January 20, 2015

Poker's Deus Ex Machina (Part II)—
How a Computer Proved Poker Is a Game of Chance

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."
But the bum hunting problem is not limited to online play. Going back even to the early WSOP days, elite brick and mortar poker players like Doyle Brunson and Amarillo Slim would seek out games with easy marks like Archie Karas and Jimmy Chagra. More recently, high stakes professional poker players have pursued games with "whales" like Cirque du Soleil founder Guy Laliberté, baseball superstar Alex Rodriguez, and Texas banker Andy Beal (immortalized in the classic book, The Professor, the Banker, and the Suicide King: Inside the Richest Poker Game of All Time). Similarly, professional poker players Phil Ivey, Andrew Robl, Dan "Jugleman" Cates, and Tom Dwan have recently posted bail money or provided support to Paul and Darren Phua, individuals who reportedly control access to Macau's famed, whale-laden high stakes poker games. Back in Vegas, a number of poker pros jealously protect their whales from poaching by other pros:
"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.

Is it any wonder the law treats poker the same as Mississippi Stud or Let It Ride?

“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


  1. Excellent two-post analysis, Grange. This all goes back to what I thought was the first rule of poker: play with players who are worse than you.

    Every so often I venture back into the world of online poker. It is not nearly as fun or profitable as it was in the glory days.

  2. @ Lightning: Every time I sit down at a poker table, I automatically look for the 1-3 players who I think will give me their money. I go out of my way to play pots with them. It certainly raises moral questions about the nature of the game. But while I wrestle with ethics, I'll stack chips.

  3. "poker skill predominately turns on game selection", buuuut.... Couldn't you make this same argument about everything? Two equally skilled chess players playing for $100/match will break even in the long run. If there's a rake being taken out, then Kasparov and Kasparov Klone will both lose in the long run. I don't think that's sufficient to say chess is a game of chance.

  4. @Jojo: Chess, as a game of perfect information, has no element of chance other than who plays white (which has been proven to have a very slight advantage). So if two GTO bots play chess, they will either always draw or the player with white will always win. Two very closely matched players--Kasparov & Karpov, say--may require a very large number of games to determine which is better. But in no case would we say chance played any role in the outcome.

    By contrast, poker has an element of chance. If you wash out skill, the game results will be almost entirely (or entirely in cases of GTO v. GTO) be the result of chance.

  5. If we added a coin flip to the beginning of every chess match to determine who gets to be white, would you say chess then is a game of chance?

    1. *would you say this new version of chess is a game of chance?

    2. No. In chess, who plays white/black is already determined by lot, so making it a coin flip or a die roll doesn't change the game. You would need to insert an element of chance into the game structure to transform it from one of skill to one of chance. For example, before each move, a player rolls a die and gets to make that many moves in a row. In that situation, chess would be a different game, but one where chance and skill are both in play. Sort of a more sophisticated version of backgammon, if you will.

    3. The concept of a "solved game" may help here. Tic Tac Toe is "solved". So long as the players both play perfect strategy, the game will always end in a draw. For other solved games, it can be proven whether the first player will always win, lose, or draw if both players play perfectly. This does NOT mean that the game is a game of chance. It just means that perfect play yields a predictable result.

      For poker, even if "solved", perfect play is still subject to an inherent element of chance that skews "winning" and "losing" over the short term. Chess and other perfect information games lack that element of inherent chance.