📗 [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?

📗 Answer: . Calculate

📗 [4 points] There are people living in the suburbs and all of them commute to work in the city. Every morning, each individual decides which way to drive to the city simultaneously: the Direct Way or the Long Way. The Long Way takes 1 hour of driving. The time spent on the Direct Way depends on the traffic is equal to \(\dfrac{n}{c}\) hours, where \(n\) is the total number of cars taking the Direct Way, and \(c\) = is the capacity. Each individual wants to minimize the driving time, and break ties by choosing the Direct Way. What is the number of people taking the Long Way in the Nash equilibrium?

📗 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 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).

📗 Answer: . Calculate

📗 [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?

📗 Answer (comma separated vector): . Calculate

📗 [4 points] Consider the following zero-sum game tree. MAX 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. 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 .

📗 Answer: . Calculate

📗 [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] 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): .

📗 [2 points] A search tree has levels (the root is at level 0, a tree with only the root has 0 levels), and every internal node has children. Suppose there is no goal node. How many goal checks (we perform a goal check every time we expand a node) will depth first search perform? Include the initial node (the root).

📗 Answer: . 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

📗 [3 points] In the following graph coloring problem, each node is either labeled as + or -. The score of the graph is the number of edges connecting two nodes with the same label (color). We are minimizing the score. If the successor function is to change the label of a single node, in hill climbing (here, valley finding), which node should we change in the following graph? Enter the index of the node (subscript in the diagram) or -1 if we are at a local minimum. Break ties by entering the node with the smaller index.

📗 Answer: .

📗 [3 points] \(N\) = firms sharing the use of a river decide whether to filter (F) or release (R) pollutants (a poisonous substance) into the river. If \(n\) firms choose to pollute the river (R), each of these \(n\) firms incurs a cost of dollars, and each of the remaining firms that choose to install filters (F) incurs a cost of (cost due to pollution plus the cost of the filter). Every firm wants to minimize costs. What is the number of firms that choose to install filters (F) in a pure strategy Nash equilibrium? Note: remember to enter an integer. 

📗 Answer: . Calculate

📗 [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: .