Comparing the Expected Value of Multiple Bonuses
One aspect of Online Casino Bonuses that some people occasionally find confusing is knowing, among various Bonus options from different casinos, and sometimes the same casino, which one to choose. Making the correct decision with respect to the best Expected Value is a complicated affair, so I intend to look at different Bonus choices from the same Online Casino and teach everyone some concepts to help identify which Bonus has the best Expected Value.
One aspect of Online Casino Bonuses that confound many players is the simple fact that there are so many of them! With that said, it is important to be able to identify the differences between one Bonus and another, how to use the math to determine an Expected Profit (if applicable) on a particular Bonus, and also to determine if pursuing a particular Bonus can be done safely within the confines of your bankroll and if you have the time for it. I mention time because, naturally, grinding out literally hundreds of thousands of dollars in Wagering Requirements (depending on the Bonus) can take some serious time!
The one thing that you should do that we are not going to do in this Article, but will do in later Articles, is compare the Bonuses of different casinos to each other. Remember, higher match percentages are irrelevant, the wagering requirements are irrelevant, the games offered are irrelevant, whether or not the Bonus is cashable is irrelevant; it is the combination of all of these factors that matters.
Just to give you an example of what I mean, let's say that you had two Bonuses and the Terms of each of the two Bonuses were as follows:
$500, 100% Match, Video Poker, 9/6 Jacks, 50xD+B
$1000, 300% Match, Slots/Keno, 30xD+B
The thing that you would want to look at here, assuming that both Bonuses are cashable, is that you will start with a bankroll of $1,000 on the Video Poker which has a House Edge of 0.46% or .0046 and an Expected Loss on the $50,000 Playthrough of $230 which results in an Expected Profit of $270 on that particular Bonus.
In contrast, that Slots Match will give you a combined Bankroll of $4,000 and you will need to make $120,000 in Wagers in order to complete that Bonus. Even if you assume an extremely liberal slots game with an RTP of 96%, then you are Expected to lose $120,000 * .04 = $4,800. You will notice that $4,800 is more than the Deposit + Bonus of $4,000, which means that you are NOT Expected to even complete the Wagering Requirements, and as a result, are Expected to simply lose $1,000.
Essentially, the smaller Video Poker bonus with the greater Wagering Requirements is worth $1,270 more than the Slot Bonus. In fact, the Slot Bonus (again, assuming a game with 96% RTP) is worth less than nothing because you are Expected to Lose.
The first thing that we have to do is select a casino that is offering different bonus types, so we need to take a look at the relevant LCB Pages:
After noticing that ThisisVegas offers both a 20% Cashback Bonus as well as a No-Deposit Bonus, I decided to use that casino for the purpose of this Article, so I went to their page on LCB to determine if there is a Deposit Bonus, which there is. I would like to state that I am not advocating this casino on a personal level, in fact, I couldn't play there even if I wanted to given the fact that they do not accept U.S. Players. I simply needed a casino that offers different types of Bonuses so that we could compare one type of Bonus to another.
The first thing that we are going to need to do is determine specifically what sort of games are available at ThisIsVegas Casino. If you're anything like me, you might not want to take the trouble of downloading an entire casino just to see what games are in their portfolio, fortunately for all of us, many of these casinos have been reviewed (including available games) on our sister site, WizardofOdds.com:
I want to know what games in each category have the lowest possible House Edge, and occasionally, one might concern oneself with a concept known as, "Element of Risk," as opposed to House Edge. For those of you wondering what the difference is between these two concepts, House Edge refers to the percentage of every bet that one is expected to lose based upon the initial bet. Alternatively, the Element of Risk refers to the Expected Loss per unit bet based on the entire game, as a result, Element of Risk is typically used when one is analyzing games that have multiple phases of betting.
If you want the quick and dirty way to do figure out the Expected Loss on your total action relative to the Wagering Requirements, then you can simply take the House Edge and multiply that (expressed as a decimal) by the total amount that needs to be wagered. This is especially relevant to deposit bonuses because very often you're not going to need to, 'Take a shot," to have a Deposit Bonus in which you have the mathematical advantage...you can often, 'Play straight up."
With that said, we learned from the LCB ThisisVegas Page that they are currently offering a 100% Match Deposit Bonus of up to $1,000 with the following Wagering Requirements:
Casino War: 150x
Video Poker: 300x
Poker Games: 150x
If we are faced with a bonus that has a positive expectation, then clearly, we would want to have an initial deposit of as much as possible in order to take advantage of said bonus. In this case, if there is an advantage to be had, we would want to deposit the full $1,000 and enjoy a bonus of $1,000 for a total bankroll of $2,000.
Now, the first thing we are going to do is determine exactly how much we are going to have to wager in order to complete the Wagering Requirements: This is a simple function of multiplying the Wagering Requirements by the total bankroll.
Another important thing to note is that the Bonus in this particular case is a non-cashable Bonus. That means, if you intend to cash out, the amount of the Bonus will be removed from your account and any difference would be cashable. Unfortunately, that also means that this Bonus sucks, you should always look for Cashable Bonuses.
NOTE: I am going to arbitrarily pretend this is a cashable Bonus as opposed to non-cashable. There are not many non-cashable Bonuses that are terribly useful because they would usually require, "Taking a Shot," in order to exponentially improve your bankroll and then grinding through the Wagering Requirements. Based on the Wagering Requirements for this Bonus, it might still be good (even as a non-cashable) but it would not even occur to me to analyze this until I had plowed through MANY, MANY much better Bonuses out there.
Okay, so now I am going to pretend that this is a cashable Bonus, as a result, the Wagering Requirements would be as follows:
Casino War/Poker Games: $300,000
Video Poker: $600,000
Here's a little trick for everyone: One thing that you will notice is that the Poker Games wagering requirement is half that of Video Poker, therefore, for a Poker Game to be the right choice here, the House Edge of the Poker Table game would have to be double (or less) the House Edge of the best Video Poker game.
We will also notice that the Blackjack game has 4/5ths (0.8) of the Wagering Requirements of the Video Poker, therefore, the House Edge of the Blackjack game would have to be less than (House Edge of Video Poker game expressed as decimal)/0.8.
The best Video Poker game offered by this casino is 9/6 Jacks or Better which, with Optimal Strategy, returns 99.54% resulting in a House Edge of 0.0046, which represents 0.46%. That being done, let's see how the House Edge of our other best choices relates to that one:
.0046/.8 = .00575 05 0.575%
As a result, we need the House Edge of the best Blackjack game to be less than 0.575%, or alternatively, it could be equal to that as that would relate the same to the Wagering Requirements. Here's the proof:
240000 * .00575 = $1,380
300000 * .0046 = $1,380
As we can clearly see, our Expected Loss on the required dollar action is $1,380, so as a result, if this were a cashable Bonus, we would be expected to finish with a bankroll of $620, which is not a positive play in either case.
While the slots may have a much lower 30x Wagering Requirement, for a total dollar action of $60,000, that does not necessarily make the Slots a better play. In fact, if you assumed an RTP of 90% on the Slot game, then that Slot game also comes with an Expected Loss of $6,000. If it were a 5% Expected Loss (95% RTP) then you are still looking at an Expected Loss of $3,000.
Regardless of the situation, in all but the rarest of occasions (which you would usually find at Brick & Mortar rather than Online Casinos) Slots are the worst thing that you can do, and with exception to Video Keno, this is no different.
Actually, I'm going to take that back. The Video Keno at many Rival Casinos is actually better than a few Slot games, in terms of RTP, but still none of them would be playable, even if the Bonus were cashable.
We would need a Table Game, in this case, with a RTP of greater than 99% given the Wagering Requirements, and being the nice guy that I am, rather than Math that one out I'll just save you a little time: There isn't one.
Remember, this type of Expected Loss would be the case in the event that the $1,000 from the Bonus were actually cashable, which it is not. The Bonus amount gets removed from the balance when you go to cash out, and furthermore, you would be cashing out for less than $1,000 anyway, so eventually the expectation is that you would play down to nothing.
If structured betting does not violate the Terms & Conditions in any way, which I am not downloading the casino (would it even let me?) to even read, one could theoretically enjoy an Advantage by starting out taking a shot at Roulette:
I'm not going to go through every possible combination of bets on the Roulette Table to illustrate all of the possibilities, but the one thing that I will do is base two bets at the $500 Maximum to illustrate how you might build your bankroll and complete the Wagering requirements with a positive expectation.
In this event, you are going to make up to four Roulette bets on a single number totaling $2,000 in the event that all of the bets lose. We have established that our best game for completing the Wagering Requirements is going to be the 9/^ Jacks or Better with an Expected Loss of $1,380, so we are going to look at the possible results (in terms of new bankroll) less the $1,380 Expected Loss and then less the $1,000 for the Bonus being removed before cash out and also less $1,000 for the original $1,000 deposit. Therefore, we shall subtract $3,380.
The first thing that we must consider is the possibility that we will lose all four Roulette bets, and that would require the Expected 36/37 probability of a loss to occur four times in a row:
(36/37)^4 = 0.89619621793
We're going to round up and call that .8962 probability that we just lose on everything and we are done. That represents the probability of losing the $1000 deposit which thereby makes our Expected Loss on that result:
(1000 * .8962) = -$896.20
Okay, so let's get to the fun stuff. Just for the record, I am going to ignore the fact that we will have put some small amount towards the Wagering Requirements already and just address this as though that didn't happen. With that in mind, let's take a look at our Expected Results (and profit) given that we hit at different times:
Our bankroll would improve to $19,500 if we hit on the first try, $19,000 if we hit on the second, $18,500 if we hit on the third and $18,000 if we hit on the fourth attempt. This represents pays of 35 to 1 for a pay of +$17,500 as well as getting the original bet amount back. What impact does this have on our Expected Profits? Let's take a look:
(19,500 - 3380) * 1/36 = $447.78 (rounded)
(19000 - 3380) * (35/36 * 1/36) = $421.84 (rounded)
(18500 - 3380) * (35/36 * 35/36 * 1/36) = $397.00 (rounded)
(18000 - 3380) * (35/36 * 35/36 * 35/36 * 1/36) = $373.20 (rounded)
If we add up all of our Expected win amounts on favorable results and subtract from that sum the Expected Loss on unfavorable results, this is what we get:
(447.78 + 421.84 + 397 + 373.20) - 896.20 = $743.62
In other words, initially playing this Bonus on Roulette (which might violate the Terms & Conditions at some Online Casinos for, 'Bet Sizing,' so make sure to read those and stay within the bet parameters, if so) you have an Expected PROFIT of $743.62 on your $1,000 Deposit which represents an advantage of about 74.362%.
I think that's a pretty decent advantage, don't you?
Remember, we have already factored in getting the initial $1,000 deposit back, as well as the non-Cashable Bonus being removed from the balance, so the number above represents profit...sexy, sexy profit!
NOTE: The above Roulette stuff IS BASED on the fact that the Bonus is not cashable, meaning, you could do that. Further, Blackjack has less Variance than Video Poker, so you might choose Blackjack to do the playthrough requirement. You would still have an Expected Profit, though not as high, but betting small would give you a greater overall likelihood of grinding out the Wagering Requirement.
Finally, there are other ways that could be done on Roulette. Making Split Bets, or even Corners for $500 could theoretically work, and hopefully you can use the above to simply modify a few numbers and do the Math on that, but I can assure you it will be at a reduced Expected Profit because you want as much Variance as possible on that Non-Cashable bankroll money.
The one thing that you will notice above is that if you add the bottom two winning results together and subtract the Expected Loss that the Expected Loss will exceed that significantly. That is the reason why the game cannot be played with an Expected Profit without the Bonus.
The Maximum Cashout on the No-Deposit Bonus is $100, the Expected Profit on the Deposit Bonus is more than $100; therefore, the No-Deposit Bonus cannot possibly be better in this case. You would only select the No-Deposit Bonus over the Deposit Bonus if you did not have any actual money to use.
Furthermore, most (but not all) No-Deposit Bonuses are structured such that the Expectation is that you will not finish the Wagering Requirements with any money left. With this No-Deposit Bonus, you can only play Slots and Keno which would have a Wagering Requirement of $69,300...you would be Expected to lose way in excess of that amount and finish with no profit.
With that said, as with any other No-Deposit Bonus, the percentage profit expectation is infinite as you have some chance of finishing with money as opposed to no chance whatsoever of losing money.
Honestly, No Deposit Bonuses are almost exclusively something that I would recommend to players who are no-bankrolled or very low-bankrolled. The reason why is that a casino that offers anything even approaching a reasonable No Deposit Bonus will generally have a Deposit Match Bonus that has a much greater positive expectation of profit.
It is important, however, to always weigh one Promotion against another. For example, if you were to encounter a casino that had a Deposit Match Promotion that would still result in an Expected Loss, (generally because of Terms & Conditions that make it impossible to play profitably) then the No-Deposit Bonus is better by default.
That said, I think I'll take that Deposit Bonus that is Expected to result in a profit of almost $750, thank you very much!
This casino offers a 20% Cash Back Bonus in which 20% of a player's losses are allegedly, 'Refunded,' but really players are just given a certain Bonus amount that is also subject to Wagering Requirements depending on which game you play.
The right way to play with these is to go for some high Variance all the way through, but in this case, it is not letting me see what the playthrough requirements are on the Cash Back because I am a USA player. I can already unequivocally say that you would be much better with the Deposit Bonus as opposed to using a Cash Back Bonus because the Cash Back Bonus percentage + the Wagering Requirements generally result in an expectation of nothing left.
That doesn't mean that you should never look at Cash Back Bonuses, they may often be the best thing that you have going at a casino at which you may already be established and are no longer getting strong Bonus offers on Deposits, but most of the first-time Deposit Bonuses you will get at the majority of casinos will be better than whatever Cash Back Bonus is offered.
With that said, at some later time (next week, maybe?) I'm going to write an Article about Cash Back Bonuses in which we will compare different Cash Back Bonuses at different casinos to give everyone an idea of what kind of Cash Back Bonuses you should be looking for.
I hope that you have enjoyed this Article and that it proves useful to you. Please always remember to read the Terms & Conditions for any Online Casino at which you may play to ensure that my betting strategies espoused here are fully legal and permitted. Thanks for reading!