it

doubles its profits to 80. If it gets caught, it pays a penalty of 120.

B pays a cost s to set the system up and a per-audit cost of c. If F complies with the

new regulations, B obtains a benefit of 50. This means that if B audits F and F complies, B’s

payoff is 50 c s  . If B catches F evading the new regulation, B’s payoff is 120 c s  .
The probability that B audits F is a, and the probability that F follows or complies with

the regulations is f.

Evade

Comply

(f)

Audit (a)

120 – c – s, -120

50 – c – s, 40

Not Audit

– s, 80

50 – s, 40

Assume c = 20 and s = 1000 for parts (a) and (b).

(a) Draw B’s best-reply.

(b) What is the mixed equilibrium of the game?

audit cost. Suppose c = 60 and s = 500. What is the mixed equilibrium of the game?

(d) Option 1 has a high set-up cost and a low cost per audit (s = 1000 and c = 20) whereas

option 2 has a low set-up cost and a high cost per audit (s = 500 and c = 60). Which option

results in greater compliance (i.e., a higher f)? Be sure to explain your answer.

(e) How do changes in the set-up cost s affect B’s and F’s equilibrium strategies? (You do

not need to consider changes in c affect B’s and F’s equilibrium strategies.)

9. The New York Times has reported that the United States Ambassador to an unnamed

country in which terrorist groups operate told members of her diplomatic staff to keep some

dice in the glove compartments of their cars. Just before coming to work in the morning and

going home at night, the diplomats should roll the dice to decide which route to take.

In the game below, a diplomat D has to choose route A, B, or C. A terrorist group, T, is

trying to kidnap this diplomat but does not have the capability to monitor routes A and B and

must choose which one to watch. Route C, by contrast, passes through a neighborhood

controlled by this group and is always monitored. As a result, route C is very dangerous for

the diplomat. Route A is the shortest route and the diplomat prefers it assuming, of course,

that it is not being watched. The size of this preference is p.

Let w be the probability the terrorists watch route A. Let a and c be the probabilities that

the diplomat takes routes A and C. (The probability the diplomat takes B is therefore

1 a c  .)

Watch A

(w)

Watch B

Route A (a)

-10, 4

4 + p , -1

D Route B

4, -1

-10, 6

Route C (c)

-12, 5

-12, 7

Assume p = 2 for parts (a) and (b).

(a) Is Route C ever a best response to what T does?

(b) What is the mixed equilibrium of the game?

is D more or less likely to take route A than when p = 4? Is T more or less likely to watch A?

Be sure to explain your answers.

10. Extra and probably hard. You also need to differentiate in order to do this problem. So if

you have not had calculus and want to try it, pair up with someone who has. In the rent-

seeking problem in class, we assumed that there were two players. Now assume that there

are N players. If player j spends

m , then the probability that player j wins is:

1 2
j

m m m 

Player j’s payoff is:

1
1 2

( ,…, )
j

j N j

m
U m m V m

m m m

 
  

  

Because the game looks the same from every player’s perspective, we will look at

symmetric equilibria. That is, every player will invest the same amount in equilibrium.

(a) What is each player’s best reply?

(b) How much does each player spend in equilibrium when there are N players?

(d) How much of the value of the prize is dissipated through the contest?

(e) As you showed in (c), the expected payoff to the game is always positive for any

number of players, i.e., for any N. Hence, additional players will always want to

compete for the prize as the payoff to not competing is 0. How much of the prize is

dissipated if the number of players N gets very large?

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