Game theory
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1. [5 Marks] (Selfish and altruistic social behavior) Two people enter a bus. Two adjacent cramped seats are free. Each person must decide whether to sit or stand. Sitting alone is more comfortable than sitting next to the other person, which is more comfortable than standing.
(a) Suppose that each person cares only about her own comfort. Model the situation as a strategic game. 1s this game the Prisoner's Dilemma? Find its Nash equilibrium (equilibria?).
(b) Suppose that each person is altruistic, ranking the outcomes according to the other person's comfort, and, out of politeness, prefers to stand than to sit if the other person stands. Model the situation as a strategic game. Is this game the Prisoner's Dilermna? Find its Nash equilibrium (equilibria?).
(c) Compare the people's comfort in the equilibria of the two games.
2. [5 Marks] [Finding Nash equilibria using best response functions) Find the Nash equilibria of the twoplayer strategic game in which each player's set of actions is the set of nonnegative numbers and the players' payoff functions are u_1(a_1,a_2) = a_1(a_2 — a_1) and u_2(a_1,a_2) = a_2(1 — a_1 — a_2).
3. [5 Marks] (A joint project) Two people are engaged in a joint project. If each person i puts in the effort x_i, a nonnegative nmnber equal to at most 1, which costs her c(x_i), the outcome of the project is worth f(x_1, x_2). The worth of the project is split equally between the two people, regardless of their effort levels. Formulate this situation as a strategic game. Find the Nash equilibria of the game when (a) f(x_1, x_2) = 3x_1x_2 and c(x_i) = (please see the attached file) for i = 1,2, and (b) f(x_1, x_2) = 4x_1x_2 and c(x_i) = x_i for i = 1, 2. 1n each case, is there a pair of effort levels that yields both players higher payoffs than the Nash equilibrium effort levels?
(please see the attached file for the remaining questions)
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1.
a) An example that models this situation is
(please see the attached file)
This game is not the Prisoner's Dilemma. If we identify Sit with Quiet and Stand with Fink then, for example, (Stand, Sit) is worse for player 1 than (Sit, Sit), rather than better. If we identify Sit with Fink and Stand with Quiet then, for example, (Stand, Stand) is worse for player 1 than (Sit, Sit), rather than better. The game has a unique Nash equilibrium, (Sit, Sit).
b) An example that models this situation is illustrated as follows, where a is some positive number.
(please see the attached file)
If a < 1 then this game is the Prisoner's Dilemma. It has a unique Nash equilibrium, (Stand, Stand) (regardless of the value of a).
c) Both people are more comfortable in the equilibrium that results when they act according to their selfish preferences.
2.
First find the best response function of player 1. For any fixed value of a2, player 1's payoff function a1(a2 − a1) is a quadratic in a1. The coefficient of a12 is negative and the function is zero at a1 = 0 and at a1 = a2. Thus, using the symmetry of quadratic functions, b1(a2) = ½a2, where b1 denotes the best response function of player 1.
Now find the best response function of player 2. For any fixed value of a1, player 2's payoff function a2(1 − a1 − a2) is a quadratic in a2. The coefficient on a22 is negative and the function is zero at a2 = 0 and at a2 = 1 − a1. Thus if a1 ≤ 1 we have b2(a1) = ½(1 - a1) and if a1 > 1 we have b2(a1) = 0.
A Nash ...
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