Homework 4 Fall 16 Econ/TIM 166a Game Theory D. Friedman & J. Musacchio DUE OCTOBER 25, 2016 1. Recall the Tragedy of the commons game played on 10/13. • The possible actions by each player i were xi = 0, … , 40 hours of fishing • The payoff function for player i (from a group of 6 players) is 6
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(In the game we played, each player chooses an integer number of hours to fish. For simplicity, in this problem we will suppose each player is choosing from the continuum of real numbers between 0 and 40) a) Find the best response function for each player i. b) Find a symmetric Nash equilibrium. c) Define the social welfare to be the sum of all 6 players payoffs. What is the social welfare in Nash equilibrium? d) Recall from problem set 1, that you computed S*, the sum of all the xi’s that optimizes (maximizes) social welfare. By what percentage less is the Nash equilibrium welfare compared to the social optimum? (This percentage is known as the efficiency loss.) e) Look at the data from the actual class play. How does the efficiency loss seen in the data compare to what you computed in d? 2. In class (Thursday 10/13) we studied a game in which players contribute to a public good. We then considered a scheme in which an amount R is deducted from the contributions and then given randomly to one of the contributors with a probability proportional to that player’s contribution. In this problem we suppose that players have a utility function that has a square root form rather than a log form as we studied in class. To keep things simple, we suppose there are only 2 players who contribute x1 and x2 respectively. 1
a. Suppose each player values the public good by x + x , one half of 2 1 2
the square root of the total amount contributed. (We choose this form since it has a diminishing returns shape.) Thus when players do voluntary contributions, they have a utility function of €
U i (x1, x 2 ) = 12 x1 + x 2 − x i . Compute the best response function of each of the 2 players. b. Find the Nash equilibria by finding the intersection of the best response functions in part a. Is there a unique equilibrium or € is the total contribution (x +x ) in each of the one or multiple? What 1
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many equilibria you find? Homework 4 Fall 16 Econ/TIM 166a Game Theory D. Friedman & J. Musacchio c. The value to society of the public good is the sum of the player’s values x1 + x 2 , thus the utility function for society as a whole after subtracting the cost of the contributions is U s (x1, x 2 ) = x1 + x 2 −x1−x 2 . € Since the utility function only depends on the total contribution (x1+x2), we can simplify it as U s (T) = T − T €
where T is the total contribution to the public good. For society, what is the optimum choice of T? d. Extra Credit: Suppose that a reward R=10 is deducted from the contributions and €
given randomly to one of the two players with a probability in proportion to their contribution. Thus each player now has the utility 10x i
U i (x1, x 2 ) = 12 x1 + x 2 −10 − x i +
. x1 + x 2
In Nash equilibrium, what is the net contribution (after deducting the reward of 10) to the public good? Hint: Set the derivative of each player’s payoff with respect to his € to 0. Add these two equations. You’ll get an equation strategy equal where x1 and x2 appear only as their sum, (x1+x2). Make a substitution G = x1+x2 -­‐ 10 and write an equation where G is the only unknown. Your equation will be difficult to solve in closed form, but you can use a computer to plot your expression and find the value of G that makes your expression 0. 3. Recall our class discussion of routing games with infinitely many players. Consider a population of total mass normalized to 1, with each member needing to go from a common source to a common destination. There are 2 paths, numbered 1 and 2. When the fraction of traffic taking the upper path is x1, its delay is x15. The delay of the lower path is always 1, regardless of its load which we call x2 . a. What is the Wardrop equilibrium? What is the average delay of driver in this equilibrium? b. Suppose now a social planner could force a particular traffic pattern in order to optimize the average delay. The average delay should be weighted according to how many drivers bear the delay of each route, so it should have the form of x1(delay path 1) + x2(delay path 1) . What is the optimum traffic pattern for minimizing average delay? c. Imagine you could charge a toll on one of the two roads, and suppose that users choose the road with smallest delay plus toll. Which road would you charge a toll on, and what would that toll be to achieve the optimum traffic pattern you found in b? 4. A batter and pitcher face each other in a baseball game. The pitcher chooses whether to throw (H)igh or (L)ow. Since the ball comes so fast, the hitter has to simultaneously choose whether to swing the bat (H)igh or (L)ow. If both Homework 4 Fall 16 Econ/TIM 166a Game Theory D. Friedman & J. Musacchio players choose the same position, the batter gets a hit, otherwise he misses the ball and gets a strike. The payoff to the hitter for hitting a low ball is 1. Since the batter hits with more power when the ball is high, he gets a payoff of x for hitting a high ball, where x>1. The payoff to the batter for a strike is -­‐
1/3. The pitcher’s payoffs are the exact opposite of the batter’s. a. Is there one or more pure strategy equilibrium? If so, find them. b. Is there one or more mixed strategy equilibria? If so, find them. Express your answer in terms of x. c. The higher the value of x, the more damage a batter can do to a pitcher when hitting a high ball. In any of the above equilibria, is there a large enough x for which the pitcher would never risk throwing the ball high? d. For your answer in part b, compute the equilibrium payoff (or payoffs if there are multiple equilibria) of the batter as a function of x. 5. Chapter 7, #14, page 260 6. Chapter 8, #4, page 297