📗 [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.

Hint

The \(n\) points on the left (or right, depending on the question) should be assigned to cluster 1. The \(n + 1\)-th point (call it \(x_{n + 1}\) from the left (or right) can be equidistant from cluster 1 center and cluster 2 center because if the distances to the clusters are the same, the point is assigned to cluster 2 due to the tie-breaking rule. Therefore, \(x_{n + 1} = \dfrac{1}{2} \left(c_{1} + c_{2}\right)\) can be used to solved for \(c_{2}\).

📗 Answer: . Calculate

📗 [4 points] There are lights in a row. The initial state is , 0 is "off", 1 is "on". A valid move finds two adjacent lights where one is on and the other is off, and switches them while keeping all other lights the same. That is, locally, you may do 01 to 10 or 10 to 01. What is the smallest number of moves to reach the goal state .

Hint

See Fall 2013 Final Q20, Fall 2010 Midterm Q4. "Moving" a light from position \(i\) to position \(j\) requires at least \(j - i\) steps. All the lights need to be moved from the current position to the "correct" position specified by the goal state.

📗 Answer: . Calculate

📗 [4 points] Imagine a population of \(N\) = individuals. Each of them simultaneously chooses between taking the vaccine and not. All individuals have the same payoffs. Suppose there are \(n\) people who choose not to take the vaccine, then the payoff from not taking the vaccine is \(- \alpha \cdot \dfrac{n}{N}\), and the payoff from taking the vaccine is \(- c - \beta \cdot \dfrac{n}{N}\), \(\alpha\) = is the herd immunity coefficient, \(\beta\) = measures the ineffectiveness of the vaccine, and \(c\) = is the cost of getting the vaccine. In a Nash equilibrium, what is the largest number of individuals who choose NOT to take the vaccine?
📗 Note: \(n\) is the number of people NOT taking the vaccine, and the question is asking for the largest number of individuals who choose NOT to take the vaccine.

Hint

📗 Answer: . Calculate

📗 [4 points] What is the projected variance of and onto the principal component ? Use the MLE (Maximum Likelihood Estimate) formula for the variance: \(\sigma^{2} = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}\) with \(\mu = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} x_{i}\).

Hint

📗 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).
\(d\) =

Hint

See Spring 2017 Midterm Q4. The resulting matrix should have 4 columns and 4 rows. Find the smallest non-zero number in the pair-wise distance matrix, suppose row \(i\) and column \(j\), merge columns \(i\) and \(j\) and rows \(i\) and \(j\) at the same time: for single linkage, take the minimum of the numbers in the two rows and columns; for complete linkage, take the maximum.

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

📗 [2 points] Consider a search graph which is a tree, and each internal node has children. The only goal node is at depth (root is depth 0). How many total goal-checks will be performed by Iterative Deepening Search in the luckiest case (i.e. the smallest number of goal-checks)? If a node is checked multiple times you should count that multiple times.

Hint

See Fall 2018 Midterm Q5. When the depth limit is \(i\), every node in the tree limited to depth \(i\) are searched, so \(\displaystyle\sum_{i'=0}^{i} b^{i'}\) nodes are searched in iterations \(i = 0, 1, ..., d - 1\). In the luckiest case, the goal node is on the first path when the depth limit is \(d\), so at least \(d + 1\) nodes are searched in iteration \(i = d\).

📗 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.

Hint

See Fall 2017 Midterm Q9. Due to the convexity of the function, hill climbing will eventually reach the global minimum. If the coefficient in front of \(i\) (or \(j\), \(k\)) is positive, then \(i\) (or \(j\), \(k\)) should be as small as possible at the global minimum. If the coefficient in front of \(i\) (or \(j\)) is negative, then \(i\) (or \(j\), \(k\)) should be as large as possible at the global minimum.

📗 Answer (comma separated vector): .

📗 [4 points] Given the following BoS (Battle of Sexes) game, what is the column (Juliet) player's (expected) value (i.e. payoff) in the mixed strategy Nash equilibrium?

Hint

📗 Answer: . Calculate

📗 [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.

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): .

📗 [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: .