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Before Alan Turing designed a machine to crack the Nazi enigma code, or laid the rules for artificial intelligence, he was an exchange student at Princeton University. He was there to check logic and computation, but additionally had the danger to wait other lectures—like the only given by mathematician John von Neumann in March 1937, at the theory of games.

Von Neumann’s work had started with a simplified version of two-player poker. Each player installed an ante of $1 then received a single card, which displayed a random number between 0 and 1. The primary player could then decide to check—which forced a showdown—or bet, during which case the second one player could decide to call or fold. Let’s suppose the sport is pot-limit. The verdict process for player 1 and player 2 can therefore be sketched out in diagram #1.

Diagram #1

The dollars on the bottom of every branch show the prospective profits for player 1. If the sport ends with a choice or a check, the winnings is determined by whose card is higher. But what's the best strategy for this game?

In poker, both players try to reduce their opponents’ profits. In 1928, von Neumann proved that during such games there would always be an optimal strategy for every player—this would guarantee them the absolute best outcome, assuming their opponent was seeking to do the same.

For the sport above, von Neumann found that the optimal strategy for player 1 is to bet in the event that they are dealt a card with a price more than 14/18—or if the price is not up to 2/18. In other words, the mathematics says they need to bluff in the event that they get dealt an overly bad card. In response, player 2 should call a gamble provided that they've a card with a price above 10/18, and fold otherwise. Even so, the sport isn’t completely fair: If both players adopt optimal strategies, player 2 will lose $2/18 (or about 11¢) per hand on average.

Turing plays poker

Von Neumann’s research certainty left an impression on Alan Turing. Within the late 1940s, Turing began to write a paper on ‘The Theory of the proper Strategy for taking part in Two-man One-card poker’. From the notes that survived after his death, apparently his main criticism of von Neumann’s version of poker was the shortage of betting rounds. ‘The restriction within the raise is an admitted blemish at the theory,’ Turing noted.

Turing tried to tackle a version of the sport during which two players could raise and re-raise as again and again as they liked. He made some inroads within the analysis, suggesting—as von Neumann did—that players should sometimes bluff with weaker cards. However, in line with the notes he left behind, he didn't discover a full strategy for the game.

Refining von Neumann’s poker

The optimal strategy for two-player poker has continued to intrigue theorists. In 2007, poker pro, Chris Ferguson published a paper in Game Theory and Applications together with his father Thomas, a math professor at UCLA, and mathematician Céphas Gawargy. Unlike Turing, who allowed multiple betting rounds, the trio keen on a more basic extension to von Neumann’s game. As before, each received a single card, with a worth between 0 and 1. But this time, the second one player could reply to a check by player 1 either with a gamble or check. The verdict tree therefore has an additional branch at the right-hand side as noted by diagram #2.

Diagram #2

What’s the optimal strategy for every player on this situation? Ferguson and his co-authors found that player 1 should again bet with a low or high card, and check otherwise. If player 1 bets initially, player 2 should call in the event that they have a high card and fold if not. If player 1 checks, player 2 must also bet with a high or low card, and talk to a middling one.

We can summarize the tactic for every player by dividing the region between 0 and 1 into intervals that designate what a player should do in the event that they have a card with a worth in that range (see diagram #3).

For example, if player 1 opens by checking and player 2 has a card with a host not up to 0.33, player 2 should respond by betting.

Diagram #3

Poker artificial intelligence

Real two-player poker games—like heads-up no-limit hold’em—are too complicated to calculate a precise optimal strategy. As a result, the most productive computer bots usually attempt to approximate the right strategy, by simulating billions of hands and refining their approach as they learn what works and what doesn’t. However, there may be still a job for the stripped-down versions of poker pioneered by von Neumann.

Last year, researchers at Carnegie Mellon University in Pittsburgh, PA pitted their heads-up no-limit hold’em bot against poker pros Dong Kim, Jason Les, Bjorn Li, and Doug Polk. Although the bot used an approximation of the optimal strategy for almost all of the game, as each round approached its conclusion it switched to an ‘endgame’ strategy, which checked out all possible options left, and identified the most productive strategy given the available information. Whereas human players often count their ‘outs’ as a hand progresses, the Carnegie Mellon bot was in a position to crunch through every possibility.

It proved a powerful strategy. In a paper reflecting at the competition, Sam Ganzfried, one of the crucial researchers at the project, said Polk told him afterwards that ‘the river process of Claudico using the endgame solver was the strongest a part of the agent’. Indeed, endgame analysis has already helped computers beat humans at checkers and chess. As Alan Turing once said, ‘we may hope that machines will eventually compete with men in all purely intellectual fields.’

Could no-limit poker be next?

Discover more concerning the science of gambling with The very best Bet: How Science and Math are Taking the Luck Out of Gambling.

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