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📗 [3 points] There are \(n\) = cookies. The brother first proposes a division of these cookies into two piles (two integers adding up to \(n\)) and then the sister take one of the two piles. Both the brother and the sister want to maximize the number of cookies they take. What is the value of the game to the brother (measured by the number of cookies he gets)? Enter an integer.
📗 Answer: .
📗 [4 points] What is the value of the solution of the following game (two values one for player 1 one of player 2)? In case of tie, the players will choose down (D).
📗 Note: in case the diagram is not clear: two players take turn and choose either to go right or go down, the payoffs from going down in periods \(1, 2, 3, 4\) are , , , , respectively, and the payoff from going right 4 times is .
📗 Answer (comma separated vector): .
📗 [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)?
📗 Note: in case the diagram is not clear, the probabilities from left to right is: , and the rewards are .
📗 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: .
📗 [4 points] Consider a zero-sum sequential move game with Chance. player moves first, then Chance, then . 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: .
📗 [2 points] Consider a game board consisting of bits initially at . Each player can simultaneously flip any number of bits in a move, but needs to pay the other player one dollar for each bit flipped. The player who achieves wins and collects dollars from the other player. What is the game theoretic value (in dollars) of this game for the first player?
📗 Note: "game theoretic value" is what we called "value of the game" in the lectures.
📗 Answer: .
📗 [3 points] Consider the following game. There are piles, each pile has sticks. A player can take one stick from a single pile, or she may take two sticks, one for each pile. The player who takes the last stick loses. Let the game theoretical value be 1 if the first player wins. What is the value of the game?
📗 Answer: .
📗 [1 points] kids are wearing either green or red hats at a party: they see every other kid's hat but not their own.
📗 Dad said to everyone: At least one of you is wearing a green hat.
📗 Dad asked: Do you know the color of your hat?
📗 Every kid said: No.
📗 Dad asked again: Do you know the color of your hat?
📗 Every kid said: No.
📗 Dad asked again: Do you know the color of your hat?
📗 Some kids (at least one) said: Yes.
📗 How many kids are wearing green hats?
📗 Answer: .
📗 [1 points] Two players, A and B, objects, each player can pick 1 or 2 each time. Pick the last object to win. If you pick first, how many should you pick?
📗 In the diagram, the node has the player name (or G for end of game) and the subscript means how many objects are left. The terminal nodes where player A wins are marked green, and where player B wins are marked blue.
📗 Answer: .
📗 [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: .
📗 [3 points] Write down the matrix normal form of the following game.
📗 Answer (matrix with multiple lines, each line is a comma separated vector): .
📗 [1 points] pirate got gold coins. Each pirate takes a turn to propose how to divide the coins, and all pirates who are still alive will vote whether to (1) accept the proposal or (2) reject the proposal, kill the pirate who is making the proposal, and continue to the next round. Use strict majority rule for the vote, and use the assumption that if a pirate is indifferent, they will vote reject with probability 50 percent. How will the first pirate propose? Enter a vector of length , all integers, sum up to .
📗 Answer (comma separated vector): .
📗 [4 points] Given the following BoS (Battle of the Sexes) game, consider a sequential move game with the same payoffs (values), in which with probability \(p\) = , Romeo (row player) moves first, and with probability \(q\) = , Juliet (column player) moves first. What is the expected (game theoretic) value of the game (two values one for Romeo and one for Juliet)?
Actions
B
S
B
\(\left(0, 0\right)\)
S
\(\left(0, 0\right)\)
📗 Note: this is a game with Chance moving first, and Romeo second Juliet third in one branch, and Juliet second Romeo third in the other branch. In case of ties, the players will choose B instead of S.
📗 Answer (comma separated vector): .
📗 [4 points] For a zero-sum game in which moves first and if the action Left is chosen, then Chance (Chn) moves Left with probability \(p\) and Right with probability \(1 - p\), and if the action Right is chosen, then Chance moves Left with probability and Right with probability . Suppose the player who moves first uses a mixed strategy \(\dfrac{1}{2}\) Left and \(\dfrac{1}{2}\) Right in a solution, what is the value of \(p\)? If it's impossible, enter \(-1\).
📗 Note: in case the diagram is not clear, the values on the leafs (each sub-branch is a row): .
📗 Answer: .
📗 [3 points] There are two players and \(k\) coins on the table. Players move sequentially with player 1 moving first. Each player chooses to take either one or two coins from the table. The player who takes the last coin wins. For which of the following values of \(k\) = {} does the first player has a winning strategy? Enter the values of \(k\), not the indices.
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