Max Bankroll for Max Variance

I know that one should expect swings of 100 units, but has anyone actually experienced being down more than 100 based not on poor play but on simple variance (natural swings in any game of chance).

In other words: if someone was to flat bet and count only so that the player/house edge is 0% and played 4 hours per day for months flat betting $1 each time, what would be the most units down at any particular point before things would cycle back down close to zero?

What is the highest number to expect theoretically? (Maximum Bankroll needed to allow for natural swings)

What is the highest number that anyone has actually experienced?

Thank you, my hand dealt tests show a variance of 80 over 5800 results but I know it must go higher.

I guess a simpler way of asking this question is:

What is the maximum lead heads can take in a coin flip contest before tails starts to make a slow comeback.

I know the theoretical answer may be quite high (200?) but a real-life answer might be more helpful.

You asked: "What is the maximum lead heads can take in a coin flip contest before tails starts to make a slow comeback?"

Once heads takes the lead, what makes you think that tails should make any kind of a comeback at all? If, at some point, heads takes, say, a lead of 50, from that moment on, is the coin more likely to start flipping tails? Should we expect tails to "catch up"? In fact, we should not.

On p. 81 of William Feller's classic text on probability theory, we find an incredible chart, illustrating the arc sine law. It shows that, if we toss a coin every second, for a full year, there is a 90% chance that the less fortunate side, or player, will actually not have the lead for more than 153 days. There is a 50% chance that he will be in the lead for 53 days or fewer, and an unbelievable 10% chance that he will not have the lead for more than two out of the 365 days!

Finally, there is one chance out of 20 (5%) that heads will hold the lead for almost 364.5 of the 365 days!!!

Don

"I know that one should expect swings of 100 units, but has anyone actually experienced being down more than 100 based not on poor play but on simple variance (natural swings in any game of chance)."

Probably everyone on the board! :-)

"In other words: if someone was to flat bet and count only so that the player/house edge is 0% and played 4 hours per day"

So, let's say 400 hands daily.

"for months flat betting $1 each time,"

Let's say a year: 365 x 400 = 14,600 hands.

"what would be the most units down at any particular point before things would cycle back down close to zero?"

See my other post. You do NOT expect things to "cycle back down to zero." Instead, the more you play, the farther from zero you would expect to be. If I play ten hands, just how far from zero could I possibly be? Ten. If I play 10,000 hands, would you expect to be within 10 of zero, or farther away?

"What is the highest number to expect theoretically? (Maximum Bankroll needed to allow for natural swings)."

After 14,600 hands, the mean result is zero. One s.d. is sqrt(14,600) = 121 units. So, you have almost a 16% chance of being down 121 units at the end of the year. However, you have double that probability, or a full 32% chance, of having been down 121 units at some point during the course of the year! (See BJA3, and the discussion of the "premature bumping into the barrier syndrome.")

There is also a 4.3% chance of being down 2 s.d.s at year's end, which is 242 units. So, there's better than one chance in 12 that, sometime during the year, you will have been down 242 units.

Get the idea? Choose bankroll accordingly. Season to taste. :-)

Don

I am not sure what you mean by �unit�. Different people measure them differently. In BJA Don suggested dividing what we sometimes call a Kelly Equivalent Bank into 400 units, and I think this is a fairly typical unit size.

Given that, a 100 unit downturn is quite common. For fixed betting levels, downturns of 400 units happen a lot. If you sized your bets that way, you would have about 13.5% chance of running a 400 unit bank. You would have about a 60% chance ruining a 100 unit bank.

Earlier this year, I had a weekend trip where I dropped 167 units. Later, just this month, I had two (longer) trips, back to back, where I dropped 82 and 88 units. This is a total 170 units. It was actually worse than that. During the last day of the last trip, I came back about 120 units. So the total downswing was somewhere around 300 units.

If you played a perfectly break-even game, then for any given bank, your risk of ruin would be 100%. Given any pre-determined amount B, you will eventually be in the hole B units. It is not a question of if; it is only a question of when.

For a 50-50 coin toss: If I have done the math right (not guaranteed), you can use the equation N=1000*B^2. This tell you about many flips are needed in order to have a 90% chance of hitting a barrier at �B. For B=100 units, this is 10^7, or about 10 Million. Of course, your finger may give out before that.

For a 50% chance or hitting B, the Number of Flips is a little more than 35*B^2.

If what you are saying is true, and I don't doubt it is, then an advantage player would either be lucky to win each year, or carry such a big bankroll to allow for the huge swings.

I don't see much difference, in any 12 month period, of someone with a 2% advantage and flipping a coin.

My new question is, how big does a bankroll need to be to allow for the large natural swings (with a less than 1% chance of ruin) for an advantage player with a small edge.

What is the maximum lead heads can take in a coin flip contest before tails starts to make a slow comeback?

I think the max lead would be infinity. There would always be the infinitesimal chance that tails would never be flipped.

kaycee

I knew I would make a mistake when I posted this.

I had incorrectly used 1/4 s the SD for a coin toss. This is not right.

Actually for this type of situation, the SD is 1. This means I was off by a factor of 4 on the SD, which translate to being off by a factor of 16 on N.

the correct figures are (approximately) 64*B^2 for 90% chance, and 2.2B^2 for 50%. Maybe another error that I missed.

I don't see much difference, in any 12 month period, of someone with a 2% advantage and flipping a coin.

BT has given you a good mathematical example, maybe an illustrative explanation would be helpful also. In the coin-flip case, you expect to win nothing on average. Because of this luck defines whether you win or lose.
If you had a 2% on the coin-flip you expect to win an extra 2 units every 100 seconds or flips. Now, you won't always win every sequence of 100 spins, but your results fluctuate around +2 units for every 100 trials rather than zero.
For example, you might get three sequences of 100 where you win 6 units, lose three units and then win three units-which gets you to +6 units, the mathematically expected win. Your luck is only average, but you are 6 units up.
If you have no edge you might get three sequences where you, say, lose 5 units, win three units and then win 2 units. You are left with no profit and break-even - but you have exactly the same amount of luck as our player with the 2% advantage in the first example.

Over time, the player with the advantage becomes less and less likely to be losing overall. The chance of losing overall, which is quite large on any small number of trials, never entirely goes away but becomes small, quite quickly in this example, so it that fails to be a consideration for practical purposes after a large number of trials.

To give a slightly more realistic (though less fun) example than BT gave, I have a small chance of losing my entire lifetime winnings from advantage play over the next year. But I am much more likely to be struck by lightning leaving my house in the morning. I have to take my chances with acts of God, so I might as well carry on gambling.

With probability 1, both heads and tails will exceed N over the other, for any given N, as the number of tosses goes to infinity.

Or at least, I think I remember my Markov processes.

"With probability 1, both heads and tails will exceed N over the other, for any given N, as the number of tosses goes to infinity."

... our player probably doesn't have that much time! :-)

Don

49/100
495/1000
4955/10,000
49555/100,000
495555/1,000,000

Look how it's possible to converge to 1/2 yet increase behind (or above) in flips with no "comming back". In 100 flips, I'm off 1 in in expectation. Up to a million flips, I'm off 4,445 flips in expectation but my win rate is still converging to .5.

You are sounding more like Clarke Can't each day.

In a period of 10 hours over several consecutive sessions I've lost over 124 units. I have another losing streak of 128 units in 9 hours. On the flipside I have won 155 units in 12 hours and 124 units in 6 hours. It can be a very streaky game.

It seems that the theoretical 'coin-flip' variance is one answer, but I'm now trying to get a sense of the worst case scenario for a real-world BJ player playing full-time.

Math Prof says the worst his bankroll dipped was 300 units before he started to recover.

Has anyone gone higher than this? Please indicate the bet spread you were using (1-8 units?).

Also, what is the average bet size on a downward swing with bets varying 1-8 units?

kenw,

My largest losing streak to date has been a particularly nasty 6-week period in which I lost over 600 units... I was hemorrhaging the Benjis like you wouldn't believe!

I played a variety of games during the stretch, some outside the U.S., but primarily I was playing DD and spreading 1:16 with an average bet of about 5.5 units.

Fortunately, I ended the streak and went on a tear, erasing the whole downturn and setting a new ATH. I guess that's the moral of the story: if you size your bets appropriately, your BR can withstand those inevitable bad streaks and allow you to get into the "long run", where your mathematical advantage will take over.

As for your question about average bet size for a 1:8 spread, that depends on your game and betting ramp. For me, during my losing streak the average was about 5.5 units with a 1:16 spread: with a 1:8 spread, the average would of course be lower, but not simply 1/2 the 1:16 spread average, because both spreads use the same "waiting bets".

Welcome to the wonderful world of blackjack!

Dog Hand

As I said above, using �units� as the measure is somewhat ambiguous.
In the post above, I did NOT say that 300 units was the worst downturn ever. It was the downturn that I had this year, I titled my post �Not That Rare� that this happens a lot.

I think the worst downturn I had was about 560 units .over 42 sessions.
That is not the worst that I have heard of. There was a player who posted on Green Chip that their team had lost several times that.