Consider the payoff matrix listed below: |1, -1 3, 0 2, 1 |R |0, 3 |1, 2 |0, 0 3, 1 5, 3 2, 1 What is the best response to "R" a. U b. C C. B d. U, C
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- A strategy for player 1 is a value for x1 from the set X. Similarly, a strategyfor player 2 is a value for x2 from the set X. Player 1’s payoff is V1(x1, x2) =5 + x1 - 2x2 and player 2’s payoff is V2(x1, x2) = 5 + x2 - 2x1.a. Assume that X is the interval of real numbers from 1 to 4 (including 1and 4). (Note that this is much more than integers and includes such numbers as 2.648 and 1.00037). Derive all Nash equilibria.b. Now assume that the game is played infinitely often and a player’s payoff is the present value of his stream of single-period payoffs, where dis the discount factor.(i) Assume that X is composed of only two values: 2 and 3; thus, aplayer can choose 2 or 3, but no other value. Consider the followingsymmetric strategy profile: In period 1, a player chooses the value 2. In period t(≥2), a player chooses the value 2. In period a player chooses the value 2 if both players chose 2 in all previous periods; otherwise, she chooses the value 3. Derive conditions which ensure…At a company, 20 employees are making contributions for a retirement gift.Each person is choosing how many dollars to contribute from the interval[0,10]. The payoff to person i is bi X xi - xi, where bi > 0 is the “warm glow”he receives from each dollar he contributes, and he incurs a personal cost of 1.a. Assume bi < 1 for all i. Find all Nash equilibria. How much is collected?b. Assume bi > 1 for all i. Find all Nash equilibria. How much is collected?c. Assume bi = 1 for all i. Find all Nash equilibria. How much is collected?Now suppose the manager of these 20 employees has announced that shewill contribute d > 0 dollars for each dollar that an employee contributes.The warm glow effect to employee i from contributing a dollar is now bi X(1 + d) because each dollar contributed actually results in a total contribution of 1 + d. Assume bi = 0.1 for i = 1, . . . , 5; bi = 0.2 for i = 6, . . . , 10; bi = 0.25 for i = 11, . . . , 15; and bi = 0.5 for i = 16, . . . , 20.d. What…Push Not Push 4,2 2,3 Not 6,-1 0,0 Would any other solution other than Nash Equilbrium approach work? Why?
- Consider the game in the image attached, which is infinitely repeated at t = 1, 2, ... Both players discount the future at rate: delta E(0, 1). The stage game is in the image attached. Suppose that the players play (C,C) in period t = 1, 3, 5, ... and plays (D,D) in period t = 2, 4, 6,... Compute the discounted payoff of each player.Consider the following game 1\2 Y Z A 10,3 3,9 B 8,5 6,1 Suppose Player 2 holds the following belief about Player 1: θ1 (A,B) = (9/10,1/10) What is the expected payoff from playing ‘Y’ ? What is the expected payoff from playing ‘Z’ ? Based on these beliefs, player 2 should respond by playing _____Assume that both Apple and Samsung have a marginal cost of 20. Apple’spayoff function is VA(PA, PS) = (PA - 20)(100 - 2PA + PS),whereas Samsung’s payoff function is VS(PA, PS) = (PS - 20)(100 - 2PS + PA).Find all Nash equilibria.
- 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 game Ms. Bennet and Mr. Darcy play in ‘First Impressions’, Selected Set V. Suppose that Ms. Bennet prefers to meet Mr. Darcy (a = 0) with probability p. Further suppose that: - The ‘meeting Ms. Bennet’ plays Ball with probability q (and Dinner with probability 1 − q); - ‘avoiding Ms. Bennet’ plays Ball with probability r (and Dinner with probability 1 − r); M - r. Darcy plays Ball with probability s (and Dinner with probability 1 − s). Write down the strategic form game and find for all values of p ∈ (0, 1) the Bayesian-Nash equilibria in mixed strategies.Translate the following monetary payoffs into utilities for a decision maker whose utility function is described by an exponential function with R 5 250: 2$200, 2$100, $0, $100, $200, $300, $400, $500
- Consider the game in the image attached below, which is infinitely repeated at t = 1, 2, ... Both players discount the future at rate: delta E (0, 1). The stage game is in the image attached. "Grim Trigger" strategies: Describe the "Grim Trigger" strategy profile of this game. Draw the finite automata representation of this strategy profile. Find the lowest value of delta for this strategy profile to form a subgame perfect equilibrium.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.Consider a game where there is a $2,520 prize if a player correctly guesses the outcome of a fair 7-sided die roll.Cindy will only play this game if there is a nonnegative expected value, even with the risk of losing the payment amount.What is the most Cindy would be willing to pay?
