# Domination

As in a second-price auction, any player i’s action of bidding b_{i }> v_{i }is weakly dominated by the action of bidding v_{i}:

•

if the other players’ bids are such that player i loses when she bids b

_{i}, then it makes no difference to her whether she bids b_{i }or v_{i }•

if the other players’ bids are such that player i wins when she bids b

_{i}, then she gets a negative payoff bidding b_{i }and a payoff of 0 when she bids v_{i}.

However, in a first-price auction, unlike a second-price auction, a bid by a player less than her valuation is not weakly dominated.

R e a s o n : i f p l a y e r i b i d s v i < v i a n d t h e h i g h e s t b i d o f t h e o t h e r p l a y e r s i s < v i , t h e n p l a y e r i i s b e t t e r o ff t h a n s h e i s she bids v_{i}. i f

# Revenue equivalence

The price at which the object is sold, and hence the auctioneer’s revenue, is the same in the equilibrium (v_{1}, . . . , v_{n}) of the second-price auction as it is in the equilibrium (v_{2}, v_{2}, v_{3}, . . . , v_{n}) of the first-price auction.

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