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📗 [3 points] Which nodes are pruned by alpha-beta pruning? The player moves first.
📗 Select the nodes:
select
Hint
See Spring 2018 Midterm Q4, Fall 2017 Midterm Q12, Spring 2017 Midterm Q3, Fall 2016 Midterm Q8, Fall 2014 Final Q13, Fall 2012 Final Q17, Fall 2008 Midterm Q5, Fall 2005 Midterm Q14. (1) Suppose the first player is max. The nodes on the left subtree cannot be pruned for sure. Compute the minimum value of the left subtree, say it is \(m\). For the nodes on the right subtree, when the minimum-so-far is smaller than \(m\), the remaining nodes can all be pruned, because when that happens, the max player will select the left branch anyways and do not have to continue searching the right subtree. (2) Suppose the first player is min. The nodes on the left subtree cannot be pruned for sure. Compute the maximum value of the left subtree, say it is \(m\). For the nodes on the right subtree, when the maximum-so-far is larger than \(m\), the remaining nodes can all be pruned, because when that happens, the min player will select the left branch anyways and do not have to continue searching the right subtree.
📗 [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.
Hint
See Fall 2018 Midterm Q7, Fall 2013 Final Q16. (1) Suppose the first player is max. \(B\) will be pruned if the minimum-so-far before reaching node \(B\) for the min player is already smaller than \(A\). If that is the case, the max player will select the \(A\) branch and not continue searching the min subtree on the right (i.e. \(B\) will be pruned.) (2) Suppose the first player is min. \(B\) will be pruned if the maximum-so-far before reaching node \(B\) for the max player is already larger than \(A\). If that is the case, the min player will select the \(A\) branch and not continue searching the max subtree on the right (i.e. \(B\) will be pruned.)
📗 Choices:
None of the above
📗 [4 points] Consider a zero-sum sequential move game with Chance. Min player moves first, then Chance, then Max. 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 .
Hint
See Fall 2019 Q14 Q15, Spring 2017 Midterm Q3, Fall 2006 Final Q5, Fall 2005 Final Q5, Fall 2005 Midterm Q7
📗 Answer: .
📗 [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.
Hint
See Fall 2018 Midterm Q8, Fall 2017 Final Q6, Fall 2013 Final Q13. Draw the game tree for \(k\) = \(1, 2, 3, ...\) to find the pattern, and note that if the remaining number of coins is divisible by 3, the player who moves in that round will always lose.
📗 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
IV
I
II
III
IV
Hint
See Fall 2012 Final Q18, Fall 2006 Final Q6, Fall 2005 Final Q6, Fall 2005 Midterm Q10. If the first numbers in one row is strictly smaller than the first numbers in another row, then that row is strictly dominated. If the second numbers in one column is strictly smaller than the second numbers in another column, then that column is strictly dominated. Remove strictly dominated rows and columns and continue this process. Strictly dominated actions will never be played in a Nash equilibrium because they are never best responses.
📗 Answer (comma separated vector): .
📗 [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
Hint
See Fall 2012 Final Q18, Fall 2006 Midterm Q12. Circle the largest number of each column and the smallest number of each row: the action pairs corresponding to the elements with two circles are the Nash equilibria. The action corresponding to a circled element is called the best response: a Nash equilibrium is when two actions are best responses to each other.
📗 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.
A \ B
I
II
III
I
II
III
Hint
See Fall 2014 Final Q5, Fall 2006 Final Q4, Fall 2005 Final Q4, Fall 2005 Midterm Q11. Circle the largest number of each column and the smallest number of each row: the elements with two circles are the Nash equilibria. The action corresponding to a circled element is called the best response: a Nash equilibrium is when two actions are best responses to each other.
📗 [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
Hint
See Fall 2014 Final Q4, Fall 2006 Midterm Q9. Find the expected payoff from playing I, II, and III, and find the maximum. The expected payoff is the sum of the products between the probabilities and payoffs.
📗 Answer: .
📗 [4 points] Given the following game payoff table, suppose the row player uses a mixed strategy playing U with probability \(p\), and column player uses a pure strategy. What is the smallest and largest value of \(p\) 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.
Hint
Draw the best response diagrams: the region where the two curves intersect are Nash equilibria (since those mixed strategies are best responses to each other). Find the smallest and largest values of \(p\) or \(q\) in that region. The way to compute the location where the blue curve is vertical (or the red curve is horizontal) is by setting the expected payoffs from U and D equal to each other (or from L and R equal to each other).
📗 Answer (comma separated vector): .
📗 [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?
Hint
This is the Volunteer's Dilemma. Suppose every other witness call the police with probability \(p\), then calling the police lead to net benefit \(b - c\) and not calling the police lead to net benefit \(0 \cdot \left(1 - p\right)^{n - 1} + b \cdot \left(1 - \left(1 - p\right)^{n - 1}\right)\), where \(\left(1 - p\right)^{n - 1}\) is the probability that no one else calls the police. A witness would use a mixed strategy only if the net benefits from calling and not calling are equal to each other, and this can be used to set up a equation to solve for \(p\).
📗 Answer: .
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