Martingale changes the House Edge

Consider a game of roulette (European, 37 numbers).

Player A is placing his chips on 27 numbers (that would mean 27 chips, but you can lower it to $9, for example, by placing $1 on 9 triplets). His 'lottery' is: lose $9 (prob 10/37) or win 11-8= $3 (prob 27/37).
His EV is $-9/37 and the house edge is, accordingly, 1/37. That is the standard reckoning of the HE of roulette.

Player B is using a classical 2-stage martingale: bet $3 on red, and if lost bet $6 on red again. She will either win $3 or lose $9, just like player A. But the probabilities here are WIN (18/37)+(19/37)(18/37) = 27.24/37 ; LOSS (19/37)(19/37) = 9.76/37. Consequently, she has better odds than player A, for the same outcomes.
Her EV is $-6.081/37 and the House Edge is 0.6757/37. Standard gambling advice would make this a better bet.

Conclusion : using a martingale effectively changes the house edge. Betting systems may be useless in the sense that they don't make the HE positive, but they DO change the terms of exchange.

Now, one would say "Hey, this is wrong because her bet is not $9." This is the position adopted by the Wizard when introducing the Element of Risk: in some cases she bets $3, in others she bets $9. The average total bet is $6.081 and we find an EoR of 1/37. Morality is saved!

Hmm.. OK, but which of the two bets would YOU prefer? It seems to me that B is objectively preferable to A. In this case HE is a better indicator than EoR.
At the onset, player B has decided she is waging $9 on her bet. It doesn't make a difference whether the "win" result happens before the second part of it has moved on the felt.

Oh my god. People have been trying to figure out a mathematical way to beat the house for centuries, if not millennia, but it took until 2014, and YOU, to finally figure it out, using nothing but common algebra!

No no no!
This in no way "beats the house".
I am just discussing the relative qualities of House Edge and Element of Risk measures. I didn't put this in the Betting Systems forum to prevent some minus habentes from believing I warrant betting systems.

Quote: kubikulann

But the probabilities here are WIN (18/37)+(19/37)(18/37) = 27.24/37 ; LOSS (19/37)(19/37) = 9.76/37. Consequently, she has better odds than player A, for the same outcomes.

Why on earth would the probabilities of winning or losing change based on how much you bet?

Because the amount of information is not the same. And THAT is what determines probabilities.

Quote: paisiello

Quote: kubikulann

But the probabilities here are WIN (18/37)+(19/37)(18/37) = 27.24/37 ; LOSS (19/37)(19/37) = 9.76/37. Consequently, she has better odds than player A, for the same outcomes.

Why on earth would the probabilities of winning or losing change based on how much you bet?

It's not based on how much you bet, it's based on considering two sequential outcomes as a single outcome.

Wx = W
LW = W
LL = L

And what information do you think has changed exactly?

Imagine player B winning her $3 in the first step, and then placing $3 each on red and black for her secound round.
That would create the same result as player A's.

What it changes? The 1/37 she leaves to the house for the Zero.

By martingaling, she spares that vigorish. This is what makes her strategy better.

I know we see martingale and we want to dog pile the poster, but looking over the math quickly OP has a point:

Bettor A has 27% chance of losing $9 and 73% chance of winning $3
Bettor B has 26.4% chance of losing $9 and 73.6% chance of winning $3.

I'm going to step away and try and come up with a good explanation for this, but for now it's an interesting observation.

(if you've been around here for a while, you know i'm not crazy. There's something here. Not house beating, but interesting)

Quote: dwheatley

I know we see martingale and we want to dog pile the poster, but looking over the math quickly OP has a point:

Bettor A has 27% chance of losing $9 and 73% chance of winning $3
Bettor B has 26.4% chance of losing $9 and 73.6% chance of winning $3.

I'm going to step away and try and come up with a good explanation for this, but for now it's an interesting observation.

(if you've been around here for a while, you know i'm not crazy. There's something here. Not house beating, but interesting)

House edge is measured in percentage of bet amount. The amounts bet are different.

Player A bets $9 once.
Player B bets $3 18 times out of 37, and bets $9 (total) 19 times out of 37. So Player B is only betting a little over $6.08 on average.

Since player B bets a little over 2/3 of what player A bets (on average), her expected loss is a little over 2/3 as much.

All this shows is that betting less money in a negative expectation game causes you to lose less.

Quote: AxiomOfChoice

All this shows is that betting less money in a negative expectation game causes you to lose less.

Exactly.

All this shows is that using a martingale makes you bet less for the same aim, wouldn't you say?

Quote: kubikulann

Exactly.

All this shows is that using a martingale makes you bet less for the same aim, wouldn't you say?

I would say that any betting strategy that gets less money on the table is a good one in a negative expectation game.

I think that the discrepancy is that you are calculating based on possible amount won or lost, while the house edge is calculated based on total amount bet. Bets with longer odds are going to favor you if you use the first calculation (because you put less money on the table). Player A is essentially laying odds, while Player B is making even-money bets. Player C could do even better by picking bets with longer odds. Why not start with a 25c bet on a single number?

Here is a similar concept. Imagine that you want to either win or lose $1000, betting on red. If you put all $1000 on a single bet, your expected loss is the house edge * 1000. If you keep making $100 bets until you are either up or down $1000, your expected loss is your house edge * the average amount bet (which will be much more than $1000).

So, yes, any strategy that minimizes the amount of money bet will minimize your losses. Which, of course, leads to the ultimate strategy of not playing.

Quote: kubikulann

I didn't put this in the Betting Systems forum to prevent some minus habentes from believing I warrant betting systems.

I don't know what we're talking about, but I voted 5 stars for the insult :)

The thing about the house edge is that it comes from the house's perspective, so the initial bet is the initial bet from their perspective. They see bets of $3 and separate bets of $6, not a bet of $3 that's raised to $6. It's a significantly higher edge than simply betting $3.

Quote: 24Bingo

The thing about the house edge is that it comes from the house's perspective, so the initial bet is the initial bet from their perspective. They see bets of $3 and separate bets of $6, not a bet of $3 that's raised to $6. It's a significantly higher edge than simply betting $3.

It seems to me that the issue with a Martingale is not that it works at small amounts on a choppy table (which is why so many people try it); it's that it rubs up against the maximum when it goes south on a losing streak for a catastrophic loss vs. very small profit when it finally wins (getting back to even, then hoping for one or more base bet wins). All this example does is illustrate that in miniature.