THE MATHEMATICS OF AUCTIONS

The word 'auction' conjures up a picture of the sale of exotic or rare articles. In fact, large volumes of sales of mundane articles are conducted through auctions. The list includes trading in foodgrains at mandis, the sale of vegetables and fruits in the wholesale markets, the sale of mineral rights by governments, and the award through the tender process of practically all Government contracts in India. Recently, licenses for cellular telephone networks were awarded in India through an auction process. In the US, Treasury Bills are routinely sold through auctions; and the Federal Communication Commission has auctioned radio spectrum frequencies to providers of mobile telephone services etc.

Auctions are, therefore, far more important in day-to-day life than the often-publicised sales of art masterpieces alone might lead one to believe. Such widespread use of auctions makes it important to understand how the mechanism works, and for this mathematical analysis and different math assignment is indispensable.

Auction Forms

There are four basic forms of auctions: theEnglish open outcry auction(also called theascending bidauction); theDutch (descending-price) auction; thefirst-price sealed bid auction; and thesecond-price sealed-bid auction.

In the familiar English auction, used in the sale of paintings or antiques, and also in the markets for fruits, vegetables and grains, bidders keep on increasing their offers, starting from a reserve price, till only one bidder is left and the hammer comes down.

In a Dutch (descending-price) auction, used in the sale of cut flowers in Holland, the price starts at a high level, and is successively decreased by the auctioneer till a bidder claims the article by shouting "Mine!" (whence the term Mining which is sometimes used to describe this auction.)

The first-price auction is used in India and many other countries for awarding government contracts. In it, bidders submit sealed bids to the auctioneer; the highest bidder wins and pays a price equal to his own bid.

In the second-price sealed-bid auction the bidders submit sealed bids and the highest bidder wins. But the price he pays is equal to the highest rejected bid or in other words the second highest bid. The second-price auction has been used in recent years in the United States for the sale of Treasury Bills.

The second-price sealed bid auction is also called the Vickrey auction, after William Vickrey who first proposed this auction mechanism, and analysed its properties. Vickrey was awarded the 1996 Nobel Prize in Economics for his pioneering work on the economics of information.

Strategic Considerations

Suppose the seller wants to sell a single unit of a good. Various buyers value the good differently; each buyer knows only his own valuation.

If the seller knew these valuations he would simply put the price just below the highest valuation. However, he suffers from theinformational disadvantagethat he does not know the valuations of the individual buyers, and hence does not know what price to set. In an auction the seller does not have to set a price as the price emerges through the bidding process. In such situations, an auction is frequently the best strategy for the seller to adopt.

Bidders want to maximize theirexpected net gain. If a buyer bids B and does get the good, his net gain is equal to his benefit from obtaining the good, V, minus the price he has to pay, c. If he does not get the good, his gain is obviously zero. The expectednet gainis therefore the probability p of winning the auction times (V - c). The probability p will depend on all the bids; it will typically increase as B increases. The question is what is the bestbidding strategy.

If you bid low, and win the auction, your gain will be large, but you run the risk of not winning the auction. On the other hand, if you bid very high and win the auction you might end up paying too much.In choosing an optimal bid there is a trade-off between the probability of winning the auction and your gain from doing so.

Bidding Strategies

Consider the bidding strategies in afirst-price sealed-bid auction. One might think that each buyer should bid the amount that the object is actually worth to him. However, it is easy to see that the buyers will underbid, and that consequently the revenue of the seller will be lower than the highest valuation.

If the buyer bids his true valuation (B=V), the net gain will be zero no matter what the other bids are. If he bids less than V ('shading' his true valuation), the probability of winning the auction decreases, but when he does win he makes a positive net profit. Hence the expected net gain if he bids less than his true valuation is strictly positive as compared to the zero net gain from bidding his true valuation.

On the other hand, in asecond-price sealed bid auctionthe highest bidder gets the object, but only has to pay the second-highest bid as the price. All the bids will now be higher than in the first-price auction. In fact, for each buyer bidding his true valuation will be the dominant strategy, that is,the strategy that is best for him irrespective of what others are doing.

Consider the problem confronting buyer k. Suppose b is the highest bid among bidders other than k. Assume that k knows the value of b (this assumption can easily be relaxed). If b is less than V, buyer k makes a positive profit by bidding V. Changing the bid to any amount higher than b does not change the net gain (he still gets the good, and pays the same price b ). On the other hand, if k bids less than b he loses the auction and is distinctly worse off. Hence bidding the true valuation V is always at least as good as bidding any other amount, and is strictly better than bidding amounts less than b . Similarly we can show that if b > V, bidding V is still the best strategy.

So in the second-price sealed bid auction each buyer bids his true valuation, and the good goes to the buyer with the highest valuation at a price equal to the second highest valuation.

Revenue Equivalence

In a seminal 1961 paper William Vickrey showed that, when there is a single object to be auctioned,all the four auction mechanisms described above result in identical outcomes. The buyer with the highest valuation gets the object, and the expected revenue of the seller is the expectation of the second highest valuation.This striking result depends on the following assumptions:

(1) The English and Vickrey Auctions

In the English auction a bidder should be willing to match any bid that is less than his valuation. The bidder's strategy is simply to choose a number (call it his reservation price) such that he will not bid above this number. Just like bidding the true valuation was the dominant strategy in the case of Vickrey auction, setting the reservation price equal to the true valuation is the dominant strategy in the English auctions. Clearly, as in the Vickrey auction, the item will be won by the bidder with the highest willingness to pay (the outcome is called "efficient" if it has this property), at a price equal to the second highest valuation. The expected benefit to each bidder, and the expected revenue for the seller, are also identical in both the auctions.

(2) The Dutch and First-price Auctions

Consider the problem facing a bidder in a Dutch auction. The price starts at a very high level and is lowered continuously by the auctioneer until the good is claimed by one of the bidders. Each bidder again has to choose the highest price at which he should shout 'Mine' and claim the good. If we call this price the "bid", it is clear that in a Dutch auction the license will go to the highest bidder at a price equal to his bid. But this is exactly what happens in a first price sealed bid auction.

Unfortunately in the first-price sealed bid or Dutch auctions there are no dominant strategies. The strategy that is optimal for bidder k depends on the bidding strategy adopted by other bidders.

Nevertheless, it can be shown that under the assumptions specified above, the first-price and the second price sealed-bid auctions lead to the same expected revenue for the seller. This remarkable result, due to Vickrey, is known as the REVENUE EQUIVALENCE THEOREM.

Consider the two items of interest for a bidder that are determined by the auction rules and the bidding strategies adopted by everybody else. These are the probability p of winning the auction and the payment c that will then have to be made. By changing his bid, a bidder can change the (p, c) pair. Given his valuation V, he will choose (p, c) to maximize his expected net gain U (p, c;V) = p. (V - c).

The optimal (p, c) pair for the bidder will depend on his valuation V. Let the optimal pair be (p*(V), c*(V)). The maximal expected profit is then U*(V) = U (p*(V), c*(V); V). Now from theEnvelope Theoremwe have

 

(1) dU*/dV = UV(p*(V),c*(V);V) = p*(V)

It is also obvious that the bidder who has the lowest possible valuation, say zero, can never win the auction and must have zero expected gain and so for both the first-price and Vickrey auctions U* (0) = 0. Integrating (1) we get

 

(2)

We have already seen that the Vickrey auction is efficient i.e., it gives the item to the bidder with the highest valuation. Vickrey showed that first-price auction is also efficient. Hence inboththe first-price and the Vickrey auctions, the probability p*(V) of winning the auction is the same. Each is equal to the probability that the valuations of the other bidders are less than V.

From equation (2) it then follows that the first- and the second- price auctions yield the same expected benefit for the bidders. Also, since the object always goes to the bidder with the highest valuation, the total surplus generated by trade is the same in both of these auctions and consequently the seller's expected revenues are identical.

Extensions

The revenue equivalence result depends on assumptions that are often not satisfied in reality. We state briefly the consequences of relaxing some of the assumptions: