By Jeff Hwang

The pyschological principle behind hit frequency is a concept called variable-ratio reinforcement, which is generally defined as delivering reinforcement after a target behavior is exhibited a random number of times.

Let's take a slot machine, for example. A gambler sits down at a slot machine and bets $1 a pull. Now as you would expect, most of the time, the gambler will bet $1 and lose, which of course is great for the casino. But if all the gambler does is bet $1 and lose every time, eventually he will quit (and/or go broke) and never want to play again. And so every few spins, the slot machine will reward the gambler with a payoff: $1 here, $1 there; $5 here, $1 there. 

And then every once in a long while, the machine will reward the gambler with a big payoff in the form of a jackpot.

Now none of this quite adds up, which is how the house wins in the long run. But the promise of the big payoff, along with the intermittent rewards, is generally enough for the casino to reinforce the target behavior, which is to have the gambler keep betting $1 a pull.

That brings us to our next topic, which is the reinforcement schedule.

Reinforcement Schedules: Variable vs. Fixed
There are two basic types of reinforcement schedules: variable-ratio reinforcement schedules, and fixed-ratio reinforcement schedules.

Let's start with the latter, which is the most basic. A fixed-ratio reinforcement schedule is a schedule in which reinforcement is delivered at fixed intervals. Let's say, for example, that you are the casino and you want the slot machine to pay out 20% of the time, or every fifth spin. That is, the gambler will lose $1 four times in a row and get a pyout on the fifth every time.

The reinforcement schedule would look something like this:

Slot Machine: Fixed-Ratio Reinforcement Shedule

Adjusted for payouts, the schedule might look more like:

Slot Machine: Fixed-Ratio Reinforcment Schedule with Payouts

In this scenario, for every 25 spins, the gambler would win $18 on the five winning spins and lose $20 on the rest, for a net loss of $2. For the house, this represents a payout rate of 92% (RTP) and thus a house edge of 8%, which isn't too far from the real thing, depending on what casino you are in.

Now all of this sounds great, but there is a major problem: Nobody would ever play a game with a payout (reinforcement) schedule like this one!

Ok, so maybe "nobody" and "ever" might be a little strong, but the point remains: It wouldn't take long for the gambler to figure out that this slot machine pays out every fifth spin, and only every fifth spin. And as a result, the gambler would eventually quit playing on the spins they know they are going to lose (assuming the payout amounts are still random, meaning that the location of the $10 payout on the schedule is either random or unknown, for example).

Using a variable ratio is the fix for this problem.

Variable-Ratio Reinforcement Schedule
A variable-ratio reinforcment schedule uses a predetermined ratio while delivering the reinforcement randomly. Going back to the slot machine, let's say that you once again are the casino and want the slot machine to pay out 20% of the time, or every fifth time on average.

Now your reinforcement schedule may look something like:

Slot Machine: Variable-Ratio Reinforcement Schedule

And adjusted for payouts:

​Slot Machine: Variable-Ratio Reinforcement Schedule with Payouts

In aggregate, the expectation is the same: Over 25 spins, the gambler will still net a $2 loss, again giving the casino a 92% payout rate and an 8% house advantage. But in reality, this scenario is far, far more likely to achive the desired result, which is to have the gambler keep playing. Because in contrast to the fixed-ratio reinforcement schedule, a variable-ratio reinforcement schedule with a 20% reinforcement ratio allows for clusters of payouts (e.g. back-to-back wins), as opposed to having spins (or blocks of spins) where the gambler can say for certain that he would lose, and quit playing as a result.

This is because the variable ratio does not specify when the payouts occur, only how often they occur on average.

That said, variable-ratio reinforcement is a concept with endless practical application. As some of you may have noticed, the above discussion came directly from the opening of my book Advanced Pot-Limit Omaha Volume II: LAG Play; in that book, the discussion was used to set the stage for how we think about adjusting c-bet (continuation bet) frequencies based on our opposition, though the concept applies to virtually any action from 3-betting pre-flop to floating the flop.

But with regard to game design, the concept of variable-ratio reinforcement applies most directly to our two basic forms of hit frequency:

1. Payoff frequency, the traditional measure of hit frequency, and
2. Betting frequency, the form of hit frequency which has largely been completely overlooked.

Jeff Hwang is President and CEO of High Variance Games LLC. Jeff is also the best-selling author of Pot-Limit Omaha Poker: The Big Play Strategy and the three-volume Advanced Pot-Limit Omaha series.

By Jeff Hwang

In S​uper Blackjack™, the player is dealt a super blackjack (A♠J♠ or A♠J♣) paying 20-to-1 on the Ante once every 675 hands, and all other suited blackjacks (paying 6-to-1 on the Ante) once every 90 hands. Combined, the player will be dealt either a super or suited blackjack once every 79 hands.

A single player playing alone might see a suited blackjack once an hour, but is unlikely to encounter a super blackjack in a given day playing alone without putting in serious hours. On the other hand, a table of four players seeing 50-60 hands per hour will collectively play 200-240 hands per hour, and will see two or three suited blackjacks per hour, while encountering a super blackjack once every few hours.

Now all of sudden, that rare, higher-payout event that a single player would be unlikely to see in a given day playing by himself occurs with some frequency, and seems that much more attainable.

This is the crux of a concept I call table-wide hit frequency, which reflects how often a given event occurs at a table as a whole rather than on a single-player basis. 

The Limitation of Communal Bets
With all of the recent talk about millennials and the desire for more social gaming opportunities (not to be confused with what's being labeled as social gaming), there has been I think a bit more interest lately in producing communal bets where everybody (or at least multiple players) are betting on the same outcome, sharing in the same results. This works great in a game like craps, where most everybody tends to bet the Pass Line, and everybody can high five each other when the point hits.

But the reason this works is because in craps, we are largely dealing with relatively flat payoffs on the pass line and odds (up to 2:1 for odds on 4 and 10), with relatively high frequency wins. However, I suspect the prospect for communal bets will be challenged when they rely on higher-payoff, lower-probability events.

Here's why:

Let's say you have a Texas hold'em table game. Let's also say you are thinking about offering a wager on the five community cards, and have constructed an event based only on those community cards, and with a top payout of a million dollars for an event that occurs once in a jillion hands. 

Now you have two problems:

1. Everybody has made the same bet. What this means is that if you have five players at the table, you have to pay five players a million dollars when that once-in-a-jillion-hand event occurs. You have in effect multiplied the risk of a Black Swan outcome for the casino operator.
2. Rare events occur significantly less often when everybody has made the same bet. If you have five players betting on the same outcome, rare events will generally occur five times less often than when everybody is betting on outcomes based on their own hands. Put differently, you have five players betting on one hand (the community cards) rather than on five hands (a combination of each player's hole cards with the community cards). And in doing so, you have essentially minimized table-wide hit frequency and the benefits that come with it.

Consequently, I would tend to shy away from using communal sidebets dependent on rare, high-payoff events.

Jeff Hwang is President and CEO of High Variance Games LLC. Jeff is also the best-selling author of Pot-Limit Omaha Poker: The Big Play Strategy and the three-volume Advanced Pot-Limit Omaha series.

By Jeff Hwang

Hand #1. The game is Three Card Poker. You make a $5 Ante wager, and are dealt A♠A♦3♥ -- a pair of aces, and more than enough hand to play. You make the $5 Play wager. The dealer turns up Q♥9♥3♠, and you win. The dealer pays $5 on both of your wagers, for a $10 net win.

Hand #2. The game once again is Three Card Poker. You make a $5 Ante wager, and are again dealt A♠A♦3♥. You make the $5 Play wager as before. The dealer turns up J♥9♥3♠, and you win again. But this time the dealer pays $5 on the Ante wager, but not the $5 Play wager, for a $5 net win.

Hand #3. Again, the game is Three Card Poker. This time you are dealt 8♦4♣3♥, which is not good enough to play (optimal strategy is to play Q-6-4 or better). But you decide to make the $5 Play wager anyway. As in Hand #2, the dealer turns up J♥9♥3♠, which beats your measly 8-high. Except in this game, you don’t lose, and the dealer correctly pays your $5 Ante wager, but not the $5 Play wager, for a $5 net win.

Why? Because like most poker variants developed since Caribbean Stud, Three Card Poker employs a qualifier.

In Three Card Poker, the dealer only “plays” when he has queen-high or better (Q-x-x+). This has a number of implications when the player makes the Play wager:

1. When the dealer has jack-high or less, the player automatically wins on the Ante wager, but pushes on the Play wager. Essentially, the dealer folds when he has less than queen-high, and forfeits against your Ante wager, but does not “call” your Play bet. This is why the dealer only pays the Ante wager when you have a pair of Aces in Hand #2. This is also why you win in Hand #3, even though technically you have a worse hand than the dealer. 
2. When the dealer has queen-high or better and the player has a better hand than the dealer, the dealer pays on both the Ante wager and the Play wager. This is why you got paid on both bets in Hand #1, when you have a pair of aces and the dealer has queen-high.

In some games using a qualifier – usually those with scalable betting – the dealer pays the play/raise/call bet regardless, and only pays the Ante when the dealer has a qualifying hand. In Crazy 4 Poker, for example, the player starts with two initial wagers – an Ante and equal Super Bonus wager – and can bet 1x-3x the Ante or fold on his five-card hand; the player can only make the 3x bet size with a pair of aces or better. The dealer qualifies with a king-high or better.

Let’s say the player has a pair of aces, and makes the 3x wager. If the dealer has queen-high or worse, the dealer will pay on the 3x Raise wager, but push on the Ante and Super Bonus bets.

Other games don’t use a dealer-based qualifier. Mississippi Stud and Let It Ride are straight paytable games; there is no dealer hand to compete against, and the player gets paid if he makes a minimum qualifying hand. In Mississippi Stud, the player gets paid if he makes a pair of jacks or better, and pushes on a pair of sixes through tens. In Let It Ride, the player is paid on a pair of tens or better. In both games, the player receives a bigger payoff for making bigger hands.

In contrast, Four Card Poker uses flat payoffs, and there is a player vs. dealer competition. However, the dealer always qualifies (i.e. there is no qualifier), while the player only needs to beat the dealer to win. 

In his telling of events as explained in an article that originally appeared in the December 2008 issue of Two Plus Two Magazine, Sklansky developed Casino Poker in 1982 (Read the article for Sklansky’s story on how his game eventually became Caribbean Stud). As in Caribbean Stud, the player starts with an ante wager and is dealt five down cards. The dealer is dealt two up cards and three down cards, in contrast to Caribbean Stud, in which the dealer is dealt one up card and four down cards.

At this point, the player can bet 2x the ante or fold, as in Caribbean Stud.

Now the challenge in making a poker-based table game with a player vs. dealer competition is that the dealer cannot play back at the player, and must have a fixed strategy. Moreover, this strategy must be known by the player, thus giving the player an advantage in that regard.

Sklansky realized that he could neutralize the player’s strategic advantage by making the dealer’s strategy unexploitable by means of game theory. In Sklansky’s Casino Poker (and Caribbean Stud) , the dealer effectively (but not physically) matches the player’s ante wager, thus putting two units in the “pot.” The player then either makes a 2x “pot-sized” bet or folds. Facing a pot-sized bet, game theory dictates that the dealer should “call” the player’s bet 50 percent of the time to avoid being exploitable.

If you’re a poker player, by now you should recognize that Sklansky’s explanation is straight out of his own book, The Theory of Poker, originally published in 1978 as Sklansky on Poker Theory.

Sklansky’s solution was to have the dealer “call” with the top 50 percent of hands, which translates into ace-king high or better (the median hand in 5-card stud is somewhere between A-K-Q-J-7 and A-K-Q-J-6). Thus, in Casino Poker and later Caribbean Stud, the dealer only calls – qualifies – when holding A-K-x-x-x or better.

So let’s say you’re playing Caribbean Stud, and ante for $5. You’re dealt A♠A♦4♦4♣3♥ for two pair, and bet $10 (the 2x raise wager). The dealer shows K♠Q♦J♥9♥3♠ for king-high, and thus does not qualify. The dealer pays $5 on your ante wager, and pushes your $10 raise bet, for a net win of $5.

But let’s say the dealer instead has K♠K♦J♥9♥3♠ for a pair of kings, and thus qualifies. In Caribbean Stud, two pair pays 2 to 1 on the raise bet when the dealer qualifies – the A-K-high math doesn’t quite add up (Roughly speaking, the player plays his top half of hands, while the dealer plays his top half of hands, or about half the time when the player also plays; on the half of hands the player folds, the player sacrifices his ante outright, while the dealer only folds and sacrifices his ante to the player about half the time the player plays, or on about a quarter of hands, thus creating a gap. And then the other quarter of hands when both the player and dealer play are a near coin flip, again roughly speaking.). This enables scalable payoffs (the ability to pay more on your play/raise bet for bigger hands). Thus the dealer pays $5 on your ante, and $20 on your $10 raise bet, for a net win of $25.

Effectively what happens is that you the player win more on your bigger hands, but only when the dealer has a big enough hand to call you with. Viewed in that context, the use of the qualifier is quite reasonable. And thus with that precedent – and for better or for worse – a qualifier has been employed in the vast majority of casino poker games developed since.

The Issues With Qualifiers
There are a number of issues associated with the use of qualifiers.

Game Design: Arbitrary Rules and Strategies
In many game designs, strategy is an afterthought. Often times the game is designed, and a qualifier is used to fix the math and make the game work.

In Three Card Poker, for example, the correct strategy is to play Q-6-4 or better. Why? Because the dealer qualifies with Q-high or better. And why does the dealer qualify with Q-high or better? Because that’s what makes the math work.

In High Card Flush, the strategy is whatever the strategy is because the dealer qualifies with a three-card 9-high flush or better. Why a three-card 9-high flush? Because that’s what makes the math work.

Even an expert poker player knowing nothing about Three Card Poker or High Card Flush could not use his skills and reason his way through these games. The strategy is what the strategy is simply because that’s what the strategy is. There is no why.

Arbitrary Outcomes
The use of qualifiers sometimes results in arbitrary outcomes. In Ultimate Texas Hold’em, the dealer qualifies with a pair or better. Now let’s examine the following two scenarios:

• Hand #1. The game is Ultimate Texas Hold’em. You post the $5 Ante and Blind wagers, and are dealt A♠K♦. You bet the 4x maximum – $20 – and the board runs Q♠J♦7♦4♣3♥, giving you A-K-high (or more technically A-K-Q-J-7). The dealer turns over A♦10♣ for A-10-high (or A-Q-J-10-7), and you win. The dealer pays $20 on your $20 Play wager, but pushes on your Ante and Blind wagers, for a net win of $20.
• Hand #2. As before, the game is Ultimate Texas Hold’em, and once again you post the $5 Ante and Blind wagers, and are dealt A♠K♦. You bet the 4x maximum – $20. This time, the board runs J♠J♦7♦4♣3♥, giving you pair of jacks with A-K-high (or more technically J-J-A-K-7). The dealer turns over A♦10♣ for a pair of jacks with A-10-high (or J-J-A-10-7), and again you win. This time, the dealer pays $20 on your $20 Play wager, pushes on your Blind wager (which pays on a straight or better), and pays $5 on you Ante wager. You show a net win of $25.

It’s completely arbitrary that you would win $20 in the first hand, but $25 in the second. In real hold’em, these situations are identical: You have A-K-high, and your opponent has A-10-high. You win. But in Ultimate Texas Hold’em, both you and the dealer are credited for having a pair because of the open pair of jacks on the board in Hand #2; and because the dealer qualifies with a pair, you now also get paid on your Ante wager.

General Confusion
If you’ve never played these games before, it’s completely confusing to show up at a Three Card Poker table, get dealt a pair of aces, and wonder why you get paid two units (one for the Ante, another for the Play wager) when the dealer has queen-high, but only get paid one unit (for the Ante) when the dealer has jack-high. Or why in some games, you get paid on the play/raise wager but not the ante wager when the dealer doesn’t qualify, while in other games you can paid on the ante wager but not the play/raise bet.

In Conclusion
On one level, qualifiers make sense in concept: The player gets bigger payouts on bigger hands, but only if the dealer has a big enough hand to call the player’s bet. But on another level, there are many issues that come with using qualifiers, and the qualifier is a relic of the original carnival games, namely Caribbean Stud and its Sklanskian predecessor, Casino Poker.

My view is that as the art and science of casino table game design progresses, the trend will be towards poker variants being designed without arbitrary dealer qualifiers.

Jeff Hwang is President and CEO of High Variance Games LLC. Jeff is also the best-selling author of Pot-Limit Omaha Poker: The Big Play Strategy and the three-volume Advanced Pot-Limit Omaha series.

If casino operators are serious about marketing to a more sophisticated generation of gamblers – millennials – there’s clearly room to offer games with both higher skill components and lower house advantages to compensate.

By Jeff Hwang

With the rapid expansion of land-based casino gaming across the U.S. over the past quarter century, along with the global poker boom and online gaming explosion of the 2000s, more people have been exposed to gambling games than ever before. And with the great advancements in gambling knowledge and game strategy over this time period, more gamblers – millennials in particular – are more sophisticated and more knowledgeable about gambling than at any point in the history of mankind.

The casino is losing its mystique, and the product of gambling is maturing. As a consequence, the tastes of the modern gambler are changing, such that there is an increasing demand for gaming activity that is more social in nature, and an increasing demand for gaming activity at least partially based on skill.

But what does that really mean for the next wave of casino games?

The first and most prominent interpretation of this trend thus far has been to develop video slot games designed to appeal to a video game generation. About a decade ago at the 2004 Global Gaming Expo (G2E), Bally Technologies (acquired by Scientific Games Corporation last year) introduced a slot game based on Pong, the landmark Atari game released in 1972. Bally’s slot adaptation – approved by Nevada gaming regulators in 2007 – included a bonus feature in which the player would play a 45-second round of Pong, and the player’s performance affected the payout; the difference in expectation between the best player and the worst player in this bonus feature was 7 percent. Other companies have followed suit, with games such as IGT’s Centipede and GTECH’s Zuma offering skill-based bonus features.

More recently, in October 2014, the New Jersey Division of Gaming Enforcement announced that it was “currently authorized to approve” and “eager to receive” skill-based game submissions for review under its New Jersey First program. New Jersey First was designed to encourage the development of casino games in New Jersey combining both the social and skill-based elements found in games like Candy Crush and Words with Friends by allowing games with such features to be fast-tracked to the casino floor within 14 days of approval, provided that the game is “submitted to New Jersey prior to or simultaneously with any other jurisdiction or testing lab.” 

The first concept to emerge from NJ First was not a video game, but rather a basketball free throw shooting contest to be held at Borgata in Atlantic City on March 21.

Oddly, completely absent from the discussion about the demand for social and skill-based games is the area of the casino floor that historically has been both the most social and most demanding of skill. That area is the table games pit.

Let’s start by first talking about the state of blackjack, the most popular casino skill game of them all.

The Rise and Slow Death of Blackjack
Ever since Ed Thorp introduced the first card counting system in 1962 in his book Beat the Dealer, blackjack has led the way as the game of choice for intelligent gamblers in the table games pit. For reference, in 1985, blackjack accounted for 81% of table games units and over half of table games revenue in the state of Nevada.

Part of the reason for blackjack’s popularity was the fact that some gamblers like to make decisions and have some level of control over the outcome. But the chief reason for blackjack’s exceptional popularity was the idea that the game was beatable, and the idea that blackjack represented a fair game.

The fact is that most people who play blackjack don’t even play perfect basic strategy; moreover, relatively few people know how to count cards, and even fewer among them can do it proficiently enough to beat the game. But the idea that a gambler can walk into a casino and beat a game that’s not intended to be beatable – and in a place in which the player is not supposed to win – is a romantic one, and has been strong enough to power the brand of blackjack for the past five decades.

And even setting card counting aside, blackjack has indeed represented the fairest gamble in the house, with house advantages under perfect basic strategy generally in the 0.50% range. The result is that in the history of legalized and regulated gambling in the United States, blackjack has been the game that smart people play.

But all of that is changing. Over the past decade, casinos have taken its most extreme steps to eliminate risk from card counters in particular, as there is no fundamental reason why the casino is obligated to offer a mathematically beatable game. The first big trend was the increasingly widespread use of continuous shuffling machines (CSMs), which speeds up the game but also eliminates card counting. Another more recent trend is that some casino operators have moved to make a more drastic step of paying 6:5 on blackjack not only on single-deck alternatives, but also on two-deck and multi-deck shoe games; paying 6:5 rather than 3:2 adds 1.39% to the house advantage.

What’s happened in the process is that the casinos are gradually (and probably unintentionally) eliminating any incentive to learn a complex basic strategy for a game that not only is no longer beatable, but in an increasing number of cases does not even offer a more favorable house advantage than can be found in games requiring no skill whatsoever. By way of comparison, the Banker bet in baccarat offers a 1.06% house advantage, while the Pass Line bet in craps has a 1.41% house advantage; neither wager requires any decision-making.

And for the most part, the modern gambler has begun to move on from blackjack. At its peak in 2000, there were 3,682 blackjack units in the state of Nevada, representing 64.4% of tables in the state, and generating $1.17 billion in revenue. By 2013, the blackjack count in Nevada was down to 2,704 units, representing 55.0% of tables, and generating $1.09 billion in revenue.

Note that the decline as a percent of all table games revenue in the table below is largely a function of the spectacular rise in predominantly high stakes baccarat play on the Las Vegas Strip over this time period.

Proprietary Alternatives: Game Strategy and the Skill-Free Rate
As blackjack continues to lose real estate on the casino floor, this real estate has in part become occupied by non-traditional, usually proprietary alternatives such as Three Card Poker and Caribbean Stud. These games are typically protected by patents and trademarks (Pai Gow is a notable exception, though several modified variations of Pai Gow are patented), and are generally leased to casinos. This is an area that has long been dominated by SHFL Entertainment (now part of Scientific Games as result of SHFL’s merger with Bally Technologies in 2013, and Bally’s subsequent acquisition by Scientific Games in 2014).

In 1985, non-traditional games represented 2.2% of units and 1.5% of table games revenue in Nevada; by 2013, these games accounted for 21.9% of units and 14.4% of table games revenue in the state.

Such new games are our area of interest. The next question is how well these new games satisfy the demand for skill games.

Let’s establish two key baseline ideas:

1. If a game is going to require any skill whatsoever, the game should have a lower house advantage than that being offered by games with bets requiring zero skill. 
   
2. The more skill a game requires, the lower the house advantage should be. 
   

 Requiring skill in a game is pointless if it means reducing the house advantage from 15% to 5%, as a gambler can always find better bets; rather, there must be an incentive for the player to acquire the skills to play a game the player knows is designed to beat him. The best example of this is blackjack, a game which has incentivized gamblers to learn a relatively complex basic strategy by (a) historically being beatable and (b) by offering a lower achievable house advantage under basic strategy than other games like craps and baccarat.

The second point is a natural extension of the first. The more decisions a player has to make, the more mistakes the player will make, and the less likely it is that the player will achieve the theoretically optimal house advantage. This leads to player value risk, or the risk that the player is losing more money faster than he should be. This needs to be compensated for with a lower theoretical house advantage.

In finance and investing, we have a theoretical risk-free rate of return, which serves as a benchmark against which all decisions are made. In casino gambling, we have what I’ll call the skill-free rate, which serves as a benchmark house advantage against which games requiring skill should be compared. At a minimum, the adjusted house advantage (better known as “element of risk,” which I’ll explain in a minute) for a game requiring skill should be lower than the 1.41% house advantage of the Pass Line wager in craps, and should likely be lower than the 1.06% house advantage of the Banker bet in baccarat. These are typically the lowest house advantages offered by the casino without requiring any decision-making or skill.

As the lowest number, the Banker bet in baccarat represents the skill-free rate.

          Skill-Free Rate = House Advantage of Baccarat Banker Bet = 1.06%

Of the traditional games – blackjack, baccarat, craps, and roulette – only blackjack requires any decision-making or any skill whatsoever.

Traditional Games: Strategy Profile

New proprietary table games are often poker-based variants, and have a range of skill requirements. Entry-level games such as Three Card Poker (SHFL), Galaxy Gaming’s High Card Flush, and Crazy 4 Poker (SHFL) have optimal strategies that can be expressed in as little as one or two rules.

In Three Card Poker, the player starts with an initial ante wager and is dealt three cards; optimal strategy is to bet with Q-6-4 or better, which is all you need to know to get to the house advantage of 3.37%. In High Card Flush, the player antes and is dealt seven cards; according to Wizard of Odds, the correct strategy is to bet the maximum allowed when dealt four or more cards of the same suit, or a three-card flush of J-9-6 or higher (9-7-5 to J-9-5 are borderline), which will get close to the house advantage of 2.64%. In Crazy 4 Poker, the player starts with two initial wagers (an Ante and equal Super Bonus wager), and is dealt five down cards; the perfectly optimal strategy is to bet the maximum (3x the Ante) with a pair of aces or better, and bet the minimum (1x) with K-Q-8-4-x or better (the best four-card hand plays).

These games are strategically simple by design, and are intended to serve as lowest common denominators. The advantage of these games is that virtually anybody can play these games perfectly or near-perfectly (in the case of High Card Flush), even if very few people who play these games seem to actually know the strategies for them. But at the same time, the limitation of these games is that they satisfy the demand for skill games only on the most cursory level.

Entry Level Proprietary Games: Strategy Profile           

 Clearly, there’s room for more sophisticated games. But how well are players being incentivized to learn more complicated strategies?

The classic smash hit Caribbean Stud (SHFL) requires five strategy rules for effectively perfect play, to approximate the theoretical house advantage of 5.22%. Even adjusting for an average bet of 2.04 units per hand (the player antes one unit, and then must bet two units or fold, which the player does on roughly 52% of hands – 1 + (2)(0.52) = 2.04), this results in relatively high adjusted house advantage* (more widely known as “element of risk”) of 2.56% per unit wagered, or more than double the 1.06% house advantage of the Banker bet in baccarat. This in large part explains why Caribbean Stud is in decline, particularly when compared to other new alternatives.

          Adj. House Advantage = House Advantage ÷ Avg. Bet Per Hand (Units)

Four Card Poker (SHFL) has a theoretically optimal house advantage of 2.79%, but this is not practically achievable; a simple strategy using only three rules will get the player to 3.40%, while an advanced strategy presented by the Wizard of Odds requires 10 rules to get to 2.85%, or within 0.062% of the theoretical house advantage. At an average of 2.14 units wagered per hand, Four Card Poker has a theoretical house advantage of 1.30%, or roughly in the range of the 1.41% Pass Line wager in craps.

Easily the two most advanced proprietary games with wide circulation at present are Mississippi Stud (SHFL) and Ultimate Texas Hold’em (SHFL). Mississippi Stud is a game that features a currently unique combination of scalable betting (the player can bet 1x-3x the ante on three successive betting rounds, for a total of up to 10 units wagered in a single hand) and scalable payoffs (all wagers pay according to a pay table, resulting in some potentially large payouts). However, the game requires 23 strategy rules to be played optimally, which gets the player to a house advantage of 4.91%; at 3.59 units wagered per hand, this results in an adjusted house advantage of 1.37% per unit – also in the range of the Pass Line wager in craps, but requiring a lot more work.

Ultimate Texas Hold’em is by far the most attractive of these games, with a theoretical house advantage of 2.19%, and only 0.53% per unit wagered (4.15 units per hand). This house advantage is not practically achievable – Stephen How of Discount Gambling presents basic strategy with 23 categories of rules that get the player to 2.3%. Michael Shackleford a.k.a. the Wizard of Odds presents a simpler strategy with 13 rules (six if you are an expert poker player and can count 33+, Ax, K2s+, K5o+, Q6s+, Q8o+, J8s+, JT as one rule), one of which is bet the river with “less than 21 dealer outs beat you”; that strategy gets to 2.43%, and adj. house advantage (termed “element of risk” by Shackleford) of 0.58%.

On a per unit basis, this puts Ultimate Texas Hold’em roughly in the same range as blackjack.

Proprietary Games: Skill Requirements and Adj. House Advantage          

In Conclusion: Incentivizing Millennials With Value
It’s tempting to want to say that millennials simply have different tastes and are more demanding of games which are social and skill-based in nature, and that’s why they don’t gamble as much as the gaming industry would like. That may be true, but it’s also a copout.

The simple fact is that even as gamblers continue to get smarter and smarter, there are fewer and fewer games for intelligent gamblers to play.

Card counting in blackjack is being systematically eliminated. This is completely understandable, in the context that there’s no fundamental obligation for the casino to offer a mathematically beatable game. But as noted earlier, with the increasing adoption of 6:5 blackjack (with an additional 1.39% house advantage), even the incentive to learn basic strategy for blackjack is also being systematically eliminated. And thusly, blackjack is dying a slow death.

Meanwhile, the new proprietary table games are not filling the void left by countable blackjack, and are not meeting the demand for skill games. Many of the most popular games currently on the market (such as Three Card Poker, High Card Flush, and Crazy 4 Poker) are extremely simplistic by design. And aside from perhaps Ultimate Texas Hold’em (with an adj. house advantage of 0.53% per unit), the modern intelligent gambler is not being properly incentivized to learn sometimes-complex strategies for games they know are designed to beat them.

The point is, if casino operators are serious about marketing to a more sophisticated generation of gamblers – millennials – there’s clearly room to offer games with both higher skill components and lower house advantages to compensate.

*Adj. house advantage is more widely known as “element of risk,” a term coined by Michael Shackleford a.k.a. the Wizard of Odds. The term element of risk is widely used in the industry, and Shackleford deserves full credit for the concept. However, the term element of risk sounds more abstract than it is, and unless you’re in the industry and know exactly what it means, it’s not obvious that the term is related to house advantage, when in fact the measures are directly related. In fact, few expert gamblers know what element of risk is, largely because few expert gamblers study unbeatable games. As such, my preference is to present the measure as adj. house advantage.

Jeff Hwang is a game inventor, and is President and CEO of High Variance Games LLC. Jeff is also the best-selling author of Pot-Limit Omaha Poker: The Big Play Strategy and the three-volume Advanced PLO series.

By Jeff Hwang

Most every gambler is familiar with the term house advantage and how it relates to traditional single-bet games like baccarat, blackjack, roulette, and craps. In craps, for example, the house advantage on the Pass Line bet is 1.41%; if you bet $10 on the Pass Line, the casino will make 1.41% x $10 = 14.1¢ on average. Similarly, the house advantage on the Banker bet in baccarat is 1.06%; for every $10 wager, the house expects to make 10.6¢.

Far more confusing is how the concept of house advantage relates to multi-stage poker games involving multiple wagers, and sometimes multiple initial forced bets. In these games, there are two basic measures of house advantage:

1. House advantage. This is house advantage expressed as a percentage of the initial ante wager.
2. Element of risk (adj. house advantage)*. Coined by Michael Shackleford a.k.a. Wizard of Odds, this is simply the house advantage per unit wagered, and is calculated as the house advantage divided by the average bet per hand. 

As noted, the house advantage on the Pass Line in craps is 1.41%. But let's say the casino offers 3x-4x-5x odds (largely standard on the Las Vegas Strip); if you bet the full 3x-4x-5x odds, you will average 3.77 units wagered per Pass Line bet, and the house advantage is 1.41% ÷ 3.77 units, or 0.37% per unit wagered -- which is also 0.37% of all money wagered.

In Three Card Poker, the player places an ante wager, and is dealt three down cards, at which point the player can either bet 1x the ante or fold. Under optimal play, the player will play Q-6-4 or better, which happens on 67.4% of hands. This is enough to recover all but 3.37% of the ante, which is the base unit house advantage. If the player antes for $10, the house expects to win 3.37% x $10 = 33.7¢ per hand.

But because the player bets an additional unit on 67.4% of hands, the player bets 1.67 units per hand on average. Dividing the 3.37% house advantage by the 1.67 units bet per hand, we get a house advantage of 2.01% per unit wagered, which is what the Wizard of Odds refers to as the element of risk.


          Element of Risk = House Advantage ÷ Average Bet Per Hand (Units)

It should be noted that the house advantage is expressed as a percentage of the initial ante only, and not all initial wagers, whether forced or otherwise.

In Crazy 4 Poker, for example, the player starts with two initial wagers – an Ante and a Super Bonus bet. In this case, the house advantage is expressed as a percentage of the Ante bet, and not both bets combined. So let’s say a player sits down at a $10-minimum Crazy 4 Poker table, and bets the minimum, which requires both a $10 Ante bet and a $10 Super Bonus bet. The house advantage in Crazy 4 Poker is 3.42%, representing 3.42% of the $10 Ante, and not the $20 total initial wager. Thus the casino expects to win 3.42% x $10 = 34.2¢ per hand when the player starts with two $10 initial forced bets.

The average bet per hand in Crazy 4 Poker is 3.14 units. This results in a house advantage of 1.09% per unit wagered – a.k.a. the element of risk.

What's the Right Way to Think About Element of Risk?
The simplest way to think about element of risk is to look at craps with odds.

Let's say you're playing $5 craps with 3x-4x-5x odds on the pass line. If you only bet $5 on the Pass Line but never take odds, the house advantage is 1.41%. This will wager will cost you $5 x 1.41% or about 7 cents.

But let's say you instead bet $5 on the pass line and always take the full 3x-4x-5x odds. In this case, you will average 3.77 units or $18.85 per pass line wager, and the house advantage is 0.37% of the total 3.77 unit or $18.85 average wager. And in this case, these wagers will cost you 0.37% of $18.85, or about 7 cents -- the same 7 cents as before.

Here's what this means. If you're a flat $5 bettor, are risk adverse and want to bet the minimum possible, then element of risk does not help you -- you are stuck betting $5 on the Pass Line at 1.41%. But if you are a bigger bettor and are willing to bet an average of $18.85 per Pass Line wager, then you are far better off betting $5 and taking the full 3x-4x-5x odds, as you are now wagering $18.85 at 0.37%, which is far better (3.77x better in fact) than simply putting $18.85 on the Pass Line with no odds at 1.41%.

What Does This Mean as Far as Target House Advantage and Element of Risk?
What this really means as far as what the target house advantage and target element of risk of a given game should be is a more complicated question. At minimum, we know that the player can bet the Pass Line with 3x-4x-5x odds at 0.37% with no skill, and that the player can bet the Banker bet in baccarat at 1.06% with zero skill and no additional wagering. And if we are going to develop games requiring skill, the 0.37% and 1.06% numbers are the ones we need to beat in order to properly compensate the player for developing the skills required to play such games.

Next: Qualifiers

*Note: The term element of risk is widely used in the industry, and Shackleford deserves full credit for the concept. However, the term element of risk sounds more abstract than it is, and unless you’re in the industry and know exactly what it means, it’s not obvious that the term is related to house advantage, when in fact the measures are directly related. In fact, few expert gamblers know what element of risk is, largely because few expert gamblers study unbeatable games. As a consequence, my preference is to present the measure as adj. house advantage.

Jeff Hwang is President and CEO of High Variance Games LLC. Jeff is also the best-selling author of Pot-Limit Omaha Poker: The Big Play Strategy and the three-volume Advanced Pot-Limit Omaha series.

By Jeff Hwang

Like other traditional casino table games, baccarat is a single-bet, no-decision game where each independent bet represents the entirety of the game; at a baccarat table with a $5 minimum wager, a player can bet the $5 minimum and get to showdown on every hand without having to place another wager. In contrast, as we’ve noted, a distinguishing feature of poker-based table games is that they are generally multi-stage games requiring multiple wagers to get to showdown. Moreover, the average amount wagered on a per-hand basis varies greatly among the different poker variants.

This average amount wagered per hand on a unit basis is the average bet per hand.

In Three Card Poker, for example, the player starts with an initial Ante wager, and is correct to make the one-unit Play bet whenever he is dealt Q-6-4 or better, which happens on 67.4% of deals. Thus, on average, the player wagers:

          1 + (.674)(1) = 1.67 units

In Four Card Poker, the player makes an Ante wager, and then can bet 1x-3x the ante or fold after receiving five down cards (the player’s best four-card poker hand plays). Played optimally, the player will make the minimum one-unit bet on 22.1% of hands, and bet the 3x maximum on 30.7% of hands, while never betting 2x. Thus the average bet per hand under optimal play is:

          1 + (.221)(1) + (.307)(3) = 2.14 units

In Crazy 4 Poker, the player starts with two initial wagers – an Ante wager and a Super Bonus bet – and can bet 1x-3x the ante or fold on his five down cards. The player can only bet 3x on a pair of aces or better, which occurs on 18.6% of hands, while betting the minimum one unit on 57.9% of hands (K-Q-8-4+). This results in an average bet per hand of:

          2 + (.579)(1) + (.186)(3) = 3.14 units

Meanwhile, in Ultimate Texas Hold’em, the player starts with two initial wagers – an Ante wager and a Blind wager – and has three opportunities to bet once: The player can either bet 3x-4x the ante before the flop, or 2x on the flop, or 1x on the river. Played optimally, the player makes the 4x maximum bet on 37.7% of hands, the 2x flop wager on 21.3% of hands, and the minimum 1x bet on the river on 21.8% of hands, resulting an average bet per hand of:

          2 + (.377)(4) + (.213)(2) + (.218)(1) = 4.15 units

The average bet per hand for some of the more prominent games currently on the market are listed in the table below.

Note: Michael Shackleford a.k.a. Wizard of Odds refers to this number as “Average Bet Size,” but I have another use for that term (referring more specifically to the size of the play or raise bet), and prefer to refer to the average bet per hand as what it is.

So what does this mean? Is a higher average bet per hand a good thing or a bad thing?

The answer is that a higher average bet per hand is neither fundamentally a good nor bad thing, but it does reflect to an extent how aggressive a game is. For instance, the top four games on this list all feature scalable betting, while the top game – Ultimate Texas Hold’em – has a max-bet frequency of 37.7%, and Four Card Poker also features a relatively high max-bet frequency of 30.7%.

Another major implication is that games with higher average bets per hand are fundamentally higher stakes games, given equal table minimums. A baccarat table with a $5 minimum wager is a true $5 minimum game, but a $5-min Three Card Poker table with an average bet of 1.67 units per hand is effectively an $8.35-minimum game. At the same time, an Ultimate Texas Hold’em table with a $5 stated table minimum has two forced bets and an average bet of 4.15 units per hand, for a true effective table minimum – that is, the minimum average wager per hand under optimal play given the stated table minimum – of $20.75 per hand.

Consequently, a $5-minium Ultimate Texas Hold’em table is closer in stakes to a $25-minimum blackjack table than a $5-minimum one, and closer in stakes to a Three Card Poker table with a $15 minimum bet (effective table minimum of $25.05) than one with a $5 minimum bet. 

And along with house advantage, the average bet per hand is also a key variable in Shackleford’s element of risk, which we’ll discuss next.

Next: House Advantage and Element of Risk

Jeff Hwang is President and CEO of High Variance Games LLC. Jeff is also the best-selling author of Pot-Limit Omaha Poker: The Big Play Strategy and the three-volume Advanced Pot-Limit Omaha series.

By Jeff Hwang

The term “hit frequency” is generally universally acknowledged in the gaming industry to refer to how often the player receives a payoff of any kind. However, as we discussed in “The Two Kinds of Hit Frequency,” this definition of hit frequency is incomplete, particularly with regard to multi-stage poker games. In such games, betting frequency (the percentage of hands a player can bet) and max-betting frequency (in games with scalable betting, how often the player can bet the max) also serve as positive reinforcement to the gambler, and should be accounted for.

Moreover, even the payoff definition of hit frequency is incomplete, as we really need two categories:

1. Win frequency: How often the player shows a net win, 
2. Payoff frequency: How often the player receives a payoff of any kind, including net wins, pushes, and fractional pays (a return with a net loss); this is the traditional definition of hit frequency. 

The distinction is extremely important. In a game like Pai Gow, for example, the player wins on about 30% of hands and ties on about 40% of hands, for a total payoff frequency – the traditional definition of hit frequency – of about 70%. And so while it’s great that the player gets something back on about 70% of hands, clearly the player feels a lot more strongly about the 30% of hands he shows a win on than the 40% of hands on which he pushes.

The table below shows the win and payoff frequencies for some of the most prominent proprietary table games on the market. In most games, win frequency and payoff frequency are essentially the same, as ties are relatively infrequent in those games. Ties occur on 3.2% of hands in Ultimate Texas Hold’em, which makes sense as a flop game in which the player and dealer share the five community cards.

Mississippi Stud presents a unique case. Mississippi Stud is a straight paytable game – there is no dealer – in which the player pushes if he makes a pair of sixes through tens, and is paid according to a pay table if he makes a pairs of jacks or better. As we noted in our discussion on betting scalability and payoff scalability, Mississippi Stud is unique among current offerings in that it features both scalable betting and scalable payoffs. In fact, the betting is extremely scalable where the player can bet up to 3x on three successive betting rounds, for a total bet of up to 10 units total (including the ante).

The tradeoff is clear, where the game has a relatively low win frequency of 28.6%. This is partially mitigated by the fact that the game pushes when the player makes a pair of sixes through tens, which occurs on 10.2% of hands, for a total payoff frequency of 38.8%.

Next: Average Bet Per Hand and Effective Table Minimum

Jeff Hwang is President and CEO of High Variance Games LLC. Jeff is also the best-selling author of Pot-Limit Omaha Poker: The Big Play Strategy and the three-volume Avanced Pot-Limit Omaha series.

By Jeff Hwang

Most poker table games start with at least one initial forced bet – generally the ante – and have at least one bet-or-fold betting round. As we’ve noted, this forced bet is essentially dead money, and all wagers that come after are made in attempt to recover the lost value of the ante.

In these games, there are two basic kinds of bets:

1. Pot odds/ante recovery bets (min-bet)
2. Offensive bets (max-bet)

Pot odds or ante recovery bets are simply pot odds plays where it’s correct to bet because of the money you already put in the pot, namely the initial ante (and sometimes additional wagers in games with multiple betting rounds after the ante, such as Mississippi Stud). These are situations in which you have too much hand to fold, but not enough to bet the max. This is similar to calling for value in poker.

Offensive bets are bets made as the money favorite. These are situations in which you have the advantage, and are looking to put as much money as possible on the table (a.k.a. bet the crap out of it). These are bets of aggression, similar to betting and raising in poker.

In blackjack, for example, doubling and most pair splitting situations are offensive betting situations in which you have the advantage and are looking to put more money on the table as the favorite (splitting 8-8 against a dealer 9 or 10 are defensive, pot-odds situations in which are splitting to lose less money rather than to win more). These situations are often far more exciting than hitting and standing, or even simply getting dealt blackjack, and provide reinforcement that keeps the player engaged – even when the player loses.

Card counting in blackjack provides an even clearer example. When you are counting cards, the game is essentially to wait for situations in which the count turns favorable, and then bet as much as possible when you have the advantage.

Anybody who’s spent much time counting cards knows that there’s little more boring than counting down a six-deck shoe. But when the count turns sufficiently positive, these situations are offensive betting situations (effectively max-bet situations) in which the player can become aggressive, providing reinforcement and keeping the player engaged (that blackjack can be beaten in this manner obviously helps).

Let’s look at some poker games.

Three Card Poker (67.4% overall betting frequency)

• Mixed: Bet 1x Q-6-4+ (67.4% of hands)
• Fixed betting size, including a mix of offensive and pot odds plays merged into the one bet size

High Card Flush (67.9% overall betting frequency)

• Offensive: Bet 3x with six- or seven-card flush (0.2% of hands)
• Offensive: Bet 2x with five-card flush (2.9% of hands)
• Pot odds/Mixed: Bet 1x with J-9-6+ (9-7-5 to J-9-5 is borderline) (64.9% of hands); some of these bets (probably at least some of the 4-card flushes) are likely offensive bets

Crazy 4 Poker (76.5% overall betting frequency)

• Offensive: Max-bet A-A-x-x+ (18.6% of hands)
• Pot odds: Min-bet K-Q-8-4+ (57.9% of hands)

Ultimate Texas Hold’em (80.8% overall betting frequency)

• Offensive: Max-bet (4x) pre-flop with 33+, any ace, K2s+, K5o+, Q6s+, Q8o+, J8s+, JT (37.7% of hands)
• Offensive: Bet 2x on flop on 21.3% of hands
• Pot odds: Bet 1x on river on 21.8% of hands
  

Three Card Poker is a perfectly flat game with a fixed bet size (1x the ante) that includes a mix of offensive and pot odds bets. High Card Flush is also a flat game by design; the player can bet 2x with a five-card flush or 3x with a six- or seven-card flush, but those situations only occur on 3.1% of hands combined.

But betting frequency and max-betting frequency are the dominant features in Crazy 4 Poker and Ultimate Texas Hold’em, both of which were designed by Roger Snow of Shuffle Master (now a part of Scientific Games). In Crazy 4 Poker, the player can play 76.5% of hands – a higher betting frequency than in either Three Card Poker or High Card Flush – while making the maximum 3x wager on 18.6% (when the player has a pair of aces or better). In Ultimate Texas Hold’em (UTH), the player can bet 80.8% of hands, while making the maximum 4x wager pre-flop on a whopping 37.7% of hands.

In both games, the ability to play a lot of hands is a strong selling point; but even stronger is the frequency with which the player can find hands with which to bet the max.

Certainly there are tradeoffs. In both Crazy 4 Poker and Ultimate Texas Hold’em, the player must start with two initial wagers. Consequently, the betting is not quite as scalable as it sounds – the maximum 4x wager in UTH is only 2x the initial two-unit starting bet, while the maximum 3x wager in Crazy 4 Poker is only 1.5x the initial two-unit wager. But such are the tradeoffs that often must be made in order to acquire certain favorable characteristics when designing – and playing – these games.

Mississippi Stud
The numbers for Mississippi Stud in the table reflect 3rd Street betting only. In Mississippi Stud, it is correct to play 68.9% of hands on the 3rd Street wager (the first two cards), and bet the maximum on all pairs, representing 5.9% of hands. Based on data published on the Wizard of Odds Mississippi Stud page:

• The player is correct to bet the 3x max on all three betting rounds on 4.1% of hands.
• The player is correct to make a 3x wager on at least one betting round on 21.2% of hands.
• The player is correct to play 68.9% of hands on 3rd Street (technically it’s 2nd street, but called the 3rd Street wager), but will fold later in the hand on 12.5% of hands, thus getting to showdown on 56.4% of hands.

Next: Win Frequency and Payoff Frequency

Jeff Hwang is President and CEO of High Variance Games LLC. Jeff is also the best-selling author of Pot-Limit Omaha Poker: The Big Play Strategy and the three-volume Advanced Pot-Limit Omaha series.

The current standard definition of hit frequency is incomplete.

By Jeff Hwang

For a given wager in a given game, the widely accepted definition of the term hit frequency is how often a player receives a payoff of any kind. However, while this definition makes sense with regard to single-bet games such as slot machine and most traditional table games, the definition is incomplete with particular regard to table games with multiple betting rounds, including virtually every poker variant aside from Pai Gow.

Why? Because in these games, monetary payoffs are not the only kind of positive reinforcement that keep the player at the table.

There are really two kinds of hit frequency:

1.     Payoff frequency
2.     Betting frequency (positive bet frequency)

Payoff frequency is the standard definition of hit frequency – how often the player receives a payoff of any kind. This includes net wins, as well as pushes and fractional pays (partial losses, as in slots machines). Win frequency is a the stronger subset of payoff frequency, and refers to how often a player shows a net win.

Betting frequency – or positive bet frequency – is how often the player can make a bet with a positive expectation. Along with payoff frequency, betting frequency is a key driver in variable-ratio reinforcement, which is the psychological principle behind hit frequency (see Variable Ratio Reinforcement).

In general, the lower the overall betting frequency, the less engaged the player will be. And in games with scalable betting, the higher the max-bet frequency – how often the player can bet the maximum as the money favorite – the more engaged the player will be.

The concepts of betting frequency – and especially max-betting frequency – are underappreciated (if not completely unaccounted for) in the current available literature. We’ll break down the concepts of betting frequency and max-bet frequency next, followed by a discussion on win frequency and payoff frequency.

Next: Betting Frequency and Max-Bet Frequency

Jeff Hwang is President and CEO of High Variance Games LLC. Jeff is also the best-selling author of Pot-Limit Omaha Poker: The Big Play Strategy and the three-volume Advanced Pot-Limit Omaha series.

By Jeff Hwang

Aside from Pai Gow, virtually every casino poker variant starts with at least one initial wager – an ante – followed by at least one bet-or-fold betting round. Most games feature a single such betting round.

• In Three Card Poker, the player starts with an initial ante, and can bet 1x or fold after receiving three down cards.
• In High Card Flush, the player starts with an ante, and can bet or fold after receiving seven down cards; the player can wager 1x-or-fold, but may also 2x with a 5-card flush, or 3x with a 6- or 7-card flush.
•  In Caribbean Stud, the player starts with an ante, and can bet 2x or fold after receiving five down cards.
•  In Four Card Poker, the player starts with an ante, and can bet 1x-3x or fold on his five down cards (best four-card hand plays).
•  In Crazy 4 Poker, the player starts with two initial wagers – an Ante and Super Bonus wager – and can bet 1x-3x the ante or fold; the player can only make the 3x wager with a pair of aces or better.

Other games feature multiple betting rounds, employed in a variety of ways.

In Ultimate Texas Hold’em, the player has three betting rounds in which to make a single bet, or otherwise fold. The player starts with two initial wagers – an Ante and equal Blind bet – and is dealt two hole cards. The player can either bet 3x or 4x pre-flop, or 2x on the flop, or 1x on the river, or otherwise fold on the river and forfeit the initial Ante and Blind wagers.

In contrast, Mississippi Stud features three distinct bet-or-fold betting rounds. The player starts with an ante and is dealt two down cards. The player can bet 1x-3x the ante or fold (the 3rd Street wager); if the player bets, the dealer reveals the first of three community cards to be shared by the players. The player can again bet 1x-3x the ante or fold; if the player bets, the dealer reveals the second community card. The player can once again bet 1x-3x the ante or fold; if the player bets, the dealer reveals the final community card to complete the player’s five-card poker hand.

Let It Ride is completely different. In Let It Ride, the player starts by placing three equal wagers, and is dealt three cards. At this point, the player can pull back a wager if he doesn’t like his hand, or let the bet ride (i.e. “let it ride”). The dealer then deals the first community card to be share by the players, after which the player can either pull back the second wager or “let it ride.”

The advantage of games with single betting rounds is that they often feature simple strategies than can be summed up in a few rules or less (The strategies for Caribbean Stud and Four Card Poker are a bit more complicated due to the fact that the dealer has an up card in both games.):

• Three Card Poker: Bet Q-6-4+
• High Card Flush: Call the maximum allowed with J-9-6+, while 9-7-5 to J-9-5 are borderline (per Wizard of Odds)
• Crazy 4 Poker: Max-bet (3x) A-A-x-x+, min-bet (1x) K-Q-8-4+

But add a few betting rounds, and proper game strategy invariably becomes more complex. According to Wizard of Odds, the strategy for Let It Ride – with two betting rounds but no bet-sizing decisions – has ten rules. Ultimate Texas Hold’em with three betting rounds to make one bet is even more complex.

 Here’s the Wizard of Odds take on the strategy for Ultimate Texas Hold’em:

• Pre-flop: Bet the max (4x) with 33+, any ace, K2s+, K5o+, Q6s+, Q8o+, J8s+, JTo
• Flop: Bet (2x) with two pair or better, any pair of threes or better, any 10-high flush draw or better
• River: Bet (1x) with any pair or better, or if “less than 21 dealer outs beat you.”

That’s 13 rules (eight for pre-flop play, though you can think of them as one rule if you are an expert poker player; three on the flop; and two more on the river), one of which requires a lot of interpretation (“less than 21 dealer outs beat you”), and is not a calculation that is necessarily natural even to expert poker players. Moreover, this strategy only gets the player to a 2.43% house advantage – still short of the game’s 2.19% theoretical house advantage under optimal play.

Stephen How of Discount Gambling has produced a strategy for Ultimate Texas Hold’em that improves to a 2.3% house advantage. This strategy has 23 categories of rules (the eight pre-flop rules represent a single category).

Meanwhile, the optimal strategy for Mississippi Stud – with three distinct bet 1x-3x-or-fold betting rounds – is similarly complex. The Wizard of Odds strategy for Mississippi Stud (credited to Joseph Kisenwether) requires 23 rules to get to a house advantage of 4.91%.

The challenge with having complex strategy is that the more rules required – the greater the skill requirement – the bigger the gap will be between the actual player disadvantage and the theoretical house advantage under optimal play. This creates player value risk, which is the risk that the player is losing more money faster than he should be, while creating perceptible barriers to entry for the player: An intelligent gambler is less likely to play a game in which he is unsure of the strategy than he is to play a game for which he is confident in his knowledge of proper game strategy.

This is especially true when the player knows the game is designed to beat him.

Consequently, as we discuss in The Demand for Skill Games, there must be an incentive for the player to acquire the skills to play a game he knows he can’t beat. What this means is that the house advantage for such games should usually be lower (as in Ultimate Texas Hold’em) rather than higher.

Next: The Two Kinds of Hit Frequency

Jeff Hwang is President and CEO of High Variance Games LLC. Jeff is also the best-selling author of Pot-Limit Omaha Poker: The Big Play Strategy and the three-volume Advanced Pot-Limit Omaha series.