Buzz, whenever I said that your mistakes in this thread were blatant, I mispoke. Truth be told, it's really easy for anyone to confuse themself by drawing up complicated examples. But really the best way to go about this problem is to express it as simply & generally as possible and then see that this covers all possibilities. Running example after example will just allow one to make an error in logic somewhere that will lead you to believe you've found a counterexample, when in fact one has honestly just made an error somewhere.

An anecdote--when I first got into gambling (period) about 5 years ago, I read up on the Martingale system, wrote my own Matlab blackjack simulator, and convinced myself that I could change the EV of blackjack by modifying my bet size, essentially doubling bets when I win repeatedly (in theory to accentuate winning streaks and downplay losing streaks). I even went to lengths to justify this falsehood in my mind by repeating something I heard somewhere about how the players' ability to change their bet size is how they can beat blackjack (not realizing at the time that this referred to card-counting). I mentioned my theory to my brother, and in 2 seconds he told me I was full of it. He explained that no matter what I did with my bet size, I was still going to have the same EV in the game, just the amount wagered would vary.

Well, I got pissed and spent about 3 hours writing down series after series of possible outcomes, trying to demonstrate to myself that my system would yield a winning result in a -EV game. In the end, I realized that I had indeed just made a mistake somewhere, and that when I thought about it my brother's direct, simply way, I saw that this clearly captured all cases, and that I was wrong. (It turns out that modifying bet size affects variance--you can change the shape of the distribution of outcomes, putting greater weight in small wins at the expense of risking an unlikely catastrophe... martingale in a nutshell.)


Now the thing is, when you draw up a very complicated example, it's very difficult to pinpoint each error in logic. I suspect that in your example above, the problem may lie in the statement:

[ QUOTE ]
But if you scoop once and don’t lose at all, you’ll end up with more money than if you win half the pot twice and don’t lose at all.

[/ QUOTE ]

Somehow I get the feeling that this statement implies that you are changing the odds... but I have to admit, I haven't thought about this carefully enough to be sure this is where things have gone awry here. I just know that something in this line of reasoning is out of whack, because a simple, general analysis shows that these two draws have identical EV & risk, just as my brother *knew* with certainty that the EV of blackjack is axiomatically fixed (barring card-counting, etc).


Finally, I apologize for dragging this thing out in the other thread. It gave me satisfaction to point out your mistake then and before because I have been on the other end of it, which is immature. I promise I won't bring this thing up again.