Tuesday, February 2, 2010

The Parable of Bob and the Bill

[letter excerpts]

Posit: Day-old bread is better than nothing. Posit: Nothing is better than chocolate. Conclusion: Day-old bread is better than chocolate.

Once there was a man named Bob who attended a charity auction. The auctioneer explained that this was a pay-as-you-bid auction, like poker: you pay as you bid, and if you lose you don't get your money back. "That's dumb," thought Bob, "but I guess if it's for charity…" Then the auctioneer unveiled the item up for bid—a twenty-dollar bill. Bob, who considered himself a very rational person, looked around the room at the other thirty-odd people and thought, "If I bid and lose, I lose everything. If I win, I get twenty dollars. I have a one in thirty chance of winning twenty dollars, so the bid is worth $(1/30 * 20) = 67 cents to me." He immediately bid a quarter, hoping to make a profit, and put his quarter out in front of him.

Another bidder bid fifty cents, and someone else bid seventy-five cents. "Going once," said the auctioneer. Bob frowned in thought. "Odd. There's only three of us bidding. That raises my expected value to $6.67, so I can afford to bid a dollar rationally." The other bidders, after some hesitation, matched his bid and soon the price was at $5. Bob shook his head, "Let's end this," and bid $6.67. Further bidding would be irrational. Bob would pocket the $20, and they could all go home.

The third bidder immediately bid $10. "Oops," said Bob, now seeing the flaw in his reasoning. "No matter what the odds are, the payoff for winning is $20, not $6.67, so of course other people will bid above $6.67. My profit is the difference between my winning bid and $20, not between my winning bid and $6.67." So he bid $15, hoping for a 33% profit on his investment. Bidder #2 bid $17, and bidder #3 bid $18. Grumbling, Bob bid $19.99. Bidder #3 sat down abruptly. Bidder #2 hesitated again, then bid $21.

Bob couldn't believe it. "Who would bid $21 for a $20 bill?" he said to himself. Then with a sudden sick shock he got it. "No," he said, "bidder #2 already bid $17. If he loses, he lost $17. If he wins at $21, he lost only $1. Clearly it is rational to bid again to reduce his losses." And with the same sick certainty, Bob knew the same logic applied to himself. Bob bid: $25. Response: $28. "Again," thought Bob, "I have the choice between losing everything, only it's $25 instead of $19.99 this time, or paying $4 for a 50% chance at winning the $20. 50% of $20 is $10, and $4 is less than $10 so clearly the only rational action is to invest another $4." Bob bid $29, hopelessly, while the rest of the audience watched in fascination. This could have gone on for quite a long time but Bob's wife hit him with her purse and told him to stop being a nit. Bidder #2 won at $30. The auctioneer thanked them all, gave $20 to bidder #2, and put their combined bids ($18 + $29 + $30 = $77) in the charity cashbox. Everyone applauded and Bob went to lie down.

Last story for today: my dad went to med school in Louisiana when I was one or two years old. Some of my earliest memories are of Louisiana and Tulane University—going for stroller rides with my mom, playing in the pool, etc. I'm not sure but I think I might even remember the flood that buried the streets in over a foot of water. (Then again, I might be getting it mixed up with the floods from my mission, which I definitely remember, including what it did to our basement rooms: it was like an indoor swimming pool, which is bad news for missionaries. Or an indoor baptismal font, which is good news I suppose.) In particular I remember a large room, which must have been at Tulane, and one wall of the room was covered with transparent glass cubes or jars, stacked one on top of the other. In each jar was yellow liquid, and in the liquid floated a dead baby. I remember two babies in particular. One had a big corkscrew of a belly button that seemed several inches long and reminded me of a screw. The other baby had two heads. I thought that was quite interesting. I don't remember if this was before or after I saw the two-headed snake at the Santa Ana zoo, which was interesting because it was alive, but I think the baby was even more interesting because it was human and had arms and legs and stuff, which snakes do not. Anyway, I don't remember feeling scared at all but I wasn't sure why the liquid was yellow and I think I drew the obvious conclusion for a two-year-old at that—especially a two-year-old familiar with swimming pools and babies—and perhaps that did disturb me a little.

"When people are married, instead of trying to get rid of each other, reflect that you have made your choice, and strive to honour and keep it." --Brigham Young

If you're so evil, eat this kitten!

No comments: