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.
Code:
- 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.
Code:
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.
Code:
[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.
I'd rather play somewhere else because i think those casinos "hate" me.
