Poker and the Nash Equilibrium

A natural strategy in poker as in any other situation is to play to win.  But is this always the best strategy?

How to play against an All-In
When a player goes all in in a poker tournament, there are two basic strategies that the other players use. Well, the other players who choose not to fold.  There is the

• let's check it down strategy, and 
• the isolate strategy.

Check It Down
Some people say there is a gentleman's agreement in poker where they check it down.  First of all, let's get it straight.  When you play poker, there is nothing gentlemanly about taking one guy's car payment so you can pay your mortgage.  There is no rule that exists to say you don't bet when someone else goes all-in.

Isolation
Another common thing seen at the poker table is a later player making a big bet, or a raise to isolate the other players out of the hand to transform the play into heads up.  Some believe this improves the raiser's chances as with less players they are more likely to win the pot.  This shows an insight into the math of poker.  Others call this same act protection.  These people either have heard someone else call it this, or the have an even better understanding of the math at play here.

The Natural Strategy
A natural strategy is that greed is good. The poker player pursues his own self interest, the pursuit of chips.  This theory is espoused by economist Adam Smith in The Wealth of Nations (1776), saying "by pursuing his own interest, [the individual] frequently promotes that of the society more effectually than when he intends to promote it."  This is the good old American Dream.  Win the chips and the world is a better place.

The Nash Equilibrium
John Forbes Nash is a Mathematician who won the 1994 Pulitzer Prize in Economics for a bargaining strategy known as the Nash Equilibrium.  His life was fictionalized in the movie A Beautiful Mind.  Nash noted that if everyone is out for themselves then sometimes everyone is playing a sub par strategy.  A Nash Equilibrium exists when each player plays his best response to the situation. In Nash's theory, each player has some understanding of what the other player might do.

Nash's theories turned traditional economics upside down, but have been very useful in a variety of game theory applications. Nash's theories suggest what if by pursuing your own self interest you may get less, but by pursuing the greater good you get more.  In practice we see this in trade cartels and shared standards (VHS vs. Betamax)

Applying this to the all-in move in poker involves stepping back for a moment.  What is your goal?  Is your goal, in a tournament, to acquire chips, or is your goal in a tournament to eliminate players?  Utilizing Nash's theory a player can postulate that he wins in the long run if he gives up chips in this all-in hand.

The Downside to Isolation
Suppose the player who goes all-in preflop has pocket Aces.  If nine players at a ten person table call him, he then has a 31% chance of winning against the nine random hands.  If the isolation succeeds and transforms this into heads up, the the all-in jumps to an 86.4% chance of winning with his pocket Aces.  The raiser has indeed improved his chances of winning, but at the same time he has improved the all-in's chance of winning.

By isolating, the raiser purses the American Dream by attempting to win more chips.  The downside is that the all-in was 2:1 favorite to lose the hand, but now is a 6:1 favorite to win.

The Equilibrium
The optimal strategy that cannot be improved upon is for all players to not bet after the all-in.  To, in practice, check it down.  By giving up the direct pursuit of chips then all players benefit against the all-in by providing the best chance of eliminating him from the game.

Variations on a Theme
Mathematically, the only situation where it makes sense to bet is when you have (flop) the nuts.  If you have the true nuts, then you do not have to be concerned about the all-in NOT being eliminated.

Collusion Warning
If everyone just happens to adopt a strategy of checking it down then that is fine and that is legal.  But as soon as someone asks the group about checking it down, then collusion is occurring and the rules of poker have been broken.

In conclusion, follow the math.  Don't bet against an all-in.