Optimal Behavior

We solve by backwards induction, beginning with the citizen’s decision to take a plea.

Plea Decision

Suppose the citizen is guilty. If caught, she accepts a plea bargain if

$$ \begin{array}{rll}u({y}_0+b-{x}_P;\alpha )\ge {\pi}_1u({y}_0+b-{x}_K;\alpha )+(1-{\pi}_1)u({y}_0+b;\alpha ).& & \end{array} $$

To see how risk aversion affects this choice, define the certainty equivalent level of consumption, c, associated with going to trial. Formally, this is the c that solves

(1) $$ \begin{array}{rl}u(c;\alpha )={\pi}_1u({y}_0+b-{x}_K;\alpha )+(1-{\pi}_1)u({y}_0+b;\alpha ),& \end{array} $$

i.e., the consumption level at which a guilty citizen with risk aversion $ \alpha $ is indifferent between obtaining c for sure, and facing the lottery of a trial. Write the solution to this equation as $ {c}_1(\alpha ) $ .Footnote ⁴ We can then rewrite the decision to accept a plea as $ {y}_0+b-{x}_P\ge {c}_1(\alpha ) $ or

(2) $$ \begin{array}{rl}{x}_P\le {y}_0+b-{c}_1(\alpha )\equiv {\overline{x}}_{P,1}.& \end{array} $$

Unsurprisingly, the guilty citizen accepts a deal only if it involves a sufficiently small punishment. Central for our purposes is how risk aversion affects the maximal accepted plea deal. By a standard result (e.g., Mas-Colell, Whinston, and Green Reference Mas-Colell, Whinston and Green1995, Proposition 6.C.2), the certainty equivalent $ {c}_1(\alpha ) $ is decreasing in $ \alpha $ . Since $ {c}_1(\alpha ) $ enters negatively into the right-hand side of Inequality 2, the greater the citizen’s risk aversion $ \alpha $ , the more punitive the plea deal at which she is indifferent between accepting the plea and going to trial. Put differently, harsher plea deals are accepted as risk aversion increases.

A similar analysis reveals that an innocent citizen accepts plea bargain $ {x}_P $ instead of going to trial if

(3) $$ \begin{array}{rl}u({y}_0-{x}_P;\alpha )\ge {\pi}_0u({y}_0-{x}_K;\alpha )+(1-{\pi}_0)u({y}_0;\alpha ).& \end{array} $$

Writing the certainty equivalent for the lottery of going to trial when innocent as $ {c}_0(\alpha ) $ , the innocent citizen accepts a plea deal if

(4) $$ \begin{array}{rl}{x}_P\le {y}_0-{c}_0(\alpha )\equiv {\overline{x}}_{P,0}.& \end{array} $$

There are two differences which determine whether the innocent or guilty are more apt to accept plea bargains, keeping risk aversion fixed. First, the probability of being convicted is at least weakly lower for an innocent citizen ( $ {\pi}_0\le {\pi}_1 $ ). Second, those who are guilty acquired an extra benefit, b, from crime, which increases their utility from any outcome and could potentially affect their tolerance for risk. This yields the following result.

Proposition 1 (Risk Aversion and Guilt Increase Plea Acceptance).

1. (i) Citizens accept a wider range of plea bargains when they have higher risk aversion ( $ \frac{\partial {\overline{x}}_{P,G}}{\partial \alpha }>0 $ ) or are more likely to be convicted ( $ \frac{\partial {\overline{x}}_{P,G}}{\partial {\pi}_G}>0 $ ).

2. (ii) If u has constant or increasing absolute risk aversion in y and $ {\pi}_0<{\pi}_1 $ , then the guilty always accept a wider range of plea deals.

3. (iii) For any u, if $ {\pi}_0 $ is sufficiently small ( $ {\pi}_1 $ sufficiently large) the guilty always accept a wider range of plea deals.

Proof . All proofs are in the Supplementary Material.

Figure 1 provides a graphical illustration of Proposition 1. The left panel illustrates how the certainty equivalent of trial changes under low risk aversion (gray line) and high risk aversion (black line) for fixed guilt status G. Consider a lottery which results in consumption 0 or 1; for comparison both utility functions are equal to 0 and 1 at these points, respectively. In particular, suppose that the citizen obtains 1 if she wins at trial and 0 if she loses, and that she wins with probability $ {\pi}_G $ . So, for either utility function, the expected utility from a trial is $ {u}_T={\pi}_G $ . The black curve is a utility function with high risk aversion, and the point $ c({\alpha}_h) $ is the certainty equivalent to the lottery of trial for a citizen with this utility function. The gray curve is a utility function with low risk aversion, and certainty equivalent $ c({\alpha}_l) $ . The black curve crosses the dotted horizontal line at a lower level of consumption, which means the certainty equivalent of trial is lower for the citizen with high risk aversion. Combined with Inequalities 2 and 4, this means that the threshold plea the citizen would accept rather than go to trial (keeping guilt status fixed) is increasing in her risk aversion $ \alpha $ .

Figure 1.

Illustration of Risk Aversion and Plea Acceptance with Constant Absolute Risk Aversion: $ u(y,\alpha )={\alpha}^{-1}(1-{e}^{-y\alpha }) $

Note: Left panel: Certainty equivalent with positive risk aversion ( $ \alpha =2 $ ; black) and risk acceptance ( $ \alpha =-2 $ ; gray). Right panel: Threshold in risk aversion to go to trial as a function of $ {x}_P $ with $ {y}_0=1 $ , $ b=0.1 $ , $ {x}_K=1 $ , for innocent individuals (gray, with $ {\pi}_0=0.4 $ ) and guilty individuals (black, with $ {\pi}_1=0.55 $ ).

The right panel illustrates how guilt status affects the critical value of risk aversion at which the citizen is indifferent between the lottery of trial and plea deal $ {x}_P $ . Higher values of $ \alpha $ indicate higher risk aversion, so a citizen offered plea deal $ {x}_P $ accepts if her risk aversion is above the relevant curve. First, notice that since the threshold for the innocent is always higher in this figure, for fixed risk aversion and a fixed plea offer, innocent citizens are always more likely to go to trial. To illustrate further, recall that in this example, the citizen anticipates payoff $ {\pi}_G $ from trial. At $ {x}_P={\pi}_0 $ , an innocent and risk-neutral ( $ \alpha =0 $ ) citizen is indifferent between pleading guilty and trial (and all risk-averse innocent citizens prefer to plead guilty)—but a risk-neutral, guilty citizen would strictly prefer to plead guilty. At $ {x}_P={\pi}_1>{\pi}_0 $ , a guilty and risk-neutral citizen is indifferent between pleading guilty and trial, while her innocent counterpart strictly prefers to go to trial.

So far, we have shown that for fixed risk aversion and a fixed plea offer, the guilty are generally more likely to plead guilty. This is consistent with the standard view. However, we have also demonstrated that for fixed guilt status, higher risk aversion increases the likelihood of accepting a plea offer. We now consider how a citizen’s risk aversion might affect her willingness to commit crimes.

Crime Decision

It is optimal for a citizen to commit a crime if

$$ \begin{array}{rll}{p}_1{\hat{U}}_1(\alpha )+(1-{p}_1)u({y}_0+b;\alpha )\ge {p}_0{\hat{U}}_0(\alpha )+(1-{p}_0)u({y}_0;\alpha ),& & \end{array} $$

where $ {\hat{U}}_G(\alpha ) $ is the (expected) utility associated with being charged with a crime with guilt status G and risk acceptance $ \alpha $ , given the plea decisions derived above.

The effect of risk aversion on this choice is complicated by the fact that the citizen’s expected utility at this stage is always a lottery, regardless of her crime choice, since she may be arrested even if she does not commit a crime. In the extreme, suppose the citizen is arrested with near certainty if she does commit a crime and has an intermediate chance of arrest if she does not commit a crime. If so, the decision not to commit a crime is “riskier” in a technical sense: the commission of a crime results in certain capture (and potentially a certain plea deal) while abstaining leads to uncertainty.

We focus on the case where the probability of an innocent person being charged with a crime in any given time period is fairly small. (There may still be a nontrivial share of innocent individuals who are charged, as it is natural to assume that the number of citizens choosing to commit a crime in any given time period is also very small.Footnote ⁵) As the probability of arrest when innocent, $ {p}_0 $ , approaches $ 0 $ , the citizen commits a crime if

$$ \begin{array}{rll}u({y}_0;\alpha )\le {p}_1{\hat{U}}_1(\alpha )+(1-{p}_1)u({y}_0+b;\alpha ).& & \end{array} $$

If the citizen (having committed a crime) would accept a plea bargain if caught, we can rewrite this inequality as

$$ \begin{array}{rll}u({y}_0;\alpha )\le {p}_1u({y}_0+b-{x}_P,\alpha )+(1-{p}_1)u({y}_0+b;\alpha ).& & \end{array} $$

If the citizen would instead prefer to go to trial if caught, the inequality becomes

$$ \begin{array}{rll}u({y}_0;\alpha )\le {p}_1{\pi}_1u({y}_0+b-{x}_K,\alpha )+(1-{p}_1{\pi}_1)u({y}_0+b;\alpha ).& & \end{array} $$

Let $ {C}_P(b;\alpha ) $ be the certainty equivalent of the lottery of committing a crime with benefit b when pleading guilty if caught, and let $ {C}_T(b,\alpha ) $ be the certainty equivalent of going to trial if caught. Notice that both are strictly increasing in b and strictly decreasing in $ \alpha $ . Combining, we can define the certainty equivalent of committing a crime and making the optimal plea choice as

$$ \begin{array}{rll}C(b,\alpha )=\max \{{C}_P(b;\alpha ),{C}_T(b;\alpha )\}.& & \end{array} $$

Since both $ {C}_P $ and $ {C}_T $ are strictly increasing in b and strictly decreasing in $ \alpha $ , C has the same properties. This yields the following result.

Proposition 2 (Crime Benefit Cutpoint).

As $ {p}_0\to 0 $ , for any risk aversion $ \alpha $ , there exists a critical $ \widehat{b}(\alpha ) $ such that an individual commits a crime if and only if $ b>\widehat{b}(\alpha ) $ , with $ \widehat{b} $ strictly increasing in $ \alpha $ .

The key takeaway from this result is the greater the citizen’s risk aversion, the larger the benefit must be to induce her to engage in crime. By continuity, this result holds as long as the chance of being charged when innocent is sufficiently low. This implies that, consistent with research in psychology, criminology, and law and economics, individuals who engage in crime are, on average, more risk-acceptant than those who do not.

Who Pleads Guilty?

Having solved the citizen’s sequential decision problem, we now consider what her derived optimal choices imply for the efficacy of plea bargaining as a sorting mechanism. Recall that the standard view is that the guilty are more likely to accept a plea deal than the innocent. So far, we have shown that, for fixed risk aversion, this is indeed typically the case. However, we have also shown that innocent citizens are on average more risk-averse than the guilty, and that the more risk-averse a citizen, the more likely she is to accept a plea deal. As a result, depending on the relative strength of these different effects, the innocent may be more likely to plead guilty.

To formalize this possibility, assume that the risk aversion parameter $ \alpha $ and the benefit from committing a crime b are independent random variables drawn from continuous distributions. Call the marginal cumulative density functions $ {F}_{\alpha } $ and $ {F}_b $ .

The three outcomes we study are then random variables at the outset of the decision problem: the decision to commit a crime ( $ G\in \{0,1\} $ ), whether a citizen is arrested and charged (call this $ A\in \{0,1\} $ ), and whether a citizen who has been charged accepts a plea ( $ P\in \{0,1\} $ ). In terms of these random variables, we are interested in whether it can be the case that

(5) $$ \begin{array}{rl}Pr(P=1|G=0,A=1)& >Pr(P=1|G=1,A=1),\end{array} $$

i.e., that conditional on being charged, the probability of accepting a plea is higher for the innocent.

From the analysis above, we can represent the probability of pleading guilty conditional on being guilty and being charged as

$$ \begin{array}{l}Pr(P=1|G=1,A=1)\\ {}\hskip2.0em =\frac{Pr(\alpha >{\widehat{\alpha}}_1(b),b>\widehat{b}(\alpha )){p}_1}{Pr(\alpha >{\widehat{\alpha}}_1(b),b>\widehat{b}(\alpha )){p}_1+Pr(\alpha <{\widehat{\alpha}}_1(b),b>\widehat{b}(\alpha )){p}_1}& & \\ {}\hskip2.0em =\frac{Pr(\alpha >{\widehat{\alpha}}_1(b),b>\widehat{b}(\alpha ))}{Pr(\alpha >{\widehat{\alpha}}_1(b),b>\widehat{b}(\alpha ))+Pr(\alpha <{\widehat{\alpha}}_1(b),b>\widehat{b}(\alpha ))},& & \end{array} $$

where $ {\widehat{\alpha}}_1(b) $ is the critical value of risk aversion that makes an individual who received benefit b from crime indifferent between accepting a plea and not. Similarly, we can write the probability of pleading guilty conditional on being innocent and charged as

$$ \begin{array}{l}Pr(P=1|G=0,A=1)\\ {}\hskip3.5em =\frac{Pr(\alpha >{\widehat{\alpha}}_0,b<\widehat{b}(\alpha ))}{Pr(\alpha >{\widehat{\alpha}}_0,b<\widehat{b}(\alpha ))+Pr(\alpha <{\widehat{\alpha}}_0,b<\widehat{b}(\alpha ))},& & \end{array} $$

where $ {\widehat{\alpha}}_0 $ is the critical value of risk aversion that makes an innocent individual indifferent between pleading guilty and going to trial. Note that $ {\widehat{\alpha}}_0 $ is not a function of the benefit from crime (b) since the innocent do not receive this benefit.

Recall that the willingness of either type of citizen to plead guilty depends in part on her likelihood of conviction at trial, and consider how each type’s critical value of risk aversion changes with this likelihood. First, notice that as the probability of conviction at trial when innocent approaches 0 ( $ {\pi}_0\to 0 $ ), the innocent always go to trial, regardless of risk preferences ( $ {\widehat{\alpha}}_0\to \infty $ ), while as the probability of conviction at trial when guilty approaches 1 ( $ {\pi}_1\to 1 $ ), the guilty always plead guilty, no matter their risk preferences ( $ {\widehat{\alpha}}_1(b)\to -\infty $ ). This implies that when trials are highly accurate, the standard logic is correct, and the guilty are more apt to take pleas.

Now consider the case where the guilty and innocent face the same chance of being convicted at trial, i.e., $ {\pi}_0={\pi}_1 $ . For simplicity, assume constant absolute risk aversion. In this case, the threshold in risk aversion at which a plea is preferred is the same for both types: $ {\alpha}_0={\alpha}_1 $ . However, once we endogenize the crime choice, the innocent are more likely to have a level of risk aversion above this threshold, and hence are more likely to accept a plea. Formally, the CDF of risk aversion conditional on being guilty lies below the CDF conditional on being innocent (when interior): $ {F}_{\alpha }(\alpha |G=1)<{F}_{\alpha }(\alpha |G=0) $ . As a result, in this part of the parameter space, there is perverse sorting where the innocent are more likely to plead guilty.

Our main result in this section is that this perverse sorting can occur as long as the parameters are not “too far” from this special case. This follows immediately from the preceding argument and the fact that the probability of pleading guilty is continuous in the relevant probability of being convicted (since $ \alpha $ is a continuous random variable and the threshold $ {\alpha}_G $ is continuous in $ {\pi}_G $ ).

Figure 2 provides some intuition. The top row presents a partition of the potential realizations of $ (\alpha, b) $ into four regions that determine whether a citizen commits a crime and whether each type of citizen pleads guilty if charged. In the left panel, the probability of conviction is much higher for the guilty than for the innocent; in the right panel, the probabilities are similar across types. In both panels, the thick line represents the precise benefit to crime $ \widehat{b}(\alpha ) $ above which a citizen with risk aversion $ \alpha $ strictly prefers to commit crime ( $ G=1 $ ) and below which she strictly prefers not to ( $ G=0 $ ). This critical benefit level $ \widehat{b}(\alpha ) $ is increasing in her risk aversion, since more risk-averse citizens require a higher benefit from crime to risk a criminal charge. The dashed vertical lines represent the level of risk aversion below which citizens go to trial when guilty ( $ {\widehat{\alpha}}_1(b) $ , above the diagonal) and innocent ( $ {\widehat{\alpha}}_0 $ , below the diagonal), and above which they plead guilty. Notice that in the left panel, where the probability of conviction is much higher for the guilty, a large proportion of the guilty accept plea deals and a large proportion of the innocent go to trial. Visually, in most of the parameter space, the plea choices “match” the guilt status: $ G=P=1 $ or $ G=P=0 $ . In the right panel, however, where the probability of conviction is similar across types, the “mismatched” regions of guilty citizens going to trial and innocent citizens taking pleas become larger, driven by their differences in risk aversion.

Figure 2.

(Top Panels) Regions of Citizen Behavior and (Bottom Panels) Distribution of Risk Aversion among the Guilty (Dark Gray) and Innocent (Light Gray)

Note: Levels of crime benefit and risk aversion (with $ u(y,\alpha )={\alpha}^{-1}(1-{e}^{-y\alpha }) $ ) determine whether a crime is committed and whether a plea would be accepted. In the left panels, there is a large difference between the probability of conviction when guilty ( $ {\pi}_1=0.6 $ ) and when innocent ( $ {\pi}_1=0.2 $ ). In the right panels, there is a small difference ( $ {\pi}_1=0.52 $ , $ {\pi}_0=0.48 $ ).

The panels in the bottom row of Figure 2 show this more precisely, by plotting the distribution of risk aversion conditional on being guilty (dark gray) and innocent (light gray). The shaded areas correspond to those with high enough risk aversion to accept a plea deal. In both panels, the distribution of risk aversion is shifted to the right for the innocent. However, in the left panel, there is again a large difference across types in the probability of conviction. As a result, here the risk aversion threshold for a guilty plea is much higher for the innocent, and so the guilty are more likely to plead guilty. In the right panel, where there is only a small difference in the probability of conviction, the risk aversion thresholds are similar across types, and so the innocent are more likely to plead guilty.

The preceding discussion sets aside the possibility that obtaining the benefit to crime b affects risk aversion by changing the citizen’s baseline wealth. It seems most plausible in our case to suppose that risk aversion does not change too much with b (i.e., that the citizen’s utility function features relatively constant absolute risk aversion). The perverse sorting effect is strengthened if the citizen’s utility function has decreasing absolute risk aversion (i.e., obtaining the benefit from crime makes the citizen even less risk-averse) and weakened if obtaining b makes the citizen relatively more risk-averse.

Combining, we have the following result.

Proposition 3 says that when trials do a fairly poor job at separating the guilty from the innocent, there is perverse sorting where the innocent are more likely to accept plea bargains than the guilty. In extreme cases, this implies that if the trial process is sufficiently noisy, those who are acquitted at trial may be more likely to be guilty than those who accepted plea bargains.

Such perverse sorting may be common in real-world justice systems, because these systems often feature noisy trial processes with non-negligible rates of error, both in convicting the innocent and in acquitting the guilty. In countries where a high burden of proof—such as proof beyond a reasonable doubt—is required for conviction at trial, the probability of conviction when guilty may be bounded well away from $ 1 $ . Similarly, in states where the quality of legal representation varies widely, or adjudicators are prejudiced or do not have the time or ability to carefully examine the evidence,Footnote ⁶ the probability of conviction when innocent may be bounded well away from $ 0 $ . These two features are not mutually exclusive: it is possible for both types of error to coexist in the same system, leading to high rates of both wrongful conviction and wrongful acquittal.

Other Confounders

Risk aversion is not the only individual characteristic that may confound the relationship between guilt and the decision to plead guilty. Various cognitive biases may play a similar role, among them overconfidence—arguably the most widely documented and robust psychological bias (e.g., Moore and Healy Reference Moore and Healy2008). An overconfident citizen could overestimate both the probability that she will get away with committing a crime in the first place and the probability that, if caught, she would be acquitted at trial, either because she cannot accurately judge her own abilities or because she engages in a high level of wishful thinking. If so, overconfidence could similarly generate an outcome where the innocent are more likely to plead guilty than are the guilty.

More concretely, let $ {\overset{\sim }{\pi}}_1 $ represent a guilty citizen’s perceived probability of being convicted at trial, and write the choice to accept a plea when guilty as

$$ \begin{array}{rll}u({y}_0+b-{x}_P;\alpha )\ge {\overset{\sim }{\pi}}_1u({y}_0+b-{x}_K;\alpha )+(1-{\overset{\sim }{\pi}}_1)u({y}_0+b;\alpha ).& & \end{array} $$

The more overconfident the citizen, the lower her $ {\overset{\sim }{\pi}}_1 $ , making her more likely to go to trial. The choice to accept a plea when innocent could be represented in a similar fashion, with $ {\overset{\sim }{\pi}}_0 $ representing the innocent citizen’s perceived probability of conviction at trial.

As before, at the crime decision stage, focus on the case where $ {p}_0 $ is small, i.e., where innocent people are arrested with low probability. Since $ {p}_0 $ is small, we can write the choice to commit a crime as

$$ \begin{array}{rll}u({y}_0;\alpha )\le {\tilde{p}}_1{\hat{U}}_1(\alpha )+(1-{\tilde{p}}_1)u({y}_0+b;\alpha ),& & \end{array} $$

where $ {\tilde{p}}_1 $ is the perceived probability of being caught when committing a crime. Since the more overconfident the citizen, the lower she perceives $ {\tilde{p}}_1 $ to be, the more overconfident she is the more likely she is to commit a crime.

Combining, overconfidence can increase an individual’s willingness to commit crimes and to go to trial, just as risk-acceptance does. Thus, by logic similar to our analysis centered on risk aversion, if trial outcomes are sufficiently noisy, the innocent may be more likely to accept plea bargains than the guilty.

Potential Offers

We now derive the plea bargain the prosecutor optimally offers to the defendant. Fix the level of evidence e. For any e such that the likelihood of conviction is higher for a guilty defendant (i.e., such that $ {\pi}_0(e)<{\pi}_1(e) $ ), there are three possible prosecutor strategies. First, the prosecutor may make a trial-inducing offer, i.e., an offer so harsh that both types of defendants reject it and go to trial. Second, the prosecutor may make a plea-inducing offer, i.e., an offer so lenient that both types of defendants accept it. Finally, the prosecutor may make a screening offer, i.e., an offer which the guilty accept, but the innocent reject.Footnote ⁹

Begin with the screening strategy, in which the prosecutor makes an offer that only the guilty accept. Because the prosecutor’s utility is increasing in sentence length, he makes the maximum offer acceptable to the guilty. This is the offer at which a guilty defendant is indifferent between taking a plea and his expected payoff at trial: $ {\pi}_1(e){x}_K. $ As long as $ {\pi}_1(e)>{\pi}_0(e) $ , this offer is not acceptable to an innocent defendant with evidence level e, because the innocent defendant expects the sentence $ {\pi}_0{x}_K<{\pi}_1{x}_K $ at trial. When making the screening offer, the prosecutor’s utility is $ {\pi}_1(e){x}_K $ if the defendant is guilty and accepts, and $ {\pi}_0(e)(w+{x}_K)-\kappa $ if the defendant is innocent and goes to trial. His total expected payoff is then

(6) $$ \begin{array}{rl}{u}_S(e)=q(e){\pi}_1(e){x}_K+(1-q(e))({\pi}_0(e)(w+{x}_K)-\kappa ).& \end{array} $$

Now consider the trial-inducing strategy. For a given level of evidence e, the trial-inducing offer is any $ {x}_P $ such that $ {x}_P>{\pi}_1(e){x}_K $ . Because both types of defendant now anticipate a lower expected sentence at trial than that attached to the plea deal, both types reject this offer and the prosecutor’s expected payoff is

(7) $$ \begin{array}{rl}{u}_T(e)=\left[q(e){\pi}_1(e)+(1-q(e)){\pi}_0(e)\right]({x}_K+w)-\kappa, & \end{array} $$

where $ q(e){\pi}_1(e)+(1-q(e)){\pi}_0(e) $ is the overall probability of winning at trial, averaging across guilt status.

Finally, consider the plea-inducing strategy. Here, the prosecutor must offer a sentence that both a guilty and an innocent defendant would accept. Accordingly, the maximum plea deal he can offer is $ {x}_P={\pi}_0(e){x}_K $ , i.e., the equivalent to the innocent defendant’s expected payoff at trial. His expected payoff from this strategy is

(8) $$ \begin{array}{rl}{u}_P(e)={\pi}_0(e){x}_K.& \end{array} $$

To determine the prosecutor’s optimal offer, we pairwise compare his payoffs from each of the three strategies above. In particular, we ask how the relative values of each offer changes as the evidence gets stronger. In the main text, we restrict attention to comparisons where each relative value has at most one crossing.

In other words, we assume that $ {u}_T(e)-{u}_P(e), $ $ {u}_T(e)-{u}_S(e) $ , and $ {u}_S(e)-{u}_P(e) $ switch signs at most once over the range of $ e\in [0,1] $ . This means that a prosecutor’s preference ranking of any two strategies cannot fluctuate back and forth as the evidence increases.Footnote ¹⁰ The Supplementary Material contains a more detailed discussion of when this assumption holds and how the results change when it does not.

Special Case 1: $ {\pi}_0(e)={\pi}_1(e) $

We start by analyzing the special case where, conditional on the strength of the evidence available at the time of plea bargaining, the probability of conviction at trial is the same for both types, and so we can write it $ \pi (e) $ . Recall that this does not mean that the innocent and guilty are equally likely to be convicted at trial in general, since the guilty tend to have more evidence against them.

Since the guilty and innocent have the same utility from going to trial, the prosecutor cannot screen the guilty: he must either make an offer all defendants accept, $ {x}_P=\pi (e){x}_K $ , or take all to trial. He prefers the former if

$$ \begin{array}{rll}\pi (e){x}_K\ge \pi (e)(w+{x}_K)-\kappa, \hskip0.3em \mathrm{or}& & \end{array} $$

(9) $$ \begin{array}{rl}\pi (e)\le \kappa /w.& \end{array} $$

If $ w<\kappa $ , then this condition holds for all e. Intuitively, if going to trial is costly for the prosecutor at any value of $ e\in [0,1] $ , he always strikes a deal to avoid paying this cost.

If $ w>\kappa $ , then Inequality 9 is not met at $ \pi (e)=0 $ , is met at $ \pi (e)=1 $ , and because $ \pi (e) $ is monotonically increasing in e, there is a critical value of evidence $ {e}^T\in (0,1) $ at which Inequality 9 is met with equality. When $ e<{e}^T $ the prosecutor makes an offer which all defendants accept, and when $ e>{e}^T $ the prosecutor makes an offer which all reject. However, because we have assumed that the evidence tends to be stronger when the defendant is guilty, the probability that $ e>{e}^T $ is also higher when the defendant is guilty. As a result, in this part of the parameter space, there is always perverse sorting.

As this special case highlights, plea offers only induce correct sorting when the prosecutor makes the screening offer, which in turn requires that the probability of conviction, conditional on the available evidence, is higher for the guilty than for the innocent ( $ {\pi}_1(e)>{\pi}_0(e) $ ).

Special Case 2: $ {\pi}_1(e)>{\pi}_0(e) $ , $ w<\kappa $

Now consider the case where $ {\pi}_1(e)>{\pi}_0(e) $ , but the cost of a trial exceeds the benefit from a win, $ w<\kappa $ . In this case, the prosecutor never takes all defendants to trial. To see why, compare the prosecutor’s payoff from the screening offer (Equation 6) to his payoff from the trial-inducing offer (Equation 7). The payoff from trial is larger if

(10) $$ \begin{array}{rl}{u}_T(e)-{u}_S(e)=w{\pi}_1(e)-\kappa \ge 0,& \end{array} $$

which never holds if $ w<\kappa $ . In other words, if the benefit from winning trials is below the cost, the prosecutor’s only two options are to make the screening offer or to make an offer which all accept. Moreover, comparing the screening offer payoff to the payoff the prosecutor obtains if all plead guilty (Equation 8), he prefers to screen if

(11) $$ \begin{array}{rl}{u}_S(e)-{u}_P(e)=\frac{q(e)}{1-q(e)}{x}_K({\pi}_1(e)-{\pi}_0(e))+{\pi}_0(e)w-\kappa \ge 0.& \end{array} $$

Notice that since $ w<\kappa $ , this condition holds neither at $ e=0 $ nor at $ e=1 $ , although it is increasing in the evidence e. Therefore, by the single-crossing condition, we assumed above, there is no level of evidence e at which the prosecutor makes the screening offer.Footnote ¹¹

Main Case: $ {\pi}_1(e)>{\pi}_0(e) $ , $ w>\kappa $

The remaining case is that in which the probability of conviction is higher for the guilty even after conditioning on the evidence $ {\pi}_1(e)>{\pi}_0(e) $ , and the benefit to a trial win exceeds the cost of trial, $ w>\kappa $ . In this case, as $ e\to 1 $ , the fraction of defendants who would be acquitted at trial, whether innocent or guilty, approaches zero, and the prosecutor’s optimal strategy is to take all to trial and obtain $ {x}_K+w>\kappa $ . Likewise, as $ e\to 0 $ , the prosecutor’s utility from both the screening offer and the trial offer approaches $ -\kappa $ , and he prefers to plead out all defendants and receive payoff $ {\pi}_0(e){x}_K\ge 0 $ . The relevant question then concerns the existence and size of an intermediate range of e where the prosecutor uses the screening offer.

To answer this question, recall the pairwise payoff comparisons we made above. First, observe that the value of the screening offer relative to pleading out all defendants, $ {u}_S(e)-{u}_P(e) $ , is increasing in e, so there exists a critical evidence level $ {e}^{SP} $ such that the prosecutor prefers screening to pleading out all defendants only if $ e>{e}^{SP} $ . Similarly, since the relative value of taking all defendants to trial compared to screening out the guilty ( $ {u}_T(e)-{u}_S(e) $ ) is increasing in e, there is a critical value $ {e}^{TS} $ such that the prosecutor prefers the trial-inducing offer to the screening offer only if $ e>{e}^{TS} $ . Finally, a comparison of the payoffs from taking all to trial and inducing all to plead guilty yields that the prosecutor prefers the former if

(12) $$ \begin{array}{rl}{u}_T(e)-{u}_P(e)=q(e)({\pi}_1(e)-{\pi}_0(e))({x}_K+w)+{\pi}_0(e)w-\kappa \ge 0,& \end{array} $$

which again does not hold at $ e=0 $ but strictly holds at $ e=1 $ , implying that there is a critical value $ {e}^{TP} $ such that the prosecutor prefers the trial-inducing offer to the plea-inducing offer only if $ e>{e}^{TP} $ .

The prosecutor’s equilibrium behavior depends on the ordering of these three evidentiary thresholds. There are two sub-cases. First, if $ {e}^{TS}\le {e}^{SP} $ , the prosecutor already prefers inducing trial to screening at a level of evidence where he still prefers inducing pleas to screening. This means that screening is never optimal, and the prosecutor’s actions depend only on whether he derives greater utility from inducing trial or inducing a plea. Here, the prosecutor tries all types of defendant when the evidence is sufficiently strong, $ e>{e}^{TP} $ , and pleads out all defendants otherwise. Because defendants with more evidence against them are more likely to be guilty, this results in perverse sorting.

By contrast, if $ {e}^{SP}<{e}^{TS} $ , then there is an intermediate range of evidence, $ e\in ({e}^{SP},{e}^{TS}) $ , where the prosecutor prefers the screening offer to both the plea-inducing and the trial-inducing offer. In this sub-case, the prosecutor tries all defendants for $ e>{e}^{TS} $ , screens the guilty for $ e\in ({e}^{SP},{e}^{TS}] $ , and pleads all defendants out for $ e<{e}^{SP} $ . Here, as shown in the proof of the following formal result, screening is possible for intermediate $ e\in ({e}^{SP},{e}^{TS}) $ only when $ q({e}^{TS})>w/(w+{x}_K) $ , i.e., when the value of trial wins is not too high.

Even if screening happens, we have not determined whether in this sub-case, the distribution of the evidence is such that on average the guilty are more likely to plead guilty: this depends on the shape of the evidentiary distribution and the size of the interval $ ({e}^{SP},{e}^{TS}) $ where screening is optimal. Recall from above that at $ {\pi}_1(e)={\pi}_0(e) $ , the screening offer and the offer which all accept become equal, at $ \pi (e){x}_K $ . By continuity, this implies that when the distance between $ {\pi}_1(e) $ and $ {\pi}_0(e) $ is sufficiently small, so is the distance between $ {e}^{SP} $ and $ {e}^{TS} $ , and the guilty are still (overall) strictly more likely to plead guilty.Footnote ¹²

Summarizing this case:

Figure 3 illustrates Proposition 6. The left panel displays equilibrium behavior in the case in which $ q({e}^{TS})<w/(w+\kappa ) $ . Here, at the critical evidence level ( $ {e}^{TS} $ ) at which the prosecutor is indifferent between the trial-inducing offer and the screening offer (i.e., at which the dark gray curve $ {u}_T-{u}_S $ crosses 0), he strictly prefers the plea-inducing offer to the screening offer (the light gray curve, $ {u}_S-{u}_P $ , is negative). He therefore makes either the plea-inducing offer (for $ e<{e}^{TP} $ ) or the trial-inducing offer (for $ e>{e}^{TP} $ ) and there is perverse sorting in equilibrium.

Figure 3.

Equilibrium Behavior

Note: The left-hand panel presents equilibrium behavior when $ q({e}^{TS})<w/(w+{x}_K) $ . The right-hand panel presents equilibrium behavior when $ q({e}^{TS})>w/(w+{x}_K) $ . The dashed vertical line in each graph represents the point at which $ q(e)=w/(w+{x}_K) $ . Parameter values: $ w=1.5 $ , $ \kappa =0.5 $ , $ q(e)=e $ , $ {\pi}_1(e)={e}^{\alpha_1} $ for $ {\alpha}_1=0.7 $ , $ {\pi}_0(e)={e}^{\alpha_0} $ for $ {\alpha}_0=1 $ . Left panel: $ {x}_K=2 $ . Right panel: $ {x}_K=12 $ .

The right panel displays equilibrium behavior in the case in which $ q({e}^{TS})>w/(w+\kappa ) $ . Here, at the critical level $ {e}^{TS} $ , the prosecutor strictly prefers both trial and screening to the plea-inducing offer (at the point where the dark gray curve, $ {u}_T-{u}_S $ , hits zero, the light gray curve, $ {u}_S-{u}_P $ , is above zero) . He therefore makes the screening offer when $ e\in ({e}^{SP},{e}^{TS}) $ . Observe that in this case, the range of evidence where it is optimal for the prosecutor to make the screening offer is quite small; therefore, for plausible distributions of evidence, there would still be perverse sorting.

The most striking implication of the results detailed in Proposition 6 is that perverse sorting may occur under most plausible empirical conditions. First, perverse sorting is likelier the higher the benefit to prosecutors from trial wins, and we have already argued that at least for the assistant prosecutors who conduct the vast majority of litigation in any prosecutor’s office, trial wins are not only “worth the cost” of trial, they are critical both to promotion within the office and (perhaps even more importantly) to obtaining a much more lucrative position at a private law firm.

Second, perverse sorting is likelier when, conditional on the evidence, the innocent and guilty face similar probabilities of conviction. Empirically, this is likely the case. In countries where the defendant has the right to know the evidence against him, the prosecutor’s evidence of guilt is known to both parties before trial. In countries like the United States, the prosecutor’s evidence of innocence is also known (by law) to both parties before trial. This suggests that only when significant “exculpatory surprises” are likely at trial will the expected rate of conviction for guilty and innocent defendants be substantially different—and defendants have numerous incentives not to keep exculpatory information secret in order to reveal it at trial. For example, revealing exonerating evidence pretrial is useful both in pushing the total evidence well below the threshold at which the prosecutor prefers trial and in substantially decreasing the expected plea offer.Footnote ¹³

One possible argument against these findings is that prosecutors are motivated by the desire to do justice. In the Supplementary Material, we briefly consider the robustness of our perverse sorting result to a situation in which the prosecutor suffers a disutility from inaccurate outcomes. Specifically, we assume that convicting the innocent generates a negative payoff of $ -\psi $ for $ \psi \in (0,1) $ for the prosecutor, while acquitting the guilty generates a negative payoff of $ -(1-\psi ) $ . An accurate outcome generates a payoff of $ 0 $ .

We show that while the accuracy motive increases the attractiveness of the screening offer and therefore broadens the range of the parameter space in which correct sorting is possible, perverse sorting remains an equilibrium outcome under circumstances similar to those elaborated in Proposition 6. If the probabilities of conviction for guilty and innocent are the same conditional on the evidence, accuracy concerns can even exacerbate perverse sorting if the prosecutor is more worried about wrongful acquittals than wrongful convictions. However, given accuracy concerns, when the conditional probabilities of conviction differ across the innocent and the guilty, higher benefits to trial wins are in general needed for perverse sorting to occur, and when it does occur, its degree may be lessened, in the sense that concern for accuracy may decrease the evidentiary cutoff above which the prosecutor takes cases to trial.

A final question is how prosecutor and defendant behavior might change if defendants varied in risk aversion. In the Supplementary Material, we briefly consider this problem, focusing on the case where the conditional probability of conviction at trial is the same for guilty and innocent defendants. We begin by assuming that the prosecutor observes risk aversion. We show that because more risk-averse defendants accept longer plea sentences to avoid trial, and the prosecutor values large sentences, risk aversion increases the appeal to the prosecutor of a plea bargain relative to a trial. Thus, the threshold level of evidence above which the prosecutor prefers trial is increasing in defendant risk aversion: if the defendant is sufficiently risk-averse, the plea sentence she accepts is so large that the prosecutor may prefer a guilty plea for almost any value of the evidence. If the probability of being guilty is lower for more risk-averse defendants, this implies that risk aversion exacerbates perverse sorting.

If risk aversion is unobserved, it is more difficult to characterize the optimal plea and determine how it changes with the strength of the evidence. This is because, if both risk aversion and evidence vary systematically with guilt, the prosecutor uses the level of evidence against the defendant to update his beliefs about both her guilt and her likely risk aversion. However, we show that many of the features that lead to perverse sorting are also present in any equilibrium to this version of the model.