Consider a group of 10 friends who are considering going on a road trip. In determining whether or not to go, each friend considers both how much they would enjoy the trip and how much it will cost them. The cost of the trip will be $500 total, and will be split equally between those who decide to go. Each friend values the trip differently, however. Specifically, the friends can be ordered in terms of how much they value the trip in dollar terms. Friend #1 values the trip at v1 = 200, friend #2 values it at v2 = 180, friend #3 has v3 = 160, and so on, with friend #10 valuing the trip at a meager v10 = 20. The payoff to friend i if they choose to go on the trip is vi ? 500 m , where m is the number of friends who decide to go. The payoff to a friend who does not go on the trip is simply 0.
(a) Is this game symmetric? Explain why or why not.
(b) Identify all Nash equilibria for the road trip game.
(c) Which of the equilibria that you identified in part (b.) are symmetric, and which are asymmetric?
(d) Select one of the Nash equilibria you identified in part (b.) and prove that it is indeed a Nash equilibrium. If you did not find any, prove that a Nash equilibrium does not exist.
(e) Now consider a more general version of the game in which the total cost of the trip is simply some number c > 0 and the friends’ valuations are simply v1 > v2 > . . . > v10. Payoffs are again vi? c m for a friend who does go on the trip, 0 for a friend who does not. How can you categorize all Nash equilibria in this case? Demonstrate how by finding them! (Hint: start by thinking to yourself, “Self, what would happen if c = 400?” Or, “Self, what would happen if v1 = 250, v2 = 230, and so on?”)