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.What is the probability of getting a king if a card is drawn from a pack of 52 cards?
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- 3. Suppose we play the following game. I give you $100 for your initial bankroll. At each time n, you decide how much of your current wealth to bet. You cannot borrow money. You can only play with the money I gave you in the beginning or any money that you have won so far. The game is simple. At each time n ≥ 1, you decide the amount to bet. I will roll a fair die. If the die comes up 1,2,3,..., or 5, you win; if the die comes up 6, then you lose. IOW, if you bet $10 on the first roll, you will either have $90 or $110 after the first roll. (a) Suppose you wish to maximize your profit on the first roll. How much should you bet? (Most of you will get this wrong.) (b) What is the expected profit on the first roll if your bet is b with 0 ≤ b ≤ 100? (c) Suppose you wish to maximize your expected profit on the first roll. How much should you bet? (d) Suppose you wish to maximize your expected profit betting on the nth roll. How much of your current wealth do you bet? (e) Let X₂, be your…2. Consider a cheap talk game in which Nature moves by choosing a sender's type, where the type space has four elements: −1, 1, 2, and 3, each occur- ring with equal probability of 1½. The sender learns his type and chooses one of three possible messages: bumpy, smooth, and slick. The receiver observes the sender's message and then chooses one of three actions: 0, 5, and 10. The sender's payoff equals his type multiplied by the receiver's action. The receiver's payoff equals the sender's type multiplied by twice the receiver's payoff. a. Find a separating perfect Bayes-Nash equilibrium. b. Find a semiseparating perfect Bayes-Nash equilibrium.16. Consider a stag-hunt game with payoffs Bob Bob stag rabbit Alice, stag 5, 5 0,4 Alice, rabbit 4,0 4,4 The most likely experimental outcome in this game should be (A) (stag, stag); (B) (rabbit, rabbit); (C) (stag, rabbit); (D) (rabbit, stag).
- Consider the following normal form game Player 2 L C R Player 1 5,4 4, 5 3,8 8,3 8,0 8,8 5,5 6,3 0,8 UMD7. Suppose the only game in town involves flipping a fair coin (so Heads and Tails are equally likely), with a $x bet. If Heads comes up, the payoff is $0.9x; if Tails comes up, you lose the $x. You have $10,000, and must win at least $5,000 by tomorrow morning to pay off a debt to a mean dude. a. Compute the likelihood of winning at least $5000 by making a single bet of $10,000. b. Compute the likelihood of winning at least $1000 by playing the game 10,0000 times and betting a dollar each time. What is the likelihood of not losing money? Message learned?1. Suppose that you and all of your classmates taking the exam now are all invited to pick an integer between 0 and 100. The average of all these guesses, say x, will be calculated, and the winner of this game will be the one who picks the number that is closest to 2. The winner will receive 10 bonus points for the final exam. Which number would you pick?
- 3. Find the saddle point, if it exists, for the following game. (b) Solve the following game by using the principle of dominance and find the probabilities of strategies for each player and the value of the game. Player B Player A II III IV V 3 4 4 II 2 4 III 4 4 IV 4 4 20 2420 8760QUESTION 9 y a 3, 5 4, 2 4, 1 2, 3 b 2, 4 6, 6 5, 1 1, 2 1,3 7,3 4, 2 3, 7 d 1, 2 5, 2 3, 4 4, 1 How many rationalizable outcomes are there in this game? Give your answer as a whole number. C.. Ayça and Barış are playing a game and following payoff matrix is for the payoffs of Ayça. Answer the questions according to the following payoff matrix. a) What is the probability that the value of the game is 10?
- 4. At the same carnival, another game is played by flipping the same weighted coin in #3 and rolling a fair dice (that is equally likely to land on each of the six sides). The player wins if the coin lands on heads and the dice lands on an odd number. What is the probability that a player wins this game? Show the model and/or calculation(s) that lead to your answer. A. 3/20 B. 1/4 C. 7/20 D. 7/10What is Ann's maximin strategy? Game Bob R Ann U 10,-1 4, 4 D 4, 1 8, -1 Select one: a. none of the other answers b. 2/5 U+ 3/5 D c. 3/5 U + 2/5 D d. 4/7 U + 3/7 D e. 2/7 U + 5/7 D1. What is the expected value of playing this game ? a. The player repeatedly rolls a six - side d die until it comes up 6. b. When the die comes up 6 , the player receives a payoff, and the game ends. The player get s precise ly one payof f. c. The initial payoff is $6, and it increases by 20%. The f irst roll payoff is $6 ; the second roll payoff is $7.20 (1.2*$6) ; the third roll payoff is $8.64 (1.2*$7.20), and so on. What is the expected value of playing this game?