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.

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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.

The rational defendant

Most of us are familiar with the Prisoner's Dilemma - a basic exercise in game theory. Two people are arrested for the same crime. There is no physical evidence and no other witnesses to the crime. The two are then placed in separate cells.

Both prisoners are given a choice. If they keep their mouths shut they can't be convicted as there would be no evidence. If one prisoner confesses he will go free while the other prisoner will receive the maximum sentence. If they both confess there will be enough evidence to convict both but the prosecutor will reduce the sentences.

	Prisoner A	Prisoner B
Prisoner A	R, R	B, T
Prisoner B	T, B	P, P

Where T>R>P>B


R represents the reward both prisoners receive if they both remain silent. P is the punishment both prisoners receive if they both confess. T is the temptation to remain silent, while B is the benefit of cooperating. 


While it is in both prisoner's self-interest to remain silent, they each run the risk that the other prisoner will confess. The choice then becomes one of maximizing one's self-interest versus minimizing the consequences. The rational choice for each prisoner is to cooperate. Economists use the term "satisficing" when a rational actor chooses the minimize his consequences rather than maximize his self-interest.

In the blog, Freakonomics, Stephen Dubner refers to a rather crude version of the Prisoner's Dilemma that one reader's children play.

My wife came up with a punishment method for my kids that I thought that you (and perhaps your blog readers) would find interesting.

When the kids get to tussling and or screaming at each other in such a way that she is finding aggravating, she will send them to their respective rooms with the stipulation that they can come out when they both agree to apologize to each other.

Game theory, I suppose, would argue that they should immediately apologize to one another to minimize the period of detention. What seems to happen, though is that one will think that the other deserves some extended detention and will give up freedom himself in order to see that the other gets it.

In both "games," the players must not only determine what is in their best interest but also what the other player is likely to do. In the first example, should one prisoner remain silent while the other talks, he will receive the maximum sentence but if he confesses, the worst he will receive is a shortened sentence while he could walk out if the other prisoner keeps silent.

In the second example it is interesting to note that one child will allow himself to be punished longer just so that his sibling gets the same punishment. It would appear that the fact his sibling is being punished is enough of a benefit to sit in silence.

The same calculus is used in the courtroom on a daily basis. For instance, a client is charged with a 2nd DWI. To complicate matters, there is a very high blood test result, but there is also a question of whether the client was actually driving the car in question. While there is circumstantial evidence that he was, there is no direct evidence.

As the defense attorney is unable to convince the prosecutor to dismiss the case, and as the prosecutor is unable to convince the defense attorney to plead his client, the matter is set for trial. On the morning of trial the prosecutor offers to dismiss the DWI in exchange for a plea to obstruction of a highway. What is the rational decision?

	Accept	Reject
Client	D, C	R, R
Prosecutor	C, D	P, P

Where D > R > C > P


R represents the reward the client receives if he is acquitted at trial. P represents the punishment the client receives if he is convicted of DWI. C represents the conviction for the reduced charge and D represents the dismissal of the DWI.

If the client accepts the deal he has a conviction on his record, but not for DWI. If the client rejects the deal the case goes to trial where the client could be acquitted or convicted. The best possible outcome for the client is to reject the state's offer, go to trial and get a not guilty verdict. But, if he chooses that path he could also get hit with a second DWI conviction (not a good thing since another DWI arrest would result in a felony charge).

The question comes down to how a rational person would evaluate the trade-off of a conviction for the reduced charge versus the weakness of the state's proof of operating. What is the risk of going to trial and being convicted of DWI? Is the loss associated with that outcome greater than the loss associated with pleading to the reduced charge?

In order to answer that question, you must assign a value to the benefit of acquittal and a negative value for the punishment of a conviction on the DWI. We would then have to assign a value to the dismissal of the DWI as well as to the conviction of a reduced offense. After assigning values we must then assign a percentage to represent the chances of an acquittal versus a conviction.

If we were to assign a value of 10 for the benefit of an acquittal, a -10 for the cost of a conviction, a 5 for the benefit of a dismissal and a -5 for the cost of a conviction for the reduced charge, and a 50% chance of conviction at trial, we come up with the following:

	Accept	Reject
Client	(5 - 5)	.5 (10)
Prosecutor	(-5 + 5)	.5 (-10)

As can be seen, if the client accepts the offer, the benefit of dismissal and cost of conviction cancel each other out while there is a greater swing in values if he declines the offer. The rational client, therefore, would be best served by accepting the prosecutor's offer. Of course as the chance of acquittal rises, it will, at some point, tip the scale in favor of rejecting the offer and trying the case.