Theory question
Say you deposit $400 somewhere, and you get a $100 bonus, clearable at a coin flipping game* that pays 1.99x the bet if you win, and 0 if you lose a bet. Thus the EV of any individual bet is .995. The $100 bonus is given up front and you can wager with it and withdraw it only once you've wagered a total of $2500. For those that remember the lingo, this is not a sticky bonus. The table limits are $1 to $100. Presume we have an infinite amount of time, thus we don't care that making 2500 $1 bets will take longer than 25 $100 bets. Presume, also, that we don't care about variance. Lastly, presume that we have an infinite amount of such bonuses being offered at different online casinos. The question is, how much should you wager to maximize your EV.
At first glance, it seems obvious that it shouldn't matter how much you bet. The EV of wagering $2500, whether it's 2500 $1 bets or 25 $100 bets, is -$12.5, thus making your expected profit for the entire venture +$87.5. Here's what I'm thinking though. If you bet $100 per game then you are obviously going to have a higher risk of ruin than if you bet $1 per game. Sometimes you may place 5 $100 bets and lose every one of them. You've lost the $400 you deposited and the $100 bonus, and that's that. Move on to the next one. So, say over a period of 1000 of these bonuses, with the $1 bets you will end up making 1000*2500 = $2.5 million in wagers (let's say you never bust), but with $100 bets you will end up busting quite a bit and over 1000 of these bonuses you will end up wagering only $2 million**. The bonus amounts to $100000 but the EV of $2.5m in bets is -$12.5k and the EV of $2m in bets is -$10k. So, by betting larger you are increasing your EV, right? Can someone point out an error here?
*The reason I'm making this a coin flipping game is because using blackjack, with similar EV, would not be good because you can end up negatively impacting your EV if you don't have a proper amount to do things like double down or split.
**These numbers aren't realistic, just some values to show that busting out more will cause you to bet less over the 1000 trials.
Quote: podcodDo you remember those old online blackjack bonuses where you'd do something like deposit, they'd give you money up front, and then you'd have to reach a certain wagering requirement to cash it out? I have a theory question based on this premise.
Say you deposit $400 somewhere, and you get a $100 bonus, clearable at a coin flipping game* that pays 1.99x the bet if you win, and 0 if you lose a bet. Thus the EV of any individual bet is .995. The $100 bonus is given up front and you can wager with it and withdraw it only once you've wagered a total of $2500. For those that remember the lingo, this is not a sticky bonus. The table limits are $1 to $100. Presume we have an infinite amount of time, thus we don't care that making 2500 $1 bets will take longer than 25 $100 bets. Presume, also, that we don't care about variance. Lastly, presume that we have an infinite amount of such bonuses being offered at different online casinos. The question is, how much should you wager to maximize your EV.
If you have an infinite supply of these bonus, and an infinte bankroll - bet table max.
With increasing bet size, you increase your variance. Hence you increase the probability that you cannot fulfill the wagering requirement, because you busted your deposit and bonus.
This might sound paradox, but: If you don't need to fulfill wagering requirements, you don't give the casino action, and you don't lose that much money on them.
Quote: MangoJIf you have an infinite supply of these bonus, and an infinte bankroll - bet table max.
With increasing bet size, you increase your variance. Hence you increase the probability that you cannot fulfill the wagering requirement, because you busted your deposit and bonus.
This might sound paradox, but: If you don't need to fulfill wagering requirements, you don't give the casino action, and you don't lose that much money on them.
OK so you agree with me and have the same rationale. Anyone else? BTW this is a question that I've posed to numerous people over the years and not one has ever come up with the same answer as me, although after I say the answer they agree. I'm just curious as to if my logic is wrong and I'm missing something along the way since it's something I came up with on my own.
It's not that much bizarre if you play this on baccarat with a friend. He will make opposing bets to yours (and you share profits), while you are bound to the wagering requirement (and he isn't). What would be the best bet size for you ?
You want to go "bust" as fast as possible. Why? When you go broke early, your friend collects the original amount and bonus, and thus you have secured the bonus.
In reality of course there is no friend playing baccarat with you. The friend is just a vehicle to eliminate variance completely. If you have an infinite number of opportunities and bankroll - you are not interested in variance. Hence in this case the same high-bet strategy must still be most efficient.
If you do need to account for variance, you would need to make a Kelly-style approach, where you maximize the EV of a log utility function rather than pure money.
Results will be different of course, but you won't need infinite bankroll ^^
I rather distrust casinos that offer tempting bonuses. It means the casino is desperate and that usually means that other players are staying clear of it which indicates they must know something.
Quote: MangoJIt's not that much bizarre if you play this on baccarat with a friend. He will make opposing bets to yours (and you share profits), while you are bound to the wagering requirement (and he isn't). What would be the best bet size for you ?
You want to go "bust" as fast as possible. Why? When you go broke early, your friend collects the original amount and bonus, and thus you have secured the bonus.
It's not legal for me to play online, so I'm not familiar first-hand with the way these bonuses work. Are you saying that if you go bust you keep the bonus amount?
Quote: MathExtremistIt's not legal for me to play online, so I'm not familiar first-hand with the way these bonuses work. Are you saying that if you go bust you keep the bonus amount?
Nope, when you go bust you lose your deposit and your bonus. What we were saying was: A betting strategy which has a higher probability of busting early performs better in the long run. This may be paradox on first view, since busting is the worst thing happening. But the frequent busts are way outweighted by the rare cases where you don't bust.
Quote: podcodDo you remember those old online blackjack bonuses where you'd do something like deposit, they'd give you money up front, and then you'd have to reach a certain wagering requirement to cash it out? I have a theory question based on this premise.
Say you deposit $400 somewhere, and you get a $100 bonus, clearable at a coin flipping game* that pays 1.99x the bet if you win, and 0 if you lose a bet. Thus the EV of any individual bet is .995. The $100 bonus is given up front and you can wager with it and withdraw it only once you've wagered a total of $2500. For those that remember the lingo, this is not a sticky bonus. The table limits are $1 to $100. Presume we have an infinite amount of time, thus we don't care that making 2500 $1 bets will take longer than 25 $100 bets. Presume, also, that we don't care about variance. Lastly, presume that we have an infinite amount of such bonuses being offered at different online casinos. The question is, how much should you wager to maximize your EV.
Yes, like others pointed out above, large bets do increase the EV. The exact formula is:
EV = Bonus - House Edge*Average wagering
Where average wagering is the average amount you will wager depending on your odds to bust early. With low bets and a low variance game, the chance to bust before completing WR might be close to zero so the average wagering would be simply equal to wagering requirement. On the other hand with initial doubling up of the whole balance on a 50/50 bet (if such were to exist), the average wagering would be half of the wagering requirement, and so on.
However if the WR is only 25x on a 99.5% game, the EV is already close to the full bonus amount (EV = 87.5% of bonus) so high risk bets won't add you much value and most people would play through this bonus with small bets. Those that would play large bets would do so to save time and effort, and not so much to increase EV.
But let's instead say that you have a 100% upfront bonus with a 30x rollover on slots, whose house edge is 5%. Such bonus would be -EV in every sense unless you made large slot bets at the beginning, and this is how many advantage players actually play these bonuses these days. I know a few who make or try to make a living out of it.
Quote: Jufo81Yes, like others pointed out above, large bets do increase the EV. The exact formula is:
EV = Bonus - House Edge*Average wagering
This whole discussion is outside my realm, but these statements appear to conflict. It looks to me as if (for a given bonus and given house edge) this formula suggests the EV is maximized by minimizing "average wagering" (except in the apparently-irrelevant cases of negative wagers or negative house edge). What am I missing?
Quote: DocThis whole discussion is outside my realm, but these statements appear to conflict. It looks to me as if (for a given bonus and given house edge) this formula suggests the EV is maximized by minimizing "average wagering" (except in the apparently-irrelevant cases of negative wagers or negative house edge). What am I missing?
Yes this is a correct statement. Where is the conflict? Of course, you have to consider that in order to be able to bust (to lose your balance) you need to wager a minimum of your initial balance (it's never possible to lose more than you stake, I hope :) ), so average wagering can't ever be zero or less than the money you start with.
Quote: Jufo81Yes this is a correct statement. Where is the conflict?
I read your formula as saying that the EV is equal to the bonus less the product of house edge times average wagering. If house edge and average wagering are both non-negative numbers, then the maximum EV is equal to the bonus and occurs when either/both the house edge and average wagering are zero. If they are both positive numbers, then the EV is less than the bonus, and (for a given house edge) the EV is progressively lower as average wagering is increased. That is what I considered to be in conflict with the statement that large bets increase EV. Do not large bets increase average wagering? I must still be missing something.
Quote: DocI read your formula as saying that the EV is equal to the bonus less the product of house edge times average wagering. If house edge and average wagering are both non-negative numbers, then the maximum EV is equal to the bonus and occurs when either/both the house edge and average wagering are zero.
Yes correct. If you could play an equal-odds game (zero house edge) then the EV of the bonus would be the value of bonus itself, no matter how much are required to wager. I actually have once played through a bonus like this.
Quote: Doc
If they are both positive numbers, then the EV is less than the bonus, and (for a given house edge) the EV is progressively lower as average wagering is increased. That is what I considered to be in conflict with the statement that large bets increase EV. Do not large bets increase average wagering? I must still be missing something.
No, the idea is that large bets decrease average wagering because the probability to bust (to lose your initial balance) increases (you start with fixed amount of balance and never add more money). Remember that once you lose both your deposit and bonus funds, your balance is zero, the game is over and you don't need to add more money to complete your requirements for the bonus.
Imagine that you start with a balance of $200 ($100 that you put in yourself plus $100 as bonus money given by the casino) and you need to wager $5000 on BJ to cash out. How much are you going to wager on average with a) $1 hands b) $100 hands, given that the maximum you need to wager is $5000? It should be very obvious that most of the time you will not make through the $5000 wagering starting with a $200 balance and $100 bets per hand. On the contrary, you are very likely to make the $5000 wagering with $1 bets.
Quote: DocI read your formula as saying that the EV is equal to the bonus less the product of house edge times average wagering. If house edge and average wagering are both non-negative numbers, then the maximum EV is equal to the bonus and occurs when either/both the house edge and average wagering are zero. If they are both positive numbers, then the EV is less than the bonus, and (for a given house edge) the EV is progressively lower as average wagering is increased. That is what I considered to be in conflict with the statement that large bets increase EV. Do not large bets increase average wagering? I must still be missing something.
Average wagering is not the wagering requirement. It is the expected wagering for a strategy heading for bust (or wagering completion).
EV is maximized if you are able to minimize average wagering for a positive house edge. You minimize average wagering by high betsize. Why does it decrease average wagering ? Because you will bust more frequently, and after each bust you don't need to wager anything!
