Martingale for SNGs

[Pokey]

[ QUOTE ]
The sum of a series of -EV wagers will never be positive no matter how you sequence them.

[/ QUOTE ]

This statement that sounds so logical is actually incorrect because of two bizarre assumptions in this problem:

1. Infinite number of possible bets.
2. Ever-increasing wagers.

Here's a thought exercise to prove the point. Let's say that we play a game. We flip a fair coin. If it comes up heads, you pay me 1/2 my bet. If it comes up tails, you win my entire bet. Every individual bet is clearly +EV for you. You promise to play as many times as I want, and you promise to offer me as large a "tab" as I want. I can guarantee myself that, with enough time, I can bankrupt you, no matter how large your (finite) starting cash is. Here's how:

1. If I haven't bankrupted you, I bet $2.
2. Flip the coin. If it comes up heads, go to step #1 ahead $1.
3. If it comes up tails, triple my previous bet and go to step #2.

Run that sequence forward in your mind. Divide it into "sets" where each set ends with a flip of heads. The probability of a set continuing to N flips is (1/2)^N. KEY POINT: since N can be infinite by (nonsensical) assumption, the probability that a set never ends becomes (1/2)^infinity, which limit theory tells us equals zero. So every set WILL end, and whenever a set ends, the sum total of all my bets and wins equals exactly $1. I can play as long as I want (by assumption), and you'll extend me all the credit I want (also by assumption), so I can bankrupt you, guaranteed.

Note that along the way, there will be plenty of times when I'm *seriously* in the hole, but because you allow me to continue increasing my bet to whatever amount I want, it doesn't matter.

Note also that everybody's intuition that -EV bets can't be bad remains true in all real-world situations; since nobody has an infinite bankroll and since no casino allows infinite betting, none of this matters in the real world, and martingale betting techniques will eventually lose you money.

Final side note: if you use a martingale betting method, you don't change your expected value, but you DO change the distribution of outcomes. A martingale betting strategy will let you win far more often than you lose, but your losses will be so staggeringly large that they wind up more than offsetting all your wins in the long run.