The large circled numbers are the bankroll, starting at 3 dollars, de/increasing by the bet amount when losing/winning. The small black numbers are the expected number of days if trying for 32 days. I win on about 24 days and only lose on eight.
STRAW-MAN: But, wanting to play less boring games, where should our Happy-Enough-Ceiling generally be?
Well, remember that gambling means to go on
a ride, a random walk of your amount of money, your ‘bankroll’, away from what
you started with. If betting small amounts, the random walk is likely to stay
close to the expected average. That expected average of your funds slowly decreases. At a
French type roulette table, 1/37 of your bet is lost on average with every spin of the
But: The larger the bets, the more will the random walk jump widely and thus wildly up and down.
Now, thinking about this random walk, our Happy-Enough-Ceiling must be closer to the amount that you bring to the table than the amount you bring to the table. Why?
The casino’s strict limit is all the money you bring to the table. Your random walk will hit most likely first whatever limit is closer to where the random walk started. If the casino’s limit is closer than your limit, you usually lose. The casino’s limit is the money you bring to the table. Therefore, you must walk away at some point well below having doubled your money. That is why my Happy-Enough-Ceiling is at a modest 4 dollars if I start Martingale with 3 dollars – not because of Zen modesty.
STRAW-MAN: But is it Zen? If you are down to 2 dollars, you are in a new Here-and-Now with a new starting amount. Why go on playing with a 4 dollar ceiling? That limit is now as far as your new “starting amount”. There should be a personal lower limit, too, one above zero.
If you stop playing at 2 dollars, your lower limit is as close to the 3 dollars at the start as the ceiling at 4 dollars. So your random walk loses most days! Remember – we started this whole series with wanting to ensure more happy days than sad ones drowned in fermented spirits.
STRAW-MAN: But look, you said “Your random walk will hit most likely first whatever limit is closer to where the random walk started.” So, why not just have my ceiling, my upper limit, closer to my starting amount than my lower limit?
If you have a personal lower limit, why bring those dollars in the first place if you will not touch them? Like getting mugged? Moreover, if you re-evaluate the Here-and-Now again every time, you would also increase your ceiling after winning. That is the same as not having a ceiling and thus losing almost always.
STRAW-MAN: Ok, so the important is that I walk away at some point well below having doubled my money.
Yes. This explains differently why putting everything on red loses most days. Winning on red pays out double your bet, therefore, the upper limit is automatically at least as far away as the amount that you bet. If I bet all I have, my upper ceiling is automatically as far away from my starting amount as my starting amount is from zero. My ceiling is then as close as the casino’s zero limit. The start of the random walk is in the middle of these two borders, and so I lose again.
So, we must bet some amount well below our starting amount.
STRAW-MAN: But you also said that we must bet large amounts in order to avoid going down with the expected 1/37 on every spin of the roulette wheel. So, bringing 1000 dollars, bets should be well below this but high, say 500 dollars, and our upper ceiling is therefore automatically at 1500 dollars. You sure you win most days this way?
Well, draw the MW/M diagram. B is your bet. Your starting amount is 2B and your Happy-Enough-Ceiling at 3B. You win about 2 out of 3 days:
You could also bet 200 dollars on red each time, and walk out when hitting 1400 dollars. Most days, you go home with 1400 dollars.
STRAW-MAN: And how much do I still bring home on the other days?
Well, if we play for only 10 rounds before getting bored, those amounts vary, but if we stay about 25 to 30 rounds, we go almost always home with either 1400 dollars or zero.
STRAW-MAN: Hmm – that sounds somehow very wrong. If you lose 1/37 of your bet on average and bet 200 dollars, then you should be down by only 200 dollars after 37 times.
Yes, this is one example for why understanding deviations Δx is so important, why averages <x> are not enough, although averages are already beyond most university students in the arts and humanities and so misunderstood as to be looked down upon as being racism and sexism.
STRAW-MAN: Oh, let’s not go there today. Back to justifying this as potentially useful. I need a 100 kilo-dollar lifesaving surgery but only have 90 thousand. What can I do at the roulette table in order to maximize the number of worlds where I survive? The Happy-Enough-Ceiling is now forced upon me; it is 100 thousand; and all I learned here otherwise is to use large amounts.
Yes, but if you bet more than what you need in order to reach your Happy-Enough-Ceiling, you have less to proceed betting in case you lose. So, it is actually very easy as far as I can see.
STRAW-MAN: Bet 10 thousand dollars! Then, after losing, re-evaluate in the new Here-and-Now and bet 20 thousand, then 40 thousand?
A desperate Martingale, yes, and finally betting the last 20 thousand fully. Always bet the amount that is still missing or, if you have already less than that, all you have. As death approaches, re-evaluating the Here-and-Now becomes more important than deceiving yourself about that you are still the one who entered the casino in order to fool yourself about being a winner relative to him.