Introductory Advantage Play (Part 2)

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Apr 03, 2015

Introduction

All Math is not Gambling, but all Gambling is Math.

In our previous article, we discussed a few common misconceptions pertaining to what advantage play is not, so now we are ready to discuss what some basic advantage plays are.

In this article, we're going to discuss a reasonably common promotion: Double Royals on Video Poker.

Every Video Poker game (other than Class II jurisdictions) is nothing more than a math problem. The overall return of a Video Poker game is determined by the following:

Determining every possible combination of starting cards.
Determining how frequently each of those combinations will occur.
Determining what the mathematical expected return of a hand is based upon what cards are kept and what cards are thrown away. AND
Determining how frequently paying hands will occur as a result of these decisions.

The combination of these determinations yields two things: An Optimal Strategy for playing the game and the Expected Return based upon said Optimal Strategy.

It sounds like a strenuous process, and it would be, except you don't have to do ANY of that.

One of, if not the best Video Poker tool ever created, is the Video Poker Strategy Calculator on our companion site, www.wizardofodds.com :

http://wizardofodds.com/games/video-poker/strategy/calculator/

All a player has to do is input the corresponding pays to the hands correctly and this tool will generate an Optimal Return and an Optimal Strategy for playing the game.

The first thing that a Video Poker player should do in determining whether or not there is an advantage to be had, and to be certain, some Video Poker paytables are so bad that even Double Royals would not yield an advantage, is to find the best available paytable for the desired denomination to be played.

(There are also going to be potential bankroll considerations, depending on a player's goal, that might make playing a lesser paytable a better decision for an individual player, but more on that later.)

For the time being, we're going to assume that the Golden Goose Casino is offering a Double Royal promotion on Video Poker this week, and the game is 9/5 Jacks or Better. We're also going to assume that $1000 coin in results in 1000 Player's Club Points which can be converted to $1 Free Play, for an added rate of 0.1% to the return.

The Golden Goose Casino doesn't have great Video Poker offerings, but they do have 9/5 Jacks or Better at any denomination from $0.05 ($0.25 Max Bet) $0.25 ($1.25 Max Bet) and $1.00 ($5.00 Max Bet). They also have 9/6 Jacks or Better at the $5 denomination ($25 Max Bet), but they're only doubling Royals UP TO $4,000. Fortunately, they will add $4,000 to a Royal on the $5 denomination.

The first thing we have to determine is what return we are looking at for each game. The comp dollars don't necessarily matter, at this point, because they would be applicable either way.

Using the strategy calculator, Double Royals on 9/5 Jacks or Better (Change the Royal return to 1600 units and the Flush return to 5 units) is:

9/5 Jacks or Better: 1.007391

We also have to look at the 9/6 Jacks or Better, normally a Royal would return 4000 * 5 = $20,000, but, with the $4,000 Bonus, it returns $24,000. 24000/5 = $4800/5 (coins) = 960 units.

9/6 Jacks or Better: 0.999559

As we can see, the 9/5 Jacks or Better is the far superior game because we realize the full value of that Double Royal, even with an otherwise inferior paytable. The 9/6 paying 960 units on a Royal still yields an advantage from the points that convert to Free Play, but barely.

We have now given due consideration of the best game to play for this promotion, so now we have to look at the strategy chart and play accordingly to maximize the value of the promotion.

One of the most immediate differences we see, at the top of the chart (the hands with the highest expected value) is that we will now throw away a high pair (J's-A's) in favor of ANY three to a Royal Flush. This chart shows that any three to a Royal Flush starting hand is included in this determination, however, one could also scroll down to, "Basic Strategy Exceptions," and see if there are any hands in which one would do better to keep the high pair.

EXPECTATION v. ACTUALIZATION

In terms of expectation, a player starting out with x amount of money will play every hand at the same overall percentage advantage. However, in terms of actualization, the player that plays more hands at a lower denomination (assuming the game is the same and that there are bankroll considerations) has a better probability of actualizing the advantage, namely, by hitting a Royal Flush.

This concept is no different than that of the House Edge in a negative expectation game, such as Craps. In Craps, the, "Bold Play," strategy on the Pass Line is more likely to double $1,000 into $2,000 than flat betting $10 with a $1,000 bankroll. The reason for that is because the player attempting to double $1,000 only bucks the 1.41% House Edge with that $1,000 on one occasion, but the player who flat bets $10 will ultimately win some bets and lose some bets ultimately causing him to make more than $1,000 in total bets. In effect, that player will expose more than $1,000 to the House Edge and faces a greater expected loss.

To wit, the player flat-betting $10 on the Pass Line will extremely rarely, if ever, double $1,000 into $2,000.

The concept of actualizing an advantage works the same way: For the player that makes a greater $$$ amount in total bets, the more likely that player is to realize the advantage.

The Kelly Criterion exists to determine the Optimal bet sizing based on the percent advantage and variance of a game, and we may touch upon that in a future article, but today we're going to discuss a relatively easy way to get an idea of the probability a player has to maximize the advantage of a Double Royal promotion. This is a solid concept that applies to any games where the pays and probabilities are known.

There is a concept that Advantage Players often call the, "Drop," between Royals. In other words, with a Double Royal promotion on 9/5 Jacks, the player is only at an advantage because of the Double Royal. The player must hit a Royal to realize the advantage. The Drop is the amount that the player is expected to lose on any non-Royal hands.

For the 9/5 Jacks, the Strategy Calculator yields a return of 1.007383 with the Double Royals, we can add .001 to that because of the Free Play from points. (Technically, the Free Play is not worth the exact face value if playing a negative expectation game or a positive expectation game, but that's splitting hairs if you are playing at an advantage) When the Free Play is considered, we yield a return of 1.008383 or 100.8383%.

The Drop between Royals can be expressed as:

1 - (x-y)

Where:

1 represents 100%

x = The Overall Return of the Play

y = The percentage value the Royal contributes to the play.

In this case, we get:

1 - (1.008383 - 0.049039) = 0.040656 or 4.0656%

That means that the player is expected to lose 4.0656% of all monies bet assuming that the player does not hit a Royal. This number is extremely relevant because it can be used (assuming all other returns happen as expected) to determine a variety of things, including:

How much money will be lost between x Royal cycles if a Royal is not hit?
How likely is the player to hit a Royal at x denomination if the player is losing 4.0656% on all hands that are not a Royal and the player has y bankroll?

Some players may be content to simply HAVE an advantage, win or lose, regardless of whether or not that advantage has a reasonable probability of being actualized. Other players, however, will put more of an emphasis on actualizing the advantage, particularly players who want to meaningfully go after the play...the player who wishes to actualize the advantage will want to know the answer to these types of questions.

Question 1: How much money will be lost between x Royal cycles if a Royal is not hit?

We're going to start by focusing on the first question, and the answer is actually quite easy. As we can see from the analyzer, Optimal 9/5 JoB strategy with the Double Royals would result in a hand cycle of:

1/.000031 = 32258.0645161 or, approximately 32,258 hands between Royals.

The only remaining step is determining how much you can expect to lose if you go x Royal cycles without hitting the Royal. You would simply take the total amount to be bet, multiply that by the drop percentage between Royals, and multiply that result by the number of hands in a cycle. We'll start off by doing half a cycle, one cycle, two cycles and then three cycles...for this example we'll say we're betting $1.25/hand:

Half Cycle: (32258/2) * (1.25 * .040656) = 819.67578

That means, if all else goes exactly according to expectation (and, it won't, for example, you're expected to hit 1.725803 Straight Flushes in that number of hands, and you can't hit a fraction of a Straight Flush) you would lose about $819.68 if the Royal goes 16,129 hands without hitting.

One Cycle: (32258) * (1.25 * .040656) = 1639.35156

Two Cycles: (32258*2) * (1.25 * .040656) = 3278.70312

Three Cycles: (32258*3) * (1.25 * .040656) = 4918.05468

One might notice, even with a Royal that pays $2,000, that one is expected to lose well over one thousand dollars if the Royal goes two cycles without hitting. That's no reason to panic as the probability of going two (or more) full cycles without hitting is fairly low:

(1-.000031)^64516 = 0.13533162915 or 13.533%

Question 2: How likely is the player to hit a Royal at x denomination if the player is losing 4.0656% on all hands that are not a Royal and the player has y bankroll?

This is a question that relates very closely to the last one and ties into the theme of Expectation v. Actualization. The fact remains that, the advantage on the hands that you play (not counting comps or potential mailers) is .8383%. That percent advantage is immutable, just like the House Edge is immutable in a negative expectation game.

This is a question mostly of interest to individuals with a minimal bankroll, or even more importantly, to individuals who gamble recreationally, but would like to take an easy (as in, easy to figure out) advantage where one can be had.

These formulas will work with any bankroll, so we're going to say that a player has a bankroll of $2,000 set aside for the purposes of going after this play. We've already determined that we don't want to play the $5 denomination, because there's no advantage there, so our choices are $1, $0.25 and $0.05:

The first thing to determine is how much we are dropping between Royals, per hand, and each denomination:

$1 Denomination: (5 * .040656) = 0.20328

$0.25 Denomination: (1.25 * .040656) = 0.05082

$0.05 Denomination: (.25 * .040656) = 0.010164

What makes this somewhat easy is that this drop is all going to be based upon initial bankroll because it is how much you are expected to lose, per hand, if you don't hit the Royal. You'll have ups and downs and will win some hands and lose some hands, but by expectation, all of that is already factored in.

At this point, we're going to determine two things:

What is the probability of hitting a Royal before busting?
What is the probability of profiting?

The way to answer the first question is simple, you take the starting bankroll and divide it by the loss per hand, and that's how many hands you get (without a Royal) before you bust:

2000/0.20328 = 9838.64620228

That was for dollars, at $5/hand, so now we are going to take that number of hands and determine the probability of not hitting a Royal in that number of hands, or more:

(1-.000031)^9839 = 0.73711325542

The inverse is the probability that we do hit the Royal in that number of hands, and that probability is roughly 26.28863%.

In other words, roughly three out of four players will bust out before they actualize that advantage if they go after this with a $2,000 bankroll on the $1.00 denomination. This is provable by taking the number of total $$$ exposed (expected hands * bet) and multiplying by that drop:

(9838.64620228 * 5) * .040656 = 2000

This is probably not a good betting amount for the player who puts a focus on actualizing the advantage. It's true that all hands will be played at the same percentage advantage, but the player is somewhat unlikely to hit the Royal in that many hands. For the player who has a limited amount of time (at 800 hands played per hour, this is still 12 hours of play) or for the player for whom a $2,000 or $400 win is meaningless, then this may be the denomination to play. For someone that wants a high probability of actualization, though, this is not ideal.

In this case, the probability of profiting and the probability of hitting the Royal are one and the same. The reason why is because if the player hits the Royal before busting, the player WILL profit. This is obvious because the player is only buying in for 1/4th of what the Royal will pay.

We will now move on to our $0.25 denomination player:

2000/0.05082 = 39354.5848091 hands

Probability of Not Hitting Before Busting:

(1-.000031)^39354 = 0.29523226011

The inverse is .70476774 or a 70.476774% Probability of hitting before busting.

We've already proven the equation, so that is unnecessary. Also, the probability of hitting the Royal before busting and the probability of profiting are almost identical, in this case, because the Royal pays exactly the amount for which the player is buying in. As long as the player hits the Royal on any except his very last hand, he will profit, if the last hand, he will break even.

Finally, our $0.05 denomination player:

2000/0.010164 = 196772.924046 Hands

Probability of NOT hitting before busting:

(1-.000031)^196773 = 0.00224273864

The inverse of that is 99.775726135 or 99.77573% to hit the Royal before busting.

We now need to determine the probability of profiting, assuming all other hands go to expectation. The easy way to do this is to simply pretend you only have a $400 bankroll, because that's how much the Royal pays:

400/.010164 = 39354.5848091

(1-.000031)^39355 = 0.29522310791

The inverse is: 0.70477689208 for a 70.477689% probability of profiting.

We notice that this amount is slightly higher than for our quarter player, and that's because the quarter player can break even on that one hand that puts him down exactly $2,000.

Conclusion:

For the purposes of an introduction to Video Poker advantage play, this is a good start. This gives a player a general idea of, when Royals are doubled:

How to determine if a paytable is positive as a result.
How to go to Wizardofodds.com and determine an Optimal Strategy.
The probability of hitting a Royal before busting out on a fixed bankroll.
The probability of hitting a Royal and finishing with a profit.

In future articles, we will look at other promotions and also take a look at how different paytables on different games may influence the value of a promotion such as Double Royals and in what way Variance of the overall paytable may play a part.

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