POLI 118: Final Exam (Summer 2020)

You must complete this exam on your own. You are not permitted to discuss the exam with anyone else

until after you have completed it and submitted it.

You are permitted to consult the textbook and your lecture notes.

1. Consider the two-player strategic game with the following payoff matrix:

L C R

T −2, 1 1, 0 1, 1

M −1, 1 2, 0 0, 1

B −2, 1 0, 1 1, 0

2. Consider the three-player version of the Voting Game:

Players Three players.

Actions Each player can cast a vote for candidate A or candidate B, and the winner of the

election is the candidate who receives the most votes.

Preferences Players 1 and 2 prefer any action profile at which A wins to any profile at which B

wins, and player 3 prefers any action profile at which B wins to any action profile at which A

wins; each player is indifferent between two action profiles if the same candidate wins at each.

3. Recall the Public Goods Provision game:

Players N = {1, 2, . . . , n} players, with n > 2.

Actions Each player’s set of actions is {0, 1}, where ai = 1 refers to player i choosing to contribute,

ai = 0 refers to the choice not to contribute.

Preferences The preferences of player i ∈ N are represented by

ui(a1, . . . , an) =

B if ai = 0 &

∑n

j=1 aj ≥ k

B − c if ai = 1 &

∑n

j=1 aj ≥ k

0 if ai = 0 &

∑n

j=1 aj < k

−c if ai = 1 &

∑n

j=1 aj < k

where k is an integer such that 2 ≤ k ≤ n, and B > c > 0.

4. Country A and B are at war. A’s army has occupied an island, and B’s army is considering whether

to invade. If it invades, A’s army can retreat over a bridge to the mainland or stay and fight. The

following extensive game models the situation:

A

B

2, 1

Retreat

0, 0

Fight

Invade

1, 2

Out

1

5. Consider a variation on the game from the previous question. Before A chooses whether to invade the

island, B can choose to deprive themselves of the option of retreating from an invasion by burning the

lone bridge that connects the island to the mainland. In that event, B has no choice but to fight if the

challenger invades.

The situation is modeled by the following extensive game. Note: B’s payoff is now listed first

and the A’s second.

B

A

B

1, 2

Retreat

0, 0

Fight

Invade

2, 1

Out

don′t burn

A

0, 0

Invade

2, 1

Out

burn

6. Consider the following extensive game:

Player 1

Player 2

2, 1

a

0, 2

b

L

Player 2

0, 3

c

1, 2

d

R

7. Consider the following extensive game:

Player 1

Player 2

Player 1

2, 2

c

1, 0

d

x

4, 2

y

a

3, 3

b

2

8. Ideological Candidates. Recall the variant of the Hotelling-Downs model of electoral competition in

which the two candidates care only about the location of the winning position (like the voters), and

not at all about winning per se. Candidate A’s favorite position is z∗A and candidate B’s is z

∗

B , and

z∗A < m < z

∗

B . (Further details in footnote.)

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9. Recall the Ultimatum Game. Player 1 can first choose any number x in the interval [0, 1], which

represents player 1’s offer to player 2. Player 2 then chooses Accept or Reject. For any offer x, if

player 2 accepts player 1’s offer, then player 1’s payoff is 1 − x and player 2’s payoff is x. If player 2

rejects player 1’s offer, then each player’s payoff is 0.

10. Recall the Agenda Control game. Congress chooses a number, x, which represents a bill. The President

then chooses whether to accept or reject the bill. If he chooses to accept, the outcome to the game

is x, and if he chooses to reject, the outcome to the game is the status quo, q. The President’s ideal

point is xP = 0 and Congress’s ideal point is a number xC > 0. Each player wants the outcome of the

game to be as close as possible to its ideal point.

11. Recall the Buying Votes game. A legislature has 3 members. Each legislator can vote for bill X or bill

Y , and the bill that receives a majority of their votes passes. Interest group X favors bill X, while

interest group Y favors bill Y . The interest groups make campaign donations to the legislators in order

to induce them to vote as the interest groups prefer. Specifically, first group X gives amounts of money

(x1, x2, x3) to the k legislators, where xi is the amount group X gives to legislator i, and then group

Y gives amounts (y1, y2, y3), where yi is the amount group Y gives to legislator i. After the interest

groups make their payments, each legislator i votes for bill X if and only if xi > yi. (So if xi = yi, the

legislator votes for bill Y .) Group X’s payoff is 0 minus the sum of its payments to the legislators if

bill Y passes, and VX minus the sum of these payments if bill X passes. Group Y ’s payoff is 0 minus

the sum of its payments to the legislators if bill X passes, and VY minus the sum of these payments if

bill Y passes.

12. Consider the following version of the Bach-or-Stravinsky game:

B S

B 4, 2 0, 0

S 0, 0 1, 4

13. Consider the following sequential version of the Public Goods Provision game with three players. First

player 1 chooses whether to contribute or not; then player 2, having observed player 1’s choice, chooses

whether to contribute or not; then player 3, having observed the choices of players 1 and 2, chooses

whether to contribute or not. The public good is provided if and only if at least two players contribute.

If the public good is provided, and a player has not contributed, their payoff is B. If it is provided, and

the player has contributed, their payoff is B − c. If it is not provided and they have not contributed,

their payoff is 0. And if it is not provided but they have contributed, their payoff is −c. Assume

0 < c < B.

14. Consider a sequential version of the Hotelling-Downs model of electoral competition. First candidate

A chooses a position xA ∈ R, then candidate B chooses a position xB ∈ R. As in the original version,

voters vote for whichever candidate has adopted the position closest to their favorite position, and the

distribution F of ideal points is such that x < y implies F (x) < F (y), and there is a number m (the

1Each candidate has a favorite position; her dislike for the other positions increases with their distance from her favorite

position. For example, if candidate i’s favorite position is z∗i , then her payoff is −|z∗i − y| if y is the position of the winning

candidate—irrespective of whether she or the other candidate wins. Let z∗A refer to the favorite position of candidate A and let

z∗B refer to the favorite position of candidate B, and assume z

∗

A < m < z

∗

B , where m is the median voter’s favorite position. If

the candidates choose positions xA and xB and these positions are equally far from m, then the outcome of the election is the

compromise position, xA+xB

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, which is the position midway between xA and xB (and equal to xA and xB if xA = xB). If one

position is closer to m than the other, then the candidate who adopted the closer position wins.

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median voter’s favorite position) such that F (m) = 1/2. Each candidate prefers winning to tying and

prefers tying to losing.

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