Sealed Bid Auctions

Scribe: Rahul Gupta, Akshat Verma

1  Sealed Bid Auctions

Auctions are a means to transfer goods from one entity to another where the price of the good is not easily determined and depends on the perception of the customer. An auction leads to transfer of goods from an entity that has a low valuation of an item to an entity, who has a higher valuation. This leads to economic efficiency and our aim is to devise mechanisms, which lead to efficient allocation, i.e., maximize the surplus in economy by allocating the item to the person who attaches the highest value to it. The outcome of an auction is determined by two parameters: the allocation, i.e., which items are allocated to which bidders and the price , which is charged for each item. Auctions are traditionally categorized into two types: Sealed Bid and Progressive Auctions.

In this lecture, we look at different auction mechanisms for a Sealed Bid auction. We take the case of single-item auctions and study First Price and Second Price mechanisms, in this context. We note that both these mechanisms lead to an efficient allocation and, under some simplifying assumptions, fetch the same revenue to the auctioneer.

2  First Price Auction

In this auction mechanism, sealed bids are invited from all the bidders. The item is then allocated to the bidder with the highest bid, at her bid price. We will now model this auction as a game where the bidders are the players. The bidders are assumed to be risk neutral - i.e. they do not place a bid which might lead to a loss. For a player i, let bi be the bid amount and vi be her valuation on item. The aim of each player is to maximize her expected profit, where profit is defined as
pi =  

vi - bi
if player gets the item
0
otherwise
(1)

We assume that the players are rational i.e their sole aim is to maximize their expected profit. Hence, we characterize the Nash Equilibrium for the auction game. For simplification, we assume that the players' valuations are derived independently from a uniform distribution on the interval (0,1). Hence, each player i aims to maximize her expected profit(Ri), , which are given by
Ri = Pri(bi)(vi - bi)
(2)
where Pri(bi) is the probability of bidder i winning, given that she bids bi.

The above can be represented as the following unconstrained optimization problem :


bi* = argmaxbi Pr(win|bi)(vi - bi)
(3)
where bi* is the optimal bid value at which the expected returns are maximized.

We now show that bi* = [(N-1)/N] vi is a solution to the problem, where N is the number of bidders in the auction. Let "j i, bj=bj* = [(N-1)/N]vj .
Pri(bi)
=
Pr(  N-1

N
vj < bi) "j i
(4)
=
Pj i Pr(  N-1

N
vj < bi)
(5)
=
(Pr(vj <  N

N-1
bi))N-1
(6)
=
(  N

N-1
bi)N-1.
(7)
Hence, the returns ri is given by
ri = (  N

N-1
bi)N-1(vi - bi).
(8)
Differentiating ri w.r.t bi, we can see that the maxima of ri (=Ri) is located at bi = [(N-1)/N]vi (=bi*).

Hence, "i, bi=bi* leads to a Nash Equilibrium, which maximizes the expected returns of each player.

3  Second Price Auctions

In second price auctions, the item is allocated to the highest bidder at price equal to the second highest bid. We will now show that this mechanism gives an incentive to the bidders to be truthful about their valuations. For this purpose, we show that the Nash Equilibrium strategy for a bidder is to bid her true value for the item. It is clear that, in this case, computing a bid is straightforward as a player simply bids independent of what the other bidders are doing. Note that in First Price auctions, bid computation can be a non-trivial task especially when the valuations are not uniformly distributed.

Expected return ri for player i is given by:
ri = Pri(bi)(vi - Second highest bid).
(9)
We now argue that "i, ri is maximized when bi = vi. One may note that bidding the true value gives a non-negative return. There are the following 3 cases:

  1. bi is not the highest bid
    In such a case, i gets a payoff of zero and hence the bidder is no better than bidding the true value.

  2. bi is the highest bid, vi is greater than the second highest bid.
    The return is the same irrespective of the value of bi

  3. bi is the highest bid, vi is less than the second highest bid.
    Bidder i gets a negative return and hence it is better to bid the true value, which would give non negative return.

Hence, bi=vi is the Nash Equilibrium strategy. One may also note that we do not assume anything about the distribution of vi. Also note that bidding the true value is a dominant strategy as it is independent of what the other bidders are doing.

4  Revenue Equivalence

The second price auctions seems to be a bad choice for the auctioneer in terms of revenue generated. However, we now show that the expected revenue generated by both the mechanisms is the same for uniformly and independently distributed vi's. The basic intuition behind this is the following. In the first price auction, the bidders report a value, which is less than their true valuation. To be more precise, a natural strategy for a bidder is to to guess the price of the second highest bid and just out-bid it, resulting in the same revenue. We now prove it more rigorously.

Lemma 1 The expected revenue generated by First Price auction is [(N-1)/(N+1)].

Proof: It is clear that the expected revenue er equals the expected highest bid(=bmax*)
er
=
E(bmax*)
(10)
=
E(  N-1

N
vmax)
(11)
=
 N-1

N
*
1

0 
x Pr(vmax (x,x+dx))
(12)

Pr(vmax (x, x +dx)) = N xN-1 dx
(13)
Hence,
er =  N-1

N

1

0 
x N xN-1 dx =  N-1

N+1
(14)

Lemma 2 The expected revenue generated by the second price auction with risk neutral bidders is [(N-1)/(N+1)]

Proof: It is clear that the expected revenue er equals the expected second highest bid(=b2max*)
er
=
E(b2max*)
(15)
=
E(v2max)
(16)
=

1

0 
Pr(v2max (x,x+dx)) * x
(17)

Pr(v2max (x, x +dx)) = N (N-1) xN-2 (1-x) dx
(18)
This is true because we have NP2 ways to choose the second and the highest bidders keeping the order into consideration. Therefore,
er =
1

0 
N (N-1) xN-1(1-x) dx =  N-1

N+1
(19)

Theorem 1 The First Price and the Second Price auction yield the same expected revenue if the bidders are risk neutral and have private, independent and uniformly distributed valuations.




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On 21 Sep 2002, 18:59.