📗 [3 points] Suppose there are \(n\) = states, \(1\) (initial state), \(2\), ..., \(n\) (goal state) and two heuristics \(h_{1}\) and \(h_{2}\), described in the table below, are both admissible. State \(i\) has successors \(i + 1, i + 2, ..., n\), for each \(i = 1, 2, ..., n - 1\). What is the minimum possible total cost from state \(1\) to state \(n\).

📗 Note: do not assume the cost between a state and its successor is 1. 

📗 Answer: . Calculate

📗 [3 points] Find the value of the mixed strategy Nash equilibrium of the following zero-sum game.

📗 Answer: . Calculate

📗 [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?

📗 Answer: . Calculate

📗 [3 points] Given the scores in the following table, if hill-climbing (valley-finding) is used, how many states will lead to the global imum? Note: the neighbors of state \(i\) are states \(i - 1\) and \(i + 1\) (if they exist).

📗 Answer: . Calculate

📗 [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 \(0, 0\).

📗 Answer (comma separated vector): . Calculate

📗 [3 points] Suppose the states are integers between \(1\) and . The initial state is \(1\), and the goal state is . The successors of a state \(i\) are \(2 i\) and \(2 i + 1\), if exist. If it is allowed to put the successors into the stack (list) in any order, what are the minimum and maximum numbers of states that will be expanded during a DFS (Depth First Search)? Enter two numbers.

📗 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

📗 [4 points] In a by grid, Tom is located at (, ) and Jerry is located at (, ). Tom uses to find Jerry and the successors of a state (one cell in the grid) are the four neighboring states on the grid (the cells above, below, to the left and to the right). What is the imum number of states that need to be expanded to find (and expand) the goal state? The order in which the successors are added can be arbitrary. Do not count repeated expansion of the same state. Include both the initial and the goal states.

📗 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

📗 [3 points] Let \(x\) = and \(v\) = . The projection of \(x\) onto \(v\) is the point \(y\) on the direction of \(v\) such that the line connecting \(x, y\) is perpendicular to \(v\). Compute \(y\).

📗 Answer (comma separated vector): . Calculate

📗 [3 points] Given the variance matrix \(\hat{\Sigma}\) = , what is the first principal component? Enter a unit vector.

📗 Answer (comma separated vector): . Calculate

📗 [4 points] Given the dataset , the cluster centers are computed by k-means clustering algorithm with \(k = 2\). The first cluster center is \(x\) and the second cluster center is . What is the imum value of \(x\) such that the second cluster is empty (contains 0 instances). In case of a tie in distance, the point belongs to cluster 1.

📗 Answer: . Calculate

📗 [4 points] Suppose the score (fitness) of a state \(\left(d_{1}, d_{2}, d_{3}, d_{4}\right)\) is \(d_{1} + d_{2} + d_{3} + d_{4}\), and only 1-point crossover with the cross-over point between \(d_{2}\) and \(d_{3}\) is used in a genetic algorithm (i.e. mutation probabilities are 0). Two states are chosen as parents at random according to the reproduction probabilities, what is the probability that one of their children is the optimal state (i.e. \(\left(1, 1, 1, 1\right)\)? Enter a number between 0 and 1.

📗 Note: the two parents are sampled with replacement, meaning the probability that two states are chosen as parents is the product of their reproduction probabilities.

📗 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): .

📗 [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: . 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] Consider the following zero-sum game tree. MIN 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] 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).

📗 Answer (matrix with multiple lines, each line is a comma separated vector): .

📗 [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

📗 [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] Which order of goal check is possible with , without specifying the order of successors when putting them in the queue (i.e. you can rearrange the order of the branches)? Enter the correct choices as a list, comma separated, without parentheses, for example, "1, 2, 4".

📗 Choices:

📗 Answer (comma separated vector):

📗 [3 points] Let \(h_{1}\) be an admissible heuristic from a state to the optimal goal, A* search with which ones of the following \(h\) will be admissible? Enter the correct choices as a list, comma separated, without parentheses, for example, "1, 2, 4".

📗 Choices:

📗 Answer (comma separated vector): .

📗 [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] 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).

📗 Answer: .

📗 [4 points] Enter the smallest integer value of \(A\) such that \(B\) will be alpha-beta pruned? Max player moves first. In the case alpha = beta, prune the node. Enter -100 if you think the answer is negative infinity.

📗 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?

📗 Answer (comma separated vector): . Calculate

📗 [3 points] Consider Iterative Deepening Search on a tree, where the nodes are denoted by numbers. Write down the sequence IDS visited in the order they are expanded (i.e. expansion path). 0 is the initial state and is the goal state. Start with depth limit 0, include the root, and include repeated nodes.

📗 Note: use the convention used in the lectures, push the rightmost successor into the stack first (i.e. expand the leftmost successor first).

📗 Answer (comma separated vector): .

📗 [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).

📗 [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: .