📗 [3 points] Given three clusters, \(A\) = {, }, \(B\) = {\(x\)}, \(C\) = {, }. Find a value of \(x\) so that \(A\) and \(B\) will be merged in the next iteration of single linkage hierarchical clustering, and \(B\) and \(C\) will be merged in the next iteration of complete linkage hierarchical clustering. Break ties by merging with the cluster with the smaller index (i.e. \(A\), then \(B\), then \(C\)).
📗 Note: there can be multiple answers, including non-integer answers, enter one of them. If there are none, enter 0.
📗 Answer: . Calculate

📗 [4 points] Given the list of states in the priority queue (frontier) and the current cost \(g\) and heuristic cost \(h\), what is the largest value of \(x\) so that state \(0\) will be removed (expanded) from the priority queue next in all three informed search strategies: UCS (Uniform Cost Search), (Best First) Greedy Search, and A Search? Break ties by expanding the state with the smallest index.

📗 Answer: . Calculate

📗 [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 \(100, 100\).


📗 Answer (comma separated vector): . Calculate

📗 [3 points] Given the variance matrix \(\hat{\Sigma}\) is a diagonal matrix, what is the smallest value of \(K\) so that the Manhattan distance between the vector \(\begin{bmatrix} 1 \\ 1 \\ ... \\ 1 \end{bmatrix}\) with ones (\(1\)'s) and its reconstruction using the first \(K\) principal components is less than or equal to ?
📗 Answer: . Calculate

📗 [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: . Calculate

📗 [3 points] Given data and initial k-means cluster centers \(c_{1}\) = and \(c_{2}\) = , what is the initial total distortion (do not take the square root). Use Euclidean distance. Break ties by assigning points to the first cluster.
📗 Answer: . Calculate

📗 [4 points] Suppose the state space has \(n\) = states that form a tree with root state \(0\). What is the shape of the tree that makes iterative deepening realize that a goal does not exist as quickly as possible (i.e. one that minimizes the number of expanded nodes)? Enter the number of nodes searched in this case.
📗 Answer: . Calculate

📗 [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): .

📗 [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?

📗 Answer: . Calculate

📗 [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): .

📗 [2 points] Consider \(n + 1\) = + \(1\) states. The initial state is \(1\), the goal state is \(n\). State \(0\) is a dead-end state with no successors. For each non-\(0\) state \(i\), it has two successors: \(i + 1\) and \(0\). We may expand the same states many times, because we do not keep track of which states are checked previously. How many states (including repeated ones) will be expanded by ? Break ties by expanding the state with the index first.
📗 Note: the tie-breaking rule may be different from the convention used during the lectures, please read the question carefully.
📗 Answer: . Calculate

📗 [4 points] Let the states be 3D integer points with integer coordinates \(\left(i, j, k\right)\) with boundary constrains and and . Each state \(\left(i, j, k\right)\) has six successors \(\left(i - 1, j, k\right), \left(i + 1, j, k\right), \left(i, j - 1, k\right), \left(i, j + 1, k\right), \left(i, j, k - 1\right), \left(i, j, k + 1\right)\) or a subset thereof subject to the boundary constraints. The score of state \(\left(i, j, k\right)\) is . Which local minimum will be reached if hill climbing is used starting from ? Enter the state, not the score.
📗 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.

📗 Answer (comma separated vector): .

📗 [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?

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): . Calculate

📗 [1 points] Please enter any comments including possible mistakes and bugs with the questions or your answers. If you have no comments, please enter "None": do not leave it blank.
📗 Answer: .