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Wagering with Zeno

Brian Hayes

Can't Win, Can't Lose, Can't Tie

I briefly discussed a version of the Zeno gambling process in an earlier column ("Follow the Money," September-October 2002). Since then I have continued to explore the game, trying to understand its long-term behavior and relate it to other models in probability theory. I've had only partial success, and so what follows is a progress report, presented in the hope that others will build on it.

A few properties of the Zeno game are easy to state. For example, the betting process appears to be fair (assuming that the coin being flipped is unbiased). Each player has the same odds of winning or losing each round, and the amount at risk is the same.

Another way of saying that the game is fair is that the expectation value for each player is 1/2. If you play many independent games, you should come out roughly even in the end. But an expectation value of 1/2 does not mean you should expect to go home with half the money at the end of a single game. Indeed, after the first coin toss, the game cannot possibly end in a tie.

But you can never go broke, either—at least not in a finite number of plays. However small your remaining wealth, the wagering rule says you can't risk more than half of it. Of course the same reasoning protects your opponent as well: If you can't lose everything, neither can you win it all.

Here's another observation: In the game-within-a-dream described above, all of the numbers mentioned have a distinctive appearance. They are fractions whose denominator is a power of 2. In other words, they are numbers of the form m /2 n , called dyadic rationals. Is this predilection for halves, fourths, eighths, sixteenths, etc., a peculiarity of that one example, or does the pattern carry over to all Zeno games?

The answer comes from an inductive argument. Suppose at some stage of the game your score is a dyadic rational, x, and is less than or equal to 1/2. Then the amount at stake in the next round of wagering is x/ 2, so that your new tally will be either x x/ 2 or x + x/ 2. But x x/ 2 is simply x/ 2, and x + x/ 2 is 3 x/ 2; both of these numbers are dyadic rationals. A similar (but messier) argument establishes the same result for values of x greater than 1/2. Thus if your score is ever a dyadic rational, it will remain one for the rest of the game. But the starting value, 1/2, is itself a dyadic rational, and so the only numbers that can ever arise in the game are fractions of the form m /2 n .

This line of argument actually yields a slightly stronger result. For a score x <1/2, the net effect of the gambling transaction is to multiply x by either 1/2 or 3/2. In either case, the denominator is doubled; as the game proceeds, the denominator increases monotonically. An important consequence is that the entire numerical process is nonrecurrent : In the course of a game you'll never see the same number twice. This is one reason the game can't end in a tie: After the first flip of the coin, the score can never find its way back to 1/2.

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