Suppose that the speed limit is 70 miles per hour on the freeway and that n drivers simultaneously and independently choose speeds from 70 to 100. Everyone prefers to go as fast as possible, other things equal, but the police ticket any driver whose speed is strictly faster than the speeds of a fraction x of the other drivers, where x is a parameter such that 0 ... x ... 1. More precisely, for a given driver, let m denote the number of drivers that choose strictly lower speeds; then, such a driver is ticketed if and only if m>(n - 1) > x. Note that by driving 70, a driver can be sure that he will not be ticketed. Suppose the cost of being ticketed outweighs any benefit of going faster.
(a) Model this situation as a noncooperative game by describing the strategy space and payoff function of an individual player.
(b) Identify the Nash equilibria as best you can. Are there any equilibria in which the drivers choose the same speed? Are there any equilibria in which the drivers choose different speeds? How does the set of Nash equilibria depend on x?
(c) What are the Nash equilibria under the assumption that the police do not ticket anyone?
(d) What are the Nash equilibria under the assumption that the police ticket everyone who travels more than 70?
(e) If the same drivers play this game repeatedly, observing the outcome after each play, and there is some noisiness in their choices of speed, how would you expect their speeds to change over time as they learn to predict each other's speeds when x is near 100 and when x is near 0? Explain your intuition.