📗 [4 points] In GoogSoft, software engineers A and B form a two-person team. Their year-end bonus depends on their relative performance. The bonus outcomes are summarized in the following table. The value of slacking to each person is \(s\) = . The total payoff to each person is the sum of the bonus and the value from slacking. What is the smallest value of \(x\) such that both players will work hard in a Nash equilibrium?

📗 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.
📗 Answer: . Calculate

📗 [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 Breadth First Search? Include both the initial and goal states.
📗 Note: use the convention used in the lectures, enqueue the states with smaller index into the queue first.
📗 Answer: . Calculate

📗 [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?

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

📗 [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] Consider a zero-sum sequential move game with Chance. Max player moves first, then Chance, then Min. 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)?

📗 Answer: . Calculate

📗 [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?

📗 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] 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] 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 \(\mu = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} x_{i}\).

📗 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 .
📗 Answer: . Calculate

📗 [2 points] Alice, Bob and Cindy go to the same school and live on a straight street lined with evenly spaced telephone poles. Alice's house is at the pole , Bob's is at the pole , Cindy's is at the pole . Where should the school set up a school bus stop so that the sum of distances (from house to bus stop) walked by the three students is minimized?
📗 Answer: . Calculate

📗 [3 points] Consider a variant of the II-nim game. There are two piles, each pile has \(n\) = sticks. A player can take one stick from a single pile; or take two sticks, one from each pile (when available). The player who takes the last stick wins. Let the game value be 1 if the first player wins (and -1 if the second player wins). What is the game theoretical value of this game?
📗 Answer: . 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: .