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MATH/MTHE 339: Evolutionary Game Theory

Homework Assignment 1

2022

1. Use iterated elimination of dominated strategies (IEDS) to reduce the following games. Recall that for each entry in the payoff matrices below, Blake’s payoff is listed first and Mark’s payoff second. Clearly indicate:

• t he order in which you eliminate strategies;

• wh ether the eliminated strategy is strictly or weakly dominated;

• and, if you find a dominant strategy equilibrium, state whether or not it must be unique.

a . (2 marks )

Mark

	W	X	Y	Z
A	40, − 11	26, 4	1, 5	8, 6
B	1 2 − 2	2 5, 0	4 1	5 , 3
C	1 1, 8	1 5, 10	− 1 11	− 2, 6
D	1 7, 5	− 11, 10	3, 13	4, − 8

b . (3 marks )

Mark

	V	W	X	Y	Z
A	0, 0	1 4	− 3 , 1	− 2, 3	− 1 − 1
B	4 − 1	3 , 0	− 3 , 1	− 1 4	− 2, 0
C	− 1 1	2 2	2 , 3	− 1, 0	2, 5
D	2 1	− 1 − 1	0 , 4	4 − 1	0 , 2
E	1 , 6	− 3, 0	− 1 4	1 1	− 1 4

2. (5 marks) Imagine two vendors (our players) who must simultaneously choose a location to position their displays. There are n possible locations that form a straight line. Further, there is one customer at each location, and customers will choose the closest vendor (and split their time at each vendor if they’re equally distanced from both). The profit for each vendor equals the number of customers they attract. Using iterated elimination of dominated strategies (IEDS), what is the dominant strategy equilibrium? Explain.

3. Laura, Alex, and Fiona are deciding on whether to go for brunch at a nearby restaurant or stay home. However, the restaurant is quite crowded. If all decide to go, then they will have to wait a long time for a table, each receiving payoffs of zero. If only two go, those two quickly get a table and receive payoffs of 1, and the one who remains home receives a payoff of − 1 (they’re envious). If only one goes, then all players receive a payoff of zero (the one who goes is lonely, and the other two aren’t envious). If none go, they all receive payoffs of zero. The three decide to take turns choosing what they’ll do: Laura first, then Alex, and

finally Fiona. All players have perfect information.

a . (3 marks ) Draw this game in extensive form (i.e. draw the game tree). And, using backward induc- tion, find the equilibrium.

b . (2 marks ) Assume that Laura stays home. Write the game in normal form (i.e. as a matrix with Alex and Fiona’s strategies and all players’payoffs).

4 . Every year before the Academy Awards, cinephiles Mike, Tom, and Crow have a movie watching marathon and mini awards show to decide on a Best Picture for the previous year. This year, there are three candi- date movies in the hunt for their Best Picture award: the spy thriller, Agent for H.A.R.M., denoted by A; the gritty character-driven drama, Brute Man, denoted by B; and the visually stunning sci-fi blockbuster, Creeping Terror, denoted by C. After watching and discussing all three movies, each person’s preferences are as follows:

• Mike prefers A to B and prefers B to C.

• Tom prefers C to A and prefers A to B .

• Crow prefers B to C and prefers C to A.

Since they seem to be stuck, they agree to have a vote. Here’s how it will work. Each person writes their

selection for best picture on a ballot. If one movie receives more votes than the others, it wins the award. If

there is a three way tie in which each movie gets a single vote, then Mike (as the host) breaks the tie. (Note:

Mike doesn’t get to vote again in the case of a three way tie, but rather whatever he voted for in a three way

tie would be the winner.) Assuming that all three movie lovers are perfectly rational, and each knows the preferences of the other two, we would like to determine how each of them would vote.

a . (3 marks ) For each individual, determine which of their strategies are strictly or weakly dominated.

b . (2 marks ) After eliminating the dominated strategy/strategies, which strategy/strategies can you

now eliminate? What will be the result of the vote?

5. (5 marks ) Suppose that there are five executives on the board of directors of a major financial corporation: the CEO and four vice-presidents. After a particularly profitable year, the five executives learn that there is a surplus of 100 million dollars to be shared among them, and they call a meeting to decide how to split up the money. After some deliberation, they agree on the following setup:

• A board member proposes how to split up the money. For simplicity, we will assume that the proposal

units are in millions of dollars. For instance, the CEO could propose that they receive 60 million and

the others receive 10. Another possible proposal is 100 million for the CEO and zero for everyone else.

• After each proposal is presented, there is a vote. In order for a proposal to pass, it needs to be agreed on by 50% or more of the board members that are voting (the proposer is allowed to vote for their own proposal).

• If the proposal passes, then the money will be split up accordingly. If the proposal is rejected, then the board member who made that proposal will lose their right to vote in the future, and the next board member in line will then make a new proposal. The process repeats.

• The ord er for proposals is as follows: CEO → VP1 → VP2 → VP3 → VP4.

Assuming that board members are perfectly rational and selfish, how will the money be divided? Ex-

plain.