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Stanford University
Department of Management Science and Engineering
Optional Problem Session 6
Winter 2010
Friday, February 26, 2010
Practice Problem 6.1 (Pure Strategy Nash Equilibrium)
Students are attending a job fair along
a 1 mile stretch of road on campus. They are uniformly distributed along this road and after the job fair
oﬃcially starts, the section is sealed oﬀ so that no students can enter or leave. The sunny California day
is suddenly interrupted by rain. There are a number of umbrella vendors on this road but the price of
an umbrella is regulated by the school. We assume that none of the students like walking in the rain so
everyone will purchase an umbrella from the nearest vendor. We also assume that there is no sharing,
thus each student will buy exactly 1 umbrella. It is permitted for more than one vendor to set up shop
at the same location, in which case they evenly split the business. Let
N
be the number of vendors on
this road.
(i) Consider the game where

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