Consider a first price auction for selling one item. There are n bidders. Each bidder j has a valuation v for the item, which is privately known and drawn independently and uniformly over the interval [0.1001, Each bidder I bids a non-negative real number
management
Description
1. (20 Points) Consider a first price auction for selling one item. There are n bidders. Each bidder j has a valuation v for the item, which is privately known and drawn independently and uniformly over the interval [0.1001, Each bidder I bids a non-negative real number b. The bidder who bids the highest number wins and if more than one bidder bid the highest. the winner is chosen uni lbrmly at random. The winner gets the item and pays her bid. All other bidders pay 0.
a. (5 points) Write down the strategy set tbr each bidder É.
b. (5 points) Assume that cveiy bidder except ¡ tises a linear strategy. i.e.. b-(v1) = a +
c v, where the constants a and c are the same for all bidders. Find the expected
payotTof bidder ¿ when she bids b. (hint: For simplicity of calculations. assume that
bidder I wins only when all the other bids are less than b1.)
e. (5 poInts) Find a Bayesian-Nash equilibrium for this auction. (hint: lecture slides).
d. (5 points) Under your equilibrium in part e. what is the expected revenue for the seller
when n = 10? What happens to the expected revenue as n —‘ co?
