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📗 [3 points] Write down the matrix normal form of the following game.
📗 Answer (matrix with multiple lines, each line is a comma separated vector): .
📗 [4 points] Consider the following zero-sum game tree. MAX 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.
📗 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.
📗 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
📗 Answer (comma separated vector): .
📗 [4 points] Suppose the states are integers between and . The initial state is , and the goal state is . The successors of a state \(i\) are \(2 i\) and \(2 i + 1\), if exist. How many states are expanded using a Depth First Search? Include both the initial and goal states.
📗 Note: use the convention used in the lectures, push the states with larger index into the stack first (i.e. expand the states with the smaller index first).
📗 Answer: .
📗 [3 points] If \(h_{1}\) and \(h_{2}\) are both admissible heuristic functions, which ones of following are also admissible heuristic functions? Enter the correct choices as a list, comma separated, without parentheses, for example, "1, 2, 4".
📗 Choices:
(1)
(2)
(3)
(4)
(5)
(6)
(7) None of the above
📗 Answer (comma separated vector): .
📗 [4 points] Run search algorithm on the following graph, starting from state 0 with the goal state being . Write down the expansion path (in the order of the states expanded). The heuristic function \(h\) is shown as subscripts. Break tie by expanding the state with a smaller index.
📗 In case the diagram is not clear: the weights are (with heuristic values on the diagonal entries): .
📗 Answer (comma separated vector): .
📗 [3 points] Let the search space be integers. Each state \(n\) has successors . Write down the shortest path (i.e. the sequence of states) from the initial state 1 to the goal state .
📗 Answer (comma separated vector): .
📗 [4 points] Suppose K-Means with \(K = 2\) is used to cluster the data set and initial cluster centers are \(c_{1}\) = and \(c_{2}\) = \(x\). What is the largest value of \(x\) if cluster 1 has \(n\) = points initially (before updating the cluster centers). Break ties by assigning the point to cluster 2.
📗 Answer: .
📗 [4 points] You are given the distance table. Consider the next iteration of hierarchical clustering using linkage. What will the new values be in the resulting distance table corresponding to the new clusters? If you merge two columns (rows), put the new distances in the column (row) with the smaller index. For example, if you merge columns 2 and 4, the new column 2 should contain the new distances and column 4 should be removed, i.e. the columns and rows should be in the order (1), (2 and 4), (3).
\(d\) =
📗 Answer (matrix with multiple lines, each line is a comma separated vector): .
📗 [3 points] When using Simulated Annealing, which value of temperature \(T\) from the list would imize the probability of moving to an inferior (worse) state? Enter one value of \(T\), not its index in the list.
📗 Answer: .
📗 [4 points] When using the Genetic Algorithm, suppose the states are \(\begin{bmatrix} x_{1} & x_{2} & ... & x_{T} \end{bmatrix}\) = , , , . Let \(T\) = , the fitness function (not the cost) is \(\mathop{\mathrm{argmin}}_{t \in \left\{1, ..., T + 1\right\}} x_{t} = 1\) with \(x_{T + 1} = 1\) (i.e. the index of the first feature that is 1). What is the reproduction probability of the state with the highest reproduction probability?
📗 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: .
📗 [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).
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📗 Answer: .
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