In Defense of Plea Bargaining

A legal fantasy

In theory, criminal conviction in the US legal system is by the unanimous vote of a jury. In practice, the overwhelming majority of felony convictions are due to plea bargaining, the defendant pleading guilty in exchange for reduced charges or an agreement by the prosecutor to ask for a lower sentence. I have criticized the system in the past, mostly on the grounds that a prosecutor can make it in the interest of an innocent defendant to plead guilty by charging him with additional offenses, not because the prosecutor believes he is guilty of them and can be convicted but to persuade him to plead guilty of the lesser offense whether or not he committed it.

It recently occurred to me that, while there are serious problems with plea bargaining as it now exists, there could be uses for it. I start with a toy model.

The prosecutor has charged a hundred defendants. He correctly believes that ninety are guilty, does not know which. A trial will tell him whether a defendant is guilty with certainty — this is a toy model — but trials are expensive.

The penalty for conviction is a hundred units — dollars, weeks in prison, lashes. The prosecutor offers each defendant a penalty of ninety units if he pleads guilty. The defendants know whether they are guilty, know that if they are guilty and go to trial they will be convicted (toy model), so all the guilty defendants accept the offer. The remaining defendants go to trial and are acquitted. That is the same result as if all hundred defendants went to trial at a tenth the cost. The system is leveraging the private information of the defendants, whether they are guilty or innocent, making it in their interest to reveal it.

The logic here is similar to that of Peter Leeson’s ingenious explanation and defense of the use of ordeals in medieval trials. A defendant could undergo an ordeal, such as being dumped, bound, into a pond. If he floated he was guilty, if he sank he was pulled out and found innocent. It was a way in which God, who knew whether he was guilty, could tell the court.

Peter’s explanation depends on defendants believing in God’s role in the process. An innocent defendant would choose to undergo an ordeal to demonstrate his innocence; a guilty defendant would structure his defense to avoid one. The priests, knowing that most of those who chose ordeals were innocent, rigged the ordeals to acquit most defendants. In support of that interpretation, Leeson offers evidence of an implausibly large number of accused criminals who succeeded in picking up purportedly red-hot iron or plunging their hands into boiling water without injury. He also points to a collection of cases where the ordeal of submersion was given only to men — because, he argued, women were more likely to float and so be convicted.

To return to my model and make it a little more realistic, assume that the trial makes mistakes in one case out of ten. An innocent defendant has a .1 chance of being convicted, a guilty of being acquitted. We again start with a hundred defendants, ten of them innocent. A guilty defendant who goes to trial has a .9 probability if conviction so an expected punishment of 90, an innocent defendant has a .1 probability of conviction, an expected punishment of 10. Any offer between the two, say 80, will be accepted by the guilty, rejected by the innocent. Ten innocent defendants go to trial, nine are acquitted, one convicted.

The outcome this time is an improvement in both cost and accuracy of over trying all of the cases: The same number of innocents are convicted but all the guilty defendants end up punished instead of nine getting off.

Punished by how much? In my initial model, a guilty defendant who went to trial got a punishment of 100, one who accepted the prosecutor’s offer a punishment of 90. This time a guilty defendant who went to trial and was convicted gets 100, one who accepts the offer gets 80. It looks as though the introduction of plea bargaining has reduced the severity of punishment.

That is true in my examples but only because I failed to adjust the punishment if convicted to allow for the effect of introducing plea bargaining to the system.

The objective is not the largest possible punishment — we don’t want to hang someone for driving a little over the speed limit. The objective is the right amount of punishment. I spend part of Chapter 15¹ of my Law’s Order on what that should be, but since that is not the subject of this post I will simply assume that the expected punishment you want to impose for this crime is 100. To do that in my first model you set the punishment if you plead guilty at 100, the punishment if convicted at trial, which, in that model, no one is — guilty defendants all plead guilty and innocent defendants are all acquitted — at something more than that, say 110. In the second model again set the punishment if you plead guilty at 100, the punishment if convicted at trial at something more than 111.²

In the first model it did not matter how much above 100 you set the punishment if convicted at trial since nobody ever received it. In the second model innocents sometimes are convicted, so you want it as low as possible, just barely above 111.

Back To the Real World

Plea bargaining in our legal system has the potential to provide the same benefits, reduced cost of trial and increased accuracy of results, by leveraging the private information of the defendant. It does produce the former, eliminates something like 95% of trials. It probably does not increase the accuracy of the system, the extent to which the guilty are punished and the innocent not punished, may well decrease it.

Ideally, the bargain offered should to be one that will be accepted by the guilty, refused by the innocent, with the result that the innocent go to trial and are usually acquitted. More realistically, in a world where defendants, innocent or guilty, have imperfect knowledge about the likely outcome of a trial and prosecutors have imperfect information about what bargain a defendant will accept, the bargain should be one that will be selectively accepted by the guilty, rejected by the innocent.

That is not the bargain it is in the interest of a prosecutor in our present system to offer. His objectives are to get as many defendants as possible to plead guilty, ideally on the least favorable terms each will accept, and to convict those who do not; a high conviction rate will both make him look good and make future defendants more willing to accept proposed deals. Offering a bargain that an innocent defendant will reject serves neither objective: It reduces the number of defendants who plead guilty and, by funneling innocent defendants to trial, increases the fraction of defendants acquitted.

A better strategy from the prosecutor’s standpoint is to offer each defendant a bargain he will accept, a better bargain to those he thinks more likely to be acquitted at trial. He can increase the probability that his offer will be accepted by stacking charges for the same offense, including ones that lead to severe punishment, even if he thinks them unlikely to succeed.

From the standpoint of the criminal justice system a further problem is that the optimal system results in most innocent defendants rejecting the offered deal and going to trial; there may be more of them than the system is equipped to handle.

To make plea bargaining work in our system we would have to change the incentives of prosecutors, make it in their interest to try to offer each defendant a bargain that he will find it in his interest to accept if guilty, reject if innocent. We could do that by treating a high acquittal rate as evidence that the official offering plea bargains was doing a good job and rewarding him accordingly. Since the result of doing that will be a high acquittal rate, which in an adversary system will be seen as evidence that the attorney prosecuting the case is doing his job badly, we would have to give the jobs of offering bargains and prosecuting cases to different officials neither of whom has power over the other.

Since the people making offers would no longer be trying to design ones that all defendants would accept there would be more trials, many of innocent defendants, some of guilty defendants more optimistic about their chances at trial than the official making the offer. We could reduce that problem by being more careful not to charge people with crimes they did not commit but would also, almost certainly, have to increase the number of cases tried, either by expanding the system, shortening trials or both.

Or, of course, we could simplify our legal system to make fewer things illegal.

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1. Following the subhead “The Theory if Optimal Punishment.”

2. Defendants are assumed risk neutral, so a .9 chance of 111 is equivalent to a certainty of 100, so anything higher makes it in the interest of guilty defendants to accept the offered penalty instead.