I always start my Competitive Strategy class by comparing economic games with real games, like poker.  At the last minute, I found this great story in the New Yorker about a champion poker player, Chris Ferguson.  Ferguson is a game theorist/computer scientist who took thirteen years to finish his PhD degree!  In the end, it all worked out.  The story reports that he has made about \$7 million playing poker and at least that much through his website Fulltiltware.com.

How did Ferguson learn to play poker?   There are two ways: experience at the poker table or using theory.  Ferguson used theory, sitting at home, practicing and playing out different moves and counter moves in his head.  He thinks both approaches lead to the same conclusion but his is less “arduous.”  Using theory, he came up with his “optimal strategy” which involves playing hands to answer the question “How do I lose the least?”

What does Ferguson’s strategy have to do with game theory? That’s the topic of this post.

Poker is zero sum game: players wins and losses add up to zero.  The most developed part of the theory of zero sum games is for the case of two players and most of our intuition about poker comes from this theory.  The most famous result in this theory is the minimax theorem due to von Neumann and Morgenstern.  I will attempt to explain the result though not the proof!

Ferguson’s optimal strategy seeks to maximize his minimum payoff – How do I lose the least?  This is called a maximin strategy.  Such a strategy is not optimal in typical games.   But the minimax theorem says: No, in zero sum games the maximim strategy is a best-response to the other player’s strategy!

In fact, it tell you even more: the maximin payoff is the same as the minimax payoff.  A minimax strategy maximizes payoffs under the assumption that other players are out to get you.  So, it is like a typical optimal strategy that says “How do I maximize my payoffs?” rather than “How do I lose the least?”  In two player zero-sum games, the two are the same so Ferguson has found the optimal strategy.  Moreover, a minimax strategy has a cool property: If other players do not play optimally, you make even more.  You are already maximizing your payoff under the assumption they are out to screw you.  If they get confused and do not screw you, you must make even more than your minimax payoff.

Unfortunately, an optimal strategy does not mean you win all the time!  Suppose two player who know the optimal strategy face each other.  What expected value can they guarantee themselves using optimal strategies?  As poker is a symmetric game, each player must be able to guarantee the same value v. But as values have to add to zero, as this is a zero sum game, we know we must have 2v=0.  This can only hold if v=0.

If Ferguson played rational players all the time, he would make zero.  Since he has made \$7m, he must be meeting dumb players a lot. His optimal strategy since it has the minimax property works well against dumb players.  Even better, if he can “read” dumb players he can play a best-response their dumbness!

This is why there is a second intuitive school of poker.  This school is based on looking at people facial expressions and body movements.  In looking at their betting to see if there are some weird regularities that help you predict their hand.  That’s the whole issue: if you can deduce their hand, your strategy is simple – fold if you have a worse hand and bet if you have a better hand.  Poker is all about bluffing and sandbagging to hide your hand.  (This is a property of many “games” like some war games where to get an advantage you should keep your strategy secret. Saddam’s bluffing about WMDs has aspects of this.)

I wish I could end this with some mind-blowing practical takeaways but I can’t.  The main thing I know from game theory is that people bluff and sandbag much less than theory says is optimal.  Binmore’s Playing for Real book has a nice chapter on Nash and von Neumann’s poker models.  Check it out if you want to know more.  The second thing is that knowing how to read people is a key part of all this.  I’m terrible at that.  Maybe we will meet across the poker table one day and you’ll be able to take all my money.