📗 [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.

Hint

IDS repeats a smaller number of state expansions for trees with a smaller depth. A tree with the minimum possible depth containing all \(n\) states has depth \(1\).

📗 Answer: . Calculate

📗 [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".

Hint

See Fall 2019 Midterm Q1, Fall 2018 Midterm Q6, Fall 2017 Midterm Q10. Since \(h_{1}\) is admissible, for any \(n\), \(0 \leq h_{1}\left(n\right) \leq h^\star\left(n\right)\). It means if \(0 \leq h\left(n\right) \leq h_{1}\left(n\right)\), then it must be true that \(0 \leq h\left(n\right) \leq h^\star\left(n\right)\), i.e. \(h\) is always admissible. Since \(h^\star\) can be arbitrarily large, only when \(h\left(n\right) < 0\) for some \(n\) or when \(h\left(0\right) \neq 0\), \(h\) is never admissible.

📗 Choices:
(1)
(2)
(3)
(4)
(5)
(6)
(7) None of the above
📗 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".

Hint

See Fall 2019 Midterm Q1, Fall 2018 Midterm Q6, Fall 2017 Midterm Q10. Since \(h_{1}\) is admissible, for any \(n\), \(0 \leq h_{1}\left(n\right) \leq h^\star\left(n\right)\). It means if \(0 \leq h\left(n\right) \leq h_{1}\left(n\right)\), then it must be true that \(0 \leq h\left(n\right) \leq h^\star\left(n\right)\), i.e. \(h\) is always admissible. Since \(h^\star\) can be arbitrarily large, only when \(h\left(n\right) < 0\) for some \(n\) or when \(h\left(0\right) \neq 0\), \(h\) is never admissible.

📗 Choices:
(1)
(2)
(3)
(4)
(5)
(6)
(7) None of the above
📗 Answer (comma separated vector): .

📗 [2 points] Consider the following version of hill climbing: at initial state \(s\) we randomly choose one of \(s\)'s neighbors with equal probability. If the chosen neighbor has a strictly better score than \(s\) we move to the neighbor; otherwise we stay at \(s\). Assume \(s\) has neighbors, and only of the neighbors has a strictly better score than \(s\). What is the chance that we move out of \(s\) in iterations or less?

Hint

See Fall 2016 Midterm Q6. The probability of moving out is 1 minus the probability of staying (seeing a worse neighbor) in all \(T\) iterations.

📗 Answer: . Calculate

📗 [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.

Hint

See Fall 2018 Midterm Q7, Fall 2013 Final Q16. (1) Suppose the first player is max. \(B\) will be pruned if the minimum-so-far before reaching node \(B\) for the min player is already smaller than \(A\). If that is the case, the max player will select the \(A\) branch and not continue searching the min subtree on the right (i.e. \(B\) will be pruned.) (2) Suppose the first player is min. \(B\) will be pruned if the maximum-so-far before reaching node \(B\) for the max player is already larger than \(A\). If that is the case, the min player will select the \(A\) branch and not continue searching the max subtree on the right (i.e. \(B\) will be 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)?

Hint

See Fall 2019 Q14 Q15, Spring 2017 Midterm Q3, Fall 2006 Final Q5, Fall 2005 Final Q5, Fall 2005 Midterm Q7

📗 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?

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.

Hint

Draw the best response diagrams: the region where the two curves intersect are Nash equilibria (since those mixed strategies are best responses to each other). Find the smallest and largest values of \(p\) or \(q\) in that region. The way to compute the location where the blue curve is vertical (or the red curve is horizontal) is by setting the expected payoffs from U and D equal to each other (or from L and R equal to each other).

📗 Answer (comma separated vector): . 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?

Hint

📗 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: .