Purification and Revelation: The Modeling of Auctions

The body sins once, and has done with its sin, for action is a mode of purification. Nothing remains then but the recollection of a pleasure, or the luxury of a regret.

Oscar Wilde (1854–1900), Anglo-Irish playwright, author. Lord Henry, in The Picture of Dorian Gray, ch. 2 (1891).¹

Only a very bad theologian would confuse the certainty that follows revelation with the truths that are revealed. They are entirely different things.

Denis Diderot (1713–84), French philosopher. Apology for the Abbé de Prades (1752) repr. in Selected Writings (ed. by Lester G. Crocker, 1966).¹

Purification:  During the lessons on mixed strategies you may have thought that the very idea was absurd.  When making a decision that may seal your fate do you leave the choice of strategy to the flip of a coin?  Suppose that the score is tied, it is the bottom of the ninth, three women on base, the count is 3 balls and 2 strikes. K. C. Sosa (known as Casey) is at bat and her team is down by one run.

Cal E. Rippem is pitching.  In her repertory of pitches are the fastball and the slider. She knows that Casey always has trouble with the slider.  On the other hand, she also knows that Casey will be paying extra special attention at this critical stage of the game and that Casey has been taking practice at hitting the slider.  What to do? On this last pitch of the game 'Callie' will go home the hero or the goat.  Does it make sense for her to use a roll of the dice to pick the pitch she will throw?  Does it make sense to say that she will throw a weighted average of the curve and the slider?  Indeed, 'Callie' throws either a slider or a curve.  Once she begins her windup she is committed to a pure strategy.

But think of this last pitch of the game from Casey's viewpoint.   Standing in the batter's box Casey has a belief about the pitch that Callie will throw.  As far as Casey is concerned, the pitcher is playing a mixed strategy, even when there is only one pitch left in the game.

To make the transition from purification to revelation suppose that you are at an antiques auction in the countryside. In America such auctions are almost exclusively English auctions.  In such an auction the auctioneer invites oral bids.  The winner of the object is that person with the highest, last bid. The highest, last bid is determined when no one responds to the call for further bids and the auctioneer brings the gavel down for the last time on 'going, going, gone!'

You are a bidder in the audience and you know perfectly well the value that you attach to the object being auctioned. Your pure strategy is to keep bidding until the going price exceeds your valuation of the object or everyone else drops out, whichever comes first.  If the price rises above your reservation price then you suffer no harm and you still have your money in your pocket.  If everyone drops out before the price rises to your reservation price then you are a net gainer; you got the object for less than you would be willing to pay. Would you ever drop out before the price rose to the level of your reservation price?

There are about 50 other people in the audience, each of whom has her own private valuation of the object. The function of the auctioneer is to induce the person in the audience with the highest valuation to reveal herself. From the auctioneer's perspective he is playing a game against a single opponent who draws her valuation of the object on the basis of a throw of the dice.

Before modeling auctions you should take a detour to a glossary page.

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1. The Columbia Dictionary of Quotations is licensed from Columbia University Press. Copyright © 1993, 1995 by Columbia University Press. All rights reserved.