From: Robert Tinker Subject: Re: Parrondo's Paradox (Science Times) Date: Sun, 30 Jan 2000 21:26:27 GMT Newsgroups: sci.math,misc.invest.stocks Summary: [missing] In article <3893bfac.4845763@news.globalcenter.net>, dcd@firstnethou.com (Dan Day) wrote: > Changing the subject slightly, my favorite counterintuitive > gambling game is the following: > > 1. You and I are sitting at the Roulette wheel, and we > decide to place side bets directly with each other, > bypassing the house percentage on table bets, so that > our money won't be siphoned off to the casino. > > 2. We bet on whether the spins on the Roulette wheel > fall red or black (we ignore any spins that hit 0 or 00). > Since there are an even number of red and black slots, > this is an even bet. > > 3. But to make things more interesting, we bet not on a > single spin of the wheel, but we instead each pick a three-spin > sequence, like red-red-black, or black-red-black, or red-red-red, > or whatever. We follow the subsequent spins of the ball, writing > them all down as they occur, and if your chosen sequence occurs > before mine does (i.e. on the three most recent spins), you win, > if mine "hits" before yours, you win. After someone wins, > each player is welcome to choose a new color sequence, or > keep playing the old one. > > 4. Since any of the eight three-color sequences is as likely to > occur as any other, the odds ought to be even. But to make > it worth your while, I'll pay you $21 if your sequence > comes up before mine, and you only have to pay me $20 if > mine comes up first. > > 5. I'll even let you choose first. And if after a while > you suspect that against all logic my sequence is somehow better > than yours (they're all equally likely to occur, of course), > you're welcome to switch to it (after someone wins the current > round), and I'll pick another sequence. > > Do you accept this seemingly can't lose proposition? > Wow. Interesting game. I struggled with this then ran a simulation. Seems like for every sequence Player A chooses, Player B can choose a sequence that crushes Player A. I still don't know why, though. Tinker Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: dcd@firstnethou.com (Dan Day) Subject: Re: Parrondo's Paradox (Science Times) Date: Fri, 04 Feb 2000 22:35:40 GMT Newsgroups: sci.math,misc.invest.stocks On Sun, 30 Jan 2000 21:26:27 GMT, Robert Tinker wrote: >> >> Do you accept this seemingly can't lose proposition? > >Wow. Interesting game. I struggled with this then ran a simulation. >Seems like for every sequence Player A chooses, Player B can choose >a sequence that crushes Player A. Yup! And by a pretty good margin, too. The amazing thing is that it *looks* even-up. (Thanks for responding -- I didn't think anyone would bite.) > I still don't know why, though. The reason is that player B, who chooses his sequence second, can "undercut" player A by choosing a three-spin sequence that ends with the same two colors that player A's sequence begins with. For example: A chooses B-R-B (starts with B-R) B chooses B-B-R (ends with B-R) For A to win, he'd (obviously) have to get B-R-B on any three consecutive spins of the wheel. But that means that "B-R" had to have ALREADY occurred, and that's two thirds of player B's winning sequence, making the odds good that B had *already* won before A could. The ONLY way that A can win is to get: 1. B-R-B on the very first three spins (only 1/8th chance) 2. R-B-R-B on the last four consecutive spins (the last three being his winning combo). He *can't* win on a series of spins ending with "B-B-R-B", because player B would then have ALREADY won (look at the first three colors in that four-spin sequence -- it's B's winning combo!) Not counting the "win immediately" case (#1) above, by "pre-empting" player A's sequence, B actual STEALS HALF of A's potential wins a single spin before A would have won the money. Heh. Heh. Heh. And if A suspects something is up and decides that B's "B-B-R" is somehow a superior sequence, and switches to it, then B just picks either R-B-B and then continues to cream A... B's rule for choosing a combo is: Slot 1: Opposite color of A's slot #2 Slot 2: Same color as A's slot #1 Slot 3: Same color as A's slot #2 The rules for the choice of slots 2 and 3 are obvious, for reasons given earlier (end with what A's combo starts with), but the rule for slot #1 needs isn't so transparent. The reason is that otherwise, it might be possible for B to accidentally put himself in the same boat that he's put A into -- having his first two colors be the same as A's last two colors. The rule for slot #1 makes that impossible, by choosing a different starting color than A's middle color. -- "How strangely will the Tools of a Tyrant pervert the plain Meaning of Words!" --Samuel Adams (1722-1803), letter to John Pitts, January 21, 1776