6. (10 points) In a competition between players X and Y, the first player to win...
a row or a 6. (10 points) In a competition between players X and Y, the first player to win three games total of four games wins. Draw a tree and determine in how many ways the competition can be played if X wins the first game and Y wins the second and third games?
Problem 3. In the game of tennis, the first player to win four points wins the game as long as the winner's total is at least two points more than the opponent. Thus if the game is tied at 3-3(Deuce"), then the game is not decided by the next point, but must go on until one player has two points more than the opponent's score. Assume that the server has a constant probability p of winning each point, independently of...
First part: Consider the following two-player game. The players simultaneously and independently announce an integer number between 1 and 100, and each player's payoff is the product of the two numbers announced. (a) Describe the best responses of this game. How many Nash equilibria does the game have? Explain. (b) Now, consider the following variation of the game: first, Player 1 can choose either to "Stop" or "Con- tinue". If she chooses "Stop", then the game ends with the pair...
explain the answer with clear handwriting please Questions: 1. [10 pts] There are two players A and B. The probability that A wins a game is 0.4, and the probability that B wins a game is 0.6. We randomly choose one player between A and B and let the chosen player play 10 independent identical games. An amount Xin dollars) is rewarded to each win, where X has a continuous uniform distribution over (0,60). Let X be the total amount...
Question 9 [10 points] Bob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 90% of their games. To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as...
Question 9 [10 points] Bob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 70% of their games To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as...
Two players take turns tossing a fair, six-sided die. The first player to roll a 6 wins the game. Determine the probability that the player who wins will be the one who tossed first.
Problem 4. (20 points) Pebbles Game. Two players play a game, starting with three piles of n pebbles each. Players alternate turns. On each turn, a player may do one of the following: take one pebble from any pile take two pebbles each from two different piles The player who takes the last pebble(s) wins the game. Write an algorithm that, given n, determines whether the first player can "force" a win. That is, can the first player choose moves...
X and Y play a series of games. X has a probability p of winning each game. To win the game, the player has to win 2 more games than the other first. (a) Find the probability that X is the overall winner. (b) Find the expected number of games played.
Q1. Suppose two players, First and Second, take part in a sequential-move game. First moves first, Second moves second, and each player moves only once. (a). Draw a game tree for a game in which First has two possible actions (Up or Down), and Second has three possible actions (Top, Middle, or Bottom) at each node. Show which nodes are terminal/decision and write down all the (pure) strategies of each player. (b). Draw a game tree for a game in...