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17-01-2023

The vanishing trial: a dynamic model with adaptive agents

Authors: Moti Michaeli, Yosef Zohar

Published in: Public Choice | Issue 3-4/2023

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Abstract

In recent years, trial waivers (plea bargains in the United States) have been introduced into many legal systems around the world. Once trial waivers (TW) were introduced, most—if not all—of these legal systems witnessed a steep increase in the usage of TW, to the extent that trials are virtually disappearing. Our model explains why introducing TW into a legal system would almost inevitably trigger a dynamic process whose ultimate result is that all defendants choose the TW. The crux of the idea is that defendants who choose the TW option impose a negative externality on other defendants: the former help the prosecution save resources, thus forcing the latter to face stronger prosecution if they choose to go to trial. This further explains the observed increase in conviction rates in trials. We also show that the introduction of TW increases total sanctioning and reduces the welfare of many, if not all, defendants. The intuition for this is that a harsh punishment in a TW, which initially attracts only the most risk-averse defendant, becomes over time attractive to more and more defendants as the expected sanction in trial becomes more severe.

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The right of defendants to a trial includes the right to file a motion to exclude wrongfully obtained evidence from being used against them at trial, to have evidence displayed in court, to hear and cross-examine witnesses, to challenge evidence introduced by the prosecutor, to appeal following a conviction and more.

Before 1990, only 19 of the 90 jurisdictions studied in the research featured TW systems in law. By the end of 2015, the number had grown to 66, reaching all six major continents and changing practice across a variety of different legal systems and traditions.

For example, TW (in the form of guilty pleas) conclude 98% of federal cases in the United States (https://www.uscourts.gov/, accessed on 08/08/2022; see also Fig. 2) and 90% of criminal convictions in the higher courts of England and Wales (UK Ministry of Justice 2014).

For example, in Georgia, 12.7% of cases were resolved through a TW system in 2005, but that figure had soared to 87.8% of cases by 2012. In Russia, deployment of TW shot up from 37% in 2008 to 64% only 6 years later (Fair Trials, 2017).

This means that, unlike the prosecutor, the defendants are adaptive. This has no effect on the existence of the equilibrium analyzed—it also exists when defendants are forward-looking—but we think this is more realistic and better portrays the observations mentioned above. The result is shown to be robust to other fictitious-play dynamics. See further discussion in Sects. 3 and 4.

This message resonates with the conclusions of others, e.g., Bar-Gill and Ben-Shahar (2009), but opposite claims are also prevalent (see, e.g., Church, 1979).

This implicit trial penalty is sometimes quite explicit, when defendants are actually threatened to face a very severe punishment in case they refuse the bargain, making the latter an offer “they cannot refuse.”

For example, the case of Bordenkircher v. Hayes, where the defendant Hayes had refused to secure a 5-year imprisonment in a plea bargain and ended up being convicted in trial and receiving a mandatory life sentence (under Kentucky’s three strikes law), is a famous demonstration of the trial penalty notion but is clearly specific to the American legal system, just like the application of severe punishments as a mandatory minimum sanction for various offenses.

The dynamic conversion to 100% plea bargains that our model produces will only be faster if we let the prosecutor’s objective function also include a component of maximizing conviction rate–see Sect. 4.

Specifically, Bar-Gill and Ben-Shahar (2009) present a model in which the prosecutor can take only one out of N defendants to trial; then, since no one wants to be that one defendant, and since the order in which the prosecutor will take them to trial is assumed to be common knowledge, all of the N defendants end up signing a plea bargain in order to avoid being that one defendant. Unlike our model, which is dynamic, their model is static—all of the defendants simultaneously decide whether to sign the plea bargain—and it portrays a situation where the \(K^{th}\) defendant correctly anticipates that the \(K-1\) preceding defendants (in the commonly known order) will sign the bargain as well.

Put differently, while Bar-Gill and Ben-Shahar (2009) blame their simplifying assumptions for this extreme result, we show that the result is likely to hold even without making such assumptions. We refer here not only to their assumptions of common knowledge of the “order of prosecution” and of perfect forward-looking on the side of the defendants, but also to the assumption that none of the defendants is risk-seeking, despite the known tendency of people to be risk-seeking in the domain of losses (as established by prospect theory, see Kahneman, 1979; Tversky & Kahneman, 1992).

In Appendix E we present a similar table for the UK (but only for the last decade), showing that also in the UK there was a parallel increase in guilty pleas and in the conviction rate in trials. Furthermore, the documented increase in the total conviction rate in Israel, from 69.5% in 1951 to 99.4% in 2011 (according to data provided by the Israel Central Bureau of Statistics), when merged with the data point of 93% guilty pleas in 2011 (Gazal-Ayal et al., 2012), implies that also in Israel the conviction rate in trials must have increased during those 60 years (as it must have started below 69.5% in 1951, and is up to [99.4–93%]/[100–93%]=91.8% in 2011).

One might think of the state prosecution as composed of a large team of specialized prosecutors, each of whom is in charge of a narrow chunk of cases with very similar attributes. In the model we study the dynamics of cases that are handled by one such specialized prosecutor. The simplifying assumption of having only one type of crime can also be found, for example, in Rasmusen et al. (2009), who explain in footnote 8 of their paper why this assumption is innocuous. In their paper, the actual strength of the cases varies, while we vary instead the perceived strength of the cases—see below.

Note that the solution may be such that some cases are not taken to court (\(i\left( x\right) =0\) for some x’s in \(\left[ 0,1\right]\)).

To see why this must hold, note that such an increase enables the prosecutor to evenly divide the added budget among all the tried cases so that \(i\left( x\right) >i^{*}\left( x\right)\) for every tried case, and hence \(sp\left( i\left( x\right) \right) >sp\left( i^{*}\left( x\right) \right)\). The optimization problem of the prosecutor implies that she might be able to do even better than this, but certainly not worse.

Reinganum (1988) shows that utilizing a uniform plea bargain across cases with similar characteristics can sometimes improve the welfare of the prosecutor and hence could be her best choice.

Formally, we assume that the mass of defendants who are not strictly risk-seeking is strictly positive.

In Sect. 4 we discuss further limitations of modeling defendants as forward-looking.

Reflecting the notion that most (if not all) criminal justice systems started from an initial state of no plea bargaining.

Potentially facing a prosecutor who invests 0 in their specific case, in which case it is dismissed.

The uniqueness of the inner steady state is guaranteed by the monotonicity assumption (Assumption 1). The multiplicity of equilibria demonstrates why a simultaneous game with perfectly forward-looking players having rational expectations is not warranted here: it will not allow us to say anything meaningful about the state of the criminal justice system if the prosecutor offers a PB of \(b\in \left( {\underline{b}},{\bar{b}}\right)\).

If b equals exactly \({\underline{b}}\), then there is another steady state at \(\alpha =0\) but this steady state is not stable. For ease of exposition we ignore this knife-edge case in the analysis.

Since b is a choice variable now, we denote by \(\alpha _{t}\left( b\right)\) the share of defendants who sign the plea bargain in period t when the offered plea-bargain sanction is b. The prosecutor in fact solves the optimization problem in two steps: first, for any possible b, she chooses the budget allocation among the defendants who do not sign the bargain (in each period t), and then she maximizes over b.

That is, there is no need to know how the prosecutor divides I among the defendants who do not sign the plea bargain.

If all defendants are risk-averse, then consider instead the offer \(b=S(B)<CE\left( 1,P\left( B\right) \right)\).

By \(b\rightarrow {\underline{b}}\) we formally mean that for any \(\varepsilon >0\), \(\exists \delta \in \left( 0,1\right)\) s.t. \(b>{\underline{b}}-\varepsilon\).

The proof shows that for any two offers \(b_{1}\) and \(b_{2}\), such that \(b_{1}<b_{2}<{\underline{b}}\), the prosecutor prefers \(b_{2}\) over \(b_{1}\).

A sufficient (and far from necessary) condition is that \({\hat{\alpha }}>{\tilde{\alpha }}\) , where \({\tilde{\alpha }}\) is the risk-neutral defendant. This condition implies that \(b_{{\hat{\alpha }}}=CE\left( {\hat{\alpha }},P\left( \frac{B-{\hat{\alpha }}c}{1-{\hat{\alpha }}}\right) \right) <CE\left( {\tilde{\alpha }},P\left( \frac{B-{\hat{\alpha }}c}{1-{\hat{\alpha }}}\right) \right) =S\left( \frac{B-{\hat{\alpha }}c}{1-{\hat{\alpha }}}\right)\). This in turn ensures that the expected sanction, which is a weighted average of the PB-sanctioning \(b_{{\hat{\alpha }}}\) and of the expected sanction in trial \(S\left( \frac{B-{\hat{\alpha }}c}{1-{\hat{\alpha }}}\right)\), is greater than \(b_{{\hat{\alpha }}}\), which is itself greater than \({\underline{b}}\), implying that offering \(b_{{\hat{\alpha }}}\) is preferred by the prosecutor over offering \(b\rightarrow {\underline{b}}\) and guaranteeing \(\alpha =1\).

Sangero (2017) suggests a few safety constraints, including that (1) judges must receive for review all of the prosecution’s evidentiary materials, (2) judges must hear a detailed explanation from defendants as to why they have confessed, and (3) judges will be required to ensure the existence of strong corroborative evidence for any sort of confession or guilty plea.

Note that if \(b={\underline{b}}\) while \(\alpha >0\), convergence again can be only to \(\alpha =1\).

Note also that \(\alpha _{1}=0\) only if \(b\ge {\underline{b}}\); hence, we start from a stable steady state (or, in the case of \(b={\underline{b}}\), we stay in the steady state where \(\alpha =0\) because \(\alpha _{1}=\alpha _{0}=0\)).

If the defendant \(x=0\) is risk-neutral himself, set \({\tilde{\alpha }}>0\) to be the largest value corresponding to a risk-neutral defendant (where the fact that \({\tilde{\alpha }}>0\) follows from the assumption that the mass of defendants who are not strictly risk-seeking is strictly positive); if all defendants are risk-averse, then consider \(b=CE\left( 1,P\left( B\right) \right)\) instead.

When \(\frac{p}{1-p}=\frac{I}{B}\), we get that without plea bargains, the probability of conviction is 1/2.

\(\alpha _{t-1}\) itself is a function of the history of bargains (the pairs \(\left\{ b_{t},\alpha _{t}\right\}\) for \(t=1,2,...t-1\)), but for simplicity we refrain from writing this explicitly in the optimization problem.

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Title: The vanishing trial: a dynamic model with adaptive agents
Authors: Moti Michaeli
Yosef Zohar
Publication date: 17-01-2023
Publisher: Springer US
Published in: Public Choice / Issue 3-4/2023
Print ISSN: 0048-5829
Electronic ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-022-01034-9

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