MathExtremist
Joined: Aug 31, 2010
• Posts: 6526
September 7th, 2010 at 9:55:07 PM permalink
It's a bit early this year to be thinking about bell ringers, but the recent flurry of discussions on betting systems has gotten me to consider a new way of approaching this. (Maybe it's an old way and I'm plagiarizing. If so, please someone let me know.) And kudos to anyone who immediately gets where I'm going with this before reading through it.

If you haven't heard a bell choir, it's pretty cool. A dozen or more players, usually wearing matching outfits and gloves, each with a few bells. Each bell plays one note only, so you need many individual bells to play a song.

For the curious, here's a video of a bell choir in a non-traditional setting:
Guerrilla Handbell Strikeforce

For the thought experiment, consider a bell choir with one bell per person. That person only ever rings that one bell, and only when its time to play it. Fred only ever plays A2, while Tony only ever plays C3, etc. Playing the right notes at the right time is critical to all music, but in the handbell case, there is only one note per bell at all. Each bell has its own characteristics, primarily pitch, but also timbre and other aspects. There are typically a finite number of individual bells (and thus, individual notes) but when you combine them in different orders and volumes, you can make an infinite number of songs. You use as many players as it takes to cover all the parts.

I propose that a betting system is like a song. It is composed of a finite number of individual bets but combined in different orders and volumes (bet sizes). The corollary of this proposition is that each betting system can be deconstructed into its component parts and given to a separate player, just like a bell choir deconstructs a song and gives its component parts to separate players. A betting system can be further deconstructed so that each component is broken down further into individual bet sizes on a single wager. In other words, Fred only ever bets \$1 on Even, Tony only ever bets \$5 on 3rd Column, etc. You use as many players as it takes to cover all the bets.

If a betting system is so deconstructed, each player's part is a sequence of bets of a single amount on a single wager, separated by some period of time >= 0 of not betting. For example, Fred might make a \$1 bet on Even, wait 9 spins, make another \$1 bet on Even, wait some more, etc. He never does anything else. All the other players act similarly with respect to their wagers and bet sizes.

It should be plain to see that Fred doesn't have an edge over the house. After all, he's just flat-betting (with pauses). But so is every other player in this scenario - each player is doing nothing but flat-betting some amount on their chosen wager. I think everyone can recognize that none of those players, individually, have an edge over the house if all they're doing is flat-betting.

Now, suppose that all of those players are betting with my money. Fred might bet \$1 on Even, and then Tony might bet his \$5 on the 3rd Column two spins later, but its all with my cash. At the end of the day, both Fred and Tony, and all the other players, are flat-betting on their wagers with my money. None of them have an edge, and since they're only using my money, my money doesn't have an edge. But having them all play with my money is equivalent to reconstructing the betting system and having one player play it. In other words, my money doesn't have an edge using the component parts of a betting system, so it can't have an edge using the sum of the component parts of that system. A betting system is nothing more than flat-betting on several different wagers in parallel, and flat-betting doesn't beat the house edge, therefore no betting system can beat the house edge.

I know many of you knew the conclusion, but is this a useful analogy? Boring? Trite? Not seasonal?