Assume the following game situation: If Player A plays UP and Player B plays LEFT then Player A gets $2 and Player B gets $4. If Player A plays UP and Player B plays RIGHT then Þlayer A gets $3 and Player B gets $6. If Player A plays DOWN and Player B plays LEFT then Player A gets $5 and Player B gets $2. If Player A plays DOWN and Player B plays RIGHT then Player A gets $1 and Player B gets $1. What is the mixed strategy expected payout for Player A?
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- Assume the following game situation: If Player A plays UP and Player B plays LEFT then Player A gets $1 and Player B gets $3. If Player A plays UP and Player B plays RIGHT then Player A gets $2 and Player B gets $5. If Player A plays DOWN and Player B plays LEFET then Player A gets $4 and Player B gets $2. If Player A plays DOWN and Player B plays RIGHT then Player A gets $1 and Player B gets $1 What is the Mixed Strategy Equilibrium for Player B? O. (LEFT, RIGHT) = (1/8, 3/8) O. (LEFT, RIGHT) = (1/4, 3/4) O. (LEFT, RIGHT) = (1/2, 1/2) O. (LEFT, RIGHT) = (3/8, 1/8)Assume the following game situation: If Player A plays UP and Player B plays LEFT then Player A gets $1 and Player B gets $2. If Player A plays UP and Player B plays RIGHT then Player A gets $4 and Player B gets $3. If Player A plays DOWN and Player B plays LEFT then Player A gets $2 and Player B gets $4. If Player A plays DOWN and Player B plays RIGHT then Player A gets $3 and Player B gets $1. What is the probability that Player A will play UP and Player B will play LEFT? 3/8 O 1/8 O 1/4 O 1/2Assume the following game situation: If Player A plays UP and Player B plays LEFT then Player A gets $2 and Player B gets $4. If Player A plays UP and Player B plays RIGHT then Player A gets $3 and Player B gets $6. If Player A plays DOWN and Player B plays LEFT then Player A gets $5 and Player B gets $2. If Player A plays DOWN and Player B plays RIGHT then Player A gets $1 and Player B gets $1. What is the Mixed Strategy Equilibrium for Player A? O (UP. DOWN) - (2/5, 3/5) O (UP, DOWN) = (1/3, 2/3) O (UP, DOWN) = (7/8, 1/8) O (UP, DOWN) - (1/8, 7/8)
- Q3. Imagine that two partners are trying to divide up $66. Imagine that this game is played only once and that player 1 gets to decide how to divide the $66. Upload a picture of the game tree. You only need to include a minimum of 5 branches.. In a gambling game, Player A and Player B both have a $1 and a $5 bill. Each player selects one of the bills without the other player knowing the bill selected. Simultaneously they both reveal the bills selected. If the bills do not match, Player A wins Player B's bill. If the bills match, Player B wins Player A's bill. a. Develop the game theory table for this game. The values should be expressed as the gains (or losses) for Player A. b. Is there a pure strategy? Why or why not? c. Determine the optimal strategies and the value of this game. Does the game favor one player over the other? d. Suppose Player B decides to deviate from the optimal strategy and begins playing each bill 50% of the time. What should Player A do to improve Player A’s winnings? Comment on why it is important to follow an optimal game theory strategy.2 Consider Anna and Joe, who value a certain good by the same amount v and can choose to either contribute (C) e to get the good or not (N) where v > e> 0. Obtaining the good only requires c from one person. The game is summarized in the payoff table below: Joe V-c,V-c V-c,V v,V-c 0,0 Find a pure strategy equilibrium and a mixed strategy equilibrium. Anna O Z
- Suppose you have $35,000 in wealth. You have the opportunity to play a game called "Big Bet/Small Bet." In this game, you first choose whether you would like to make a big bet of $15,000 of a small bet of $5,000. You then roll a fair die. If you roll a 4, 5, or 6, you win the game and earn $15,000 for the big bet or $5,000 for the small bet. If you roll a 1, 2, or 3, you lose and lose $15,000 for the big bet and $5,000 for the small bet the game Utility U₂ U₁ BEL 0 11 LATE EE ARTE Are the Small Bet and Big Bet considered fair bets? O Big Bet is fair, but Small Bet is not. No, both are not fair. Yes, both are fair. 20 OSmall Bet is fair, but Big Bet is not. G HA 1 35 D E 1 1 1 1 1 F 1 U 50 Income (thousands of dollars)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…Consider the following sequential game: Player 1 plays first and can choose between L and R. Player 2 plays second and can choose between U and D. • If player 1 chooses L, player 2 does not get to play and both players get £1. If player 1 plays R, player 2 gets to play. If player 2 plays U then both players get £2. If player 2 plays D then player 2 gets £3 and player 1 gets £0. What the the outcome of the subgame perfect equilibrium? O a. £3-£0 O b. £1-£1 O c. This game does not have a subgame-perfect Nash-equilibrium. O d. £0-£3 O e. £2-£2
- 6 Two people will select a policy that affects both of them by applying a "veto" in a sequential and alternate manner, that is: person 1 begins to veto a policy and then person 2 exercises his "veto" with the remaining policies; the process repeats until only one policy remains. Assume that there are 3 policies: X,Y,Z, and that person 1 prefers X to Y to Z and person 2 prefers Z to Y to X. a. Represents the game extensively b. Give the number of subgames C. Indicate the total strategies of the players d. find all subgame perfect nash equilibria e. Find all Nash Equilibriums.6. The owner of an antique piece of furniture is looking to sell their good to a known buyer. The seller has a reservation value r whereas the buyer has valuation v > r. Suppose the buyer incurs a one-off transportation cost of t from travelling to the seller to purchase the good. Assume that r, vand t are known to both parties. The game proceeds in two stages: First, the buyer decides whether or not to travel to the seller's location in order to purchase. Second, if they travel then the seller makes a take-it-or-leave-it offer (ultimatum) of price p to the buyer, which the buyer then can either accept or reject. (a) Represent this game in extensive form (b) Find the seller's optimal offer in Stage 2 of this game given that the buyer has already travelled. Is it accepted or rejected? (c) Find a Subgame Perfect Equilibrium of this game. (d) Are there any Nash equilibria which are not subgame perfect? Give an example if one exists. (e) Suppose the seller could offer free delivery at a…The following table contains the possible actions and payoffs of players 1 and 2. Player 2 Cooperate Player 1 Cooperate Not Cooperate 15, 15 20, -10 Not Cooperate -20, 20 10, 10 This game is infinitely repeated, and in each period both players must choose their actions simultaneously. If both players follow a tit-for-tat strategy, then they can Cooperate in equilibrium if the interest rate r is ✓. At an interest rate of r=0.5, If instead of playing an infinite number of times, the players play the game only 10 times, then in the first period player 1 receives a payoff of