Monday, March 21, 2011

Game theory, poker and trial strategy

When playing poker, is it better to bluff when you think your hand has a chance of winning or is it best to bluff when you think your hand stands no chance of winning?

In other words, do you shove your chips into the middle of the table with a pair of jacks or with nothing better than a ace-high?

According to game theory, the optimum bluffing strategy is to bluff when you have the weakest hand.

Chris "Jesus" Ferguson has a Ph.D. in computer science and relied on game 
theory in winning the 2000 World Series of Poker Main Event.

To see why, let's pretend we are playing one-card poker with a deck of an ace, a deuce and a trey (with the ace being the lowest card). When it's your turn to act you have the choice of betting or checking. Your opponent then has the choice of calling the bet, checking behind you or folding.

Logic dictates that you will alway bet when you have the trey. You know your opponent can't beat you because you have the highest card. If you don't bet and your opponent checks behind you, you win nothing. If you bet and your opponent folds then you win nothing. But, if you bet and your opponent calls the bet, you win his bet. The only way to make any money when holding a trey, then, is to bet first.

If you have the deuce there is an equal chance your opponent has a trey or an ace. If you bet with the deuce, you will win half the time and you will lose half the time. If you check with the deuce you are forced to fold your hand if your opponent bets. If your opponent holds the trey, he will bet. If he holds the ace, he will fold. The best play, then, if you hold the deuce is to check.

On the other hand, if you have an ace, you know you have the worst hand. If you don't bet, but your opponent bets you are forced to fold and lose nothing. If you bet and your opponent bets then you  lose the bet. But, if you bet and your opponent folds, then you win his bet.

Let's say, for instance, you hold the ace. Your opponent therefore holds either the trey or the deuce. If you bet and he holds the trey, your opponent will call your bet and win the hand. If you bet and he holds the deuce he will be forced to fold his hand because he can only beat a bluff. The best play, then, is to bet whenever you hold the ace - since the only way you can win the hand is if your opponent folds in the face of your bet.

By following this strategy you will win every time you hold the trey and half the time you hold either the deuce or the ace. Your opponent will be forced to fold whenever he holds the ace or deuce meaning you have a 2-to-1 chance of winning every hand when it's your turn to act first.

Whenever your opponent acts first and bets out you will call if you hold a trey or a deuce - since you can never win calling a bet with an ace. If your opponent checks you will bet if you hold an ace or a trey, since he can only call your bet if he has a trey.

Now that's all well and good, you say, but what on earth does that have to do with defending folks accused of committing crimes? Just imagine all of your cases could be sorted like cards. Some cases are strong, others are weak and the rest fall somewhere in the middle. What is the optimum strategy for defending these cases?

According to game theory you push the prosecutor to trial on your best cases. That forces the prosecutor to evaluate the case and should, in most instances, result in dismissals (or at the very least, reductions). But you already knew that.

What do you do with the bad cases and the cases that fall somewhere in the middle (the aces and deuces)? Based on our poker game example, you push the bad cases to trial as well and try to work out the rest. Why you might ask would you do such a thing?

You do it because it's the only way you're going to get a dismissal on the worst cases. If you push a case to trial then the prosecutors have to deal with witnesses, some of whom are reluctant or live out of town, and evidentiary issues. You never know what's going to happen. Maybe the prosecutor thinks twice about whether her case is a whale (a trey) or a dog (an ace).

You should win (or get dismissals) on your best cases. The only chance you have of winning your weak cases is to go to trial. If the case falls in the middle, however, you have to weigh the benefit or winning with the risk of losing. These are the "coin flip" cases that could go either way. These cases have to be "played for value."

If you think about it, you should already be doing this intuitively. We tell some clients their cases are slam dunks. We tell others that they have nothing to lose by going to trial. It's the ones in the middle that are the most difficult to handle.


For more information on Chris Ferguson, game theory and the World Series of Poker I recommend you check out Positively Fifth Street by James McManus.

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