David Chalmers (insane brilliant philosopher of mind in Austrailia) has apparently written not just one paper on the two-envelope thing, but two papers on the two envelope thing! (Cross-reference: How to make people crazy with probability.)
I have it on good authority from my Mathematician Friend (TM) that the answer is that there’s no probability distribution on the real numbers. I suspect he wanted to add “so shut up about it already!!”
Will this be another 300 comment post? At Marginal Revolution, there’s a discussion of the Tullock Lottery. The summary from wikipedia is as follows:
The setup involves an auctioneer who volunteers to auction off a dollar bill with the following rule- the dollar goes to the highest bidder, who pays the amount he bid. The second-highest bidder also must pay the highest amount that he bid, but gets nothing in return. Suppose that the game begins with one of the players bidding 1 cent, hoping to make a 99 cent profit. He will quickly be outbid by another player bidding 2 cents, as a 98 cent profit is still desirable. However, a problem becomes evident as soon as the bidding reaches 99 cents. Supposing that the other player had bid 98 cents, they now have the choice of losing the 98 cents or bidding a dollar even, which would make their profit zero. After that, the original player has a choice of either losing 99 cents or bidding $1.01, and only losing one cent. After this point the two players continue to bid the value up well beyond the dollar, and neither stands to profit.
Here’s my two cents. I don’t think it’s really a paradox. Some chap named Keith says the following:
If there is no equilibrium, avoiding the game altogether is
not an equilibrium, either. After all, if nobody bids at
all, then you should a penny and win.
It’s a paradox. Playing is a mistake. If nobody plays,
then not playing is a mistake, too.
But that can’t be right. If we assume complete information and rationality on behalf of the bidders, any given bidder A will know the extent of the universe of other potential bidders. So A’s utility calculuation while considering her first bid will be as follows. If she fails to bid, her expected gain is zero. If she does bid, then she must assume that everyone else has exactly the same utility calculation, and conclude that her expected gain is – all her money.
Can someone who knows more game theory than I chime in here? (Is anyone actually reading this blog yet?) I don’t think I understand how nonparticipation is not an equilibrium. The claim that there is no equilibrium seems to rely, sub rosa, on each actor’s not having access to the utility calculations undertaken by the others.
Indeed, for once it looks like wikipedia gets it right. From the wikipedia page again:
The actual expected value of bidding again is not Zero cents due to the unterminated nature of the game; the value of the bid is actually zero cents multiplied by the possibility of the other player giving up at that point, added to the value of losing two cents multiplied by the probability of the other player giving up at that point, in an infinite series with unbounded loss.
Exactly! The operative phrase there is “multiplied by the possiblity of the other player giving up at that point.” Which is zero, for the reasons given. If A has no reason to give up, then neither does the other player. It is concluded that the expected value of bidding is negative at all points. I’d suspect the experimental results to the contrary result from subjects irrationally failing to equate the opponent’s expected actions with their own and concluding that their opponent will give up at some point. I daresay it’s necessary to the bid decision to conclude that your opponent will give in sooner than you will.
Although this is all uninformed speculation. Someone clue me in? Please?
There’re a bunch of probability puzzles that seem to drive people crazy even today. Beyond the classic Monty Hall problem, there’s the puzzle about picking between two envelopes, where one has twice the amount of money as the other. The second of those problems gave rise to over 300 debating comments in its recent appearance in the Volokh Conspiracy. It is also broken down here.
If you ever want to really bust up a party full of people who are intelligent and aggressive, but don’t know probability, try that one. It’s guaranteed to turn a group of otherwise normal people into snarling math-maniacs.