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📗 [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).
Hint
📗 Answer: .
📗 [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] 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).
A \ B
I
II
III
IV
I
II
III
IV
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 action pairs corresponding to 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.
📗 Answer: .
📗 [4 points] Consider the four points: \(x_{1}\) = , \(x_{2}\) = , \(x_{3}\) = , \(x_{4}\) = . Let there be two initial cluster centers \(c_{1}\) = , \(c_{2}\) = . Use Euclidean distance. Break ties in distances by putting the point in the cluster with the smaller index (i.e. favor cluster 1). If a cluster contains no points, do not move the cluster center (it stays at the initial position). Write down the cluster centers after one iteration of k-means, the first cluster center (comma separated vector) on the first line and the second cluster center (comma separated vector) on the second line.
📗 Note: the red points are the cluster centers and the other points are the training items.
Hint
See Fall 2019 Midterm Q22, Spring 2018 Midterm Q7, Fall 2017 Final Q22, Spring 2017 Midterm Q5, Fall 2014 Final Q20, Fall 2013 Final Q14, Fall 2006 Final Q14, Fall 2005 Final Q14. Find which cluster each \(x_{i}\) belongs to (call it \(k_{i}\)): it's the cluster center that is the closest to the point. Compute the new cluster centers \(c'_{1}, c'_{2}\) as \(c'_{k} = \dfrac{1}{\displaystyle\sum_{k_{i} = k} 1} \displaystyle\sum_{k_{i} = k} x_{i}\).
📗 Answer (matrix with multiple lines, each line is a comma separated vector): .
📗 [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.
Hint
Fall 2008 Midterm Q5
📗 Answer: .
📗 [4 points] Consider a zero-sum sequential move game with Chance. Min player moves first, then Chance, then Max. The values of the terminal states are shown in the diagram (they are the values for the Max player). What is the (expected) value of the game (for the Max player)?
📗 Note: in case the diagram is not clear, the probabilities from left to right is: , and the rewards are .
Hint
See Fall 2019 Q14 Q15, Spring 2017 Midterm Q3, Fall 2006 Final Q5, Fall 2005 Final Q5, Fall 2005 Midterm Q7
📗 Answer: .
📗 [4 points] Suppose \(n\) = witnesses heard a gunshot near 221B Baker Street. The benefit from at least one witness calling the police is \(b\) = and the cost of calling the police is \(c\) = . If no witness calls the police, everyone gets 0. In a Nash equilibrium in which every witness uses the same mixed strategy, what is the probability that no one calls the police?
Hint
This is the Volunteer's Dilemma. Suppose every other witness call the police with probability \(p\), then calling the police lead to net benefit \(b - c\) and not calling the police lead to net benefit \(0 \cdot \left(1 - p\right)^{n - 1} + b \cdot \left(1 - \left(1 - p\right)^{n - 1}\right)\), where \(\left(1 - p\right)^{n - 1}\) is the probability that no one else calls the police. A witness would use a mixed strategy only if the net benefits from calling and not calling are equal to each other, and this can be used to set up a equation to solve for \(p\).
📗 Answer: .
📗 [2 points] You performed PCA (Principal Component Analysis) in \(\mathbb{R}^{3}\). If the first principal component is \(u_{1}\) = \(\approx\) and the second principal component is \(u_{2}\) = \(\approx\) . What is the new 2D coordinates (new features created by PCA) for the point \(x\) = ?
📗 In the diagram, the black axes are the original axes, the green axes are the PCA axes, the red vector is \(x\), the red point is the reconstruction \(\hat{x}\) using the PCA axes.
Hint
See Fall 2018 Midterm Q13, Fall 2017 Final Q10. Coordinate \(i\) is given by the projection of \(x\) onto the principal component \(v_{i}\). If the principal component is a unit vector \(u_{i}\), use the simplified formula: \(u_{i^\top} x\); otherwise, use the formula: \(\dfrac{v_{i^\top} x}{v_{i^\top} v_{i}}\).
📗 Answer (comma separated vector): .
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