Overview: Traditionally, auctions are used to sell assets such as a painting, a car, or even a single share of the stock of a company. In its simplest form, an auctioneer describes the item to be sold, and interested parties bid for it. The highest bidder wins, minus the auctioneer’s commission, with the net proceeds going to the seller, and the item goes to the winner. Auctions can also be used to sell liabilities. In an auction, assets have positive values, and liabilities have negative values. Either way, highest bidder wins, but note that for negative numbers, -200 is higher than -300.
History
Auctions have been used in commerce for well over 2000 years dating back at least to the Babylonian auctions of women for wives. Modern auction theory today is inextricably tied to modern game theory which dates at least back to 1928 and published in Von Neumann and Morgenstern’s 1944 book Theory of Games and Economic Behaviour. Since then, academic work has focused on auctions of assets with positive values rather than the negative values used for liabilities. It is amusing to note that the Babylonians auctioned beautiful women for positive values and required less beautiful women to come with dowries (negative values) funded in part with profits from selling the beautiful women!
Auction Theory
Let’s next discuss the difference between oral, shout-out auctions and sealed bid auctions. Well, oral auctions obviously need people to do the shouting. While this is a time honored tradition, everyone admits that computers are faster and can process sealed digital bids with ease. In addition, sealed bids can be kept secret. In fact even the winners can be kept secret which can be useful when auctioning works of art, for example.
The classic oral bid starts low, perhaps at the lowest acceptable amount for the seller, the so-called reserve price, and the auctioneer or people (perhaps agents representing others on the phone) in the audience gradually raise the proposed amount and indicate a willingness to buy at that price. The auction ends when no one is willing to buy at a higher price than the last bidder, who is declared the winner of the auction. This is called the English ascending oral auction. If no one is willing to buy at the reserve price at the beginning of the auction, then nothing is sold.
The Dutch descending oral auction starts with the auctioneer proposing a very high price. The auctioneer gradually lowers the proposed selling price until the reserve price is reached or someone shouts out their willingness to buy at that price. This person is the winner of the auction. If no one is willing to buy when the reserve price is reached, then nothing is sold.
There is an important difference between these two types of auction. The English auction discloses information about the losers of the auction, while the Dutch auction does not.
It is easy to see how to implement the Dutch auction with sealed bids. The auctioneer simply picks the highest bid, called the first price, among the sealed bids, and the losers depend on the discretion of the auctioneer for their privacy. All offer prices must be greater or equal than the published reserve price. Ties can be handled by order of submission (as in the oral case) or by picking the winner randomly. Again the honesty of the auctioneer is paramount to handle ties.
It is perhaps a little less obvious how to implement an English auction with sealed bids. If you like puzzles, put down this paper and spend a little time trying to figure it out. The answer is NOT that people get to submit multiple bids!
Canadian William Vickrey, a Columbia University professor, wrote a series of papers starting in 1961 with “Counterspeculation, Auctions, and Competitive Sealed Tenders”. This work earned Vickrey a share in the 1996 Nobel prize in economics. In particular, he solved the puzzle about emulating an English auction with sealed bids. His solution is astoundingly simple. Among the sealed bids the highest bidder wins the auction; however, the winner pays the second highest bid, called the second price. Ties are handled as above, and since the second highest bid in a tie equals the highest bid, that is what the winner pays in a tie. This type of auction is called a Vickrey Auction.
Auction Strategies
If a player wants to program a computer to partake in an auction, then this player’s computer program needs an algorithm to compute its next bid. Such an algorithm is called a (bidding) strategy. The inputs to the algorithm can be not only the known state of the auction, but also external information. For example, if the auction is for shares of risk on hurricane insurance, then weather information will certainly be an input. Intrinsic knowledge about the subject (e.g. construction knowledge, weather knowledge, etc.) can also be programmed into the algorithm. Thus the algorithm is highly proprietary.
Optimal Strategies
The mathematician John F. Nash, Jr., subject of the excellent book and movie, A Beautiful Mind, wrote in 1950 a one-page paper “Equilibrium points in n-person games”. This seminal paper was cited by the Nobel Committee 44 years later when Nash was awarded a share in the Nobel prize in economics. Nash proved in his one-page paper that there always exists an n-tuple of strategies
(n = #players) such that for each player, his/her strategy yields the highest obtainable expectation against the n-1 strategies of the other players. Such an n-tuple of strategies is today called a Nash equilibrium point. Nash’s paper shows these exist, but the paper doesn’t give constructive methods to discover them.
Nash’s result applies generally to n-person games, and in particular to n-person auctions, independent of the type of auction. Since 1950, economists have been analyzing auctions knowing that Nash equilibrium points of optimal strategies always exist. This means that players will spend considerable resources looking for and developing optimal strategies, relative to other players, to be successful on auctions.
Counter-intelligence
Aaron Brown, in his book Red Blooded Risk, The Secret History of Wall Street, writes "...much more quant effort is devoted to studying how other investor/bettors act than to estimating fundamental value." This is quite likely true of other games, e.g., poker and war games. Thus, it is most likely that players on any auction exchange will do the same. In fact, success on an exchange comes from being better than the competing players and not necessarily from absolute competence.
Modern Portfolio Theory (MPT)
Now MPT, like Auction Theory, has historically been studied for portfolios of assets, although holdings of shorts are definitely part of MPT analysis. That said, any player will surely apply its own spin of MPT within its strategy function. For liabilities, this is fraught with peril, however, primarily because investment risk for liabilities is very different from investment risk for assets, and the auction exchange must put into place bidding policies to mitigate such risk. This makes for an interesting trade-off between how a player uses MPT to diversify its portfolio and how it uses its knowledge relative to its business model to optimize its portfolio.
Auctioning liabilities
Associated with any liability is the risk or probability of the liability occuring in a given time frame. If a home or business owner is worried about the destruction of the home or business occuring due to a hurricane or flood, then this owner will want to pay a reasonable price to an insurance company to assume all or part of this liability for the coming year. The kind of liabilities that are conducive to being auctioned off are those which have little, if any, acturarial data. The owner of such liabilities, usually an insurance company, bundles a number of them into a contract. All or part of such a contract can be auctioned off with the auction determining the contract's value. The auction exchange has many ways to divide up a contract into “shares” to be auctioned off. At the same time, much as with assets, a small number of shorts can be created. Shorts of a share of a liablity valued at -P will have price P. If an event occurs triggering the liability, then the holder of a share pays P dollars, and the holder of a short receives P dollars. Both shares and shorts can be auctioned off independently. The actual face values of the shares and shorts are pre-determined by the owners before the auction, and there are many techniques to do so.
Example: A $20,000,000 collection of hurricane insurance policies along the Atlantic coast of the U.S. could be divided via geography (state, inland, and coastal) into 20 “series” each valued at -$1,000,000. To divide such a series into -$20 shares would take 50,000 shares. To add, say 10,000 shorts each valued at $20/short, we would need to increase the number of shares to 60,000. The total value still being auctioned is -$20,000,000.
Sealed Bids
The concept here is that the auctions of liabilities are performed in the cloud. If all bids for shares or shorts are submitted electronically, they are at least initially sealed. It makes no sense to unseal the bids while other bids are being submitted.
Dutch Auction versus English Auction versus Vickery Auction.
With all bids being submitted in the same time interval for a round of auctions, it makes no sense to “start at the bottom” of the bids. This eliminates the possibility of an English Auction, leaving the Dutch Auction, although an exchange could have the winner of the auction pay the second price, and thus run a type of Vickrey Auction.
Methods to implement an auction exchange for liabilities will be discussed in a future post.
Bibliography
- Equilibrium points in n-person games, JF Nash
- Counterspeculation, Auctions, and Comparative Sealed Tenders, W. Vickrey
- Theory of Games and Economic Behavior, von Neumann and Morgenstern
- Theory of Games and Economic Behavior – (Excellent) Book Review by Alan Copeland
- Modern Portfolio Theory and Investment Analysis, 9th edition, Ed Elton, Martin Jay Gruber, pub Wiley
- Auctions: Theory and Practice, Paul Klemperer. An on-line draft can be found here: http://www.nuff.ox.ac.uk/users/klemperer/VirtualBook/VirtualBookCoverSheet.asp
- Putting Auction Theory to Work, Paul Milgram, Churchill Lectures in Economics. ISBN 0-521-55184-6,
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