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📗 [3 points] Let the search space be integers. Each state \(n\) has successors . Write down the shortest path (i.e. the sequence of states) from the initial state 1 to the goal state .
Hint
See Fall 2018 Midterm Q2, Fall 2017 Final Q13. Suppose the successors are \(a n, a n + 1, ...\), then start from the goal state \(g\), look at the remainder when \(g\) is divided by \(a\) to figure out which integer is its "predecessor" and do this repeatedly until the initial state is reached.
📗 Answer (comma separated vector): .
📗 [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: .
📗 [3 points] Consider Iterative Deepening Search on a tree, where the nodes are denoted by numbers. Write down the sequence IDS visited in the order they are expanded (i.e. expansion path). 0 is the initial state and is the goal state. Start with depth limit 0, include the root, and include repeated nodes.
📗 Note: use the convention used in the lectures, push the rightmost successor into the stack first (i.e. expand the leftmost successor first).
Hint
See Fall 2018 Midterm Q5, Fall 2006 Final Q1. Start with 0, then DFS on the tree with depth 1, then DFS on the tree with depth 2, ...
📗 Answer (comma separated vector): .
📗 [3 points] Four individuals (i.e. candidate solutions) in the current generation are given by -digit ( dimensional) sequences: . Individual 1: ; Individual 2: ; Individual 3: ; Individual 4: . The fitness function is . What is the result of performing 1-point crossover for the sequences with the highest fitness (break ties by preferring the sequence that appears earlier in the list) with a cross-point between digit and digit .
📗 Note: the first line representing the first child should start with the sequence with the highest fitness, and the second line representing the second child should start with the sequence with the second highest fitness.
Hint
See Spring 2017 Midterm Q2. Compute the fitness of all four sequences and find the two with the highest fitness say \(d^{\left(1\right)}, d^{\left(2\right)}\). The crossover of the two sequences between digits \(i\) and \(i + 1\) are \(d^{\left(1\right)}_{1}, ..., d^{\left(1\right)}_{i}, d^{\left(2\right)}_{i+1}, ..., d^{\left(2\right)}_{6}\) and \(d^{\left(2\right)}_{1}, ..., d^{\left(2\right)}_{i}, d^{\left(1\right)}_{i+1}, ..., d^{\left(1\right)}_{6}\).
📗 Calculator: .
📗 Answer (matrix with 2 lines, each line is a comma separated vector): .
📗 [4 points] Enter the largest integer value of \(A\) such that \(B\) will be alpha-beta pruned? Min player moves first. In the case alpha = beta, prune the node. Enter 100 if you think the answer is 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. 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] 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?
Row \ Col
L
R
U
D
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): .
📗 [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: .
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