[GrekeHaus]

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

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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.

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Apparently, you didn't read (or understand) my previous point. I'll elaborate.

The problem again here is that you're talking about a series which doesn't converge. Here's a simple example:

1-2+3-4+...

Person A sees this series and says

1+(-2+3)+(-4+5)+...=1+1+1+...=infty

Person B sees the series and says

(1-2)+(3-4)+(5-6)=...=-1-1-1-...=-infty

What you're looking at here is an example similar to this. It's a series that doesn't converge, so summing it has no meaning. You're taking the "Person A" approach. Here's the "Person B" approach to summing the same series.

Let the person flip until they are behind. For the same reasons as stated in the previous argument, they will be behind, with probability 1. Group these flips together and call the sum a_1. Now, let the person keep flipping until the sum is less than a_1. Call this next group of flips a_2. Keep doing this for each n until a_{n+1}-a_n-...-a_1<0.

In this way, we get a_n <=-1 for every n. So taking the sum gives

a_1+a_2+...<=-1-1-...=-infty

So now your winning strategy is actually losing you an infinite amount of money.

The *key* when calculating EV is that you're actually taking a limint as n->infinity. In this case, your EV will be -1/4*(average bet size). Simply summing the numbers in a way that supports your hypothesis doesn't work since you are summing a series that doesn't converge.