Las Vegas February 2007
We met Dad and kayigo in Las Vegas for our yearly mini-family reunion. (And as usual, I'm more than a week late posting about it.) Though relaxing, our trip was, on the whole, unexceptional. We stayed at the Orleans, an off-Strip casino on Tropicana Avenue at the border between an ugly industrial zone, mostly warehouses, and an equally ugly residential/commercial area of alternating strip malls and trailer parks, intercalated with narrow rows of brand-new but still cheap-looking townhouses. As you've probably inferred, nothing even remotely interesting lay within comfortable walking distance. (Maybe if the temperature were 15 °F (8 °C) warmer, or the prevailing winds 40 kt calmer, we'd have enjoyed walking the mile and a half (2.5 km) east to the Strip, but under the circumstances we didn't even consider it.) That worked partially in our favor, because the hotel, faced with the necessity of complete self-sufficiency within a vast wasteland, offered a vast array of casino games (at denominations we could stomach, unlike most of the Strip) and restaurants.
I have more to say about gambling, but because it's rather technical I'll start with the unexpected highlight of our vacation.
Last Christmas we bought kayigo a knitting book entitled Oddball Knitting: Creative Ideas for Leftover Yarn. We didn't think much about it afterward, until she surprised us in Las Vegas with a pair of incredibly kawaii knitted cats, patterned after Altair's (Plates I and II, left) and Yuki's (right) markings. Length: slightly less than 4 in (10 cm) nose to tail.
The bows are functional as well as pretty: they hold the heads on. Note that the tiny ears are appropriately flattened and cupped (best seen in Plate II). These woolen kitties have earned a place of honor amongst the stuffed animals in our bedroom (safely out of reach of the real cats they depict), and together will grace our Christmas tree next year as the star ornaments.
While we played video poker in the Orleans Casino, kayigo asked me an intriguing statistical question; one that I lacked the materials to answer right away. I'll have to provide some background so that my analysis makes sense to people who don't spend horrendous amounts of time chasing that elusive royal flush.
Video poker made its debut in casinos not long before I turned 21, and since has surpassed in popularity just about every game except slot machines. The play is very simple, following the rules for draw poker: in the canonical game, five cards are dealt from a newly-shuffled, standard 52-card deck, and the player may choose to keep any, all or none of them. Desired cards are "held"; the others are discarded, and new cards are drawn to replace them. The final hand wins money, or not, according to a pay table. A standard set of payoffs for the original "Jacks or Better" game appears below. Returns are expressed in number of coins returned per coin bet: a pay of 1 means that we simply get our money back (which for our purposes still counts as a "win").
Pair of Jacks or Better1Two Pair2Three of a Kind3Straight4Flush5Full House8Four of a Kind25Straight Flush50Royal Flush800
A more detailed discussion of video poker mechanics, with examples of each type of winning hand, may be found here. Myriad variations have sprouted up, some counting deuces as wild or adding a joker to the deck. Most follow the tried and true Jacks or Better pattern, but many offer spectacular wins for certain hands, such as Four Aces, at the expense of the less exciting ones.
In video poker there is no "bullshit" element as is so important in live poker: the goal is to construct the most valuable hand according to the pay table, rather than to beat one's opponent(s). A startlingly large percentage of players don't realize this. Often I'll see people do silly things like keeping a "kicker" along with a pair, presumably to fool the (omniscient) computer into believing they have a better hand.
It is possible to apply statistical principles to playing video poker because, at least in the state of Nevada, poker machines are required by law to deal from a "fair deck." In other words, each of the cards has an equal probability of appearing, regardless of the context. Thus, a game that deals four cards to a royal flush on every hand, but never allows a winning hand, is illegal in Nevada. (Other gambling jurisdictions may not hold their video poker machines to the same standards. Not to worry, though: most poker games are manufactured in Nevada, and therefore automatically deal a fair game.) Here we see a huge advantage of video poker over slot machines. Casinos are not required to post the expected return of their slot machines. They may advertise that a bank of slots pays "up to 99% return," but by doing so are only required to set one machine in the bank to pay that well. And of course slot machines lead the bettor on by continually just missing that jackpot-winning combination. In contrast, the pay table of a video poker machine defines the return. For example, a machine paying off according to the table above returns 97.3%: not particularly good compared to live blackjack (> 99.5%) or craps (99.4%, strictly betting the line with double odds), but far better than nickel slot machines (typically about 92-94%) and incomparably better than live keno (75% or less).
One recent trend amusing to students of probability is the increasing popularity of "multiple-play" video poker. The first variation on this theme to appear was "Triple Play" draw poker. Here, the first five cards are dealt as usual, but the cards held are applied to three hands, and new cards are drawn from three independent decks missing the original dealt cards. Thus, if I were dealt four cards to a royal flush, I'd have three independent chances to pull the fifth card for a huge jackpot, and if I were really lucky, I'd draw the fifth card more than once.
Two statistical properties of this game are apparent. First, the expected return is exactly the same as on a regular game using the same pay schedule, but because the three hands are correlated (on account of having the same first five cards), the variation in pay from hand to hand is greater. Hence, the player will experience more exciting winning streaks and more costly losing streaks. And, perhaps, some thrilling moments of victory: imagine the joy of being dealt a royal flush! Several years ago I did precisely that on a Triple Play nickel poker machine, and received $600 in nickels for the hand.
That brings up the second property: in Triple Play, the player's outcome depends heavily on receiving a strong initial (dealt) hand, and even more so in the more recent multiple-play variations. Soon after Triple Play, Five and Ten Play video poker appeared, followed by Fifty and even Hundred Play! In the last version, the player's fortune is almost completely determined by the initial hand, since drawing 100 times independently to the same held cards will usually return very close to the expected value given that particular opening hand. In other words, the draw becomes almost deterministic as the number of multiple hands grows very large-and, as a result, the excitement is actually diminished. Nobody plays Hundred Play Poker for anything but pennies per hand, but even then, a pat royal flush would be worth an outrageous fortune.
The ridiculous but natural extension of this is Infinite Play video poker, in which the player wagers an infinitesimal sum on an infinite number of hands, all linked through the original five cards. Drawing to the entire limitless set of hands would generate a perfect "probability distribution"-in this context, a list of the exact probabilities of all possible outcomes-and hence would pay precisely the overall expected return given which cards were dealt and saved. Statisticians often perform similar exercises when studying very complex probabilities: if a probability distribution is too complex to characterize mathematically, they'll sample from it instead, many many times, to obtain an approximate, empirical distribution that converges to the true distribution as the sample size approaches infinity. Not surprisingly, this technique is called the "Monte Carlo" approach.
Anyway, the reason I'm telling you all this is that on this trip we encountered a new variation on the multi-play theme, called Multi-Strike. This game involves four poker hands that are not played simultaneously, and do not share the same dealt cards. The player bets all four hands, but doesn't necessarily have the opportunity to play all four. She begins by playing the first hand to completion; this hand pays off as usual. If the hand wins something-anything-she progresses to the second hand, totally independent of the first, but which pays double. If that hand wins, a 4X-paying hand comes next, and with yet another win she proceeds to the final, 8X-paying hand. In short, "losing" the first hand, by failing to get at least a pair of jacks, costs four hands' worth of coins, but making it all the way to the 8X hand, whether or not this hand "wins," will almost certainly net the bettor a considerable sum (since on the way, three hands paying 1, 2 and 4X have returned at least that much times the original bet on one hand).
At first I balked at playing this game, on account of one additional feature. Occasionally one or more of the first three hands would be marked "Free Ride," meaning that the player would progress to the following hand even without winning anything. The Free Rides appeared randomly with no clear governing mechanism. To me this chaotic behavior indicated a departure from the standard video poker model-and hence that the game was essentially a slot machine masquerading as a video poker machine. There appeared to be no way to calculate the expected return, what with those occult random events with unknown likelihoods.
However, kayigo, upon scanning through the games online help, discovered a probability table for the Free Ride mechanic! That immediately restored my confidence in the game's fairness, simply because such disclosure is utterly anathema to the philosophy of slot machines. It was revealed that the chances of receiving a Free Ride on the first, second and third hands were 7.7%, 7.0% and 6.4%, respectively. So, her question to me was this: based on the payoffs and Free Ride probabilities, what is the expected return for the entire game?
The answer is straightforward to calculate, if we assume that the best strategy for single-hand draw poker is also optimal for Multi-Strike. (I'll describe why I don't think it is in a moment.) Under the optimal Jacks or Better strategy for the pay table displayed above-identical to the one offered on the Multi-Strike machines at our hotel-a hand will score at least a pair of jacks or better 45.5% of the time. Thus, in the absence of the Free Ride feature, the bettor has a probability of 45.5% of advancing from any one level to the next. The total probability of advancing, then, is the likelihood of either "winning" with at least a pair of jacks or being awarded a Free Ride (or both). Evidently the Free Rides occurred independently of the poker hands; after all, they appeared before all the cards were dealt. Hence, we can figure the overall likelihood of advancing as 1 minus the probability of not being paid (which is 54.5%) and not receiving the coveted Free Ride, and because these two events are independent, the chance of both occurring is simply the product of the individual probabilities.
In more precise notation,
Pr(advancing) = 1 - Pr(losing hand)Pr(no Free Ride)
= 1 - (0.545)(1 - Pr(Free Ride)),
where Pr = probability and Pr(Free Ride) = 0.077, 0.070 or 0.064 at the 1X, 2X and 4X levels, respectively.
Overall return on Multi-Strike is the sum of the payoff for each hand multiplied by the probability of playing that hand. The first hand pays just like a regular Jacks or Better hand, and of course the probability of reaching the first hand is 1. The second hand pays double, but the player advances to the second hand only about half the time (more accurately, with a probability of 1 - (0.545)(0.923) = 0.497). And the chance of reaching the quadruple-paying third hand is the chance of advancing from hand 1 to hand 2, multiplied by the chance of advancing from hand 2 to hand 3-a little under one-quarter. And similarly for the final hand.
In summary, the expected return for Multi-Strike poker is just the return for Jacks or Better (= 0.973) multiplied by
[1 + 2 × Pr(Play 2X hand) + 4 × Pr(Play 4X hand) + 8 × Pr(Play 8X hand)] / 4
where the 4 in the denominator reflects the initial bet of four hands' worth of coins. Plugging in the probabilities of advancement as calculated by the described method,
Return = 0.973 × [1 + (2 × 0.497) + (4 × 0.245) + (8 × 0.120)] / 4
= 0.973 × 0.984
= 0.957
Using the basic, single-hand strategy, then, Multi-Strike poker returns 95.7%, somewhat less than the 97.3% for regular old Jacks or Better. (This type of computation-summing over discrete events with known probability to obtain an overall average, or "expected value"-is ubiquitous in statistics. I didn't think I'd wind up writing about my job, but this kind of thing is exactly what I spend much of my time at work doing.)
From this analysis we seem to be at a disadvantage playing Multi-Strike, compared to its parent game. However, the optimal strategy for single-hand poker is not ideal for Multi-Strike. We clearly wish to maximize our chance of reaching the top, high-paying hands. Consequently, early on we will benefit from holding combinations of cards that confer the greatest probability of winning something, even if such choices are suboptimal in terms of overall expected return for that particular hand. Some combinations of cards are held because they offer a small probability of a large payout; an example is three cards to a straight flush. In the first hand of Multi-Strike, we may, therefore, discard the partial straight flush-especially if none of the cards is a jack or higher-and instead save a single jack or ace, because although we don't expect to win as much holding a single high card, we do expect to win far more often. Still, we shouldn't abandon all opportunities for big payoffs at the 1X level: we must weigh the probability of winning a huge prize early on, such as a royal flush, against the less princely but more attainable payoffs from the 4X and 8X hands.
Formulating a comprehensive strategy for Multi-Strike poker is, alas, well outside the scope of what I really would like to do with my next six months' worth of free time. Instead, I'll leave it as an exercise for the reader.
I have more to say about gambling, but because it's rather technical I'll start with the unexpected highlight of our vacation.
Last Christmas we bought kayigo a knitting book entitled Oddball Knitting: Creative Ideas for Leftover Yarn. We didn't think much about it afterward, until she surprised us in Las Vegas with a pair of incredibly kawaii knitted cats, patterned after Altair's (Plates I and II, left) and Yuki's (right) markings. Length: slightly less than 4 in (10 cm) nose to tail.
The bows are functional as well as pretty: they hold the heads on. Note that the tiny ears are appropriately flattened and cupped (best seen in Plate II). These woolen kitties have earned a place of honor amongst the stuffed animals in our bedroom (safely out of reach of the real cats they depict), and together will grace our Christmas tree next year as the star ornaments.
While we played video poker in the Orleans Casino, kayigo asked me an intriguing statistical question; one that I lacked the materials to answer right away. I'll have to provide some background so that my analysis makes sense to people who don't spend horrendous amounts of time chasing that elusive royal flush.
Video poker made its debut in casinos not long before I turned 21, and since has surpassed in popularity just about every game except slot machines. The play is very simple, following the rules for draw poker: in the canonical game, five cards are dealt from a newly-shuffled, standard 52-card deck, and the player may choose to keep any, all or none of them. Desired cards are "held"; the others are discarded, and new cards are drawn to replace them. The final hand wins money, or not, according to a pay table. A standard set of payoffs for the original "Jacks or Better" game appears below. Returns are expressed in number of coins returned per coin bet: a pay of 1 means that we simply get our money back (which for our purposes still counts as a "win").
Pair of Jacks or Better1Two Pair2Three of a Kind3Straight4Flush5Full House8Four of a Kind25Straight Flush50Royal Flush800
A more detailed discussion of video poker mechanics, with examples of each type of winning hand, may be found here. Myriad variations have sprouted up, some counting deuces as wild or adding a joker to the deck. Most follow the tried and true Jacks or Better pattern, but many offer spectacular wins for certain hands, such as Four Aces, at the expense of the less exciting ones.
In video poker there is no "bullshit" element as is so important in live poker: the goal is to construct the most valuable hand according to the pay table, rather than to beat one's opponent(s). A startlingly large percentage of players don't realize this. Often I'll see people do silly things like keeping a "kicker" along with a pair, presumably to fool the (omniscient) computer into believing they have a better hand.
It is possible to apply statistical principles to playing video poker because, at least in the state of Nevada, poker machines are required by law to deal from a "fair deck." In other words, each of the cards has an equal probability of appearing, regardless of the context. Thus, a game that deals four cards to a royal flush on every hand, but never allows a winning hand, is illegal in Nevada. (Other gambling jurisdictions may not hold their video poker machines to the same standards. Not to worry, though: most poker games are manufactured in Nevada, and therefore automatically deal a fair game.) Here we see a huge advantage of video poker over slot machines. Casinos are not required to post the expected return of their slot machines. They may advertise that a bank of slots pays "up to 99% return," but by doing so are only required to set one machine in the bank to pay that well. And of course slot machines lead the bettor on by continually just missing that jackpot-winning combination. In contrast, the pay table of a video poker machine defines the return. For example, a machine paying off according to the table above returns 97.3%: not particularly good compared to live blackjack (> 99.5%) or craps (99.4%, strictly betting the line with double odds), but far better than nickel slot machines (typically about 92-94%) and incomparably better than live keno (75% or less).
One recent trend amusing to students of probability is the increasing popularity of "multiple-play" video poker. The first variation on this theme to appear was "Triple Play" draw poker. Here, the first five cards are dealt as usual, but the cards held are applied to three hands, and new cards are drawn from three independent decks missing the original dealt cards. Thus, if I were dealt four cards to a royal flush, I'd have three independent chances to pull the fifth card for a huge jackpot, and if I were really lucky, I'd draw the fifth card more than once.
Two statistical properties of this game are apparent. First, the expected return is exactly the same as on a regular game using the same pay schedule, but because the three hands are correlated (on account of having the same first five cards), the variation in pay from hand to hand is greater. Hence, the player will experience more exciting winning streaks and more costly losing streaks. And, perhaps, some thrilling moments of victory: imagine the joy of being dealt a royal flush! Several years ago I did precisely that on a Triple Play nickel poker machine, and received $600 in nickels for the hand.
That brings up the second property: in Triple Play, the player's outcome depends heavily on receiving a strong initial (dealt) hand, and even more so in the more recent multiple-play variations. Soon after Triple Play, Five and Ten Play video poker appeared, followed by Fifty and even Hundred Play! In the last version, the player's fortune is almost completely determined by the initial hand, since drawing 100 times independently to the same held cards will usually return very close to the expected value given that particular opening hand. In other words, the draw becomes almost deterministic as the number of multiple hands grows very large-and, as a result, the excitement is actually diminished. Nobody plays Hundred Play Poker for anything but pennies per hand, but even then, a pat royal flush would be worth an outrageous fortune.
The ridiculous but natural extension of this is Infinite Play video poker, in which the player wagers an infinitesimal sum on an infinite number of hands, all linked through the original five cards. Drawing to the entire limitless set of hands would generate a perfect "probability distribution"-in this context, a list of the exact probabilities of all possible outcomes-and hence would pay precisely the overall expected return given which cards were dealt and saved. Statisticians often perform similar exercises when studying very complex probabilities: if a probability distribution is too complex to characterize mathematically, they'll sample from it instead, many many times, to obtain an approximate, empirical distribution that converges to the true distribution as the sample size approaches infinity. Not surprisingly, this technique is called the "Monte Carlo" approach.
Anyway, the reason I'm telling you all this is that on this trip we encountered a new variation on the multi-play theme, called Multi-Strike. This game involves four poker hands that are not played simultaneously, and do not share the same dealt cards. The player bets all four hands, but doesn't necessarily have the opportunity to play all four. She begins by playing the first hand to completion; this hand pays off as usual. If the hand wins something-anything-she progresses to the second hand, totally independent of the first, but which pays double. If that hand wins, a 4X-paying hand comes next, and with yet another win she proceeds to the final, 8X-paying hand. In short, "losing" the first hand, by failing to get at least a pair of jacks, costs four hands' worth of coins, but making it all the way to the 8X hand, whether or not this hand "wins," will almost certainly net the bettor a considerable sum (since on the way, three hands paying 1, 2 and 4X have returned at least that much times the original bet on one hand).
At first I balked at playing this game, on account of one additional feature. Occasionally one or more of the first three hands would be marked "Free Ride," meaning that the player would progress to the following hand even without winning anything. The Free Rides appeared randomly with no clear governing mechanism. To me this chaotic behavior indicated a departure from the standard video poker model-and hence that the game was essentially a slot machine masquerading as a video poker machine. There appeared to be no way to calculate the expected return, what with those occult random events with unknown likelihoods.
However, kayigo, upon scanning through the games online help, discovered a probability table for the Free Ride mechanic! That immediately restored my confidence in the game's fairness, simply because such disclosure is utterly anathema to the philosophy of slot machines. It was revealed that the chances of receiving a Free Ride on the first, second and third hands were 7.7%, 7.0% and 6.4%, respectively. So, her question to me was this: based on the payoffs and Free Ride probabilities, what is the expected return for the entire game?
The answer is straightforward to calculate, if we assume that the best strategy for single-hand draw poker is also optimal for Multi-Strike. (I'll describe why I don't think it is in a moment.) Under the optimal Jacks or Better strategy for the pay table displayed above-identical to the one offered on the Multi-Strike machines at our hotel-a hand will score at least a pair of jacks or better 45.5% of the time. Thus, in the absence of the Free Ride feature, the bettor has a probability of 45.5% of advancing from any one level to the next. The total probability of advancing, then, is the likelihood of either "winning" with at least a pair of jacks or being awarded a Free Ride (or both). Evidently the Free Rides occurred independently of the poker hands; after all, they appeared before all the cards were dealt. Hence, we can figure the overall likelihood of advancing as 1 minus the probability of not being paid (which is 54.5%) and not receiving the coveted Free Ride, and because these two events are independent, the chance of both occurring is simply the product of the individual probabilities.
In more precise notation,
Pr(advancing) = 1 - Pr(losing hand)Pr(no Free Ride)
= 1 - (0.545)(1 - Pr(Free Ride)),
where Pr = probability and Pr(Free Ride) = 0.077, 0.070 or 0.064 at the 1X, 2X and 4X levels, respectively.
Overall return on Multi-Strike is the sum of the payoff for each hand multiplied by the probability of playing that hand. The first hand pays just like a regular Jacks or Better hand, and of course the probability of reaching the first hand is 1. The second hand pays double, but the player advances to the second hand only about half the time (more accurately, with a probability of 1 - (0.545)(0.923) = 0.497). And the chance of reaching the quadruple-paying third hand is the chance of advancing from hand 1 to hand 2, multiplied by the chance of advancing from hand 2 to hand 3-a little under one-quarter. And similarly for the final hand.
In summary, the expected return for Multi-Strike poker is just the return for Jacks or Better (= 0.973) multiplied by
[1 + 2 × Pr(Play 2X hand) + 4 × Pr(Play 4X hand) + 8 × Pr(Play 8X hand)] / 4
where the 4 in the denominator reflects the initial bet of four hands' worth of coins. Plugging in the probabilities of advancement as calculated by the described method,
Return = 0.973 × [1 + (2 × 0.497) + (4 × 0.245) + (8 × 0.120)] / 4
= 0.973 × 0.984
= 0.957
Using the basic, single-hand strategy, then, Multi-Strike poker returns 95.7%, somewhat less than the 97.3% for regular old Jacks or Better. (This type of computation-summing over discrete events with known probability to obtain an overall average, or "expected value"-is ubiquitous in statistics. I didn't think I'd wind up writing about my job, but this kind of thing is exactly what I spend much of my time at work doing.)
From this analysis we seem to be at a disadvantage playing Multi-Strike, compared to its parent game. However, the optimal strategy for single-hand poker is not ideal for Multi-Strike. We clearly wish to maximize our chance of reaching the top, high-paying hands. Consequently, early on we will benefit from holding combinations of cards that confer the greatest probability of winning something, even if such choices are suboptimal in terms of overall expected return for that particular hand. Some combinations of cards are held because they offer a small probability of a large payout; an example is three cards to a straight flush. In the first hand of Multi-Strike, we may, therefore, discard the partial straight flush-especially if none of the cards is a jack or higher-and instead save a single jack or ace, because although we don't expect to win as much holding a single high card, we do expect to win far more often. Still, we shouldn't abandon all opportunities for big payoffs at the 1X level: we must weigh the probability of winning a huge prize early on, such as a royal flush, against the less princely but more attainable payoffs from the 4X and 8X hands.
Formulating a comprehensive strategy for Multi-Strike poker is, alas, well outside the scope of what I really would like to do with my next six months' worth of free time. Instead, I'll leave it as an exercise for the reader.