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📗 [3 points] Select the values of \(A\) such that \(B\) will be alpha-beta pruned? The player moves first. In the case alpha = beta, prune the node.
📗 Choices:
None of the above
📗 [3 points] Consider the following game. There are piles, each pile has sticks. A player can take one stick from a single pile, or she may take two sticks, one for each pile. The player who takes the last stick loses. Let the game theoretical value be 1 if the first player wins. What is the value of the game?
📗 Answer: .
📗 [2 points] Consider a game board consisting of bits initially at . Each player can simultaneously flip any number of bits in a move, but needs to pay the other player one dollar for each bit flipped. The player who achieves wins and collects dollars from the other player. What is the game theoretic value (in dollars) of this game for the first player?
📗 Note: "game theoretic value" is what we called "value of the game" in the lectures.
📗 Answer: .
📗 [4 points] Consider the following zero-sum game tree. player moves first. Draw a new game tree by re-ordering the children of each internal node (including the root), such that the new game is equivalent to the tree above, but alpha-beta pruning will prune as many nodes as possible. (You do not have to submit the drawing.) Enter the number of nodes pruned.
📗 Note: in case the diagram is not clear, the values on the leafs (each sub-branch is a row): .
📗 Answer: .
📗 [4 points] Consider a zero-sum sequential move game with Chance. player moves first, then Chance, then . The values of the terminal states are shown in the diagram (they are the values for the Max player). What is the (expected) value of the game (for the Max player)?
📗 Note: in case the diagram is not clear, the probabilities from left to right is: , and the rewards are .
📗 Answer: .
📗 [4 points] There are lights in a row. The initial state is , 0 is "off", 1 is "on". A valid move finds two adjacent lights where one is on and the other is off, and switches them while keeping all other lights the same. That is, locally, you may do 01 to 10 or 10 to 01. What is the smallest number of moves to reach the goal state .
📗 Answer: .
📗 [3 points] Consider a variant of the II-nim game. There are two piles, each pile has \(n\) = sticks. A player can take one stick from a single pile; or take two sticks, one from each pile (when available). The player who takes the last stick wins. Let the game value be 1 if the first player wins (and -1 if the second player wins). What is the game theoretical value of this game?
📗 Answer: .
📗 [4 points] For a zero-sum game in which moves first and the value to the MAX player is given in the diagram below, consider the static board evaluation (heuristic function) at the internal states provided in the table below. What are the smallest and largest possible values of \(x\) above and below which IDS (iterative deepening search) with depth limit \(1\) will find the correct solution for the game? You can assume all values are between \(-100\) and \(100\). Enter two numbers between \(-100\) and \(100\) (possibly including \(-100\), \(100\)).
📗 Note: for example, if you think \(10 < x < 20\), enter \(10, 20\); if you think any \(x > 10\) works, enter \(10, 100\); if you think any \(x < 20\) works, enter \(-100, 20\); if you think every \(x\) is okay, enter \(-100, 100\); if you think no such \(x\) exist, enter \(-100, -100\) or \(100, 100\).
State (Action)
Left
Middle
Right
Static Board Evaluation
\(x\)
📗 Answer (comma separated vector): .
📗 [3 points] There are two players and \(k\) coins on the table. Players move sequentially with player 1 moving first. Each player chooses to take either one or two coins from the table. The player who takes the last coin wins. For which of the following values of \(k\) = {} does the first player has a winning strategy? Enter the values of \(k\), not the indices.
📗 Answer (comma separated vector): .
📗 [4 points] For a zero-sum game in which moves first and if the action Left is chosen, then Chance (Chn) moves Left with probability \(p\) and Right with probability \(1 - p\), and if the action Right is chosen, then Chance moves Left with probability and Right with probability . Suppose the player who moves first uses a mixed strategy \(\dfrac{1}{2}\) Left and \(\dfrac{1}{2}\) Right in a solution, what is the value of \(p\)? If it's impossible, enter \(-1\).
📗 Note: in case the diagram is not clear, the values on the leafs (each sub-branch is a row): .
📗 Answer: .
📗 [3 points] There are \(n\) = cookies. The brother first proposes a division of these cookies into two piles (two integers adding up to \(n\)) and then the sister take one of the two piles. Both the brother and the sister want to maximize the number of cookies they take. What is the value of the game to the brother (measured by the number of cookies he gets)? Enter an integer.
📗 Answer: .
📗 [3 points] Given the following game matrix (zero-sum game), suppose A (row) knows that B (col) will use the mixed strategy on I, II, III. What is the expected payoff for A if A plays optimally?
A \ B
I
II
III
I
II
III
📗 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.
📗 [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).
A \ B
I
II
III
I
II
III
📗 Answer: .
📗 [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] 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: .
📗 [4 points] Given the following BoS (Battle of Sexes) game, what is the row (Romeo) player's (expected) value (i.e. payoff) in the mixed strategy Nash equilibrium?
Romeo \ Juliet
Bach
Stravinsky
Bach
Stravinsky
📗 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: .
📗 [4 points] Suppose \(n\) = witnesses heard a gunshot near 221B Baker Street. The benefit from at least one witness calling the police is \(b\) = and the cost of calling the police is \(c\) = . If no witness calls the police, everyone gets 0. In a Nash equilibrium in which every witness uses the same mixed strategy, what is the probability that no one calls the police?
📗 Answer: .
📗 [4 points] Given the following game payoff table, suppose the row player uses a pure strategy, and column player uses a mixed strategy playing L with probability \(q\). What is the smallest and largest value of \(q\) in a mixed strategy Nash equilibrium?
Row \ Col
L
R
U
D
Note: the following is a diagram of the best responses (make sure you understand what they are and how to draw them). The red curve is the best response for the column player and the blue curve is the best response for the row player.
📗 Answer (comma separated vector): .
📗 [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] Write down the matrix normal form of the following game.
📗 Answer (matrix with multiple lines, each line is a comma separated vector): .
📗 [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: .
📗 [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] 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] Find the value of the mixed strategy Nash equilibrium of the following zero-sum game.
MAX \ MIN
A
B
A
B
📗 Answer: .
📗 [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?
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