Game Theory in Economics, Finance and Business

It is the morning commute in Congestington, DC. There are 100 drivers, and each driver is deciding whether to take the toll road or take the back road. The toll for the toll road is $10, while the back road is free. In deciding on a route, each driver cares only about net income, denoted y, and his travel time, denoted t. If a driver’s net income is y and his travel time is t, then his payoff is assumed to be y – t (where we have made the dollar value of one unit of travel time equal to 1). A driver’s expected income per day before the trip is $1,000. If k drivers are on the toll road, the travel time for a driver on the toll road is assumed to be k (in dollars). In con¬trast, if k drivers take the back road, the travel time for those on the back roads is 2k (again, in dollars). Drivers make simultaneous deci¬sions as to whether to take the toll road or the back road.
(a) Derive an individual player’s payoff function (i.e., the expression that gives us a player’s payoff as a function of her strategy profile.)
(b) How many drivers will there be on both types of roads in a Nash equilibria?
(c) Use a graph to illustrate your result.
(d) What would be the situation if there were 101 drivers?