auction theory

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.

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