Another caricature of escalation is a game devised by Douglas Hofstadter. It is known as the “luring lottery” or “largest-number game.”
Many contests allow unlimited entries, and most of us have at some time daydreamed about sending in millions of entries to better the chance of winning. The largest-number game is such a contest, one that costs nothing to enter and allows a person to submit an unlimited number of entries.1 Each contestant must act alone. The rules strictly forbid team play, pools, deals, or any kind of communication among contestants.
All entries have an equal chance of winning. On the night of the drawing, the contest’s sponsors pick one entry at random and announce the lucky winner. He or she wins up to a million dollars, according to rules of the contest.
When the game’s generous sponsors say “unlimited” entries, they mean it. A million entries – a billion entries – a trillion entries – all are perfectly okay. The number must of course be finite, and it must also be a whole number. Just so you understand how it works: suppose that 1 million other people have entered the contest, and all of them – meek souls – have submitted a single entry. Then you submit a million entries yourself. That makes 2 million entries in all. Your chance of winning would be 1 million/2 million, or 50 percent. But if someone else came along and submitted 8 million more entries just before the deadline, there would be 10 million entries total, and your chance of winning would drop to 1 million/10 million, or 10 percent. The person with the 8 million entries would then stand an 80 percent chance of winning.
Obviously, it’s to your advantage to submit as many entries as possible. You want to submit more than anyone else; ideally, a lot more than all the other contestants put together. Of course, everyone wants to do this. There is a practical limit to how many entries anyone could fill out and send in. Besides, the postage would be prohibitive...
No problem! The rules say that all you have to do is send in a single 3 by 5-inch card with your name and address and a number indicating how many entries you wish to submit. If you want to submit a quintillion entries, just send in one card with “1,000,000,000,000,000,000” written on it. It’s that simple. You don’t even have to write out all those zeros. You can use scientific notation, or the names of big numbers (like “googol” or “megiston”), or even high-powered exotica like the Knuth arrow notation. Qualified mathematicians will read each card submitted and make sure everyone gets proper weight in the random drawing. You can even invent your own system for naming big numbers, as long as there is room to describe it and use it on the 3 by 5 card.
There is one catch.
A minuscule asterisk on the lottery posters directs your attention to a note in fine print saying that the prize money will be $1 million divided by the total number of entries received. It also says that if no entries are received, nothing will be awarded.
Think about that. If a thousand people enter and each submits a single entry, the total entries are 1,000 and the prize money is $1 million/1000, or $1,000. That’s nothing to cry in your beer over, but it’s no million.
If just one person is so bold as to submit a million entries, the maximum possible prize money plummets to 1 lousy dollar. That’s because there will consequently be at least a million entries (probably many more.) If anyone submits 100 million entries, the prize can be no more than a cent, and if more than 200 million entries are received, the “prize,” rounded down to nearest cent, will be nothing at all!
The insidious thing about the largest-number game is the slippery way it pits individual and group interests against one another. Despite the foregoing, the game is not a sham. Its sponsors have put a cool $1 million into an escrow account. They are fully prepared to award the full million – provided, of course, that just one entry is received. They are prepared to award any applicable fraction of the million. It would be a shame if someone doesn’t win something.
How would you play this game? Would you enter, and if so, how many entries would you submit?
You’ll probably agree that it makes no sense even to think about submitting more than 200 million entries. That unilateral act, on the part of anybody, effectively wipes out the prize. Fine. Extend this reasoning. Maybe you feel that a dollar is the smallest amount worth bothering over. Or you reason that people’s time is worth something. Estimate how long it takes to fill out a 3 by 5 card (including the time spent deliberating) and multiply this by a minimum wage or a billing rate. Add in the price of a postage stamp. The result is what it costs to enter the contest. It makes no sense to submit a number of entries that would itself suffice to lower the prize money below the threshold of profit.
Taking this to an extreme, maybe the best plan is to submit just one entry. Maybe everyone else will do the same. Then at least everyone will have a “fair chance” – whatever that means in a largest-number game.
Maybe you shouldn’t enter the contest at all. Then no one could say you weren’t doing your share to keep the prize money high. The thing is, the largest-number game is like all lotteries: if you don’t play, you can’t win. Worse yet, if everyone decided not to play, no one would win.
In the June 1983 issue of Scientific American, Douglas Hofstadter announced a largest-number game open to anyone who sent in postcards before midnight, June 30, 1983. The prize was $1 million divided by the total number of entries. The management of Scientific American agreed to supply any needed prize money.
The result was predictable. Many of the correspondents sought to win rather than to maximize winnings. The game became, in effect, a contest to see who could specify the biggest integer and thereby get his name in the magazine (as “winner” of an all but infinitesimal fraction of a cent). Nine people submitted a googol entries. They had their chances wiped out by the fourteen contestants who submitted a googolplex. Some people filled their postcard with nines in microscopic script (designating a number bigger than a googol but much, much smaller than a googolplex). Others used exponents, factorials, and self-defined operations to specify much bigger numbers, and some crammed the postcard with complex formulas and definitions. Hofstadter couldn’t decide which of the latter entries was the biggest, and no one got his name in the magazine. Of course, it didn’t matter financially who won. The prize amount would round to zero; and had Scientific American issued a check for the exact amount, it would have had to employ the same mathematical hieroglyphics the winning contestant had used, and no bank would accept it!
Greed is part and parcel of the largest-number game, but not the incentive to be thought clever by naming the biggest number. In a true largest-number game, the players are motivated only by maximizing personal gain. The real question of how to play the game remains.
No matter what, only one person can win. The best possible outcome of the game, then, is for the full million dollars to be disbursed to the winner. This occurs only when there is a single entry. If the contestants were permitted to work out some share-the-wealth scheme and coordinate their actions – which they are not – they would surely arrange to have just one person submit an entry.
The largest-number game is almost a mirror image of the volunteer’s dilemma. You want everyone except one person to volunteer not to enter. The ideal would be to draw straws so that exactly one person gets the go-ahead to enter. This unfortunately requires explicit communication among the players, which is not permitted.
The game appears to be one of pure defection, with no cooperative way to play. That’s not so. There is a collectively rational way to play the largest-number game (or a volunteer’s dilemma). It is a mixed strategy: for each and every person to decide whether to volunteer based on a random event such as a roll of dice.
Each person can roll his own dice independently of the others, without any need of communication. If there are thirty-six players, then each might roll two dice and submit a single entry only if snake eyes came up (a 1 in 36 chance).2
This would work just fine. But the mind reels at such “rationality.” This isn’t what happened in the Scientific American lottery, and it is hard to imagine any mass of living, breathing human beings acting as prescribed. The temptation to cheat is superhuman. You roll your pair of dice and it comes up snake eyes – oops, no, the one die hit the edge and now it shows a three! You don’t get to enter. Who would know if you helped Dame Fortune along and flipped the die back over? No one! Everyone rolls the dice in the privacy of his home and acts independently. Who would know if you rolled a fair and square four and entered anyway? No one!
This rationalization doesn’t change things one bit. If everyone reasons like this, everyone will enter and wipe out the prize. It’s like the volunteer’s dilemma, only worse. In a volunteer’s dilemma, everyone is penalized according to the proportion of defectors among the players. In the largest-number game, even one defector can ruin it for everyone.
So what do you do? Do you go through the almost pointless exercise of calculating odds and rolling dice (knowing full well that most others are doing no such thing, and that some are already filling cards with as many nines as the points of their pencils will permit), or do you fill a card with nines yourself? The only reasonable conclusion is that the largest-number game is a hopeless situation.
1. Stories of misguided people who try to win state lottery jackpots by buying a huge number of tickets occur in the news now and then. The most publicized case of massive multiple entries in a free lottery occurred in 1974. Caltech students Steve Klein, Dave Novikoff, and Barry Megdal submitted about 1.1 million entries on 3 by 5 pieces of paper to a McDonald’s restaurant sweepstakes. The Caltech entries amounted to about one fifth of the total entries, winning the students a station wagon, $3,000 cash, and about $1,500 in food gift certificates. Other contestants were outraged. Many quoted in the press were sure they had been cheated out of something, even though the rules explicitly allowed unlimited entries. “We were disappointed to hear of the sabotage of your efforts by a few ‘cuties’!” a man from Orange, California, wrote McDonald’s. “Undoubtedly all giveaway contests in the future will have to be safeguarded in some way from similar tactics by ‘punks’ such as these.” Those sharing this man’s feelings are advised to avoid the largest-number game!
2. Actually, it’s best to adjust the odds slightly. The trouble with giving each of n persons a 1/n chance is that there’s still a fairly big chance that no one will get the go-ahead to enter. By the laws of probability, this chance is about 37 percent when n is a few dozen or more. This is balanced by the chance that two or three or more people will get the go-ahead.
A better plan is to allow each person a 1.5/n chance of getting the go-ahead. This decreases the chance of no one entering at the expense of increasing somewhat the chance of extra entries.
William Poundstone, Prisoner’s Dilemma, Doubleday, NY 1992, pp. 272-276.