EXERCISE \(85.1\) (First-price sealed-bid auction) Show that in a Nash
equilibrium of a first-price sealed-bid auction the two highest bids are the
same, one of these bids is submitted by player 1, and the highest bid is at
least \(v_{2}\) and at most \(v_{1}\). Show also that any action profile
satisfying these conditions is a Nash equilibrium.
In any equilibrium in which the winning bid exceeds \(v_{2}\), at least one
player's bid exceeds her valuation. As in a second-price sealed-bid auction,
such a bid seems "risky", because it would yield the bidder a negative payoff
if it were to win. In the equilibrium there is no risk, because the bid does
not win; but, as before, the fact that the bid has this property reduces the
plausibility of the equilibrium.
As in a second-price sealed-bid auction, the potential "riskiness" to player
\(i\) of a bid \(b_{i}>v_{i}\) is reflected in the fact that it is weakly
dominated by the bid \(v_{i}\), as shown by the following argument.
\- If the other players' bids are such that player \(i\) loses when she bids
\(b_{i}\), then the outcome is the same 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 her payoff is negative when she bids \(b_{i}\) and zero when she
bids \(v_{i}\) (whether or not this bid wins).
However, in a first-price auction, unlike a second-price auction, a bid
\(b_{i}b_{i}\) because if the other
players' highest bid is less than \(b_{i}\) then both \(b_{i}\) and
\(b_{i}^{\prime}\) win and \(b_{i}\) yields a lower price.
Further, even though the bid \(v_{i}\) weakly dominates higher bids, this bid is
itself weakly dominated, by a lower bid! If player \(i\) bids \(v_{i}\) her payoff
is 0 regardless of the other players' bids, whereas if she bids less than
\(v_{i}\) her payoff is either 0 (if she loses) or positive (if she wins).
In summary,
in a first-price sealed-bid auction (with perfect information), a player's bid
of at least her valuation is weakly dominated, and a bid of less than her
valuation is not weakly dominated.
An implication of this result is that in every Nash equilibrium of a first-
price sealed-bid auction at least one player's action is weakly dominated.
However, this property of the equilibria depends on the assumption that a bid
may be any number. In the variant of the game in which bids and valuations are
restricted to be multiples of some discrete monetary unit \(\epsilon\) (e.g. a
cent), an action profile \(\left(v_{2}-\epsilon, v_{2}-\epsilon, b_{3}, \ldots,
b_{n}\right)\) for any \(b_{j} \leq v_{j}-\epsilon\) for \(j=3, \ldots, n\) is a
Nash equilibrium in which no player's bid is weakly dominated. Further, every
equilibrium in which no player's bid is weakly dominated takes this form. When
\(\epsilon\) is small, each such equilibrium is close to an equilibrium
\(\left(v_{2}, v_{2}, b_{3}, \ldots, b_{n}\right)\) (with \(b_{j} \leq v_{j}\) for
\(j=3, \ldots, n)\) of the game with unrestricted bids. On this (somewhat \(a d\)
hoc) basis, I select action profiles \(\left(v_{2}, v_{2}, b_{3}, \ldots,
b_{n}\right)\) with \(b_{j} \leq v_{j}\) for \(j=3, \ldots, n\) as "distinguished"
equilibria of a first-price sealed-bid auction.
One conclusion of this analysis is that while both second-price and first-
price auctions have many Nash equilibria, yielding a variety of outcomes,
their distinguished equilibria yield the same outcome. (Recall that the
distinguished equilibrium of a second-price sealed-bid auction is the action
profile in which every player bids her valuation.) In every distinguished
equilibrium of each game, the object is sold to player 1 at the price \(v_{2}
.\) In particular, the auctioneer's revenue is the same in both cases. Thus if
we restrict attention to the distinguished equilibria, the two auction forms
are "revenue equivalent". The rules are different, but the players'
equilibrium bids adjust to the difference and lead to the same outcome:
the single Nash equilibrium in which no player's bid is weakly dominated in a
second-price auction yields the same outcome as the distinguished equilibria
of a first-price auction.
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