auction theory

Exercise 1 – Auction Theory (96573)

1. Give a formal proof that truthfulness in the second-price auction for n players is a dominant strategy.

2. Find a Bayesian-Nash equilibrium in the first-price auction, when players’ values are independently drawn from the uniform distribution on [a,b], for

any b > a > 0. Hints for one possible solution:

a. Assume that the equilibrium bid function is b(z)=+·z, and that b(a)=a.

b. Start by writing the utility function u(x,z) that denotes a player’s utility when her value is x and she bids b(z), for arbitrary x,z.

c. Since b(z) is an equilibrium, it follows that a player’s utility is maximized when she bids b(x). This should give you a first-order

condition on the bid function that will lead you to get an exact

expression for  and , which gives you the bid function.

3. In the setting of the previous question, suppose there are 3 players with values independently drawn from the uniform distribution on [2,14]. Suppose player

1 has value v1=9. What is her equilibrium bid? Show that she will not

improve her expected utility by bidding 6 or by bidding 8.