It is the morning commute in Congestington, DC. Of 100 drivers, eachdriver is deciding whether to take the toll road or take the back roads. Thetoll for the toll road is $10, while the back roads are free. In deciding ona route, each driver cares only about income, denoted y, and his traveltime, denoted t. If a driver’s final income is y and his travel time is t, thenhis payoff is assumed to be y - t (where we have made the dollar value ofone unit of travel time equal to 1). A driver’s income at the start of the dayis $1,000. If m drivers are on the toll road, the travel time for a driver onthe toll road is assumed to be m (in dollars). In contrast, if m drivers takethe back roads, the travel time for those on the back roads is 2m (again,in dollars). Drivers make simultaneous decisions as to whether to take thetoll road or the back roads.a. Derive each player’s payoff function (i.e., the expression that gives usa player’s payoff as a function of her strategy profile.)b. Find a Nash equilibrium.
It is the morning commute in Congestington, DC. Of 100 drivers, each
driver is deciding whether to take the toll road or take the back roads. The
toll for the toll road is $10, while the back roads are free. In deciding on
a route, each driver cares only about income, denoted y, and his travel
time, denoted t. If a driver’s final 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 income at the start of the day
is $1,000. If m drivers are on the toll road, the travel time for a driver on
the toll road is assumed to be m (in dollars). In contrast, if m drivers take
the back roads, the travel time for those on the back roads is 2m (again,
in dollars). Drivers make simultaneous decisions as to whether to take the
toll road or the back roads.
a. Derive each player’s payoff function (i.e., the expression that gives us
a player’s payoff as a function of her strategy profile.)
b. Find a Nash equilibrium.
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