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.
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