Russell Crowe’s performance in A Beautiful Mind brought game theory media attention, but in the poker world experts like Chris ‘Jesus’ Ferguson have already employed it to win millions.

When it comes to poker players everyone knows the stereotypes – fresh-faced Scandinavians, internet millionaires and grizzled old road gamblers – but in the modern poker world another significant group is the so-called ‘maths geeks’. Most celebrated among these players are those who specialise in an area called ‘game theory’. Game theory is a branch of mathematics that deals with decision making and strategy in situations where competing parties attempt to achieve a goal through optimal behaviour, and where the success of one side can impact on another negatively.

The classic example of this is the ‘prisoners’ dilemma’, where two suspects are arrested and the police, not having sufficient evidence, separate them and offer both a deal. If one testifies and the other stays silent the latter gets ten years and the first goes free (and vice versa), whereas if they both keep silent the punishment is two years and if they both confess it’s six months. In an ideal world, co-operation would be the best option, but assuming human self-interest it is safer for the prisoners to each betray the other.

Game theory was pioneered by John Von Neumann in Theory of Games and Economic Behaviour (1944) and developed further by John Nash, whose work on n-person games revolutionised the field. When his story was told in the film A Beautiful Mind (2001), it gained even more popularity, and this has been reflected in poker. Recently we have seen countless online programs designed for maths-oriented poker players, and analyses of poker as a negative-sum game, where one person’s wins are another’s loss while the house continues to take its share.

In reality, though, game theory has been around for some time, and its figureheads in the poker world – who learnt to play through early computer simulations – are walking, talking proof of its usefulness. David Sklansky wrote about it in his seminal work The Theory of Poker, but the real king of the game theorists is Chris ‘Jesus’ Ferguson. Ferguson used his research and expertise to snare five WSOP bracelets, including the title of World Champion in 2000, and two WSOP Circuit events in 2005 alone, while hiding his mathematical background carefully behind long hair, beard, sunglasses and a trademark cowboy hat.

Analyse this

Poker tournaments are a place where game theory is relevant, but how exactly can it be used? Well, thinking like a game theorist in poker involves reducing the game itself to a series of mathematical problems that can be ’solved’, either with a definite solution like ‘fold’ or ‘move all-in’ or a balance of the possible options that creates the best outcome. A simple example of this can be limping in with Aces 20% of the time under the gun and raising the other 80%.

The correct play in any given situation, of course, depends on how your opponents play, and mixing it up with aces would only be sensible against intelligent opponents. If they were raising all-in every hand anyway you would always just call. The latter is called an exploitative strategy, and there are also many plays in poker that are mathematically unexploitable and therefore always correct. For example, on the bubble in a sit & go a very big stack can profitably move all-in from the small blind with virtually any hand when the blinds get big and everyone has already folded because of the prize-structure implications.

This is the simple side of game theory in poker, but against skilled opposition, who play near perfectly, you would need to employ an optimal strategy – making winning impossible even if they knew what you were doing. For example, suppose you are playing a game where there is $200 in a heads-up pot by the river, you have a 20% chance of hitting your hand and the optional bet is $50. Given all these factors and the fact that your opponent is getting 5/1 on a call ($50 to win $250) you should bluff another 4% of the time, duplicating the odds that your opponent gets on the call into your bluffing strategy, which makes his decision equally bad either way.

Similarly, when you are facing top-class opposition you need to mix up your play throughout a hand to make it difficult for your opponents to read you. Limping with aces a certain predetermined amount of the time is one simple example, but against the best this needs to be carried through into the rest of your play. For example, when most cautious players flop a set they check in the hope of inducing action, but more aggressive players will tend to bet out with it in the hopes of disguising the hand and winning a big pot.

Money talks

A game theorist wouldn’t ask which play was right, but what blend of the two was the most profitable. This would also involve factors like their player image and be adjusted based on whether the opponent had any exploitable tendencies. But if they decided that the split was 2/1 in favour of checking the hand (assuming the flop brought no flush or straight draws) the decision after that point should be as random as possible.

For this reason, players have been known to use devices like the second hand on their watch or a credit card number to help them choose randomly the course of action they should take. So think carefully before you act the next time a cowboy looks at his watch before betting into you!

Game Theory in Practise

The secret of making game theory work for you is in keeping your opponents guessing.

You are playing against a table of world-class opponents in a WPT event, and late on you call a raise from one of them with 10-10 on the button to see the flop come a beautiful 10-9-9 with two suited cards. You only have 15-20 big blinds left in your stack and the raiser fires out a bet of about five big blinds.

Do you:

• a) fold
• b) call
• c) raise
• d) all-in?

The answer is b) call or c) raise all-in. If you only have one predictable move per situation, a great player will see them reading you like a book. Calling with a monster here and only raising all-in with a big draw or pair is what would be expected. But against this player you need to mix it up to create an optimum strategy, and part of this play should be occasionally moving all-in in a situation like this with the nuts to keep them guessing!