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Game Theory

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?
(Hint: Let m be the number of drivers using the toll road, n the number of drivers using back roads, so that m + n = 100. Consider an individual driver’s payoff if driving on the toll road or the back road respectively. Equilibrium obtains if no driver has an incentive to change his decision to drive on either road. As a default option, let us stipulate that a driver does not change his decision if payoffs from changing and not changing are equal, that is, a driver only switches to the other road if this gives a higher payoff.)
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