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📗 [4 points] You will receive 4 points for this question and you can choose to donate x points (a number between 0 and 4). Your final grade for this question is the points you keep plus twice the average donation (sum of the donations from everyone in your section divided by the number of people in your section, combining both versions). Enter the points you want to donate (an integer between 0 and 4).
📗 Answer: (The grade for this question will be updated later).
📗 [3 points] There are \(n\) = students in CS540, for simplicity, assume student \(0\) gets grade \(g = 0\), student \(1\) gets grade \(g = 1\), ..., student \(n - 1\) gets grade \(g = n - 1\). The payoff for each student who drop the course is \(0\), the payoff for the students who stay is if the student has the lowest grade among all students who decide to stay in the class, and the otherwise. If each student only uses actions that are rationalizable (i.e. survive the iterated elimination of strictly dominated actions), how many students will stay in the course? If there are multiple correct answers, enter one of them.
📗 Answer: .
📗 [4 points] There are people living in the suburbs and all of them commute to work in the city. Every morning, each individual decides which way to drive to the city simultaneously: the Direct Way or the Long Way. The Long Way takes 1 hour of driving. The time spent on the Direct Way depends on the traffic is equal to \(\dfrac{n}{c}\) hours, where \(n\) is the total number of cars taking the Direct Way, and \(c\) = is the capacity. Each individual wants to minimize the driving time, and break ties by choosing the Direct Way. What is the number of people taking the Long Way in the Nash equilibrium?
📗 Answer: .
📗 [3 points] The following BoS (Battle of the Sexes) game has a Nash equilibrium where the row player uses \(B\) with probability and the column player uses \(B\) with probability . What are the values \(\left(x, y\right)\)? If there are multiple possible values, enter one of them, if there are none, enter \(-1\).
Actions
B
S
B
\(\left(x, y\right)\)
\(\left(0, 0\right)\)
S
\(\left(0, 0\right)\)
📗 Answer: .
📗 [4 points] Perform iterated elimination of strictly dominated strategies (i.e. find rationalizable actions). Player A's strategies are the rows. The two numbers are (A, B)'s payoffs, respectively. Recall each player wants to maximize their own payoff. Enter the payoff pair that survives the process. If there are more than one rationalizable action, enter the pair that leads to the largest payoff for player A.
A \ B
I
II
III
IV
I
II
III
IV
📗 Answer (comma separated vector): .
📗 [4 points] Perform iterated elimination of strictly dominated strategies (i.e. find rationalizable actions). Player A's strategies are the rows. The two numbers are (A, B)'s payoffs, respectively. Recall each player wants to maximize their own payoff. Enter the payoff pair that survives the process. If there are more than one rationalizable action, enter the pair that leads to the largest payoff for player A.
A \ B
I
II
III
I
II
III
IV
📗 Answer (comma separated vector): .
📗 [3 points] Perform iterated elimination of strictly dominated strategies. Player A's strategies are the rows. The two numbers are (A, B)'s payoffs, respectively. Recall each player wants to maximize their own payoff. Enter the payoff pair that survives the process (i.e. payoffs from rationalizable actions). There should be only one such pair.
A \ B
I
II
III
I
II
III
📗 Answer (comma separated vector): .
📗 [4 points] The Nash equilibrium of the following simultaneous move zero-sum game is (U, L): the entry marked by \(x\). What is the smallest and largest possible integer values of \(x\)? Enter two numbers. (U, L) can be one of possibly many Nash equilibria.
📗 Note: if there is only one possible value, enter the same value twice; and if no values are possible, enter \(0, 0\).
MAX \ MIN
L
C
R
U
\(x\)
M
R
📗 Answer (comma separated vector): .
📗 [3 points] Consider the standard PD (Prisoner's Dilemma) game in the following table with two prisoners that belong to the same criminal organization, and the criminal organization punishes whoever confesses which decrease the prisoner's value by \(x\). What is the smallest value of \(x\) so that (deny, deny) is a Nash equilibrium?
A \ B
Deny
Confess
Deny
Confess
📗 Answer: .
📗 [4 points] In GoogSoft, software engineers A and B form a two-person team. Their year-end bonus depends on their relative performance. The bonus outcomes are summarized in the following table. The value of slacking to each person is \(s\) = . The total payoff to each person is the sum of the bonus and the value from slacking. What is the smallest value of \(x\) such that both players will work hard in a Nash equilibrium?
-
B works hard
B slacks
A works hard
\(x, x\)
A slacks
📗 Answer: .
📗 [3 points] \(N\) = firms sharing the use of a river decide whether to filter (F) or release (R) pollutants (a poisonous substance) into the river. If \(n\) firms choose to pollute the river (R), each of these \(n\) firms incurs a cost of dollars, and each of the remaining firms that choose to install filters (F) incurs a cost of (cost due to pollution plus the cost of the filter). Every firm wants to minimize costs. What is the number of firms that choose to install filters (F) in a pure strategy Nash equilibrium? Note: remember to enter an integer.
📗 Answer: .
📗 [1 points] Two identical antiques are lost. The airline only knows that its value is at most dollars, so the airline asks their owners (travelers) to report its value (non-negative integers larger than or equal to . The airline tells the travelers that they will be paid the minimum of the two reported values, and the traveler who reported a strictly lower value will receive dollars in reward from the other traveler. If you are one of the travelers, what will you report?
📗 Answer: .
📗 [1 points] Write down an integer between and that is the closest to two thirds \(\dfrac{2}{3}\) of the average of everyone's (including yours) integers.
📗 Answer: .
📗 [4 points] Imagine a population of \(N\) = individuals. Each of them simultaneously chooses between taking the vaccine and not. All individuals have the same payoffs. Suppose there are \(n\) people who choose not to take the vaccine, then the payoff from not taking the vaccine is \(- \alpha \cdot \dfrac{n}{N}\), and the payoff from taking the vaccine is \(- c - \beta \cdot \dfrac{n}{N}\), \(\alpha\) = is the herd immunity coefficient, \(\beta\) = measures the ineffectiveness of the vaccine, and \(c\) = is the cost of getting the vaccine. In a Nash equilibrium, what is the largest number of individuals who choose NOT to take the vaccine?
📗 Note: \(n\) is the number of people NOT taking the vaccine, and the question is asking for the largest number of individuals who choose NOT to take the vaccine.
📗 Answer: .
📗 [3 points] Identify the pure strategy Nash equilibria in the following zero-sum game. A (row) is the max player, B (col) is the min player.
📗 [4 points] What is the row player's value in a Nash equilibrium of the following zero-sum normal form game? A (row) is the max player, B (col) is the min player. If there are multiple Nash equilibria, use the one with the largest value (to the max player).
A \ B
I
II
III
IV
I
II
III
IV
📗 Answer: .
📗 [2 points] What is the row player's value in a Nash equilibrium of the following zero-sum normal form game? A (row) is the max player, B (col) is the min player. If there are multiple Nash equilibria, use the one with the largest value (to the max player).
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