Bonus update (23 Nov 2016): My good friend Poker Grump wrote a great article over at Poker News on the same issue of selective attention a few weeks back. His article is better because he has a Penn & Teller magic video. So go read that article then come back here for dancing gorilla videos.
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After several stressful months at the office, not to mention a mind-boggling Presidential election, a trip to Las Vegas came at just the right moment. A few days to unplug from the world were just what the doctor ordered. Poker. Good food. Cocktails. Friends. The Vegas Rock 'N Roll Half Marathon. Well, the last one was more the excuse for Vegas than the highlight, yet there is something awesome about being mere yards from a stage where Snoop Dogg is rapping about peace, love, and weed, before running for two hours in the neon glow of the Vegas Strip at night.
As has become the norm for my Vegas trips, I made my headquarters at Aria; still the best value for upscale hotel rooms, awesome poker room, and great mid-Strip location. But there may have been a donkalicious late night drinking session of 2-6 spread limit Hold Them at Monte Carlo. Allegedly. Unquestionably there was a side trip for a session of Pot Limit Gambooool at the new Wynn poker room (actually in Encore). This is a fantastic room, with restrooms and a sports book window conveniently located in the room, lots of space between tables, top shelf drinks, and a cool, upscale vibe. And, of course, no Vegas poker trip is complete without a late night session of Poker With The Drunks at Planet Hollywood, where I cashed out for nearly a grand in profit. P-Ho remains the gold standard for lucrative late night poker.
Monday morning rolled around, and I decided to squeeze in a last three hour poker session at Aria. My legs were a little sore from the race, but I was rested, caffeinated, and sober. I should have been ready to play my A-game. Instead, I ended up looking like a total idiot.
But, for a moment, let's digress. Take a quick look at this video.
You may well have already seen this video, which was the centerpiece of an exceptional bit of psychological experimentation. The point of the experiment was to test how people observed an overall situation when they were focused on one aspect of the situation. Here, where people were focused on the task of counting the number of passes made by the people in white t-shirts, half of the observers completely missed the gorilla walking through the scene. That's right. People who were intently focused on tracking one part of the scene were utterly oblivious to another part of the scene, even something as absurd as a gorilla.
The researchers called this psychological phenomenonselective attention. Essentially, when your brain is focused on one task, it mutes or outright ignores information unrelated to the task at hand. And it can manifest itself in a wide range of daily activities. Including poker. And in a game where observation is a key skill, overlooking important information can be a costly leak.
Back to my session at Aria. I got into a game where most of the players had been at the table together since early in the morning. And it was quickly obvious why. Most of the players were over $500 deep, with several having over $1,000 stacked behind. The table economy ran through a total maniac across the table who raised preflop more than two out of three hands. His standard play was to raise preflop by splashing a random handful of chips into the pot, usually $30-$80. Then, he would c-bet nearly every flop by jamming a big stack of $80-$150 into the middle. Of course, the maniac attracted multiple callers every hand, with players looking to catch a hand and take a bite out of the maniac's stack. For his part, the maniac, as maniacs are wont to do, caught improbable hand after improbable hand to vacuum up chips from tilty nits.
So how many times did I screw up in this session? The number of the counting shall be three ....
Hand #1: Early in the session, I had roughly my $300 starting stack and was in the small blind. Maniac raised to $20, and I called with 75 soooted along with three other players. Flop was a gorgeous 7-7-5 with two hearts. Catamaran! We checked it around on the flop. Turn was another 5. Boo! I led out for $50, and got called by the big blind. Maniac and another guy folded, but the cutoff—a fairly standard older nit—raised to $150. Damn, pretty clear he has the other 7 and we're chopping the pot. So, I shoved, expecting the big blind to fold, the nit to call, and to run out the board.
Except the big blind didn't fold. Instead, he kept looking at his cards and thinking. He cut out chips for a call, and kept looking back at his cards and the board. Eventually, he sighed and mucked. I rolled my cards and said, "I flopped it, but guess now we'll chop it."
It was only then that I realized the old nit in the cutoff hadn't snap-called! That was ... awkward. And seconds ticked by as the nit stared at my hand, the board, and his hand. Finally, he reluctantly folded. Obviously he didn't have the last 7, so he might have had something like an overpair or possibly an open-ended straight flush draw. In any event, I likely cost myself his $100 or so call.
Hand #2: Later in the session, I was on the button with the Spanish Inquisition—6h3s. The table maniac raised to $30 and I called, along with three other players. The flop was interesting—9h5h4h—giving me an open-ended straight draw, but presenting the danger of drawing dead, I was prepared to fold to any bet, on the theory there is always a better place to get it in bad. But instead it checked around, and I gladly asked the dealer for a free card.
The turn was even more interesting—the 2s—giving me the straight. This time, the maniac threw out $50, basically 1/3 of the pot. Two players flat called, and I made the reluctant crying call.
The river was the 2h. Although I doubted there had been a slow-played set or two pair that boated, the fourth heart on the river was almost certainly the nail in the coffin for my straight. The other players took turns looking at their hole cards and checking. I checked and waited to see the inevitable showdown between big single hearts in the hole. As everyone stared at each other, I rolled my cards and dramatically announced, "I have a straight to the six!" hoping to prod the other players to show down and move things along.
Knowing my hand wasn't good, I looked over at the TV, waiting for the next hand. I heard the dealer announce "Flush wins." Well duh. But then, I saw the dealer pushing the pot to me. What the heck??
Oh yeah. I had a baby heart in my hand, so I had a flush, not a straight. Of course, I could only beat a naked 3h in the hole, but that was the only other heart held by anyone in the hand at showdown. Cha-ching! Feel like I missed a value bet there with that monster ....
Hand #3: Once again, maniac opened in middle position for $35. Once again I called on the button with 8s7s. But to my surprise, the rest of the table folded. The flop was pretty good—Ah9s6c—giving me an open-ended straight draw. Maniac bet $50, I raised to $130, and the maniac auto-called. Hmm, he might have a hand this time. Turn was interesting—9c. Maniac checked, I bet $200, he called. At this point, if I didn't hit my straight, I was done with the hand. It was far too probable maniac had an Ace or 9.
River was not just a blank, it was a killer card—6h. Basically, if maniac has any Ace, 9, 6, or pocket pair, he wins. Maniac checked, and I checked. Maniac says, "You win," I respond, "I have eight high" and flashed my hand. Maniac goes, "Oh, I can beat that!" and tables QcJc. Ahhh, so he chased a flush draw, missed, and still has me beat. Sucks to be me.
I went to muck my loser hand when the guy next to me says, "Wait, you chop!" I paused, then realized that maniac and I were both playing the board because the Ace on board was the kicker for each of our hands. I tabled my cards, and we chopped the pot.
I ended up stacking maniac when I slow-played QhQd preflop, and we got it all-in on a flop of AsQcTc. Maniac showed Ac2c, which was stronger than I hoped. But the board ran out blanks and my set held up for a monster pot.
So back to selective attention. In each of these hands, I was so focused on one thing—a player's action or chasing my draw—that I missed other important developments. Of course, having it happen three times in a three hour session is not something I am terribly proud of. I'm certain that the circumstances—a last quick session, some residual fatigue, playing a maniac—contributed to the problem. But it's also a phenomenon that happens even to the best trained professionals; for example, a decent percentage of radiologists failed to detect a gorilla shape when reviewing CT scans for tumors. And for you smug folks who saw the gorilla in the first video and who are laughing at my stupidity, try this follow up test:
In any event, being aware of this psychological phenomenon will hopefully make it less likely to recur in my future poker sessions.
AUTHOR'S NOTE: This post is the second of two related posts. Part I is HERE.
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As discussed in my last post, the recent news that a team of researchers has "essentially weakly solved" heads-up limit hold' em poker (HULHE) should be considered significant support for the legal argument that poker is a "skill game"—i.e., a game where skill, rather than chance, is the "dominant factor" in the game. In fact, the Cepheus computer program's ability to play a non-exploitable, game theory optimal (GTO) strategy does advance the skill game argument by showing that the skill-chance analysis cannot be confined to a single hand, demonstrating the importance of making long-term strategic decisions (e.g., balancing ranges). Further, Cepheus proves that, at least for the HULHE variation of poker, a player can use a GTO strategy that is indifferent to the role of chance over the long-term (i.e., the strategy will not lose to a non-GTO strategy over a statistically significant number of hands).
So, has Cepheus resolved the skill game legal argument in favor of poker as a game of skill? Unfortunately, the opposite may well be the case. As one of Cepheus' researchers explained, an important implication of a GTO strategy is that it is designed to be impervious not only to the effects of chance, but also to any counter-strategy (emphasis added):
"Since poker is a symmetrical game, the end strategy which Cepheus plays is an unbeatable one. While chips can, of course, be won from Cepheus in the short term, there is no decision which can be made against it which will be a winner in the long term. If a perfect opponent, either human or computerized, were to play a semi-infinite number of hands against Cepheus the best possible result would be for them to break even. Any imperfect opponent, which unfortunately includes all human players, would make mistakes along the way and lose. That being said, what Cepheus cannot do is maximize its winnings against weak opponents, a skill [at] which humans excel. Cepheus is simply an invincible, immovable bunker, a Maginot Line that actually works."
Thus,if two players face each other and both play a GTO strategy, then neither player will be able to exploit the other player, neither player will have a strategic advantage, and the result of the game will be left entirely to chance. In other words, the fact HULHE has a GTO strategy necessarily implies that the game can be played in such a way that the sole determining factor in the outcome is the effect of chance (i.e., which player gets luckier).
Now the possibility of a GTO v. GTO showdown may appear only theoretical. But let's consider the opposite situation, where two HULHE novices are matched up. Assuming neither player has any knowledge of proper strategy, again the results of the match are determined solely by chance. Now, let's take the next leap—two equally experienced, talented players who try to exploit the other player's flaws. In order to exploit those flaws, each player will necessarily make flawed strategic decisions (i.e., deviate from GTO strategy). However, over time, the constant back-and-forth of play should result in a series of game adjustments by both players which leaves each of them playing a close approximation of GTO strategy, such that neither player has a significant strategic edge. Again, the long-term results of such an even-skill match would be governed mostly (predominantly) by chance.
This implication for GTO strategy—that skilled players will eventually reach a close approximation of a GTO equilibrium strategy—is real and not theoretical. As was noted by Cepheus' researchers (emphasis added):
"So what does the availability of Cepheus’ data mean to limit hold’em play, particularly in the online environment where there are no effective checks against referencing Cepheus while play is ongoing? Not a great, deal unfortunately. While Cepheus would have undoubtedly had a detrimental and traumatic effect on a competitive online environment there is effectively no environment left to traumatize. Due to the rake, which is the share of the pot which the house claims as its fee, poker is a negative sum game. As the fundamental of heads up limit hold’em became better understood and the skill gap between competitors narrowed, many players found themselves in a position where they were able to beat their opponents but not both their opponents and the rake. More and more often, competition between players began to result in both players losing and the situation was exasperated [sic—exacerbated] by the decline of the online poker industry, which shifted a large portion of competitive play to lower stakes where the rake represents a larger percentage of a player’s potential winnings. Poker players, being rational people, did the only sane thing they could do, which was decline to play anyone who appeared to be of even remotely similar skill. At of the time of writing this article on a Saturday evening there are, on Pokerstars, the current market leader, thirty-five heads-up limit hold’em tables above the one dollar level where players are waiting for an opponent and one table at which two players are actually competing. Cepheus will undoubtedly prove a valuable sparring partner and research tool for casino players and enthusiasts looking to sharpen their skills, but the heyday of heads-up cash play has, unfortunately, already passed."
This concern about relative skill between players is common within the poker community. Online poker players have long engaged in the process of "bum hunting"—looking for games with known weak players to exploit. As poker professional Paul Ratchford explains:
"Poker is a zero sum game minus a cut that the house takes. So in an environment where all players are good and everybody plays game theory optimal poker EVERYBODY loses. The house takes money out of pots at an enormous rate especially at lower games. In fact, versus a bunch of skilled regulars (with zero recreational dollars in play) it may be impossible at a 6-max or full ring table for even some of the best in the world to win…. The bottom line is that if you are a professional poker player you need to be bum hunting / table selecting."
"This game started about a week before the $1 million One Drop tournament and ran daily. Though a security guard kept gawkers and potential short-buys at bay, recognizable faces included One Drop Founder Guy Laliberte, Rick Salomon (the movie producer most famous for his Paris Hilton tape), and self-described model / actor / astronaut / asshole” Dan Bilzerian (@DanBilzerian). When this game runs, even the pros who play the regular 300-600 mix at Aria move elsewhere. 'Crazy' Mike Thorpe, who organizes many high-stakes mix games in town, says the regular 300-600 players at Aria, which include David 'Viffer' Peat and Ivey Room host Jean-Robert Bellande, have to move to Bellagio because Bobby Baldwin himself (Bobby’s Room namesake) would rather host his nosebleed no-limit game in The Ivey Room without pros."
Ironically, Bellande has himself been criticized by other poker players, including WSOP Main Event champion Greg Merson, for setting up high stakes poker games filled with whales, then excluding other poker pros from those games. Of course, this pattern of skilled "sharks" seeking out less talented "fish" to exploit isn't limited to high stakes play.
The irony of "bum hunting" or targeting "whales" and "fish" is that these weaker players generally play a highly flawed poker style that diverges markedly from GTO strategy. Although a GTO strategy would profit off these weak players over time, a non-GTO style will actually exploit weak players faster and for greater profit. So, in essence, poker's best players generally profit off of weaker players by utilizing a non-GTO strategy. Poker professional Paul Ratchford explains this irony (albeit in the context of no-limit hold 'em):
Maximum exploitive NLHE occurs when a player chooses the most exploitive line to maximize his/her expected value. Most players do not play balanced ranges and, therefore, we should seek to maximize our edge by playing appropriately unbalanced in response. In a Rock, Paper, Scissors example, where we know that our opponent will throw rock 100% of the time, we would simply use paper 100% of the time. Even if we knew that our opponent threw rock 40% of the time, 30% paper, and 30% scissors, the maximum exploitive play would still be paper 100%. In NLHE, if you play heads up versus an opponent who folds 100% of the time to three-bets, your response would be to reraise 100%. It is important to note that if you are playing against a GTO opponent, the maximum exploitive strategy will be GTO. The appropriate response to a perfectly balanced Rock, Paper, and Scissors range is to be perfectly balanced yourself.
Or, as Cepheus' own researchers admit, "what Cepheus cannot do is maximize its winnings against weak opponents, a skill [at] which humans excel." In other words, maximizing profits in poker requires deviation from Cepheus-style GTO poker strategy.
The analytical takeaways from the discussion above can be distilled into these Poker Postulates:
A poker player's relative skill advantage over his opponent matters more than his absolute skill level—i.e., "In the land of the blind, the one-eyed man is king."
As the difference in skill between poker players increases, the effect of chance on game results decreases, but is never eliminated altogether.
In poker games between players of substantially similar skill, results will be determined predominately by chance.
In poker games between players of substantially dissimilar skill, results will be determined predominately by skill, even though over a short period of time chance may permit a lesser-skilled player to prevail.
Skill in poker is more readily demonstrated by utilizing a non-GTO strategy to exploit weaker players than in utilizing a non-exploitable GTO strategy, at least insofar as success is measured by profits.
To be blunt, then, poker skill ultimately is not measured by how well a player selects starting hands, calculates pot and implied odds, or balances ranges. Likewise, poker skill is not measured by degree of similarity to or deviation from a GTO strategy. Rather, poker skill predominately turns on game selection; that is, being able to get into a game with weaker opponents whose flawed strategies can be exploited via a non-GTO strategy.
Returning, then, to the skill game legal argument, the clear implication of the Cepheus GTO strategy research is that poker advocates are left defending the awkward proposition that poker "skill" has little to do with game-related strategy and mostly means "preying on weak players" (or bum hunting, or fleecing fish—pick your own metaphor). Presented in this context, poker players begin to look less like mathematical savants and more like casino operators luring patrons to a -EV table game. In fact, for many poker players, their odds of winning money would actually be enhanced dramatically if they gave up poker for a seat at a house-run table game.
“The creatures outside looked from pig to man, and from man to pig, and from pig to man again; but already it was impossible to say which was which.” ~~ George Orwell, Animal Farm
One of my favorite TV shows is the CBS drama, Person of Interest. The show's plot is driven by the concept that a fully functional artificial intelligence (AI) computer program has actually been created. The AI system—known as "The Machine"—was originally created as an omnipresent surveillance tool to detect terrorist plots for the government. The Machine's creator feared those in power would abuse its abilities and went underground, programming The Machine with an ethical code and using it to predict and prevent criminal acts.
In last week's episode—the aptly titled "If-Then-Else"—the show's protagonists were caught in a conundrum, needing both to hack into a computer system to prevent a stock market crash while also escaping a trap meant to capture or kill them. The show flashed back to when The Machine was first created and its creator was teaching it to play chess. The Machine would calculate thousands of possible moves ahead, and when a chosen line of attack ultimately failed, would alter its strategy for future games, thereby learning how to play the game better. The rest of the show involved watching The Machine run through multiple alternative game plans for the team of heroes, with many of them ending in the team's demise. The Machine ultimately found a solution which gave the team a slim but real chance of survival. Of course, the show ended in a shocking cliffhanger—the apparent death of one of my favorite characters—which initially angered me, but ... well, as they say, "Spoiler Alert".
Just a few days after watching that Person of Interest episode, news broke via Science magazine that a team of scientists in Canada had "essentially weakly solved" heads-up limit hold 'em poker (HULHE). The methodology used to develop the Cepheus poker program is uncannily similar to that used with Person of Interest's fictitious Machine. Essentially, Cepheus played billions of billions of hands of poker against an identical program, beginning with random trial and error as to the proper strategy—bet, raise, call, or fold—at each decision point. Once the hand concluded, Cepheus would assign a "regret factor" to each decision based on the hindsight knowledge of the actual hand results. As tens of thousands of similar hands and situations accumulated, Cepheus would adjust its strategy to lessen or avoid decisions with higher regret factors, instead pursuing the balance of decisions which, overall, caused the least regret. For example, Cepheus might initially only check or call on most flops with a pocket pair higher than any card on the board, but would over time learn that betting or raising a high percentage of the time is a better (i.e., more profitable) strategy. Eventually, the program reached a point where further adjustments created more regret, meaning that the program had developed a non-exploitable, Game Theory Optimal (GTO) strategy.
"At first, I was roundly stuffed by the computer’s non-stop aggression. Any bluffs I made failed miserably. To counteract this, I became more aggressive preflop and stopped bluffing almost entirely. Cepheus’s game did not adapt to my play and it made what I would consider several questionable plays. The program was reluctant to ever give up any sort of hand in a large pot making it easier to get lots of value from moderately weak hands."
Hall's claim is utterly at odds with the claim that Cepheus has solved HULHE, and reflects a lack of understanding of what GTO strategy means in game theory. If Cepheus in fact is playing a GTO strategy, then by definition Hall cannot play a style which attacks a flaw in Cepheus' strategy because GTO strategy has no flaws to exploit. As Cepheus' creators explain (emphasis added):
"Since poker is a symmetrical game, the end strategy which Cepheus plays is an unbeatable one. While chips can, of course, be won from Cepheus in the short term, there is no decision which can be made against it which will be a winner in the long term. If a perfect opponent, either human or computerized, were to play a semi-infinite number of hands against Cepheus the best possible result would be for them to break even. Any imperfect opponent, which unfortunately includes all human players, would make mistakes along the way and lose."
Rather than exposing a supposed flaw in Cepheus, Hall's short-term positive results were purely a matter of short-term variance—that is, Hall got lucky. Now, this is not to say that Hall's change in tactics had no effect; either:
Hall originally was playing sub-optimal poker and correctly adjusted toward GTO strategy, improving his results (which were augmented by short-term variance); or,
Hall incorrectly adjusted away from GTO strategy to exploit a perceived (but illusory) flaw, won over the short-term because of variance, but would in fact lose over the long-term utilizing that strategy.
To be fair, Hall acknowledged that the 400 hands he played—essentially one or two decent cash game sessions—were an insufficient sample size to evaluate Cepheus. Still, Hall doubled-down on his irrational doubting of Cepheus:
Perhaps the best way to show off Cepheus would be to issue a challenge over a fixed amount of hands to a world-class professional player like Daniel Negreanu or Phil Ivey. This could create poker’s own version of Deep Blue v Garry Kasparov and would certainly be interesting for poker junkies like myself. I’d probably still take man over machine, though.
Assuming a statistically significant number of hands were played, Hall picking a human to defeat a computer playing GTO strategy? GTFO!
Hall's article did indirectly point out one crucial distinction between the Cepheus GTO strategy and the strategies employed by skilled human poker players: Human players will often deviate from GTO strategy in specific situations against specific opponents in order to maximize their profits through exploitation of weak players' worst errors, even though that particular non-GTO strategy would lose money over the long run against most opponents. As Cepheus' creators readily admit (link added):
"[W]hat Cepheus cannot do is maximize its winnings against weak opponents, a skill at which humans excel. Cepheus is simply an invincible, immovable bunker, a Maginot Line that actually works."
Although Cepheus is an impressive achievement in its own right, my thoughts immediately turned toward how Cepheus would play in the legal world. Does the development of a computer program capable of GTO poker play decisively prove that poker is a game of skill rather than a game of chance?
Ah, yes, our old friend, the "skill game argument", which you might recall from such classic crAAKKer posts as: "Tilting at Poker Windmills", "Why Poker Litigation Fails", and "Garnishing a Turd (Part IV): DiCristina Ends, Not With a Bang, But a Whimper". The foundation for the skill game argument is that, for purposes of gaming law in some states, "gambling" is determined by an evaluation of whether skill or chance is the "dominant factor" in determining the outcome of the game in question. The skill game argument seeks to prove that poker is a game in which skill predominates over chance, and therefore is not illegal under applicable state gaming laws. Unfortunately, every appellate court to have considered the skill game argument to date has rejected the argument or found it irrelevant (see Section E and FN3 of my discussion of the DiCristina appellate decision for the full summary of this litigation futility).
The skill game argument reached its apex in the DiCristina litigation, where a federal district court judge found poker to be a game of skill for purposes of the federal Illegal Gambling Business Act. As discussed in my analysis of the DiCristinadistrict court decision, the court's analysis of the skill game issue was driven in large part by Dr. Randall Heeb's sophisticated statistical analysis of millions of online poker hand histories. Dr. Heeb was able to demonstrate that winning players displayed a skill edge greater than expected variance within a few thousand hands of play, and also that winning players won more money than losing players even when playing the same starting hands.
Cepheus advances the skill game argument by demonstrating that there is a theoretical strategy for playing HULHE which is optimal, in the sense of being non-exploitable over a sufficiently large number of hands. In fact, Cepheus' creators note that there may be multiple GTO strategies for HULHE: "different Nash equilibria may play differently". (p. 9).
Cepheus makes two significant contributions to the skill game argument. First, Cepheus demonstrates that individual poker decisions must be evaluated in the aggregate, over time. To this point, many of the examples of poker skill used to support the skill game argument have focused on individual or tactical poker plays—for example, how pot odds, stack size, or starting hand strength can be used to determine correct game decisions. Cepheus shows that the game is substantially more complex than any one play or hand, and that a strategic approach to game decisions is both necessary and possible. Although successful poker players recognize the importance of long-term strategic game theory concepts such as range-balancing, Cepheus is a rigorous mathematical and logical proof of the importance of poker players thinking beyond the immediate play or hand. In other words, Cepheus is a refutation of the superficial legal argument that poker is a game of chance because, regardless of skill, players are still "subject to defeat at the turn of a card" in a particular hand.
Cepheus also makes another, more significant contribution to the skill game argument. Some of the legal arguments made in favor of poker as a skill game overreach, trying to establish that nearly nothing in the game is beyond the control of the player; these arguments fall flat against the easily observable elements of chance in play. Cepheus, however, takes the element of chance head on and renders it irrelevant. Cepheus does not deny the presence of a significant element of chance in the game. Cepheus simply is indifferent—impervious even—to the effects of chance over the long-term. Regardless of what cards may fall by chance, Cepheus will over the long-term win against opponents playing a non-GTO strategy. For purposes of the legal skill game argument, Cepheus serves as the embodiment of the triumph of skill over chance.
So, does Cepheus mean that the legal skill game argument is over? Hardly. Cepheus is a proof of HULHE only. One of the Cepheus researchers, Neil Burch, is participating in a couple of threads in the Two Plus Two poker forums where he describes some of the limits of the Cepheus results. First, Burch does not think heads up no-limit poker is solvable using current methods and technology because of the exponentially greater number of decision points in play. Second, Burch points out that moving from a heads up to even a three-person game adds significant layers of complexity to the analysis, including the interesting possibility that two players could collude to exploit a third player, even if that third player was playing an equilibrium strategy. Consequently, Cepheus is better viewed as a "proof of concept" of the degree of skill involved in poker, not a proof that every version or permutation of poker has a GTO strategy impervious to the effects of chance.
Unfortunately, what Cepheus giveth, Cepheus also taketh away. Like a demon from a bad horror flick, Chance does not want to stay dead and buried. Stay tuned, true believers, for our next episode when Cepheus resurrects Chance to haunt the skill game argument once again.
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AUTHOR'S NOTE: This post is the first of two related posts. Part II is HERE.
Last month I was in Vegas for a work conference and stayed at the Encore just to check it off my list of Vegas hotels. The rooms were comfy and classy, and allowed me a few convenient sessions of poker in the Wynn poker room. The Wynn poker room was the unofficial host room of the first Ironman of Poker outing, and it has consistently been one of my favorite rooms in Vegas. Classy, reserved, filled with Euro-donks and high-end booze. A little slice of poker heaven.
During one session, I was seated at a $1/$3 NLHE game with the typical cast of Vegas characters—a couple of hoodie wearing "pros", a couple of drunks looking to kill some time, a couple of businessmen with more money than skill, a couple of LAGgy Euros ... and a solid TAG younger gal. In what will be a surprise to my readers, I actually played less than two hands per orbit for the first few hours as I got a read on the table and tried to zombify myself back from card death with the occasional semi-bluff. The two hoodie guys obviously knew each other, and from their chatter they made it clear they were vastly superior to the other players at the table, and most of the other players in the room. Frankly, they made it sound like they were slumming it while waiting for a seat in the $5/$10 or $10/$20 NLHE games. Hmmm, wonder why they didn't head down the Strip to Bellagio or Aria?
Anyway, late in the evening an interesting hand developed. I had built my stack to just over $500, and the other players in this hand all had $300-$800. TAG Gal limps UTG, hoodie raises to $12, Euro calls, other hoodie calls. I'm OTB with QdTs and call. Yes, it is a marginal call, but I had position and an uber-tight image to my credit, and I had some reads on the other players. Plus, I was bored. TAG Gal called as well, and we see a flop of:
Jd9d8d
Holy suited flop, Batman! The other players quickly check to me, and I bet $50 into the $60 pot. I figured my nut straight was good here most of the time, and I wanted to give poor odds to a naked Ad or Kd chasing the flush draw. If raised, I would go with my read, but fold most of the time despite my outs against a baby flush—my Qd was really just a blocker and an emergency draw, not a real flush draw. TAG Gal thought, then called. All the remaining yahoos postured, then folded.
The turn was interesting, to say the least:
Jd9d8d 7d
TAG Gal checked quickly. Hmmm, that kind of sucked. If TAG Gal had called with Ad, Kd, or Td, I was now drawing either dead (if she held the Td for the straight flush) or to one out (Td again, but for a gutterball straight flush of my own). About the only legit hand she could hold that I beat here was a set, or possibly top two pair. But I discounted those hands a bit since she did not bet or raise the flop; her hand felt like a draw or combo draw, and the turn made most of those hands good. Of course, she was definitely good enough to be making a move with a weak hand, hoping to represent the flush, but I wasn't sure she would try that move on the flop with three other villains behind her. On balance, I saw no reason to bet, so I checked and planned to call a small river bet with my now nut-straight bluff catcher.
The river served the pickle on this sh*t sandwich of a hand:
Jd9d8d 7d Js
Yowzer! So most two pair hands and all sets just improved to a full house or quads. TAG Gal thought, then checked yet again. Hmmm, what in hell could I beat? Her entire flop range of flush draws, two pairs, and sets all now beat me. About all I could beat were a pure bluff and a baby flush that I counterfeited on the turn. I thought about throwing out a big bet as a bluff, but decided most of the hands she held that could beat me would call, and the hands I could beat would fold, sort of the opposite of a Sklansky-approved play.
So I checked behind. TAG Gal sighed, grinned, and rolled over Td9h for the straight flush. I laughed and rolled over my hand as well, just to let the table know I hadn't been bluffing the flop. Hoodie & Hoodie, Inc. immediately start jabbering like Monty Python's three-headed knight:
Hoodies: "You didn't bet the river? You have to value bet that river!"
Me: "I figured I was beat."
Hoodies: "She checked the turn and the river! You have to bet for value!"
Me: "I checked for value."
Hoodies: "Checked for value? Huh?"
Me: [pointing to my stack] "I still have all these chips."
Hoodies: "But she checked! It's stupid to check behind on the river."
I just smiled and let them continue to lecture me on proper poker strategy until they both busted out and left in search of a "better game". Probably green-chip War at Bellagio.
In all seriousness, I guess I can see some tournament or higher stakes games with a lot of meta-game factors in play where betting the river might be a decent play. But in low stakes games, in my experience players usually have what they represent, particularly with coordinated boards in a multi-way pot.
But what do you think? Obviously in this case TAG Gal was check-raising any river bet, but is her range wide enough to make a river bet a decent play to consider? Was I a total donkey on this hand? Inquiring minds want to know!
"What difference would it practically make to anyone if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle. Whenever a dispute is serious, we ought to be able to show some practical difference that must follow from one side or the other’s being right. ...
It is astonishing to see how many philosophical disputes collapse into insignificance the moment you subject them to this simple test of tracing a concrete consequence. There can be no difference anywhere that doesn't make a difference elsewhere—no difference in abstract truth that doesn’t express itself in a difference in concrete fact and in conduct consequent upon that fact, imposed on somebody, somehow, somewhere and somewhen."
As a philosophy major, I was always strongly drawn to pragmatism because of its emphasis on linguistic clarity and logical consequences. The core principle of pragmatism, stated by philosopher William James above, can be paraphrased like this: A difference, to be a difference, must make a difference.
I was reminded of this core principle of pragmatism this past week when an interesting debate erupted on the All Vegas Poker discussion forums over the relative merits of a common poker maneuver—checking in the dark. [FN1] For those unfamiliar with the term, checking in the dark (or "checking dark") simply means that the player first to act on a subsequent betting round exercises his betting option by checking before the next board card(s) are put out by the dealer, rather than waiting to check until after the player is able to see the board card(s). Checking in the dark has been popularized by the televised antics of players like Phil Hellmuth (who at times seems almost addicted to the play). But I have to wonder, have any of the players who routinely employ the maneuver actually analyzed whether checking dark is a smart play?
Applying the principles of pragmatism—or Sklansky's Fundamental Theorem of Poker, if you prefer—if checking dark is a good maneuver, it will give a poker player some advantage over the player who employs the more traditional check. Now the player checking dark invariably will be playing the hand out of position, placing them at a distinct disadvantage. A player first to act will typically end up checking a rather high percentage of the time, either out of weakness from missing the board, or out of strength when disguising a hand that strongly connected with the board. However, a player first to act will find that leading out with a bet is advantageous in a variety of situations, including:
Flopping two pair or a set on a draw heavy board.
Betting a middle pair or bluffing on a dry, junky board.
Blocking a bet the first to act player cannot call.
Putting the opponent to the test on a scary board (e.g., a flop with an Ace, or a turn or river card that completes an obvious draw).
Now this isn't to say that a player will lead out with a bet on all of these boards on all occasions. The point is that checking dark deprives a player of the option of making these plays. Clearly giving up the option to lead out with a bet on hands where doing so is the correct play is -EV. So, if it has any value, the dark check must be a sufficiently powerful, profitable maneuver to overcome the disadvantage of giving up the ability to bet boards where doing so is the correct play.
Given the obvious disadvantages caused by checking dark, what are the possible advantages of checking dark? In all of the discussions I've heard and read on the subject, three reasons for checking dark come up the most frequently: a) checking dark reverses the position of the players when heads up, b) checking dark disguises the strength of a player's hand, and c) checking dark prevents a player from giving out information on the strength of his hand. On closer examination, all three justifications are spurious.
With respect to "reversing position" of the players, checking dark simply does not reverse the positions of the players, either actually or effectively. The player checking dark still acts first, he simply exercises that option prior to the next board card being put out. Here we return to the principles of pragmatism—what is the difference between checking dark and checking normally? In terms of position, there is no difference. If the player first to act checks, the other player can bet or check behind as usual, and it doesn't matter if the first check was dark or not. If the first player to act in a heads up situation checks dark, and the other player checks behind, the dark-checking player does not miraculously gain the ability to act, as they would if they were in position. So, as there is no difference in action whether the opening check is dark or normal, there is no positional advantage gained by checking dark.
So what about the claim that checking dark disguises the strength of a player's hand? This argument essentially states that because the player first to act usually checks, a dark check prevents the other player from knowing whether a flop (or turn or river) was helpful to the dark-checking player. Let's assume a player acting first would normally check 70% of the time out of weakness, check 10% of the time out of disguised strength, bet 10% of the time out of strength, and bet 10% of the time as a bluff. A dark check has no effect on the check out of weakness play or the check out of strength play, as a dark check and a regular check each look the same to a rational opponent. But a dark check deprives the player of the ability both to bet strong hands and to bluff. So, a dark check gives no advantage to hands to a player who wants to check, but hurts a player by depriving him of the chance to bet certain hands. Assuming a player balances his donk bets between betting strong hands and bluffs, a dark check that takes both plays out of the equation simply does nothing to disguise the player's hand that cannot be accomplished by normal, solid play. But if a player is stupid enough only to bet his strong hands out of position, then a dark check might in fact be a decent play, though the move is likely more than offset by those hands where a donk bet is the correct play and the dark check leads to a check behind. Of course, such a player has a whole other kettle of fish issues to worry about.
Finally, let's look at the claim that a dark check prevents a player from giving out information as to the strength of his hand. We just looked at situations where a player checks his weak hands and bets his strong hands, but what about the more sophisticated player whose betting patterns do not reveal his hand strength? In these situations, advocates of checking dark maintain that the maneuver prevents the player from revealing whether they are strong or weak by taking out of consideration elements such as facial reactions and timing of response. Of course, this argument assumes the dark-checking player never looks at the board before their opponent acts. But if a player is bad enough to give off reliable physical tells about the strength of his hand, then checking dark is almost certainly the least of his worries—any decent player will simply eat the dark checker alive based on his physical tells which he will eventually have to give off when he inevitably looks at the board.
In summary, then, for most players checking dark is almost certainly a -EV maneuver compared to simply playing a hand normally and checking when appropriate. For a small handful of players—multiple WSOP bracelet winners and tell-spewing donkeys—checking dark might occasionally be a decent play. In other words, although plenty of players have convinced themselves they are making a sophisticated play by checking dark, their thought process is flawed. Good players don't check dark. [FN2]
"A great many people think they are thinking when they are merely rearranging their prejudices."
~William James
-------------------------------------------------------------------------------------- [FN1] The debate was reminiscent of a similar debate on the AVP forums a few years back. Also, as is usually the case, Poker Grump has already weighed in on the debate with this thoughtful post.
[FN2] Of course, players like Phil Hellmuth are sufficiently better than most of their opponents that they can overcome the -EV of using the dark check. That doesn't mean that checking dark is a good play on its own merits.
Gin card: In poker, a card that gives two players strong but different hands. Usually, one player will make the strongest possible hand (often referred to as the "nuts"), while the other player will make a very strong but losing hand (e.g., a card gives one player a flush and another player a straight or smaller flush, or one player makes quads while another player makes a full house).* Alternatively, getting the specific card(s) one needs to make one's hand (e.g., hitting a set or an inside straight draw).
Last week, I made my Ali-like return to the Meadows ATM, where I hadn't played in several months. But, my buddy Santa Claus was in town for work, so we met up for Jethro's BBQ and some poker. After stuffing myself with smoked brisket, pulled pork, and andouille sausage, it was off to the Meadows poker room.
The crowd was typical for a Wednesday night, with eight or nine tables in action for the mid-week tournament. Santa and I had to wait only a few minutes before getting into a new $1/$2 NLHE cash game with several tournament bustouts. Seat selection is a key skill for poker success, so I made the important strategic decision to sit in the 3 seat. Santa, however, unwisely chose the 2 seat.
The game started rather tight, typical for a mid-week game. After a couple of orbits, I found As5s in the big blind. Shockingly, a bunch of us all limped. The flop was junky with a couple of hearts and one spade. A bad player two to my left bet $10, and I called along with the hijack, thinking my Ace might be live and figuring I could represent the flush if a heart hit. The turn was a big spade, giving me the backdoor flush draw. I checked, bad player bet $25, hijack called, and I called. River was a baby spade. Gin! I bet out $50, bad player called, and hijack folded. I rolled over the nuts and hilarity ensued. My opponent stared at the board and my hand, then commenced angry, non-stop muttering until he busted out a few hands later. As Dusty Schmidt says, "Just like in the porn industry, you need to backdoor it if you really want to get paid."
An orbit later, I was back in the blinds. A couple of aggressive guys who had busted out of the tournament had joined the game. Most of the table limped preflop, and I closed the action checking my option with JTo. The flop came down 9-8-3 rainbow. I checked, aggro guy in middle position bet $10, aggro in hijack called, and I called. Turn came a Queen. Gin! Believing in the theory that the best way to get money in the pot is to put money in the pot, I led out with a $25 bet. I was hoping to get one caller. Instead, first aggro guy raised to $50, then the next aggro guy pushed all-in for roughly $150. With the action back on me, I paused a moment, trying to figure out what was going on. The turn had put a backdoor flush draw on board, but I had one of that suit, so I couldn't be up against a freerolling straight with a flush redraw. I decided the worst case for me was to be dodging a flush draw and a set, and there's no way I could fold the current nuts even though those draws were live. The other guy had roughly $200 left behind, and I decided if he could call the current raise, he could call my push. So, I pushed, and he snap-called. I rolled my hand, and both opponents rolled over ... Q-9 for top two pair. Ruh roh Rooby! That's about as good as I could hope for. Variance was kind, and the river rolled off a blank. I scooped a nice pot, and a few hands later, racked up and cashed out with a tidy profit.
Santa, meanwhile, stuck to his silly Seat 2 strategy. I headed home to celebrate Gin Night:
* I've used the term "gin card" for years, as have several of my poker buddies. Interestingly, I was unable to find a definitive origin for the phrase, but did find several references going back to 2006 using the term, including United Poker Forum (May 2007), Full Contact Poker (August 2007), Two Plus Two (September 2009) (though the forum archives reference the term much earlier in strategy posts dating back at least to 2006), Poker News (November 2009), and the Durrrr Challenge website (December 2010).
The earliest reference I could find was in the Two Plus Two archives where there is discussion in 2005 about a blog post by Daniel Negreanu where he reports hitting his "gin card" and losing:
From his blog he says, "The flop came A-A-10 and I was pretty sure that my opponent had A-K, K-K, Q-Q, or maybe even AA or JJ. He checked and I checked. The turn was my gin card, an 8. Or not... the dude had four aces! Goodbye."
In any event, although the exact moment where "gin card" crossed over into the poker lexicon is probably lost to the mists of time, I think it's safe to say the phrase probably came into vogue sometime around the Moneymaker boom.
An anonymous commenter from my post about a recent poker trip to Kansas City inquired:
Curious as to why you do not play 2/5 [NLHE]? Seem to have the experience and game to win at this level.
I do, in fact, play 2/5 NLHE and 1/2 and 2/5 PLO from time to time, though in the past three years I have only played PLO in Vegas (1/2 at Aria and Venetian, and 2/5 at Aria). In Iowa, I used to regularly play 2/5 NLHE at the Meadows, the Horseshoe in Council Bluffs, and Riverside near Iowa City. In Vegas, I have played 2/5 NLHE mostly at Bellagio and Venetian, with a couple of sessions at Wynn. Frankly, back when 2/5 NLHE was the smallest game spread by Bellagio, that room was consistently my most profitable place to play in Vegas (though I hated the crowded room layout and snooty management).
However, I have definitely played a lot less 2/5 NLHE over the past year and a half. There are a number of reasons:
Bankroll: There have been a number of home projects, vacations, etc. that I have paid for by dipping into my poker funds (and yes, I have kept a dedicated poker bankroll for several years now). So, with a smaller bankroll, 1/2 NLHE is a better fit, since $1,000 is four or five buy-ins at most games, while it would be two buy-ins at most 2/5 games.
Variance: Closely related to bankroll, the swings at 2/5 NLHE are proportionally bigger. The Iowa 2/5 games can play loose and wild at times, with several regulars at each poker room willing to gamble with marginal hands and draws for big stacks. About a year and a half ago, I was regularly playing 2/5 at all of the Iowa rooms, and went on a serious cold streak, losing numerous huge pots and buy-ins when I was run down by a series of suckouts, bad beats, and coolers, including a streak of six consecutive sessions where I flopped the nut straight, got it all in on the flop against two pair, and got run over by a boat. Seeing my then $12K roll cut in more than half in two months made me a little gun shy for a time.
Mental game: 2/5 NLHE plays at a higher level—or at least on a different level—than 1/2 NLHE. My playing time has been cut the past two years from two or three times a week to roughly once a week. It's simply tough to keep my game sharp enough to play 2/5 at a strong level without playing more often, which is compounded by the fact that 2/5 games aren't always running in Iowa rooms. 1/2 games, by contrast, play at a generally basic level, and I can jump into most 1/2 games and knock the rust off in short order.
The biggest reason why I play 1/2 as my standard game, however, is the most basic—1/2 NLHE is consistently a profitable game for me regardless of where I play. In the past month or so, I've played seven sessions, with six of those profitable, including five sessions with a profit over two buy-ins, and two sessions with a profit over three buy-ins. Now, I'm clearly enjoying a little positive variance recently, but multiple buy-in profits are fairly common at the 1/2 level. There are several reasons why:
Novices: For most players, 1/2 NLHE is the entry level no-limit game. It's easy to spot the newbies, and easier to take advantage of their particular weaknesses.
Bad players: Calling stations, bad bluffers, loose-passive players, scaredy cat nits, you name the leak, there's a player with it at most 1/2 tables. What's even better is that the lower stakes and buy-ins at 1/2 keep these folks coming back for more losses, without any real improvement in their games.
Gamblers: When folks who are at the casino to gamble decide to give poker a whirl, they often will treat poker like any pit game, and throw away buy-in after bad bluff after loose call. Often, they augment the gambling rush with a liquor buzz. These folks view the game as pure entertainment, and don't think twice about feeding the game as part of the fun. If you have ever sat in a late night 1/2 game at the Venetian when the drunk trust fund babies come down after enjoying bottle service at Tao nightclub, you'll know exactly what I mean.* Cha-ching!
Of course, the downside to playing mostly 1/2 NLHE is that there is a ceiling to the profit potential. Pots are generally smaller and there is less money in play. I play for fun and "fun money", so the stakes and the profits don't mean as much to me as to serious players who are playing as a primary or supplemental source of income. For those more serious players, 1/2 games are simply too small to provide steady income at a reasonable level of profitability. Frankly, the Poker Grump is the only poker player I know who is able to generate consistent profits primarily from 1/2 games sufficient for supporting a comfortable lifestyle (of course, he isn't exactly living a "balla" life).
So, while 2/5 NLHE is unquestionably more intellectually challenging, when it comes to consistently combining fun and profit, 1/2 NLHE is the nuts.
"Why did I rob banks? Because I enjoyed it. I loved it. I was more alive when I was inside a bank, robbing it, than at any other time in my life. I enjoyed everything about it so much that one or two weeks later I'd be out looking for the next job. But to me the money was the chips, that's all.
Go where the money is ... and go there often."
—Willie Sutton, Where the Money Was: The Memoirs of a Bank Robber (Viking Press 1976)
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* Of course, a similar "drunkathon" phenomenon can be observed at many Vegas casinos. In addition to the Venetian, my personal favorite late night fishing holes include Bally's, Planet Hollywood, and TI. Other folks I know swear by the MGM poker room for drunkfest poker, though I have never run particularly well at that room. Frankly, though, if a casino serves alcohol at the poker table, you will likely find a steady stream of heavily lubricated, heavily bankrolled gamblers feeding the action in the 1/2 NLHE games. This phenomenon is notably less common at the higher stakes games.
If you die in an elevator, be sure to push the "Up" button.
—Sam Levenson
During my past couple of trips to Vegas, I played a lot of hours at various Harrah's / Caesars Entertainment conglomerate poker rooms. Many of these rooms now permit a button straddle, which allows the button to post a blind raise ($5 at Bally's, $10 at Planet Hollywood, and any amount at Harrah's) which is live, meaning the straddler retains the right to raise. Action begins with the small blind, and proceeds with the button getting last action.
I happen to love the button straddle. Rather than a regular under the gun straddle, where the straddler plays the hand out of position, the button straddle lets the straddler play the hand with the best position. In other words, when on the button, a player can raise the effective stakes of the game for the one hand in which he holds the best position. Limped pots that check to the button can be stolen with greater value. Also, if a table is passive or fears big bets, a button straddler can build and steal pots with preflop raises combined with a strong c-bet or two-barrel bet. I find that many players at the 1/2 NLHE level don't mind an initial bet of $10-$12, but fear the big flop bet. Button straddling simply makes stealing from these types of players easier and more lucrative.
However, my favorite reason for using the button straddle arises when one of the blinds on my button complains that they hate when I button straddle. Thank you, sir, for telling me what to do to make you play poorly! Also, players who whine about the button straddle pretty much telegraph their hand when they call a button straddle and/or a raise from a button straddle. Again, thank you sir for making it easy to play correctly against you!
Next time you get the chance, hit that button straddle and see the money roll in.
But it’s a totally different story in a big game. If I raise it $3,000 or $4,000 and the other guy and I have a lot of chips on the table, he’ll be a little more hesitant about raising me now because he knows there’s a very good chance I’ll play back. The guys I play with know that when I put my children out there, I don’t like to let them drown.
A few days ago, I played a cash game session at the Meadows ATM. During the game, I saw an example of what is probably among the top five poker leaks, at least among recreational players—calling with a hand you are fairly sure is behind solely because of the amount of money already invested in the pot.
The hand went roughly like this. There was a straddle, and just about everyone in the room limped into the pot. Straddler raised to $24, and there were four callers. The flop came out highly coordinated—something like 9h8h8d. Straddler led out for $75, and got two callers. The turn was the Jh. Straddler led out for $125, with both players immediately going all-in for ~$800 more and ~$100 more. Now the big stack had never once shown down less than the second nuts when putting big money in the pot, while the other player was an uber-tight older nit. Straddler thought a minute, then said, "Well, I've already put most of my money in the pot, I have to call," and called for his last $150. Sure enough, the big stack had 98 for the flopped boat, while the old nit had the nut flush. The boat held up for another monsterpotten to the big stack. Straddler claimed to have held AhAd, and left after busting on the hand.
Now the big leak in this hand is the Straddler's thought process. He was fixated on the size of the pot and the fact he had committed a lot of his chips to the pot. The problem is that this fixation on the pot size and prior action blinded him to the reality that he was likely either drawing thin or dead. In economics terms, the money already committed to the pot is regarded as a sunk cost. A sunk cost is essentially a prior expenditure that cannot be recovered. For example, a developer decides to build a casino on the Las Vegas Strip. Building commences, and the project is 50% completed, but then the economy tanks. The projections for income from gambling, conventions, shopping, and other resort activities are no longer valid and must be revised. The new figures show that the projected income stream from the completed casino will not meet the additional expenses needed for completing and opening the casino; in other words, the project cannot be finished at a profit. A rational developer would abandon the project, despite having already invested tens or hundreds of millions into the project. The money invested prior to the decision point is irrelevant to the decision regarding whether to complete or abandon the project. If the additional investment needed to complete the project will not result in a profitable venture, then there is no point in making the additional investment.
Turning back to poker, the amount you personally have invested in a pot is never a valid consideration governing your future play in the hand. Either you have correct odds to call vs. your opponents' range(s), or you don't. The amount you have "invested" or put in the pot up to the decision point is irrelevant; it is a sunk cost. The past is past, and that money is now part of the pot; it is no longer "your" money or "your" investment. Reasoning that you "have to call because you already put in $X" or "it's an easy fold because you've only invested $Y" is a major leak. A smart poker player will look only at the total pot versus the wager he is facing (and future implied wagers, where relevant) when making his decision.
The problem is that many recreational poker players, because they are loss-averse, will improperly factor the sunk cost of their prior wagers into their decision making process. These players often improperly invoke the concept of being "pot-committed" to justify their ultimate decision to make a bad call. Let's look at an example. You flop a monster draw with QJs, but get to the river and have Queen-high after missing your draws (let's ignore your ineptitude in failing to get all-in on the flop). The river paired the board, so you decide to take a stab at the $500 pot, betting $300. Your opponent, a rock, goes all-in for $400. It's $100 to call into a pot of $1,200. You're getting 12:1 on the last call, but is your hand ever good here, let alone one out of every thirteen hands? Are you truly pot-committed here? Calling in these situations is a terrible leak, yet many recreational players make that last call because they're fixated on all the money they've already put into the pot.
There are circumstances where loose calls with marginal hands are warranted, but those loose calls must be justified by relevant considerations: your opponent's likely range based on his betting line and your reads, coupled with your knowledge of your opponent's playing style (particularly whether he is prone to bluffing or making value bets with marginal hands). Basing your decision to make a loose call on the fact that you've already put a lot of money into the pot is throwing good money after bad. Once you've paid enough money to know you're beat, why voluntarily pay even more for the privilege of losing the pot? Making a habit of making these bad loose calls will eventually sink your bankroll.
"Don’t make loose calls and hopeless bets to avoid giving up on the pot. It’s okay to leave your kids out there sometimes. Maybe they have soccer practice today."
During my recent Festivus/WPBT trip to Vegas, I played a fun session of Pot Limit Gamboool!(TM) at the Venetian. I've played in this game during several recent trips, and it has been uniformly entertaining and generally profitable.
Although I generally don't post hand strategy posts here (reserving those mostly for the VPN or AVP discussion forums), I do want to cross-post one PLG hand for comment from my readers, partially because I'm hoping there are some PLG savants who follow crAAKKer, and mostly because the hand involved the infamousKatkin.
The game plays 9-handed, $200-$500 buy-in. The blinds are $1/$2, but count as $5 total for purposes of the pot (i.e., the first "pot" raise with no limpers can be to $15 total--$5 blinds, $5 "call" + $10 pot raise). The game was pretty sane by PLG standards. Most pots were limped preflop or one raise with 1-3 callers; preflop 3-bets almost always meant suited Aces or Kings, or suited Broadway wrap type hands. Stacks were mostly in the $300-$800 range.
On this hand, I was in the BB. Katkin was UTG and limped, as did a Canadian player in LP, and another Canadian in the SB. I was in the BB and completed. My hand was AdKh9h7d; not too shabby for a blind hand.
Flop ($20): QdTd7s
SB checked to me. I bet $15. Katkin raised pot, to $65 total. Canadian LP raised all-in, for $120 total. Canadian BB insta-raises pot to $340 total.
Katkin had about $400 more behind. He had been playing solid PLG, showing down reasonable starting hands for position and preflop betting. He had picked off one bluff with two pair against an aggressive player's missed draws, but otherwise was not putting money into a pot without a good hand or good draw.
Canadian LP had been a little more loose, was playing with a hyper-aggressive buddy (they were a couple of college age kids on vacation, FWIW), but had not been nearly as wild as his friend. His stack had dwindled after paying off with a couple of non-nut draws that hit, but hit someone else harder. He liked to see flops with suited middle / low cards ("rundown" hands).
Canadian SB was a moderately aggressive guy, seemed to be a mid-30s guy on vacation. He had maybe $200 more behind. He had played a lot of hands, but when he made a big bet, he had a real hand or real draw.
Action is on me, I have about $750 more behind (covering everyone in the hand). Obviously I have nut draws, but with the multiway action with a player left to act behind, is this a raise all-in, a flat call, or a fold?
*** Note: Results are posted after the jump. ***
Katkin really made this a tough decision for me. Against the other two guys, I was pretty sure Canadian SB had a set of Qs or Tens; if he had a big draw he would likely have flatted. I had one blocker to his boat redraw, in the event I would call and hit one of my draws. There were three problems I had, in order of increasing importance:
My draws, although to the nuts, weren't as wide as a monster wrap with flush draw. Someone with KJ98dd has better straight draws with diamond blockers.
Canadian LP likely had either a set of Tens or Sevens, or a two-way draw of some sort. So, he likely had some of somebody's outs, but it wasn't clear whose.
Katkin. Freakin' Katkin. He had shown the initial real aggression on the flop, and was yet to act. The hand felt like he had a big draw, but there was a possibility he held the set of Qs and Canadian SB had the monster draw. But assuming he held the monster draw, I was drawing thinner than usual in a hand where I must improve to win.
If the action were three-way without Katkin, this seems to be an easy auto-shove. But with Freakin' Katkin gumming up the works, I was stymied. Finally, I folded very reluctantly. Freakin' Katkin pushed, Canadian SB called. The players showed:
Katkin: AhKdJh4d
Canadian SB: QcQsXX (set of queens, no redraw; frankly, this hand is functionally equivalent to a set of Tens in terms of stealing boat outs from Canadian SB.
Canadian LP: QT98 (top two pair plus a wrap, but no diamond draw)
The turn put the 6d on the board, and the river was a blank that did not give Canadian SB his boat redraw. Freakin' Katkin raked the monsterpotten.
In hindsight, I think the correct play here was for me to push, knowing Katkin can't call with a lesser, non-nut draw. If Katkin has a set, so be it, hope to hit the draw and miss the boat. But most of the time I think this action means Canadian SB has the big set, and Katkin the draw.
What do you PLG experts think? Inquiring minds want to know!
Twenty-odd years ago, I was a teen in a tiny farm town in western Nebraska. Video games were popular, but the only one in town was in the local bar, a place that was verboten to me. But on school trips and summer camps, I managed to sneak away with friends to the occasional arcade or hotel game room, where we'd pump quarters into all the classics: Pac-Man, Ms. Pac-Man, Donkey Kong, Centipede, Frogger, Defender, Galaga, Space Invaders, Missile Command, Mortal Kombat, Asteroids, Duck Hunt, Double Dragon, Street Fighter ... OK, so there were a lot of video games back in the day.
One of my personal favorites was Gauntlet, a multiplayer game with a Dungeons & Dragons-esque theme. Up to four people could play at once, with each player selecting his or her own type of character: Wizard, Warrior, Elf, or Valkyrie. Each type of player had its own strengths and weaknesses, and the players needed to work as a group to be successful. There really was no ultimate objective, just lots of killing of ghosts and demons, avoiding the Death wraiths, collecting treasure, and trying to eat food to stay alive for another level (though the game would helpfully take more quarters if you couldn't find a snack onscreen). One of the alternately cool and obnoxious parts of the game was an announcer with a deep and oddly-accented computerized voice that would intone various warnings:
"Blue Wizard needs food, badly!"
"Do not shoot the food!"
"Use magic to kill Death!"
"Red Warrior is about to die!"
I got to thinking about Gauntlet recently because I have noticed a marked uptick in bad players in the low-stakes cash games I play. Although my poker buds and I have generally observed games getting tougher the past couple of years, recently there has been a notable—and welcome—influx of new bad players. So what's the Gauntlet connection? Well, most of these bad players seem to be "Pay Off Wizards"—players who simply feel compelled to call value bets, particularly on the turn and river, even though they know they are likely to be behind. Pay Off Wizards seem to play in mortal fear of being bluffed off a hand, often convincing themselves their modest holding has a real chance of winning. Here are just a handful of the most egregious examples I've collected over the past three cash game sessions:
I flopped the Queen-high flush (in clubs, natch). I was called down for pot-sized bets on all three streets by ... AhJh for Ace-high.
I hold ATs, flop trips on a T-T-7 board, turn is a small blank, river is a 7, and I get called down on all three streets ($75 turn and $125 river) by .... 72o.
I turned the nut straight, got called. I bet the river, was raised, and then had my reraise all-in called by ... second pair, no kicker.
I play 87 sooooted, flop the stone cold nuts with 4-5-6 rainbow. I got called on the flop and turn in three spots, and got two players to call all-in on the river trey with ... K7o and J7s.
I flop a flush draw with A3s, miss, but hit a trey on the river. I bluff a 3/4 pot-size bet, and get called by .... AK unimproved.
I flop a flush draw with A5s, miss, but turn a 5. I bluff the river and get paid off by ... KQ unimproved.
I play 64 sooooted OTB for a raise. Player calls me on blank flop. I turn a flush draw and keep firing; only a call. I river the flush, bet the pot, get called by .... second pair.
I play A6 soooted, float the flop in position with bottom pair. Turn trips, raise the turn and bet the river, get called by ... TPTK.
I raise OTB with pocket ducks, flop a set on an Ace-high board. I get called in two spots all the way to showdown ... by A9 (rivered two pair) and A5 (top pair no kicker).
It's not just me and my weird LAG playing style, either. I've seen similar payoffs in favor of other players as well, including bizarre calls of river value bets by rocks and nits who wouldn't voluntarily risk a redbird without having the near-nuts. Well, I'm not one to object to draining players of their cash (and life force). Just call me Poker Death.
"Pay Off Wizard is about to rebuy!"
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Here's a Poker After Dark episode where Phil Laak acknowledges his extraordinary Pay Off Wizard skills:
Medical students are taught to "think horses, not zebras" to remind them that in most patients, a common diagnosis is more likely than a rare condition. This concept occurred to me recently when I saw a couple of hands posted in the strategy forum over at All Vegas Poker (AVP), and a similar hand on Vegas Poker Now (VPN). In one hand, a player flopped a baby flush and ran into a bigger flopped flush, and in the other two hands, a player flopped the idiot end of a straight. Of course, in each thread, the poster wanted to know how to avoid these potential coolers (well, other than by folding preflop, natch).
If you play poker regularly, you will run into the occasional monster cooler, something tougher to lay down than two pair versus a better two pair. Sometimes you win, sometimes you lose, and sometimes you get to enjoy the slow motion carnage as a bystander to the train wreck. The thing is, coolers tend to be memorable:
In Confessions of a Winning Poker Player, Jack King said, "Few players recall big pots they have won, strange as it seems, but every player can remember with remarkable accuracy the outstanding tough beats of his career." It seems true to me, 'cause walking in here, I can hardly remember how I built my bankroll, but I can't stop thinking of how I lost it.
Although set over set is bad enough, I have seen flopped set-over-set-over-set on two occasions, once with me involved; in both cases, bottom set turned quads for the win. In a memorable stretch of runbad last fall, I got stacked three times in a single session of 2/5 NLHE, with a flopped King-high flush versus a flopped Ace-high flush twice, and a flopped set over set on the third hand. Or, on my Festivus trip to Vegas last winter, I was involved in a hilarious hand with a couple of total yahoos, flopping a Yak-high flush against BWoP'sLe Dawn-ed flush, and Poker Grump's Set O' Presto. My hand did finish in a strong second place; unfortunately, I had bet the win, rather than the exacta box.
In any event, although weird coolers seem to happen frequently in poker, I decided to figure out the actual odds for the most common coolers: set over set, flush over flush, and straight over straight. Here's the executive summary (the "show your work" segment is below the jump). Now, there are actually a couple of different ways of looking at the probability problem, with different relevance to our decision-making at the table:
The "Perfect Storm" calculation—Determining the odds that, as a deal begins, the right confluence of events will occur to create a cooler hand that will sink us or our opponent (or any two players at the table).
The "Doomswitch / Boomswitch" calculation—Determining the odds that, upon hitting a monster flop, an opponent has the necessary hole cards to complete the cooler hand.
Note that the odds below are "pure" odds, calculated without regard to player hand selection. So, assuming players tend to muck certain hands preflop (e.g., J2s, 74, 43) as "junk", or fold small pairs to preflop raises, the realistic odds of flopped coolers are much longer. Also, the odds below are merely a calculation of how often we will find ourselves in these flopped cooler situations, not how often we will be on the top or bottom side of these cooler flops. The relative tightness/looseness of our and our opponent's starting hand ranges will dramatically affect how often we are winning or losing in these situations.
Set over set:
Perfect storm odds—Thankfully, Brian Alspach has already calculated these probabilities in great detail for a Poker Digest article. The perfect storm probabilities for a set over set situation vary somewhat based on the number of players in a hand, as more players mean a higher probability of players being dealt pocket pairs. However, for a ten-handed game where all players take pocket pairs to the flop, flopped set over set should occur in roughly 1 out of every 167 hands. Since we will only be dealt a pocket pair once every 17 hands, the odds we will be involved in a flopped set over set situation (assuming all pocket pairs see the flop) are 1 in 2,839.
Doomswitch / Boomswitch odds—We flopped a set, so what are the odds our opponent's random hand also flopped a set?—are 1 in 90.
NOTE: Not all flopped set over set hands will wind up as coolers. There will be cases where a draw heavy board will slow down the action. I haven’t factored these situations out of the calculations above, because those hands typically will still end up as coolers if both players play their sets aggressively versus draws. Also, there will be rare cases where the flop will be full house versus quads, which again are not factored out, as they are also coolers (just much colder).
Flush over flush:
Perfect storm odds—Unlike with sets, where more players in a hand mean more possible pocket pairs, which increases the odds of a flopped set over set situation, the odds of flush over flush situations decrease with more players, as more cards of the suit in question will be distributed to players, rather than being available for the flop. To put it another way, having more than one opponent with suited hole cards decreases the odds of a flush flopping, while having more than one opponent with distinct pocket pairs increases the odds of sets flopping. So, we need to look at the perfect storm odds of two players with random hands flopping flushes—1 in 19,491.
Doomswitch / Boomswitch odds—We flopped a flush, so what are the odds our opponent's random hand also flopped a flush?—1 in 39.
NOTE: Not all flopped flush over flush hands will wind up as coolers. There will be a few rare cases where the lower flush will in fact flop a straight flush or open-ended straight flush draw. I haven’t factored these situations out of the calculations above, because those hands typically will still end up as coolers.
Straight over straight:
Perfect storm odds—To keep consistent with our flush odds calculation, what are the odds of two players with random hands flopping straight over straight? This takes a little more thought, as there are three ways two players can get straight over straight—"bookend" straights with connectors (e.g., 34 vs. 89 on a 567 flop); "gapper" straights with a shared middle card (e.g., 35 vs. 58 on a 4-6-7 flop); and "bookend-gapper" straights with a shared middle card (e.g., 34 vs. 48 on a 5-6-7 flop, or 37 vs. 78 on a 4-5-6 flop). The final odds—1 in 24,038.
Doomswitch / Boomswitch odds—We flopped a straight, so what are the odds our opponent's random hand also flopped a straight?—1 in 82.
NOTE: Not all flopped straight over straight hands will wind up as coolers. A monochrome flop may easily slow down one or both players. Also, in the shared middle card straights, there will be a few rare cases where the lower straight will in fact flop a straight flush or open-ended straight flush draw. I haven’t factored these situations out of the calculations above, but it does mitigate somewhat the effect of straight over straight coolers.
Conclusions:
Realistically, the Doomswitch / Boomswitch odds are the ones we care about the most, since we rarely think about cooler hands if we aren't already in a potential cooler situation. So, what can we conclude from these odds?
Flopped cooler hands are quite rare. Flopping bottom/middle set, a small flush, or the idiot straight should be a happy occasion, not a time to start seeing monsters under the bed.
It is much more likely to see a flopped set over set situation, both because it is easier to hit the requisite perfect storm situation, and because player self-selection makes it more likely players will play pocket pairs to the flop, while many suited or connected/gapped cards get folded preflop as "junk" (e.g., J2s, 74o, 43).
Because flopped cooler hands are rare, if we encounter aggression after flopping bottom/middle set, a small flush, or the idiot straight, we are most likely up against a hand we can beat. If we flopped a set, we are usually looking at two pair or a draw. If we flopped a straight or flush, we are usually up against two pair, a set, or a draw (including pair-plus and combo draws).
Because we are usually ahead on the flop with these hands, it might make sense, absent truly deep stacks, to play fast and aggressive with smaller sets, straights, and flushes.
Conversely, with top sets, nut/big flushes, and nut/big straights, it may pay to raise smaller for value, or to slowplay, hoping our opponent will commit more chips on a safe turn card. If our opponent in fact is on the wrong side of a cooler, he will likely let us know by getting his chips in on the flop without our help.
Although flopped coolers are rare, the odds of a cooler materializing greatly increase as the turn and river change the board texture. Although we may flop the best hand, if the chips don't go in on the flop, we may well get run down on later streets.
Now, although the numbers tell us that flopped coolers are rare, sometimes those hoofbeats you hear are in fact from a herd of zebras. Or gazelles. Maybe wildebeests. Anyway, good poker intuition can still play a valuable role in helping sniff out the trap hands where you seem destined to go broke. For example, about a year ago I was playing $2/5 NLHE at the Meadows ATM. There were a few limpers, and one of the regular maniacs raised to $25. There were a couple of callers, all standard. I found Yaks in the BB, so I popped it to $150 straight. To my surprise, a young guy UTG smooth called, as did the original raiser. The flop came out Qh-Jh-7d. Yahtzee! But, it was a busy board, so I decided to play aggressively, and bet $350. The young guy UTG thought, then pushed all-in, and the maniac snap-folded. Now, the young guy is a regular, and a solid player. He isn't necessarily rock-tight, but his play smelled a lot like QQ. I just couldn't see him playing 77 for $150 preflop, and I would have expected him to reraise with AA/KK. AhKh made some sense, while AQ and QJs were longshots. On the other hand, he is capable of a big move if he smelled weakness, though I couldn't think of a hand he might think I had that I would lay down, other than AK or an underpair. We were each pretty deep, around $1,500 at the start of the hand. So, calling would cost me ~$1,000. My heart told me I was beat, but in the end, I just couldn't lay down the Yaks. Sure enough, he rolled over QQ, and it was "good hand, good night" time for me.
A similar hand occurred at the 2009 WSOP, the infamous Billy Kopp blowup hand. Essentially, Kopp went from being one of the dominating top three stacks with the final table bubble in sight, to busting out short of the final table, thanks mostly to gacking off a huge stack to Darvin Moon when both players flopped flushes. Although there are many better poker players who have dissected and analyzed that pivotal hand, it seems to me that the idea the Moon had flopped a higher flush never even occurred to Kopp. Although flush over flush was improbable, once Moon showed serious interest in the hand on the flop and the turn, alarm bells should have been going off. I'm not saying Kopp should have laid his hand down, but at some point, a baby flush turns into a bluff catcher, and a deep stacked player needs to think about protecting his stack.
So, even though we should expect horses, not zebras, it pays to remember that sometimes:
Detailed, boring math below the jump (feel free to point out math errors in the comments or via email!):
Set over set:
The basic calculations for the "perfect storm" odds are in the Alspach article. Simply multiply the odds of any two players flopping sets (1/167) by the odds we will be dealt a pocket pair (1/17) to get the odds we will find ourselves in a flopped set over set situation—1 in 2,839.
Now, for the "doomswitch / boomswitch" odds, we need to: a) assume we flopped a set, and b) calculate the odds our opponent has a pocket pair that also made a set on the same board. Since we made a set with one of the board cards, our opponent must have a pocket pair that matched the rank of either of the remaining board cards (assuming they are distinct ranks). Let's say the flop is J-9-5, and we have a set of 5s. Our opponent can have JJ or 99 to be ahead. Once this board flops, there are 6 ways for our opponent to hold a pocket pair of Js, and 6 ways for him to hold a pocket pair of 9s. There are C(47,2) possible starting hands (after we know our hole cards and the flop), so the odds of our opponent holding a pocket pair that also flopped a set is:
Chances of being dealt 2 suited cards = 312/1326 = 23.53%
Once you have 2 suited cards chances of seeing a flop that gives you a made flush:
Number of possible flop combinations: = C(50,3) = 19,600
Flop combinations containing all of your suit = C(11,3) = 165
165/19600 = 0.008418367347 or about .84%
So, assuming you play any 2 suited cards, chances you'll get a flopped flush would be 312/1326 * 165/19600 =0.00198079232 or, as you figured, about .2%
The more important number here is the .84% chance that you'll get a flopped flush.
To adapt this methodology to two flopped flushes, just work in the odds for a second hand having been dealt two suited cards of your same suit out of the remaining 50 cards:
However, once you flopped a flush, what are the odds your opponent with a random hand also flopped a flush? This is a little easier. There are 8 cards of our suit remaining after we flop our flush. To get two hole cards dealt of our suit, we calculate:
(8/47) * (7/46) = 0.0259 = 2.59% = 1 in 39
Straight over straight:
First off, there are only 8 ways to make a “bookend” straight vs. straight (ignoring suits for the moment):
Low Hand / Flop / High Hand
A2 / 3-4-5 / 67
23 / 4-5-6 / 78
…
89 / T-J-Q / KA
Let’s start by finding the odds of being dealt the low side of one of these straights—“low bookend connectors”. Now, the ranks 2-8 each work in two starting hand combos, while the Ace and 9 only work in one combo each. There are 9 ranks for the first card, with four suits for each rank, giving us 9*4 = 36/52 odds of being dealt a qualifying first card. Then, for each qualifying first card 2-7, there are 8 corresponding second cards that will give us connectors able to make a low bookend straight, while for the Ace and 9 there are 4 corresponding second cards that will give us connectors able to make a low bookend straight. Thus, the odds of getting a connecting card for the second card is 7/9(8/51) + 2/9(4/51) = [(7*8) + (2*4)] / (9 * 51) = 64/459. This gives us odds for getting low bookend connectors of (36/52 * 64/459) = 0.0965 = 9.65% = ~1 in 10.4.
Next we need our opponent to get the corresponding high bookend connectors, which fortunately are exactly one hand (disregarding suits). So, for their first card, they can be dealt any of the eight available cards that will make the corresponding high bookend connectors (8/50), and the second card dealt to complete the high bookend set must be one of the four cards of the other rank (4/49). So, the odds of our opponent getting dealt the high bookend connectors that match our low end are (8/50 * 4/49) = 0.0131 = 1.31% = ~1 in 77.
Finally, we need the gin flop, with cards of exactly the three ranks needed to complete the straight for both sets of bookend connectors. Needing one card out four from each of three ranks on the flop, means there are 64 (4*4*4) flop combinations (order doesn’t matter) that can complete the bookend straight. There are C(48,3) total flops = 17,296, giving us odds for hitting a gin flop of 64/17,296 = 0.0037 = 0.37% = ~1 in 270 (note: you could also calculate the flop odds as (12/48)*(8/47)*(4/46) = 0.0037).
Thus, the final odds for flopping the idiot end of bookend connector straights: 0.0965 * 0.0131 * 0.0037 = 0.0000047 = 0.00047% = ~1 in 213,796.
But wait, there’s more! (actually, a LOT more). There are also straights where the two starting hands share a middle card and flop straight over straight. These straights occur in the following pattern:
9T / J-Q-K / TA
9J / T-Q-K / JA
9Q / T-J-K / QA
9K / T-J-Q / KA
So, ignoring suits, for each rank Ace (one) through 9, there are 4 starting hands that can flop the idiot end of a straight over straight (note that half are gapper vs. gapper, while the other half are gapper vs. connector, though each version has a shared middle card, so the distinction is not important). So, we have 36 starting hands in total, without reference to suit. Adding in suits, there are 16 (4*4) ways to be dealt each starting hand, and there are a total of C(52,2) = 1,326 starting hands, so the odds of being dealt an eligible idiot straight hand is (16*36)/1,326 = 0.4344 = 43.44% = ~1 in 2.3.
Now, for the high hand. Once the low hand is set, our opponent needs the exact matching high hand. These odds are slightly lower than with the bookend connectors, as there is one shared middle card between the low and high hands. So, the odds of our opponent getting dealt the matching high hand is (7/50 * 3.5/49) = 0.0100 = 1.00% = 1 in 10.
Finally, we again need the gin flop, with cards of exactly the three ranks needed to complete the straight for both hands. The flop odds calculation is identical to the flop odds we calculated for the bookend connectors: there are 64 (4*4*4) flop combinations (order doesn’t matter) that can complete the straight, and there are C(48,3) total flops = 17,296, giving us 64/17,296 = 0.0037 = 0.37% = ~1 in 270.
So, the odds for a flopped straight over straight where both hands share a middle card is: 0.4344 * 0.0100 * 0.0037 = 0.0000161 = 0.00161% = ~1 in 62,217.
Finally, to get the total odds of a flopped straight over straight, we add the bookend straights to the shared-middle card straights, and get 0.0000047 + 0.0000161 = 0.0000208 = 0.00208% = ~1 in 48,077. But, this was calculated as merely being on the idiot end of straight over straight. We could just as easily be on the high end of the equation, so we need to double these figures to get the final odds for being in any hand with a flopped straight over straight: 0.0000208 * 2 = 0.0000416 = 0.00416% = 1 in 24,038.
Onto the doomswitch/boomswitch odds—if we flop a straight, what are the odds our opponent has also flopped a higher or lower straight (identical straights are not included, only the coolers)? Depending on whether it is a bookend vs. bookend situation, or a shared middle card situation, our opponent can have either 8 or 7 cards for the first hole card, and either 4 or 3 cards for the second hole card. So, the odds become (7.5/47) * (3.5/46) = 0.0121 = 1.21% = 1 in 82. For a bookend vs. bookend situation, the odds are slightly better: (8/47) * (4/46) = 0.0148 = 1.48% = 1 in 68. If a gapper straight is involved, the odds are: (7/47) * (3.5/46) = 0.0113 = 1.13% = 1 in 88.
Now, why are the odds of a flopped straight over straight so much lower than the odds of a flopped flush over flush? There are several factors in play. First, it is easier to get an eligible straight starting hand than flush starting hand. Conversely, it is easier for our opponent to get an eligible flush hand than it is to get an eligible higher straight hand. But the real kicker is that the gin flop is easier (nearly 24% easier) to hit in a flush over flush hand than in a straight over straight hand: 84/17,296 flush flops rather than 64/17,296 straight flops. Considering it is slightly easier to hit a flush draw than a straight draw certainly makes it “feel” correct that it should be tougher to flop straight over straight than flush over flush, even if a straight is an easier hand to make than a flush, considered ab initio.
