P90 Joined: Jan 8, 2011
• Posts: 1703
December 7th, 2011 at 5:20:58 AM permalink
This is a semi-practical challenge I've come across recently. While I could really ignore it, the challenge itself has sparked my interest, as this is clearly a mathematical problem that could be solved.

You have a virtual game of blackjack with known rules, minimum and maximum bets. However, individual bets are not tracked: you only know how many chips the player came in with, how many he left with, and how many hands were played. Using only this information, you need to estimate player's action, i.e. the total amount bet.
Clearly you can't get an accurate read, but you don't have to, as the results can average out over multiple sessions.

You can tell at least, ignoring doubles and splits, that it's no more than hands*max.bet, no less than hands*min.bet, and no less than total change in bankroll. But can you get something more specific, estimate the most likely bet size, relying on variance and change in bankroll, and how specifically would you go about it?
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boymimbo Joined: Nov 12, 2009
• Posts: 5994
December 7th, 2011 at 6:56:57 AM permalink
I think the variance in blackjack is too wide to get an answer to that question. You know what the house advantage is as well. So, the only thing you can do is match the real loss with the expected loss.

Real loss = Chips started with - chips left with.

Expected loss = House advantage * number of hands * average bet.

The only assumption you can make is expected loss = real loss.

Average bet = (Chips started with - chips left with) / (House Advantage * Number of hands).
where Average bet (min) = minimum bet
and
Average bet (max) = maximum bet.

Real world. Person comes with \$500, leaves with \$300. Plays 200 hands. Assume 1.5%.

Average bet = (500 - 200) / (.015 * 200) = 300 / 30 = \$10.

Of course, the variance is far too wide for this assumption to hold true. For example, what is the average bet when the person comes out a winner (less than 0, apparently).
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DJTeddyBear Joined: Nov 2, 2009
• Posts: 10200
December 7th, 2011 at 7:33:18 AM permalink
Boy -

I didn't look too deeply into your equation, but I'd say that's not the average bet, but the average loss per hand.

And, if the person is a winner, then the same formula produces the average win per hand.
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boymimbo Joined: Nov 12, 2009
• Posts: 5994
December 7th, 2011 at 8:00:45 AM permalink
No, the average loss per hand is unknown. It's expressed as the house advantage x the average bet.

Let's try this again

Let A = Average Bet, H = House Advantage, S = Starting chips, E = Ending Chips, and N = number of hands.

L = Total Loss = S - E

Q = Total Expected Loss = H * A * N

You *must* assume that Q = L. That is the best assumption you can make absent any other information. I can't figure out any other method to estimate average bet.

Therefore (S-E)= H*A*N, and

A = (S-E) / (H*N).

Another example, player starts with \$10,000, leaves with \$8,000, plays 100 hands. House advantage is the assumed 1.5% for blackjack (real advantage + 1% for player error).

A = (10,000 - 8,000)/(.015*100) = 2,000 / 15 = \$133.33 = Average Bet.

Of course, the variance is large enough that you could win, in which case the Average Bet would be below zero, meaning that you would have to set Average Bet to the minimum bet.

Because variance and bet sizes differ so much, that's why it's important for the pit to record the bet size from time to time.

There may be a way to estimate average bet based on a statistical model of bankroll, time of day, gender and age of the player. For example, you might have a known graph of average bet for blackjack based on a male 30 year old bringing \$1,000 to a \$10 dollar table on a Saturday night (=\$25). Whereas it might be more likely that on a Tuesday afternoon, a 65 year old female bringing \$100 to the table on a \$5 dollar table is probably betting the minimum.... but the problem doesn't state that information is known.

I guess that as a person estimating comps, I would make the assumptions based on the greatest of the following:

(1) the minimum bet
(2) a percent of bankroll (such as 2%).
(3) the formula above
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P90 Joined: Jan 8, 2011
• Posts: 1703
December 7th, 2011 at 9:41:58 AM permalink
Quote: boymimbo

I think the variance in blackjack is too wide to get an answer to that question. You know what the house advantage is as well. So, the only thing you can do is match the real loss with the expected loss.
The only assumption you can make is expected loss = real loss.

That would only hold for an extremely large number of hands, say 10,000. In this case the player can be expected to play 10-100 hands per session (what's more, . The result will either be zero, or being up or down by at least one bet, a lot more than actual EV.

I was rather thinking of using the variance of blackjack to estimate the bet size. For instance, you could map the probability of betting \$1 and ending a session of N hands with +/- \$1, \$2, \$3...\$N. It's very unlikely for you to get \$N, and I think the most likely result would be 1 standard deviation from 0. Weighing these probabilities, it would be possible to determine the most likely bet size range for given loss or win.

That's the principle I'm hoping to exploit, averaging out over multiple sessions, just not sure how specifically best to go about it, where to get an approximating formula or tables to approximate, and whether it's feasible to apply asymptotic corrections in place of hard limits.

Estimating by bankroll size itself is not a viable alternative, since it's a simulation in the first place, so betting patterns aren't likely to reflect real life. Although the probability of busting a bankroll might be useful as an additional input, if it hasn't been busted.
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P90 Joined: Jan 8, 2011
• Posts: 1703
December 9th, 2011 at 7:48:12 AM permalink
OK, so let me be more clear. Here's what I'm going for:

Deviation = variance_function (N);
Bet = bankroll_change / deviation;
if (bet>max_bet) bet=max_bet;
if (bet<min_bet) bet=min_bet;

First of all, I presume I should take SD for the variance function. Where can I find the variance or SD formula for playing N hands of blackjack with given rules (or just any)?
Then, I correct for house edge. My guess is SD-(edge*N) for wins and SD+(edge*N) for losses.

And a more difficult question, is there a smoother alternative to flooring/capping it at min_bet<bet<max_bet? More importantly, should it be floored at all, or is it a mistake and is the correct way when using SD rather to let it average out?
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