Plea-Bargaining and Credible Signalling

OverviewThis class looks at the role of economic analysis in understanding pre-trial bargaining. This is a topic that has been explored to some extent in earlier seminars, but we do so in a more direct and interactive way in this seminar by actually playing a number of games that are broadly analogous to the pre-trial bargaining process.

We adopt the model of pre-trial bargaining originally proposed by Mnookin and Kornhauser (“Bargaining in the Shadow of the Law”). As we all know, trials are costly and somewhat unpredictable. If you can offer your opponent a pre-trial settlement that is less expensive than their expected loss at trial they should, if they follow the prescriptions of expected utility theory, accept.

Crucial to this process is the role of signals. Signals are what inform the litigants about their expected losses at trial. If I can convince my opponent that I am more likely to win at trial than they are, they should be more willing to accept my proposed settlement offer. But how can I do this? What strategies can I employ to make my signals more believable?

We explore these issues by looking, in particular, at pre-trial bargaining in criminal litigation.

1. The DuelWe start by looking at a simple, classic game: the duel. The question here is “Who should fire first?” Although not directly analogous to the litigation process, it lays the foundation for understanding games that are.

2. Symmetrical Duel with Exit OptionsWe build our first model game which resembles pre-trial litigation. It is like the classic duel, only that players advance toward a firing line (“trial”) and have certain exit options (“offers of settlement”).

4. Plea-Bargaining and InnocenceWe look at the plea-bargaining process in more detail and consider the suggestion that even innocent defendants should, if they are rational, plead guilty to some offence.

3. Asymmetrical Duel with Exit OptionOur second model of the litigation process is much like the first, only this time most of risk rests on one side. It is argued that this is analogous to the plea-bargaining process.

5. Mechanisms for SignallingThe innocence problem is a type of signalling problem. We look at signalling problems more generally and consider the importance of credible signalling mechanisms.

1. The Classic Duel - When should the first shot be fired?

Player A

Has two options: (i) take a shot; or (ii) take a step forward?

Player B

Has two options: (i) take a shot; or (ii) take a step forward?

Step 1a Step 1bStep 2a Step 2bStep 3a Step 3bStep 4a Step 4b

Analysis

This is a classic and well-studied game. It has a simple structure. The players take alternating steps toward each other, deciding at each step whether to fire or not. We assume that the probability of hitting the target goes up as the players get closer to one another (though we need not assume that each player is an equally good shot).

The question is when should the first shot be fired. This all depends on the probabilities of hitting the target. If both sides know these probabilities there is a straightforward mathematical argument for determining when the first shot should be fired. Answer: It should be fired whenever the probability of the player whose turn it is hitting the target is equal to his/her opponent’s probability of missing the target at the next step.

There is a formal proof of this, but we won’t go into it in this module. An interesting question is what happens when you don’t know your opponent’s probability of hitting the target? What should you do then?

Analogy to law?

This game isn’t directly analogous to the litigation process. Though it does provide a foundation upon which we can build a more realistic representation of the litigation process (next two pages). Hence why we are discussing it.

There is, however, something litigious about it. Litigation does involve two (or more) sides trading blows. The difference is that whereas a duel is all about who makes the first move; litigation is more about who can stay going for the longest. This is sometimes referred to as a war of attrition, which is the inverse of a duel.

The duel itself is a good model of product wars between companies, where the prize goes to the company that first releases a viable product onto market. For example, Apple was the first company to release a viable touchscreen smartphone, and profited accordingly. The viability of the product can be likened to the probability of hitting the target in the game.

Player A

Has two options: (i) take step forward; or (ii) take the exit option

Firing Line Firing Line

Player B

Has two options: (i) take step forward; or (ii) take the exit option

If the player’s reach the firing line, they each get to throw three sponges at each other. The player who lands the most sponges wins

Exit 1Get hit by one sponge

Exit 2Get hit by two sponges

Exit 1Get hit by one sponge

Exit 2Get hit by two sponges

Analysis

This game is a variation on the classic duel. We will play it in class using sponges rather than bullets. Players advance toward a firing line. At each step they have an option to escape the game by paying a certain penalty (being hit by a prescribed number of sponges). When they reach the firing line, they each get a chance to shoot at each other.

The key question in this game is not when should I fire, but, rather, when should I drop out of the game. Again, the answer depends entirely on the probabilities of success once you reach the firing line.

Analogy to law?

This game is much more like the litigation process, albeit litigation of a certain type. Since the game is symmetrical (i.e. both sides have essentially the same options, rewards and costs) it is most like a suit-countersuit in civil litigation. For example, imagine Apple and Samsung suing one another for violation of patents.

Here, the firing line is like the trial; the sponges being thrown are like the legal arguments in the trial. The exit options are akin to offers (and counteroffers) for settlement prior to trial.

2. Symmetrical Duel with Exit Options - Countersuits

Player A

Has two options: (i) take step forward; or (ii) take the exit option

Firing Line Firing Line

Player B

Has two options: (i) take step forward; or (ii) leave the game (without getting hit)

If the players reach the firing line, then B gets to throw three sponges at B; A simply has to stand there and hope that B misses

Exit 1Get hit by one sponge

Exit 2Get hit by two sponges

Leave the game

Leave the game

Analysis

This game is much like the previous one, only this time it is weighed heavily against one side. Player B is free to leave the game at any time prior to the firing line without paying any major costs. Furthermore, it is only player B who gets to fire once the firing line is reached. A’s decisions are limited. He/she must decide whether to ‘tough it out’ to the firing line, or escape with a penalty of some sort.

Whether A should do so or not depends on what his/her’s expected penalty is one he/she reaches the firing line, which in turn depends on how good a shot B is.

Analogy to law?

This game is very like what obtains in the criminal justice system. Player A is like the defendant; Player B is like the prosecution. The prosecution threatens trial for a major charge against A (e.g. murder). They then offer A certain exit options (lesser charges) prior to trial.

A has to decide whether to accept liability for one of the lesser charges, or to tough it out until trial. A should tough it out if B’s probability of success at trial is such that A’s expected punishment from trial is less than either of the two exit options.

3. Asymmetrical Duel with Exit Options - Plea-Bargaining

4. Plea Bargaining and the Innocence Problem

Plea bargaining is the common practice whereby someone who is being charged with a criminal offence will agree to plead guilty to a lesser offence in order to avoid the risk of being tried for the more serious offence.

Plea bargaining is often rational from both the prosecutor's and the defendant's perspective. The prosecutor wants to maximise the amount of punishment per unit of prosecutorial resources; trials are costly (the resources spent in one trial could be used to secure other convictions); and their results are unpredictable. Accepting a lesser plea helps to maximise convictions and minimise costs. Contrariwise, from the defendant's perspective, the goal is to minimise the amount of punishment, but, again, trials are unpredictable and pleading to a lesser offence is often the best way to minimise expected punishment.

The plea-bargaining process follows very much the structure of the duelling game outlined on the previous page.

The diagram opposite simplifies the structure of the decision-making problem from the defendant’s perspective. The defendant can choose to go to trial or to accept a plea deal. If he/she goes to trial there are two outcomes: (i) they are found guilty (cost C1) or (ii) they are found innocent (cost 0 - in reality, of course, going to trial might have some cost). If they accept the plea deal they incur another cost (C2, where C1 > C2).

As long as C2 is less than p(C1) (i.e. the expected cost of trial is greater than the cost of the plea deal) the defendant should accept the deal. (Arguably he/she should accept the deal if the values are equal too?).

What is plea-bargaining?

N

P

Go to trial

Accept plea deal

Found GuiltyP

Not Guilty1 - p

C1

0

C2

Should the innocent plead guilty? Innocence as a Signalling Problem

One common argument is that pleading guilty to a lesser offence is a rational choice even when you are not guilty. This is called the “innocence” problem. The argument rests on a couple of assumptions. First, we must assume that there is some non-trivial risk of an innocent defendant being found guilty at trial (i.e. trials are not perfect at determining guilt/innocence). This is a plausible assumption. Second we must assume that the prosecutor has the ability to offer a wide range of potential alternative forms offence and punishment (i.e. varying sentence length, amounts of fines, community service orders or whatever). This might be less plausible. Mandatory sentencing laws often interfere with what prosecutors can offer during plea-bargaining (and this, in turn, is sometimes used to argue against mandatory sentencing laws).

Granting those two assumptions, the argument is as follows. If there is always some non-trivial risk of being found guilty at trial (even when innocent) then the expected punishment from trial will always be greater than zero. Consequently, there must always be some amount of punishment (however small) that you could be offered that would make pleading guilty to some lesser offence more attractive than proceeding to trial. The trial, after all, is risky and could be even more costly.

The innocence problem largely stems from a signalling problem. We will discuss signals and their role in bargaining and decision-making in more detail on the following page. For the time being, we can focus on some general concepts.

One thing makes a plea deal more attractive is when prosecutors can send signals to the defendant suggesting that their risk of being found guilty at trial is high. This comes in the shape of witnesses and forensic evidence. Conversely, one thing a defendant can do to make the prosecutors drop the charges is to send a credible signal of innocence. This can come in the shape of a solid alibi or evidence that refutes the prosecutor’s case.

One thing that can hamper the defendant is when they have purely private knowledge of innocence (i.e. knowledge that is not accessible to anyone else). For example, I know that I was at home alone on the night of incident in question, but no one can confirm this story. The problem that arises then is that no signal of innocence will seem credible. Merely insisting on one’s innocence is no good since even guilty defendants will be inclined to do that.

So what can be done?

5. Mechanisms for SignallingThe Importance of Credible Signalling

The law, like many other social institutions, is a signalling system. A signalling system can be defined as any system with the following three elements:

Message Set: A set of possible signals that could be sent through the system, e.g. “I am innocent” and “I am guilty”. Typically, these signals are propositions about an actual or likely state of the world or state of mind.

Sender: An agent who selects one of the possible signals from the message set and conveys it to another agent (the receiver).

Receiver: An agent who must interpret the signals from the sender and assign some truth value to them (e.g. "P is innocent").

In law, the sender and receiver roles are played by a variety of people, most commonly lawyers and witnesses (senders) and judges and juries (receivers). People can also change back and forth between these roles. In order for receivers to do their jobs properly, credible signals must be sent through system, i.e. signals that reliably indicate whether a proposition is true or false.

Credible signals are important in many walks of life. When you purchase a product with a warranty, the warranty is essentially a signal that is sent by the manufacturer to you to encourage you to buy the product. How does it work?

The warranty is a costly and hard to fake signal. If the product fails, the manufacturer would end up losing money by fulfilling their promise under the warranty. This is what makes the warranty a good signal: the willingness of the manufacturer to pay up in the event of a faulty product indicates that the product must be of reasonably high quality. Otherwise, why would the manufacturer risk losing money? Other examples of costly signals abound. For example, criminal gangs often get their members to tattoo themselves in particularly awkward or conspicuous ways (e.g. on the eyelids) because this is a costly signal of commitment to the criminal gang.

Other signals are cheap and easy to fake. Consider a typical example from job interviews. Interviewers often ask people whether or not they are interested in working for the particular company. Most people say “yes”, and many interviewers are encouraged by this. But why? It is easy to say that you are interested, even when you are not. Indeed, it is easy to have a long speech about how interested you are, which you could deliver while dancing on the interview table. This is certainly more costly than merely saying “yes” (think of the embarrassment factor). But it is still relatively easy to fake. Educational achievement is a more costly, difficult to fake, signal.

The upshot is this, no matter who you are in the legal system, you have to try to send credible signals to other people. How are you going to do this?

How to Signal CrediblyLet’s bring this back to pre-trial bargaining in general and plea-bargaining in particular. We now know that one of the best ways to get the other side to avoid the trial is to send credible signals to the other side about their (im)probability of success at trial. Thus, if I’m the defendant, I need to convince the prosecution (plaintiff) that I’m going to win; if I’m the prosecution (plaintiff), I need to convince the defendant that they are going to lose.

How might this work in the context of the duelling games covered earlier? Well, you might imagine each player engaging in a shooting display at each step, trying to convince the other player that they are the better shot and hence it would be wise to step out of the game earlier. Or, if many rounds of the game are played, a player could develop a reputation for success that would scare the other player out of the game. Another possibility, would be for one player to appear reckless or unhinged. This could convince the other side that he/she doesn’t value life in the same way and so would be a dangerous opponent. All of these are merely suggestions. You could probably think of other signalling strategies in this game.

How would this work in a legal context? Well, there a several strategies depending on the nature of the trial. In the suit-countersuit case, where two large companies are suing one another, a common strategy is simply to throw lots of money at the case: hire more lawyers, file more motions and so on. This might convince the other side to drop out, particularly if their claim seems weak. It is also possible to develop a reputation for success from previous trials and bargains. Indeed, this is arguably what most lawyers spend their careers doing (relatively few cases are actually litigated, most are settled out of court.

How about in plea-bargaining in the criminal justice system? How can we resolve the innocence problem? As we saw on the previous page, one of issues here is that protestations of innocence are often cheap and easy to fake. This makes it difficult for the defendant with private knowledge of innocence to convince the prosecution to drop the case, or to offer them a more lenient deal.

One suggested solution to this, from the work of Covey, is to introduce a “subwager” into the pre-trial bargaining game between prosecutor and would-be defendant. The subwager is like a gamble that the defendant can choose to take, but which would only really make sense to take if the defendant is in fact innocent. He gives the example of voluntary optional interrogation as a subwager. His reasoning being that it is far more difficult to “survive” robust interrogation and maintain innocence if you are actually guilty, than if you are actually innocent.

You can think of the decision to submit yourself to interrogation as being akin to a really high stakes bet in a game like poker: a guilty defendant could try to bluff, if they like, but they would risk a lot if they get caught. Better then to remain silent. As Covey notes, his idea only really works in systems where the right to silence is protected.