Find the optimal strategies, P and Q, for the row and column players, respectively. -5 3 -8 P = Q = Compute the expected payoff E of the matrix game if the players use their optimal strategies. (Round your answer to two decimal places.) E =
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- Two athletes of equal ability are competing for a prize of $10,000. Each is deciding whether to take a dangerous performance enhancing drug. If one athlete takes the drug, and the other does not, the one who takes the drug wins the prize. If both or neither take the drug, they tie and split the prize. Taking the drug imposes health risks that are equivalent to a loss of X dollars. a) Draw a 2×2 payoff matrix describing the decisions the athletes face. b) For what X is taking the drug the Nash equilibrium? c) Does making the drug safer (that is, lowering X) make the athletes better or worse off? Explain.Problem 3. Consider the following two-player game. Player 1 (the row player) has three strategies {U, F, D}, and Player 2 (the column player) has four strategies {L, M, N, R}. In each cell, the first number is the pay-off of Player 1, and the second is that of Player 2. There is an unknown x in the table. Consider pure strategies only. U F D L (0, 3) (6,0) (3, 1) M (1,0) (5, 1) (4,2) N (2,5) (x, 2) (1, 1) R (4,2) (5, 1) (1, 3) (1) Find a number x such that {F, N} is the unique Nash equilibrium. (2) Find a number x such that the game has more than one Nash equilibria. What are these Nash equilibria in this case? (3) Is it possible to find a number x such that the game has no Nash equilibrium? For each of your answers above, an explanation needs to be provided.Consider a game between player A with a choice between moves d₁, d2 and d3, and player B with a choice between 81, 82 and 83, and the following pay-off matrix: 81 82 83 d₁ (3,2) (2,0) (1,-1) d2 (1,0) (3,1) (2,-2) d3 (0,0) (2,3) (1,2) Is that game solvable? (We consider strong Pareto optimality for that question.) Oa. The game is solvable in the strict sense, with solutions (d₁, 1, (d2, 62) O b. The game is solvable in the strict sense, with solution (d₁, 81) О с. The game is solvable in the completely weak sense, with solutions (d₁, 1), (d2, 82) The game is solvable in the strict sense, with solutions (d1, 61), (d3, 82), (d2, 82) Od. O e. The game is solvable in the completely weak sense, with solutions (d₁, 81), (d3, 82), (d2, 62), (d3, 83) O f. The game is not solvable Og. The game is solvable in the completely weak sense, with solutions (d₁, 61), (d3, 62), (d2, 82) Oh. The game is solvable in the strict sense, with solutions (d1, 81), (d3, 82)
- Now answer the following questions: Perform iterative elimination of dominated strategies on the game. Draw the new table for each iteration as shown in the lecture video. Write corresponding actions at left and up of the table. Explain why are you removing each row or column. You should reach a 2*2 game. Rewrite the table separately. From the 2*2 game you get in question 1, find a mixed strategy σ1=(p,1−p) for Player 1 that will make Player 2 indifferent about two actions. Find a mixed strategy σ2=(q,1−q) for Player 2 that will make Player 1 indifferent about two actions. If the players are playing the mixed strategies σ1 and σ2, they both will indifferent about their strategies, hence the strategy profile (σ1,σ2) will be a Nash Equilibrium of this game. Find the expected utility for both Player 1 u1(σ1,σ2) and Player 2 u2(σ1,σ2) at this Nash Equilibrium.You are given the payoff matrix below. B1 B2 B3 A1 1 Q 6 A2 P 5 10 A3 6 2 3 a. Determine the range of values of P and Q in order for Player A to choose strategy A2 and PlayerB to choose strategy B2.b. Solve the problem in the perspective of the opponent. Are the ranges the same?c. What are the values of the game for (a) and (b)?Consider the Stackelberg game depicted below in which you are the row player. R U 4,0 1,2 3,2 0,1 0,0 2,0 You may choose whether you want to be the leader (and commit to a possibly mixed strategy) or the follower in a game against the course staff (column player). You may trust that we maximize our expected payoff. The points awarded to you will equal one half of the expected payoff you obtain. If you want to be the leader, please submit your commitment strategy. For example, if you want to commit to [0.5: U, 0.2: M, 0.3: D], then submit: 0.5 0.2 0.3 If you want to be the follower instead, just submit: F
- What is the payoff to player 2 under the strategy profile (AK,D,FL) in this game?What is the payoff to player 3 under the strategy profile (BK,C,FM) in this game?Consider the following Guessing Game. There are n = 10 players simultaneously choosing a number in {1, 2, 3}. The winners are those closest to 1/2 the average guess (they evenly split the prize between the winners if there is more than one). Find the set of rationalizable strategy profiles. Justify your answer. please no handwriting and this course about game theory (topic Rationalizability) answer with all steps, pleaseTwo athletes of equal ability are competing for a prize of $12,000. Each is deciding whether to take a dangerous performance-enhancing drug. If one athlete takes the drug and the other does not, the one who takes the drug wins the prize. If both or neither take the drug, they tie and split the prize. Taking the drug imposes health risks that are equivalent to a loss of XX dollars. Complete the following payoff matrix describing the decisions the athletes face. Enter Player One's payoff on the left in each situation, Player Two's on the right. Player Two's Decision Take Drug Don't Take Drug Player One's Decision Take Drug , , Don't Take Drug , , True or False: The Nash equilibrium is taking the drug if X is greater than $6,000. True False Suppose there was a way to make the drug safer (that is, have lower XX). Which of the following statements are true about the effects of making the drug safer? Check all that…
- Design the payoffff matrix of a game with no Nash Equilibria. The game should have 2 players, 2 strategies for each player, and the payoffffs for each player should be either 0 or 1.Game theory: Consider a collective action game with thirty individuals (N = 30). When the number of participants in the joint project is n, each individual, including shirkers, receives a benefit of B(n) = 18n and each participant incurs a cost of C(n) = 32 − 2n. 1. Find all of the Nash equilibria, both stable and unstable ones. 2. Find the socially optimal outcome. 3. Check if any of the Nash equilibria is socially optimal. Explain your answer.For Question 13, consider a game in which the "Best Response Arrows" are as illustrated below. Further, payoffs for each player from the strategy choices of "E/G" and "F/H" are also indicted: if Player 1 chooses E and Player 2 chooses G, then both players get a payoff of x>0; if Player I chooses F and Player 2 chooses H, then both players get a payoff of y> 0. 13. Player 1 E F Player 2 H #E ↓ G x,x y.y Based upon the Best Response Arrows and payoffs indicated above, this game A. does not fit the definition of a Prisoner's Dilemma for any values of x and y. B. fits the definition of a Prisoner's Dilemma for all values of x and y. C. fits the definition of a Prisoner's Dilemma for x y.