How to calculate the EV of a sequence of bonuses.

[3Dice]

A lot of misinformation is available on this subject - partly because it is not as trivial as it may seem at first glance, partly because there is no clear definition of what we are trying to calculate.

So lets first try to define what we are exactly trying to calculate. The formula you'll find all over the web is not 'mathematically wrong' .. to be precise, it just doesn't calculate the number you need. To avoid confusion I wont repeat that formula here, but what it calculates is the EV of a bonus, _if you only take just one bonus your entire life_. If that is the masterplan, take a bonus once and then never again, then you can use the formula you'll find on many websites. (that formula and the strategy that goes with it does not maximize the EV, it minimizes the risk of ruin by 'grinding' which is playing low bet low variance.).

If however, you are a player that values the entertainment value of gambling, and are wondering whether or not on the long term you should or rather should not take bonuses, then you need an entirely different formula.

Unfortunately, the formula to mathematically determine, ahead of time what the EV of a sequence of bonuses is is complex, and moreover it requires information (like e.g. the numerical variance of the game) that a player typically does not have access to.

That doesn't mean you cannot calculate the EV of a sequence of bonuses. In fact, its really easy to do on past data. From a player point of view, you can use a simple algorithm that will update the EV of every next bonus taking into account all the past bonuses you have already received. Allow me to skip the explenation for a second and get right down to the stuff you need.

Algorithm to calculate the running EV of a sequence of bonuses.

 - Log all your sessions. For each session note :

 - B = bonus.
   S = stake.

 - after each session, calculate the following numbers.
   TB = total of all bonus.
   TS = total stake over all sessions.
   
 - after each session you can calculate the EV of the sequence up
   to that point. (assuming a 5% houseedge here).

   TOTAL_RUNNING_EV = TB - TS*0.05

An example ..

So lets look at a sample. We'll simplify things and have a player lose 9 times in a row, then win 1 time. He'll deposit $50 every time, and get a $50 bonus on top of it. He spins a reel of fortune that has 10 slices. 9 win nothing and one wins 9.5 times betsize.

 deposit 1 : 50D + 50B. total stake = 100.

 The EV after deposit 1, using formula above : 50 - 5 = 45

 deposit 2 : 50D + 50B. total stake = 100.

 The EV now is 90

 deposit 3,4,5,6,7,8,9 : 50D + 50B. total stake = 100 each time.

 The EV now is 9*50 - 9*5 = 405

 deposit 10 : 50D + 50B. total stake = 100, total win = 950

 The EV now is 10*50 - 10*5 = 450

 The player cashes out at this point. He deposited 500, cashes
 out 950 .. exactly the EV of the bonus over his own deposit.

 The total stake on the machine is now 1000, total win 950 .. i.e. 95% machine.

Also notice that if our player had decided to not take a bonus on that last deposit, but just play his own 100, he would still have an EV of 400 from past bonuses ...

Also notice what a max-cashout would do here .. a 50 bonus with 10x max cash sound reasonable ? .. in this case it would COMPLETELY eliminate the player advantage ..

In fact, in this scenario a maxcash of 10x (or 500) would be the equivalent of a WR of more than 200x. (stay away from maxcash bonuses!!)

The rule of thumb.

[COLOR="Red"][SIZE="4"] [CENTER]For as long as 5% of your totalstake is lower than your
total bonus, you  are playing at positive EV.

[/CENTER] 
[/SIZE][/COLOR]

Playing without bonus in a sequence of bonuses.

What is the effect of making deposits where you don't claim bonuses inbetween those where you do take bonuses ? Well, the formula stays the same .. total bonus will not increase, but total stake will. In the example above, if our player had not taken 10 bonuses, but only 5, alternating a deposit with a bonus and a deposit without a bonus, then at the tenth session he would have an EV of :

EV = 5*50 bonus - 1000*0.05 = 200

And so when he cashes out on the tenth session (after 5 deposits of 50 on which he claimed bonuses and 5 of 100 without bonus), he cashes out 950 on a total deposit of 750 .. or again exactly the EV ahead when the machine is at 95% (so he didn't get lucky - he got the exact average RTP).

In other words depositing without a bonus doesn't instantly mean you have no EV from past bonuses anymore. As long as you follow the rule of thumb above - you are playing at a positive EV.


Closing thought.

It's lady luck that gives out the best bonuses. It's not the best mathematicians that win the most - its the luckiest players.

Cheers,

Enzo

 

[Wibbler]

Um, if you're playing without a bonus then you're just gambling. And your example is just plain ludicrous - you pulled the "lose 9 then win 1" sequence out of thin air, and I've only heard of a handful of bonuses that allow you to cash out after just wagering 2xB.

My rule of thumb - you're only playing at +EV if you're playing with a bonus. Any other form of play (unless you're an expert card-counter or poker player) will on average just be handing money into the casino's pockets.

 

[hitobito2005]

My rule of thumb - you're only playing at +EV if you're playing with a bonus. Any other form of play (unless you're an expert card-counter or poker player) will on average just be handing money into the casino's pockets.

Of course playing without a bonus is -EV, but what he's saying is that playing without a bonus does not necessarily cancel out the times when you did play with a bonus, ie over all your past play (bonus and non-bonus) your advantage may still be positive.

 

[Jufo]

Of course playing without a bonus is -EV, but what he's saying is that playing without a bonus does not necessarily cancel out the times when you did play with a bonus, ie over all your past play (bonus and non-bonus) your advantage may still be positive.

Of course the advantage can still be positive as long as the total -EV from non-bonus play is in total smaller than the +EV from all bonus play but why play without bonuses if the EV maximized by only playing with bonuses?

Also the EV doesn't apply to past results. EV means "Expected value" of a result that has not yet happened.

An example: You flip a fair coin ten times. Before you start flipping, the expected number of heads is 5. Suppose that after 9 flips the coin has been tails 9 times. Before the 10th flip the expectation of heads is not 5 anymore but 0.5.

Similarily if you repeat a bonus ten times ($50 D, $50 B) with EV of $45 per one bonus, then the total EV before starting playing is $450. But if you lost your deposit first 9 times then at this point the expectation is not $450 anymore but instead it is -405$.

 

[3Dice]

Um, if you're playing without a bonus then you're just gambling.

Not quite .. explain to me what the difference is between :

1. deposit $100, get 100% bonus, then deposit another $100

2. deposit $200, get 50% bonus.

By making the followup deposit, you effectively transform the 100% bonus on the previous deposit to a 50% bonus on both deposits. Again, to know if you still have EV use the rule of thumb printed above.

And your example is just plain ludicrous - you pulled the "lose 9 then win 1" sequence out of thin air, and I've only heard of a handful of bonuses that allow you to cash out after just wagering 2xB.

Would you care to give a reason why this is a "ludicrous" example ? It is just a simplified version of a slot if you like, at a 95% RTP. And the wagering requirement is irrelevant. You don't have to cashout on a deposit that came with a bonus even to take advantage of the EV.

The only given is that on the long term the machine pays out 95% of what was put in. So the casino makes 5% of the total stake. If 5% of your total stake is less than what the casino gave you in total bonus, then you have a positive EV. You will continue to have that EV on subsequent bonus-free deposits until such point in time where 5% of your totalstake becomes bigger than your total bonus.

My rule of thumb - you're only playing at +EV if you're playing with a bonus. Any other form of play (unless you're an expert card-counter or poker player) will on average just be handing money into the casino's pockets.

The point of my post is exactly to eliminate this wrong train of thought. Perhaps you should work out some number samples for yourself to see where you are wrong.

Kindest Regards,

Enzo

 

[3Dice]

Similarily if you repeat a bonus ten times ($50 D, $50 B) with EV of $45 per one bonus, then the total EV before starting playing is $450. But if you lost your deposit first 9 times then at this point the expectation is not $450 anymore but instead it is -405$.

That's not correct Jufo. You should've heard some alarm bells when you say -405$ EV on a $100 deposit . I could've understood you (wrongly) thinking the EV at that point is just $45, but -405$ ?

You can't just ignore the fact that most of the money you lost up to that point was bonus .. that's why I demonstrated the reverse effect. If we repeat the scenario indefinitely, the player will cashout $950 for each $500 he deposits. Hence the EV is $450.

TotalWithdraw - TotalDeposit = TotalBonus - Totalstake * 0.05

This relationship needs to be maintained for the machine to be a 95% machine.

Kindest regards,

Enzo

 

[Jufo]

Not quite .. explain to me what the difference is between :

1. deposit $100, get 100% bonus, then deposit another $100

2. deposit $200, get 50% bonus.

If no play takes place between the deposits in option 1, then options 1 and 2 are mathemtically equal. However the option 1 without the follow-up deposit is better for the player than option 2 so the only effect that the follow-up deposit has, is to decrease the bonus value, in other words it is -EV to make such a follow-up deposit.

By making the followup deposit, you effectively transform the 100% bonus on the previous deposit to a 50% bonus on both deposits. Again, to know if you still have EV use the rule of thumb printed above.

Yes, player still has +EV after the follow-up deposit but less +EV than without making it.

You will continue to have that EV on subsequent bonus-free deposits until such point in time where 5% of your totalstake becomes bigger than your total bonus.

Theoretical EV on past outcomes has no real value to the player. The player either finished below or above EV and made a gain or loss. The result is already known and there is no expectation left in past results.

My rule of thumb - you're only playing at +EV if you're playing with a bonus. Any other form of play (unless you're an expert card-counter or poker player) will on average just be handing money into the casino's pockets.

The point of my post is exactly to eliminate this wrong train of thought. Perhaps you should work out some number samples for yourself to see where you are wrong.

Could you clarify what exactly your point is? Are you saying that the player has +EV compared to his present situation by depositing and playing without bonus? And if not then why is Wibbler's train thought wrong?

 

[Jufo]

That's not correct Jufo. You should've heard some alarm bells when you say -405$ EV on a $100 deposit . I could've understood you (wrongly) thinking the EV at that point is just $45, but -405$

Nope, it is correct. The player has deposited $50 nine times and lost all of the deposits so he is standing at -450$. The expectation of the 10th bonus is +45$, making his expectation at this point -405$. See the coin flipping example if you didn't get it.

You can't just ignore the fact that most of the money you lost up to that point was bonus .. that's why I demonstrated the reverse effect. If we repeat the scenario indefinitely, the player will cashout $950 for each $500 he deposits. Hence the EV is $450.

The EV is $450 only before any playing takes place, not after the fact.

This relationship needs to be maintained for the machine to be a 95% machine.

The player might need to play infinitely many times to reach the expected profit of $45 per bonus, and never without a bonus.

 

[3Dice]

If no play takes place between the deposits in option 1, then options 1 and 2 are mathemtically equal. However the option 1 without the follow-up deposit is better for the player than option 2 so the only effect that the follow-up deposit has, is to decrease the bonus value, in other words it is -EV to make such a follow-up deposit.

It decreases the value of that bonus - obviously - but its still +EV towards not having taken a bonus at all.

Theoretical EV on past outcomes has no real value to the player. The player either finished below or above EV and made a gain or loss. The result is already known and there is no expectation left in past results.

Surely you can't be serious. There is +EV until houseedge*totalstake > bonus given. The time paused between spins (wheter it be minutes or weeks) is absolutely irrelevant.

Could you clarify what exactly your point is? Are you saying that the player has +EV compared to his present situation by depositing and playing without bonus? And if not then why is Wibbler's train thought wrong?

I'm saying the player has +EV compared to the situation of taking no bonus at all.

Surely if all you care for is EV then you should probably only take signup bonuses, never deposit without bonus, switch casino's more often than underpants. In doing so you'll have a load of installing/uninsatalling/administration - not to mention heartache and stress trying to cashout from the shady places this strategy undoubtfully takes you.

If however you care for other parameters like smooth cashouts, good service, no hidden or difficult T&C, quality of entertainment, then you will find that the occasional inbetween deposit without bonus is a requirement, and as demonstrated doesnt mean you can't still be playing at a positive EV.

This is not written for those who would like to abuse these bonuses and are playing only for that advantage. This is written for those people who value gambling as entertainment and to give them the reference frame they need to judge bonuses.

Kindest regards,

Enzo

 

[3Dice]

Nope, it is correct. The player has deposited $50 nine times and lost all of the deposits so he is standing at -450$. The expectation of the 10th bonus is +45$, making his expectation at this point -405$. See the coin flipping example if you didn't get it.

Jufo, your machine doesn't end up at 95%. The coin flipping example does not apply. I'm not trying to predict a future outcome. I'm just saying on average player wins 1 in 10 sessions. Consider it all in the past if you like. I'm just looking at the full sample. A new outcome adds to the full sample. The full sample will end up at 95% payout. When it does, the player will have withdrawn $450 of the total of $500 in bonus given. Anything you suggest that doesn't end at a total EV of +$450 for a total stake of $1000 half funded with bonus - is not correct.

again .. as long as totalstake * 0.05 < bonus .. you are at +EV and with neutral luck (ie machine at 95%) will still end up cashing out.

Cheers,

Enzo

 

[just play]

Enzo, can you "dumb down" your first post for people like me that don't understand?

 

[Wibbler]

I'm saying the player has +EV compared to the situation of taking no bonus at all.

So you accept, then, that playing without a bonus is -EV

 

[blankley]

Suppose I deposited 50 get 55 bonus, wager 3150 at 5%.

Should I never play there, because I'm starting at -ev which I can never get back?

 

[3Dice]

Suppose I deposited 50 get 55 bonus, wager 3150 at 5%.

Should I never play there, because I'm starting at -ev which I can never get back?

That depends - are you in it for just the EV or do you also consider bonus a means to prolongue your play and thus increase the entertainment value ?

Looking at it from that point of view one could state that the only way to lose the EV of a bonus is by replacing it with entertainment.

Kindest regards,

Enzo.

 

[blankley]

I believe that bonuses are for entertainment value only and people should never cash out after having been credited with one as that would be wrong.

I also believe the children are our are future, teach them well and let them lead the way.

 

[KariRunk]

I don't see the point of this thread. Since there's almost always wagering requirements for bonus, you need to take that to account when counting EV. Which makes almost all bonuses -EV for player and +EV for the casino.

And if the slots really are truly random, the past results doesn't have anything to do with the future EV.

 

[BingoT]

I don't see the point of this thread. Since there's almost always wagering requirements for bonus, you need to take that to account when counting EV. Which makes almost all bonuses -EV for player and +EV for the casino.

And if the slots really are truly random, the past results doesn't have anything to do with the future EV.

Nominated The Following User Has Nominated This Post:
Casinomeister (Today)

I would think it's a good point if The Casinomeister Nominated it.:clap:

 

[Jufo]

Suppose I deposited 50 get 55 bonus, wager 3150 at 5%.

Should I never play there, because I'm starting at -ev which I can never get back?

One correct point in Enzo's first post was the sentence "For as long as 5% of your totalstake is lower than your total bonus, you are playing at positive EV."

So the bonus you mentioned above can be made +EV playing it so that on average you wager 55 / 5% = 1100 or less. So if you play the bonus with large initial bets in a manner that you have at least 65% chance of busting your bankroll at the beginning (so that you finish the WR only 35% of time), then even a bad bonus like this will actually have positive expectation.

Looking at it from that point of view one could state that the only way to lose the EV of a bonus is by replacing it with entertainment.

On the other hand if you only play with positive expectation you get much more action with the same initial bankroll and I think you get more entertainment that way. Also many people feel that winning from casinos is more entertaining than losing to them as there is this "I beat the house" psychological reward feeling. Playing bonuses in a manner that they have positive EV also requires some strategical thinking, which might give a new dimension to the otherwise tedious and repetetive casino games, and it actually gives the player some level of control in a situation where he normally has no control over the outcomes dealt to him.

 

[blankley]

'They will be denied, declined, or revoked from players who are deemed to abuse the spirit of this offer. '

So that is a redundant term?

Entertainment can include winning as well as losing?

Entertainment can include a few large bets as well as a lot of small bets?

I fear that casino operators do not accept this as a valid.

 

[Jufo]

'They will be denied, declined, or revoked from players who are deemed to abuse the spirit of this offer. '

So that is a redundant term?

Entertainment can include winning as well as losing?

Entertainment can include a few large bets as well as a lot of small bets?

I fear that casino operators do not accept this as a valid.

That quoted term is against Casinomeister policies (see https://www.casinomeister.com/accredited-casinos/) so you should avoid playing at all casinos with such unfair terms. I think casinos should accept the fact that offering bonuses have the potential to cause them losses and if they are not willing to take this then they should simply stop offering bonuses instead of revoking such rogue FU clauses.

 

[blankley]

The accredited list will be one short very soon if that is the case.

Do you know where I copied that term from amigo?

 

[KariRunk]

I would think it's a good point if The Casinomeister Nominated it.:clap:

I mean what's the point of posting all those calculations when they're not valid on majority of cases of bonuses. They're valid only if you get bonus with no or 1X wagering requirements!

And since those calculations was posted by casino representative, I have to question his motives about posting those.

 

[skiny]

I can fully understand the math behind determining the expected value of a bonus but we're dealing with games of chance with random outcomes. Most often these outcomes are not to our benefit. The sample worked perfectly over 10 sessions and I realize that this is very simplified version of it but it's highly unlikely that anything is going to work out over 10 sessions. In fact show me any casino that will even look at just 10 sessions as a valid sample of any game's payout. It might take hundreds of sessions before a player comes close to the expected payout on a machine. It might take one session to be well over that and a player may never see more than 75 or 80% regardless of what kind of bonus the player takes. That's just the reality.

Now putting all that aside, taking a bonus doesn't exactly increase the value of your deposit because of the wage requirements attached. In fact it decreases the value of every wager because every dollar you deposit will have to be wagered 10, 20 or even 30 times before you can take any money back out.

If you deposit 50 dollars and just play with your deposit you might wager 300 before you finally get lucky (In fact these days if you can even wager 300 you're already lucky) and maybe your balance is now 200. You can cash it out and walk away.

Even with a 10x D+B WR on a 100% bonus which is better than anything you'll find most places that I know of you now start with 100 but before you can touch it you have to wager 1000. If you have exactly the same game play described above you've hit your 200 dollar balance, you've wagered 300 and before you can cash out you still have to wager almost 4x what you have in your account. If you're anything like me, you've already used up all the magic in your little rabbit's foot just winning the 200. It'll take a miracle to wager another 700 before I go broke. And that too is the reality.

None of this makes deposit bonuses a good or bad thing. It's as simple as this...

If you deposit 50 and take a 50 dollar bonus -

If you lose 50 before you hit anything, you're lucky you took the bonus because you can keep playing.

If you hit something good before you lose 50, you might as well have not taken the bonus because now you have the task of grinding out the WR.

And you never know until after you start playing so I guess we're already gambling before we even open a game. We're gambling on whether or not we need the extra 50 bucks to cash out or if the WR will break us before we can.

 

[KasinoKing]

I mean what's the point of posting all those calculations when they're not valid on majority of cases of bonuses. They're valid only if you get bonus with no or 1X wagering requirements!

And since those calculations was posted by casino representative, I have to question his motives about posting those.

I know why you think that - and I did too for a VERY long time.
But I can assure you that Enzo is not only totally honest, but also a mathematical genius compared to you, me and most other people on this forum.
You need to open your mind to possibilities other than what just seems "logical".
VERY hard to do, believe me I know!

You probably wont agree with me right now, but given some time you will eventually see that I, and therefore Enzo, is correct.

KK

 

[hitobito2005]

IBut I can assure you that Enzo is not only totally honest, but also a mathematical genius compared to you, me and most other people on this forum.

Probably worth mentioning that the maths in Enzos post is UK Year 7-8 level at best (12 to 14 year olds).

Maybe even primary school level (under 11 years old).

Things like gamblers fallacy probably get in the way, but it's far from complicated.

 

[Jufo]

I know why you think that - and I did too for a VERY long time.
But I can assure you that Enzo is not only totally honest, but also a mathematical genius compared to you, me and most other people on this forum.
You need to open your mind to possibilities other than what just seems "logical".
VERY hard to do, believe me I know!

I got the feeling from Enzo's post that it was an attempt to lobby non-bonus play by mixing "positive expectation" and "non-bonus play" somehow together, which made me question his motives. However, not really knowing his intentions and the actual point he was trying make, I might have misunderstood and if so, then apologies.

You probably wont agree with me right now, but given some time you will eventually see that I, and therefore Enzo, is correct.
KK

Obviously a statement "If you are ahead $2000 overall at our casino and you play today at lose $1000 back then that's really okay because you are still ahead overall" is correct in the sense that the person is still ahead overall. It doesn't take a PhD in Mathematics to know that. So why even make this kind of statement?

 

[skiny]

Obviously a statement "If you are ahead $2000 overall at our casino and you play today at lose $1000 back then that's really okay because you are still ahead overall" is correct in the sense that the person is still ahead overall. It doesn't take a PhD in Mathematics to know that. So why even make this kind of statement?

Actually the statement usually is more like "You lost 50 dollars every week for the last 4 and a half months but you won 1k last may so you're still up a hundred bucks.... why complain?"

 

[XOXOXOXOXOX]

Do all these rules for EV calculations only apply if playing at the same casino? the same slot?

There are some casinos where i never able to win after many many deposits playing mostly with bonuses(100% only). So based on the said here about EV/variance, should i continue dumping money and hoping to win big one day?? I'd rather play somewhere else because i think those casinos "hate" me.

 

[Rusty]

Deposit 500 get 100% bonus = 1000
playX1 95% return = 950.
Win 450.
Easy money

May as well say that or deposit the 50 ten times with 100% bonus get 95% return on each=same result.ev 45x10

Like Jufo I do not understand why the ev of the last deposit is not 45 because you are not depositing 500 and getting 500 bonus with an ev of 450 you are depositing 50 with 50 bonus with an ev of 45 surely?

A slot has no memory (supposedly ) so each session is a separate event not because of any delay but because of your starting balance.
It is the same as depositing the 500 and having a balance of 1000, losing the first 900 and then saying the ev on the last 100 is 450.Why?
You can not undo the money you lost and pretend you are playing with it on the 10th deposit or on your last 100.
Or to put it another way why do we now expect an ev of 450 on a deposit of 50D+50B all of a sudden?

You say you can not ignore that half the money lost was bonus money but if the slot is not influenced by past results it does not matter if the all the previous money was bonus money surely - it is a past event and can not effect on the future. The only way we can say the ev on deposit ten is 450 is if the slot has guaranteed 95% return over the exact amount of spins used through all 10 deposits and slots are not designed that way.
What you are saying in effect is that each time you deposit and lose you are more likely to win next deposit - how can that be?

Maybe you are not saying any of that and I have missed the point - I'm scratching my head.

 

[skiny]

Do all these rules for EV calculations only apply if playing at the same casino? the same slot?

There are some casinos where i never able to win after many many deposits playing mostly with bonuses(100% only). So based on the said here about EV/variance, should i continue dumping money and hoping to win big one day?? I'd rather play somewhere else because i think those casinos "hate" me.

Actually if you keep depositing and never win these casinos love you very much.

 

[XOXOXOXOXOX]

well seriously i think this EV thingy only works if spinning the same slot for millions of spins... unless slot B remembers how much was lost playing slot A. prove me wrong.

 

[greasemonkey]

well seriously i think this EV thingy only works if spinning the same slot for millions of spins... unless slot B remembers how much was lost playing slot A. prove me wrong.

If the payout% is the same then what does it matter if its the same slot or same casino or same software or same planet?

 

[The Dude]

I would think it's a good point if The Casinomeister Nominated it.:clap:

It's an excellent thread initiated by an excellent post. It's provocative, informative, somewhat controversial, and has generated good discussion on EV and bonus play - so yeah, it got nominated

...And since those calculations was posted by casino representative, I have to question his motives about posting those.

Like KK said, Enzo has a history here of explaining casino math. Most of us eagerly welcome his input - it has nothing to do with generating traffic to his site, it has to do with making us think. I would guess his motives are to generate an interesting discussion and perhaps to explain a few things that may be elusive to many.

Casino reps are members, just like you. What is your motive for questioning his motives?

 

[skiny]

Deposit 500 get 100% bonus = 1000
playX1 95% return = 950.
Win 450.
Easy money

May as well say that or deposit the 50 ten times with 100% bonus get 95% return on each=same result.ev 45x10

Like Jufo I do not understand why the ev of the last deposit is not 45 because you are not depositing 500 and getting 500 bonus with an ev of 450 you are depositing 50 with 50 bonus with an ev of 45 surely?

A slot has no memory (supposedly ) so each session is a separate event not because of any delay but because of your starting balance.
It is the same as depositing the 500 and having a balance of 1000, losing the first 900 and then saying the ev on the last 100 is 450.Why?
You can not undo the money you lost and pretend you are playing with it on the 10th deposit or on your last 100.
Or to put it another way why do we now expect an ev of 450 on a deposit of 50D+50B all of a sudden?

You say you can not ignore that half the money lost was bonus money but if the slot is not influenced by past results it does not matter if the all the previous money was bonus money surely - it is a past event and can not effect on the future. The only way we can say the ev on deposit ten is 450 is if the slot has guaranteed 95% return over the exact amount of spins used through all 10 deposits and slots are not designed that way.
What you are saying in effect is that each time you deposit and lose you are more likely to win next deposit - how can that be?

Maybe you are not saying any of that and I have missed the point - I'm scratching my head.

If you deposit 50 and take a 100% bonus and lose it, there is no "expected" value on this bonus anymore. There is a known value. The value is zero. If you do this 9 times in a row you now have a known value of all 9 of those bonuses. 9*0.

 

[KasinoKing]

A slot has no memory (supposedly ) so each session is a separate event not because of any delay but because of your starting balance.
It is the same as depositing the 500 and having a balance of 1000, losing the first 900 and then saying the ev on the last 100 is 450.Why?
You can not undo the money you lost and pretend you are playing with it on the 10th deposit or on your last 100.
Or to put it another way why do we now expect an ev of 450 on a deposit of 50D+50B all of a sudden?

You say you can not ignore that half the money lost was bonus money but if the slot is not influenced by past results it does not matter if the all the previous money was bonus money surely - it is a past event and can not effect on the future. The only way we can say the ev on deposit ten is 450 is if the slot has guaranteed 95% return over the exact amount of spins used through all 10 deposits and slots are not designed that way.
What you are saying in effect is that each time you deposit and lose you are more likely to win next deposit - how can that be?

Maybe you are not saying any of that and I have missed the point - I'm scratching my head.

I scratched my head over this for a long time too - and got splinters in my fingers!
What it boils down to is what Enzo said in his first post:

The rule of thumb.

[B][COLOR="Red"][SIZE="4"] [CENTER]For as long as 5% of your totalstake is lower than your
total bonus, you  are playing at positive EV.

[/CENTER] 
[/SIZE][/COLOR][/B]

It seems totally bizarre I know - but it actually benefits you if you bust-out quickly on several bonuses, because the amount of play-through is very small.
It took me years to realise that Enzo is correct, but at the end of the day all it is, is very simple mathematics.

KK

 

[thelawnet]

A lot of misinformation is available on this subject - partly because it is not as trivial as it may seem at first glance, partly because there is no clear definition of what we are trying to calculate.

So lets first try to define what we are exactly trying to calculate. The formula you'll find all over the web is not 'mathematically wrong' .. to be precise, it just doesn't calculate the number you need. To avoid confusion I wont repeat that formula here, but what it calculates is the EV of a bonus, _if you only take just one bonus your entire life_. If that is the masterplan, take a bonus once and then never again, then you can use the formula you'll find on many websites. (that formula and the strategy that goes with it does not maximize the EV, it minimizes the risk of ruin by 'grinding' which is playing low bet low variance.).

If however, you are a player that values the entertainment value of gambling, and are wondering whether or not on the long term you should or rather should not take bonuses, then you need an entirely different formula.

It depends what you mean by 'values the entertainment value of gambling'.

If you value that above all else, you won't care about formulas at all, you'll just play.

Unfortunately, the formula to mathematically determine, ahead of time what the EV of a sequence of bonuses is is complex, and moreover it requires information (like e.g. the numerical variance of the game) that a player typically does not have access to.

Well not really. Most of the time bonuses are essentially independent events - one doesn't affect the next. So if you get $100/month at a casino, the EV over the year is simply 12x the EV of a single bonus. The variance only affects the EV of a single bonus, and then only if the bonus is given up front rather than post-wager - it doesn't releate to the *series* of bonuses at all - the complexity of the series is of the same order as the complexity of the single bonus. The only time it becomes complex is when you have things like 'get bonuses on your first 5 deposits; if you make a cashout you will not be eligible for any further bonuses'.

That doesn't mean you cannot calculate the EV of a sequence of bonuses. In fact, its really easy to do on past data. From a player point of view, you can use a simple algorithm that will update the EV of every next bonus taking into account all the past bonuses you have already received. Allow me to skip the explenation for a second and get right down to the stuff you need.

There is a bit of a logic error here. There is no such thing as 'past EV'. 'Expected value' refers to 'expectation' - i.e. the future. Once it is no longer 'expected', it is either a win or a loss - +V or -V, there is no longer in +EV.

Algorithm to calculate the running EV of a sequence of bonuses.

 - Log all your sessions. For each session note :

 - B = bonus.
   S = stake.

 - after each session, calculate the following numbers.
   TB = total of all bonus.
   TS = total stake over all sessions.
   
 - after each session you can calculate the EV of the sequence up
   to that point. (assuming a 5% houseedge here).

   TOTAL_RUNNING_EV = TB - TS*0.05

Not relevant. You want to know how much you might make in the future, but how much you might have made in the past is no more useful than saying 'if I only I'd studied harder at school, I'd now be making $1m/year as a lawyer'. It *might* have happened but it didn't.

An example ..

So lets look at a sample. We'll simplify things and have a player lose 9 times in a row, then win 1 time. He'll deposit $50 every time, and get a $50 bonus on top of it. He spins a reel of fortune that has 10 slices. 9 win nothing and one wins 9.5 times betsize.

 deposit 1 : 50D + 50B. total stake = 100.

 The EV after deposit 1, using formula above : 50 - 5 = 45

 deposit 2 : 50D + 50B. total stake = 100.

 The EV now is 90

 deposit 3,4,5,6,7,8,9 : 50D + 50B. total stake = 100 each time.

 The EV now is 9*50 - 9*5 = 405

 deposit 10 : 50D + 50B. total stake = 100, total win = 950

 The EV now is 10*50 - 10*5 = 450

 The player cashes out at this point. He deposited 500, cashes
 out 950 .. exactly the EV of the bonus over his own deposit.

 The total stake on the machine is now 1000, total win 950 .. i.e. 95% machine.

Also notice that if our player had decided to not take a bonus on that last deposit, but just play his own 100, he would still have an EV of 405 from past bonuses ...

Well no, he would be $450 down and choosing to make a play of -$5 EV.

It is here that you see the real danger with your reasoning, the gambler's fallacy.

Let's say that you've just received July's pay cheque, $1500, and then a casino sends you an email saying 'Nine Times Lucky, get $100 bonus when you deposit $100, on your next NINE deposits'.

So you play each deposit and unfortunately lose nine straight. Tough luck.

So you think 'damn, I've lost nine times already, my luck can't continue like this'.

WRONG. Not only is the tenth deposit no more likely to win than the first, by playing without a bonus your EXPECTATION is to lose. In the long term, if you keep playing, you can expect to lose all of your money.

The danger of saying 'My EV is +405' is that when you are actually $450 down, you feel you should bet higher to compensate, to get back to expectation. THe problem with this is that it simply makes you even more likely to lose, so our player ends up running through his entire pay cheque, trying to fulfil his EV.

Given that by losing 9 bonuses in a row you have a smaller bank than when you started, then making the 10th, bonusless, deposit is less likely to be rational, because you have less money to put at risk. OTOH, had you won two of the previous nine deposits, then the 10th deposit makes more sense, not for any EV, but simply because you can better afford to play, afford the entertainment of trying to win again.

Also notice what a max-cashout would do here .. a 50 bonus with 10x max cash sound reasonable ? .. in this case it would COMPLETELY eliminate the player advantage ..

Well if you play at $100 stake, yes. But there's no mention of a WR here, so let's say you played at $40/spin.

So:

spin 1 - bet $40, if you win, cash out $60 (remaining balance) + $40*9.5 = $440
spin 2 - bet $40, if you win cashout $20 + $40 *9.5 = $400
spin 3 - bet $20, if you win cashout $20 *9.5 = $190

So the EV here is 1/10 * $440 + 9/10*1/10*400 + 9/10*9/10*1/10*190 - 50 =
44 + 36 + 15.39 - 50 = $45.39

This is actually BETTER EV than the $100 bet you suggest. Why? Because on average you are betting less money on the 5% edge game, because you might simply cashout after the first spin (having only bet $40).

Of course this is hopelessly simplistic, as not many casinos will be happy about you betting $40 on a $50 bonus (but then neither would they accept a $100 wager, so perhaps the point is moot).

In fact, in this scenario a maxcash of 10x (or 500) would be the equivalent of a WR of more than 200x. (stay away from maxcash bonuses!!)

As per the illustration above, the way around this is drastically smaller bets. The problem comes either in that the maxcash is so low that you are likely to reach it at any reasonable bet size OR that the WR is so high that you cannot win without high risk bets.

The rule of thumb.

[COLOR="Red"][SIZE="4"] [CENTER]For as long as 5% of your totalstake is lower than your
total bonus, you  are playing at positive EV.

[/CENTER] 
[/SIZE][/COLOR]

Absolutely not.

You are playing at positive EV when you are playing at positive EV. That's all there is to it.

If someone gives you a $1000 bonus on a $1k deposit, and you have to wager $10k at 5% to cash it out (expected loss of $500), and you reach $9k after $10k of wagering, then decide to try and hit either $8k or $10k, then all that extra wagering is -EV, but you likely don't care, because you are up $8,000. (of course if you take it too far and lose your whole $9k then that would be rather bad, but for the purposes of this example let's assume we just stick with risking $1k)

OTOH, if you lose your deposit + bonus and then redeposit and try to win back the money you have lost, then that does matter because you are likely to end up losing even more money. It is this kind of behaviour that makes gamblers lose their shirts, and casinos make their money - the thinking that because the player 'should' have won, he can keep on playing until he does.

From a player protection perspective sending out this kind of message is less than ideal..... The past is the past and has no memory. The future should be considered on its own merits along with what you can afford to gamble (which is impacted on by past WINS, not past EV). In other words, if I buy a lottery ticket, which has a 50-55% HA every week for 5 years, I have an expected loss of about £130. If I win the jackpot of £10m, my EV is still -£130, but I now have £10m in the bank. My past EV is completely irrelevant to the fact that I might enjoy, and can certainly now afford, to deposit £1k at a time into online casinos with no bonus. OTOH, had you played the lottery on a double jackpot week (let's say PA 10%), and bought £100k of tickets, and not won, and that £100k was all your worldly wealth, despite the fact you 'earned' £10k of EV, you would be unable to afford any kind of gambling at all.

So when a player takes and loses with several bonuses in a row, correct behaviour is to just shrug and get over it, not fret about how much EV he lost or try and chase it back.

Playing without bonus in a sequence of bonuses.

What is the effect of making deposits where you don't claim bonuses inbetween those where you do take bonuses ? Well, the formula stays the same .. total bonus will not increase, but total stake will. In the example above, if our player had not taken 10 bonuses, but only 5, alternating a deposit with a bonus and a deposit without a bonus, then at the tenth session he would have an EV of :

EV = 5*50 bonus - 1000*0.05 = 200

And so when he cashes out on the tenth session (after 5 deposits of 50 on which he claimed bonuses and 5 of 100 without bonus), he cashes out 950 on a total deposit of 750 .. or again exactly the EV ahead when the machine is at 95% (so he didn't get lucky - he got the exact average RTP).

In other words depositing without a bonus doesn't instantly mean you have no EV from past bonuses anymore. As long as you follow the rule of thumb above - you are playing at a positive EV.

As above, you never have any past EV. A future play is +EV or it is -EV, and a past play is either a winner or a loser.

Closing thought.

It's lady luck that gives out the best bonuses. It's not the best mathematicians that win the most - its the luckiest players.

Well yes and no. If you deposit $100 every day and play roulette with no bonus, then in the long term you will NOT be lucky, in fact you are guaranteed to end up a loser. In fact, in the very long term this is true of all gambling on games without a bonus. The way to 'get lucky' is to play big stakes, high risk games, or low house edge games (or maybe all three). In any of these cases what you are doing is playing so that the variance exceeds the expected loss. You can lose $1m into a slot machine with a $10m jackpot, but if the jackpot pays off you are suddenly 'the luckiest player'. Whereas if you lose $1m playing roulette at $10/spin, there's NO WAY that you will ever win that back, because roulette isn't risky enough.

 

[heador112]

Whereas if you lose $1m playing roulette at $10/spin, there's NO WAY that you will ever win that back, because roulette isn't risky enough.

That is not correct you can win Millions at Roulette with high table limits if you have a good run . Most Ocs have low standard limits for Roulette but this may increase with a VIP status . I heard from other People Ladbrokes and 32RED have pretty big limits.

There are several landbased Casinos where you can win more then a Million with a lets say 40K limit.

There are also strategys which allow you to play Roulette with a positive EV without a bonus or something like that but those strategys work only in landbased Casinos . So if you play long enough can can even win $1M with $10 per spin .( But winning and loosing that amount of money take ages with this betsize )

 

[Jufo]

That analysis was pretty much spot on, thelawnet. I think Enzo is somewhat of an authority figure here and probably the most active casino rep. For him to give such mathematically unjustified advice, whose primary purpose seems to be making players lose more money at the expense of casinos, does seem rather worrying to me. Again I apologize if I have somehow understood his intentions wrong.

 

[KariRunk]

Casino reps are members, just like you. What is your motive for questioning his motives?

My main concern is that some naive newbie will get suckered into believing that there is any substance to this.

BTW, this is quote from you from rouletteguy thread. Here's another:

The problem with discussing roulette "systems" on boards like these, is that in most cases those who tout that systems work usually have a vested interest.

Isn't rouletteguy also just member, just like me?

 

[jelsmith]

My main concern is that some naive newbie will get suckered into believing that there is any substance to this.

BTW, this is quote from you from rouletteguy thread. Here's another:



Isn't rouletteguy also just member, just like me?

Enzo is a respected forum member and casino manager.....He didnt post it to sucker anyone into anything...

 

[thelawnet]

That is not correct you can win Millions at Roulette with high table limits if you have a good run . Most Ocs have low standard limits for Roulette but this may increase with a VIP status . I heard from other People Ladbrokes and 32RED have pretty big limits.

There are several landbased Casinos where you can win more then a Million with a lets say 40K limit.

There are also strategys which allow you to play Roulette with a positive EV without a bonus or something like that but those strategys work only in landbased Casinos . So if you play long enough can can even win $1M with $10 per spin .( But winning and loosing that amount of money take ages with this betsize )

It is correct that if you continue to play at $10 you cannot win 1m. In a land-based casino you could only win if you had a biased wheel, there is NO 'strategy'

 

[NASHVEGAS]

Enzo is a respected forum member and casino manager.....He didnt post it to sucker anyone into anything...

Fine.......perhaps you can address the substance that legitimately concerns KR.

Keep in mind some believe in long term gambling success (i.e.winning) based on the law of averages, money management, disclipline, etc (and continue to post of such). It is simply gambler's fallacy just like many other claims, posts, etc. in this forum.

 

[jelsmith]

I mean what's the point of posting all those calculations when they're not valid on majority of cases of bonuses. They're valid only if you get bonus with no or 1X wagering requirements!
And since those calculations was posted by casino representative, I have to question his motives about posting those.

She stated earlier about low or no WR....thats what i get at 3dice low or no WR so I can totally see the possibility. Thats why I dont think he's suckering anyone.

Do I have bad runs at 3dice YES! (currently havin one now..lol) but i cant count the multitude of times Ive been given a managers bonus with NO WR. I also have had ample opportunity to cashout with a match bonus there...do I ? No not usually but thats my own fault. Overall Im in the hole there, so Im not a 3dice brown noser either. I was just stating my opinion based on the facts as I see them.

 

[tencardcharlie]

Hi everyone, first post in a few years from me I think.
I hope the post makes sense despite my lack of english skills.

Of course, you are right that it is possible to turn a bonus with negative expectation to a positive one by busting out often and therefore wagering less than the requirements. This is why some casinos have certain maximum bet sizes when you play with a bonus. However instead of:

[COLOR="Red"][SIZE="4"] [CENTER]For as long as 5% of your totalstake is lower than your
total bonus, you  are playing at positive EV.

[/CENTER] 
[/SIZE][/COLOR]

I would rather say: as long as 5% of your expected total stake is lower than your total bonus, you are playing at positive EV. There is an important difference. If you bust really quick on a few bonuses, and therefore wager less, that doesn't necessarily mean you had an advantage, you could just as well have been unlucky. At the same time: just because you wagered more, doesn't mean that your expectation for that session was negative, since you could have wagered a lot less, should you have been unlucky and busted early.

Of course, if you track your gameplay for a long time, you will converge towards your true rate of wagering per bonus dollar, so with a large sample you could estimate your expected wagering with this method. (This is basically the same idea as estimating your EV by calculating your actual winnings.) However, that could take a good while, especially if you are playing high variance games like slots. Until you have a large enough sample you will have no clue. In fact you will be clueless anyways if you don't know when your sample size is large enough. (Note that I havent made any estimations on the sample sizes needed so I may be exaggerating the problem, it would be interesting to see some numbers).

Therefore, I somewhat doubt that a recreational player will be able to make much use of this thumb rule, although it does illustrate an important point: increased risk of busting -> increased value of bonus.

Even if you reach a large enough sample, your calculation will be useful if the conditions for the bonuses you take doesn't change too much, which seems a wee bit optimistic considering how bonuses have changed in the past.

Also I think this:

So lets first try to define what we are exactly trying to calculate. The formula you'll find all over the web is not 'mathematically wrong' .. to be precise, it just doesn't calculate the number you need. To avoid confusion I wont repeat that formula here, but what it calculates is the EV of a bonus, _if you only take just one bonus your entire life_. If that is the masterplan, take a bonus once and then never again, then you can use the formula you'll find on many websites.

Is somewhat confusing. The risk of busting out will affect your EV the exact same way if you are playing with just one bonus as if you are going to play series of bonuses.

However, the basic idea of increasing variance when playing with a bonus is a good one, as it increases the value of the bonus by busting out, as well as the fun (variance=fun).

 

[just play]

but i cant count the multitude of times Ive been given a managers bonus with NO WR

I can....once....yup, once, which was a few weeks ago.

If I send them an email about having bad play (which has happened for the last few months, and I mean BAD play) they will toss me $10 or sometimes $20, but never just because, never.

I have read many many many players get these bonuses put in their account, I am starting to wonder why I don't receive them? Favoritism? More chat people? Who knows. Maybe I AM the unluckiest person.



I'm still waiting for the "dumb down" version of all this math talk, I just don't understand it.

 

[tencardcharlie]

I'm still waiting for the "dumb down" version of all this math talk, I just don't understand it.

I'll give it a try: Since the casino has an advantage when you play: the less you play, the less you are expected to lose. This loss can be compensated by bonuses. In order to know whether you get enough bonuses to compensate for the losses you need to compare the size of the bonuses you recieve to the amounts you wager.

If the bonuses you have recieved totals less than 5 % of the amount you wager, (assuming 5% house edge) you have probably been playing in a way that is expected to lose in the long run (according to Enzos rule of thumb). If the bonuses are more than 5 % of the amount you wagered, you are probably playing in a way that will make a profit.

Why? Because if you bet, say $1000, you are expected to lose $50 with 5% edge. So if you have received more than $50 of bonuses for each $1000 of wagering, the bonuses will have compensated for the expected losses.

That said, I find the use of such a thumb rule somewhat problematic.

 

[skiny]

I think the best rule of thumb is to forget what you lost. It's gone. If you think any formula is going to help you win it back you're only going to end up chasing those loses down the rabbit hole. You can write down everything you deposited, every bonus you took and everything you lost for the last 20 years and it won't mean a bloody thing. None of it has any bearing on what's going to happen on your next spin.

For me the expected value of a 50 dollar bonus is 50 more dollars that I can wager this session. Decide what you can afford to lose today and if you don't think that will get you much play time take the deposit bonus and hopefully you'll get what you feel is an adequate amount of entertainment for your money.

And if you do get lucky enough to win something don't dump it all back in trying to get rich. You won. Now go buy yourself a wide screen TV or some new tires or a hula skirt if you got the hips for it and when you spent it all come back tomorrow and decide what you can afford to lose on your next session.

 

[tencardcharlie]

I think the best rule of thumb is to forget what you lost. It's gone. If you think any formula is going to help you win it back you're only going to end up chasing those loses down the rabbit hole. You can write down everything you deposited, every bonus you took and everything you lost for the last 20 years and it won't mean a bloody thing. None of it has any bearing on what's going to happen on your next spin.

Dang, now I find myself defending an idea which I don't even agree with... but:

This isn't about predicting results, it's about evaluating if you get enough bonuses to compensate for your expected loss on wagering. (basically, do I take more bonus money than HA*wagering) I do agree that the wording of the first post is misleading though.

 

[jod5413]

Glad you stopped in, hope you stay a while.

Hi everyone, first post in a few years from me I think.
I hope the post makes sense despite my lack of english skills.

Of course, you are right that it is possible to turn a bonus with negative expectation to a positive one by busting out often and therefore wagering less than the requirements. This is why some casinos have certain maximum bet sizes when you play with a bonus. However instead of:


I would rather say: as long as 5% of your expected total stake is lower than your total bonus, you are playing at positive EV. There is an important difference. If you bust really quick on a few bonuses, and therefore wager less, that doesn't necessarily mean you had an advantage, you could just as well have been unlucky. At the same time: just because you wagered more, doesn't mean that your expectation for that session was negative, since you could have wagered a lot less, should you have been unlucky and busted early.

Of course, if you track your gameplay for a long time, you will converge towards your true rate of wagering per bonus dollar, so with a large sample you could estimate your expected wagering with this method. (This is basically the same idea as estimating your EV by calculating your actual winnings.) However, that could take a good while, especially if you are playing high variance games like slots. Until you have a large enough sample you will have no clue. In fact you will be clueless anyways if you don't know when your sample size is large enough. (Note that I havent made any estimations on the sample sizes needed so I may be exaggerating the problem, it would be interesting to see some numbers).

Therefore, I somewhat doubt that a recreational player will be able to make much use of this thumb rule, although it does illustrate an important point: increased risk of busting -> increased value of bonus.

Even if you reach a large enough sample, your calculation will be useful if the conditions for the bonuses you take doesn't change too much, which seems a wee bit optimistic considering how bonuses have changed in the past.

Also I think this:



Is somewhat confusing. The risk of busting out will affect your EV the exact same way if you are playing with just one bonus as if you are going to play series of bonuses.

However, the basic idea of increasing variance when playing with a bonus is a good one, as it increases the value of the bonus by busting out, as well as the fun (variance=fun).

 

[skiny]

Dang, now I find myself defending an idea which I don't even agree with... but:

This isn't about predicting results, it's about evaluating if you get enough bonuses to compensate for your expected loss on wagering. (basically, do I take more bonus money than HA*wagering) I do agree that the wording of the first post is misleading though.

This thread is entitled "How to calculate the EV of a sequence of bonuses." It was not entitled "Does your bonuses compensate for an average RTP of less than 100%"

When the term "Expected Value" is used it gives the impression that an assumption can be made by looking at past stats. When I take a bonus it has an "expected value" of it's face monetary value. It doesn't have a known value until I'm done spinning. Usually it's now known to be worth zero.

And so when he cashes out on the tenth session (after 5 deposits of 50 on which he claimed bonuses and 5 of 100 without bonus), he cashes out 950 on a total deposit of 750 .. or again exactly the EV ahead when the machine is at 95% (so he didn't get lucky - he got the exact average RTP).

Anytime a player cashes out he (or she) got lucky. You can make 10,000 deposits at any casino and it doesn't matter if you take bonuses or not. You may never see the average RTP.