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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.
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
📗 [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] 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.
📗 [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. Hint
See Fall 2017 Midterm Q3, Fall 2010 Final Q6. The first player will always flip all the bits in the first round to collect the payments and pay of the costs (as long as the payments are larger than the costs).
📗 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
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: .
📗 [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.
📗 Choices:
(I, I)
(I, II)
(I, III)
(II, I)
(II, II)
(II, III)
(III, I)
(III, II)
(III, III)
None of the above
📗 [2 points] What is the game theoretical value of the following matrix 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
📗 Note: here, I am using "game theoretical value" and the "row (max) player's value in a Nash equilibrium" interchangeably, which is slightly different from Professor Zhu's definition in Fall 2012 Final Q18. In the case there is only one Nash equilibrium, the value should be the same. 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 element with two circles is the value of the game. 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] 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
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] 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.
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): .
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📗 Answer: .
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