Q1 Let Ø^F : [v; v¯] R₊ be the symmetric equilibrium bidding function in a first price auction.

1. Let U ^F (w v) denote the interim utility of a type v bidder who pretends to be of type w when the bidding is done according to Ø^F . (U ^F (v) = U ^F (v v) is then the interim equilibrium utility of type v bidder.)

2. Now consider the all-pay auction, whose equilibrium we derived in class to be

Ø^A(v) = Ø

A(v  ) +

v

s f (s)ds:

v

We also noted that in this auction the interim utility is

U ^A(wj v) = w F (v) ¡ Ø^A(w):

Use the fact that from Payoff Equivalence, we must have U ^A(v v) = U ^F (v v) for all v, to compute Ø^F .

Q2 Consider an auction environment with two buyers, where the value of buyer 1

v₁ [0; 1] and that of buyer 2 is v₂ [1; 2], both drawn according to a uniform distribution on the two intervals. Assume sellers value v₀ = 0.

1. If you run an efficient auction, such as a second price auction, who gets the object? Depict the appropriate regions in the following picture.

2

1

(0,0)

1

v₁ ¡!

Figure 1.