“All in.” Your opponent slides a stack of chips across the high-stakes poker table. You glance back at your cards, a pair of sixes. The game is Texas Hold’em. Only two of you remain, and no community (face-up) cards have been dealt yet. Things rarely get simpler than this in poker, and you have a binary decision to make: call (match your opponent’s bet) or fold (give up). To a professional player, though, every detail demands consideration. What was the betting pattern before the all-in push? Who acted first? How many chips does each player have, and how many are in the pot? When will the blinds, or forced bets, increase? And of course, how likely are sixes to win the hand? You’ve studied poker strategy, memorized probability tables and run the numbers in your head. It all points to folding as the objectively best decision. But you’ve noticed over a prolonged tournament that your opponent has a tendency to overbet with mediocre hands. Do you stick with your training and fold or adjust your strategy on the fly to exploit the weakness you’ve observed?
This question of whether to use what’s known as “game theory optimal versus exploitative play” captures a central conversation in high-level poker. Its mathematical underpinnings trace back 80 years, but rapid advances in AI have brought mid-20th-century math to the forefront of modern gaming. New tools teach poker players optimal strategy for the game, so why would they ever decline to use it?
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Optimal Play?
Objectively optimal play might seem counterintuitive in a game based on randomly dealt cards and messy human psychology. Take bluffing—when a player pretends they hold an unbeatable hand to scare an opponent into folding. Lying about one’s cards feels inherently psychological and resists a rigorous mold of objectivity. But we should never underestimate mathematicians’ knack for turning human behavior into tidy equations.
In fact, the foundational 1944 book on mathematical game theory by mathematician John von Neumann and economist Oskar Morgenstern, Theory of Games and Economic Behavior, highlighted poker as a central example. The authors analyzed a simplified variant that distilled the game down to its most fundamental dynamics: two players would each receive a number between 0 and 1, with higher numbers representing stronger hands, and then engage in a single round of betting. Von Neumann and Morgenstern proved not only that an optimal strategy exists but also that bluffing is an essential part of that strategy.
Of course, Texas Hold’em packs a great deal more complexity than this toy example. Who’s to say that an optimal strategy even exists in full-fledged multiplayer poker? The late mathematician John Nash, that’s who. In the 1950s Nash, who went on to win a Nobel prize in economics in 1994 and was later depicted in the 2001 biopic A Beautiful Mind, propelled the then nascent field of game theory. His most famous discovery, now called a Nash equilibrium, occurs when no player of a game would benefit by deviating from their chosen strategy (assuming others don’t deviate from theirs).
Game theorists consider this condition optimal because if you and I play a game where we each begin with any old strategy, and then I adapt mine to take advantage of what I see you doing, and then you counter-adapt to my change, and so on, we will eventually reach a steady state in which neither of us can keep improving. With a Nash equilibrium, players can even announce their strategies in advance, and still everybody’s best course will be to stick with the equilibrium. In a one-page paper in 1950, John Nash proved that every finite competitive game—from mahjong to Magic: The Gathering—has at least one Nash equilibrium.
Despite its name, game theory applies to a broad spectrum of topics beyond traditional games, including economic systems, nuclear deterrence and evolutionary biology. To researchers in this field, games refer to any interactions among rational decision-makers whose actions and payoffs can be rigorously defined and analyzed. So Nash’s theorem has wide-reaching implications. In poker, it justifies the search for optimal strategies in a game once thought to rely on gut instinct and reading tells.
An AI Poker Revolution
Just because we know that Texas Hold’em has a Nash equilibrium, that doesn’t mean we know what it looks like. As games ratchet up in complexity, their optimal strategies tend to become harder to figure out. Anyone could learn how to play perfect tic-tac-toe in one sitting by memorizing a few move sequences. For a more elaborate game such as checkers, which always ends in a draw with perfect play, humans could never memorize enough variations to implement the optimal strategy. Scientists have created unbeatable algorithms that play optimally, however, because computers can store massive databases of positions and extensively search the game tree in a way that humans cannot. Meanwhile chess computers have dominated the best human players since around 1997 (when world champion Garry Kasparov lost a historic match to IBM’s Deep Blue), yet chess computers still don’t exhibit optimal play—the next generation of chess engines will crush today’s.
Unlike chess, poker involves imperfect information. Players know their own cards but not their competitors’, which makes the game more daunting to model computationally. This explains why the algorithmic revolution in poker didn’t come until the recent AI boom. In 2015 computer scientists announced an algorithm that displayed essentially perfect play for a restricted version of the game with only two players and constrained bet sizes. Only four years later, we got the first superhuman AI for multiplayer Texas Hold’em. A flurry of commercially available software tools called “solvers” followed, and in the span of a few years every rounder (person who plays poker for a living) with a few hundred dollars to spare had a card shark at their fingertips who could tell them how to play in almost every situation.
“The game went from being this fuzzy art to a hard science,” says Liv Boeree, a former professional poker player. To stay ahead in today’s environment, advanced players study the game by using computer programs such as PioSOLVER, which approximates optimal strategies. For simple and common situations, pros will memorize the machine’s recommendations, whereas they glean more high-level lessons from its behavior in rare and more complicated situations. For any elite poker player, studying with these solvers is essential. “If you want to play high stakes against the best, absolutely ... you’d get eaten alive [if you didn’t use solvers],” says Boeree, a World Series of Poker champion. “There were some players who just rejected the entire notion, and they didn’t work with solvers..., and for the most part, they got left behind.”
AI has both confirmed some common wisdom about Texas Hold’em strategy and overturned some maxims that players had gotten wrong. For example, computers find success in “donk betting”—initiating the first bet on a round of betting after merely calling another player’s bet on the previous round—despite the folk belief that donk betting is an amateur move. AIs also play a wider variety of hands in situations where expert humans tend to fold. Like chess engines, multiplayer poker solvers don’t literally play optimally, but they dominate humans thoroughly enough that we have a lot to learn from them.
How to Win
In defining the Nash equilibrium, I smuggled in a critical detail: equilibrium occurs when no player would benefit by deviating from their chosen strategy (assuming others don’t deviate from theirs). When other players do deviate despite this, however, it’s often wise to deviate in response.
Take rock-paper-scissors as an illustrative example. What is its Nash equilibrium? Think for a moment: What strategy from both players would leave no incentive to deviate? Answer: players should toss rock, paper and scissors perfectly at random; each has a one third chance of appearing, regardless of all previous rounds. You can announce this strategy to your opponent in advance, and they will be helpless to take advantage of your candor.
If you and your opponent both play this equilibrium strategy, you can expect to win half of the decisive rounds (ignoring ties). Now suppose your opponent deviates. In the extreme case, imagine they always play paper. If you stick with the equilibrium strategy, then still you’ll win half of the decisive rounds because you play the winning scissors and the losing rock with equal frequency. But you can instead exploit your opponent’s deviation by always playing scissors and cutting their paper on every round. Less dramatic deviations still give you opportunities to exploit. For example, empirical research on rock-paper-scissors shows that when people win one round, they are slightly more likely to repeat the throw that they just won with. Knowing this can give you an edge. If you just lost to rock, for example, then play paper next because your opponent is likely to throw rock again. The Nash equilibrium is the only strategy that is not susceptible to exploitation.
The same dynamics play out in poker at a much more complicated scale. As players learn more optimal techniques from their AI collaborators, they also learn how to sniff out when their opponents fall short of optimal play and how best to punish them.
You might think there’s a catch here. If your opponent deviates, isn’t the optimal decision to exploit them ruthlessly rather than to blindly stick to the Nash equilibrium and leave potential money on the table? If you discover that an opponent deviates from the Nash equilibrium in predictable ways, then deviating yourself to exploit their weakness may net you more money. As soon as you exploit them, however, you’re now veering from the equilibrium and opening yourself up to exploitation. If your opponent always throws paper and you start only throwing scissors, eventually they’ll catch on and start rocking your scissors.
As former poker pro Igor Kurganov puts it, “any time you pick up on a mistake by your opponent, you improve your model of how they think about the game, adjust how you play against them to account for that mistake and, by that, become exploitable yourself.”
Most players agree that to stay competitive at the top levels of poker they must use a blend of game theory’s optimal and exploitative play. Optimal is more defensive, whereas exploitative is more offensive. Some teachers recommend that you should begin a tournament by emulating optimal play—and only after you’ve had time to observe your opponent’s weaknesses should you sprinkle in your exploits. The flexibility to switch between strategies separates the fish from the sharks. “This whole process works better the more certain you are that you’re smarter than [your opponent] about the game,” Kurganov says, adding that “you do less exploitative adjustments when you feel like they’re as good or better than you.”
For some, the emergence of superhuman poker engines has sapped the game of its intrigue, while others contend that computers have added a new layer to the game. Boeree, who retired from pro poker in 2019 and now works as a science communicator, philanthropist and podcast host, falls more into the former camp. “It felt like it took a little bit of the magic out of the game, like, ‘Oh, okay, the mystery has been solved,’” she says. But Boeree acknowledges that the new age of poker has no shortage of enthusiasts. “Since COVID it’s been booming,” she adds. “The World Series of Poker got more players than ever before last year. Records are getting smashed. So clearly it has not killed the game.” Instead we might say that the changing landscape of poker is still finding its equilibrium.