B. Basic Models of Settlement Bargaining
17. Outcomes and Payoffs
17.4 Other Payoffs: Plea Bargaining
As an example of a significantly different payoff measure, consider negotiations between a defendant (D) in a criminal action and a prosecutor (P). This type of settlement bargaining, called plea bargaining, has been addressed in a number of papers. Early papers in this area are Landes (1971), Grossman and Katz (1983), and Reinganum (1988). In Landes’ model the payoffs are expected sentence length for P versus expected wealth (wealth in two states: under conviction and under no conviction) for D, who is guilty. In Grossman and Katz’ model D may be innocent, and he knows (privately) whether he is guilty or innocent (a two-type model). D seeks to minimize the
disutility of punishment (he is risk averse) while the uninformed prosecutor maximizes a notion of justice that trades off the social losses from punishing the innocent versus freeing the guilty; the Grossman and Katz analysis is a screening model with innocent defendants choosing trial.
Reinganum's model involves two-sided asymmetric information: D knows (privately) whether he is guilty or innocent (two types), while P knows (privately) the strength of the case; that is, the probability that the case will yield a conviction at trial (a continuum of types). Guilt and evidence are correlated, so there is a relationship between the two sets of types. A special case of this relationship appears in Grossman and Katz and was also employed in Rubinfeld and Sappington, discussed in Section 15.2: innocent defendants can more readily obtain supporting evidence than can guilty ones. P's payoff is social justice minus resource costs while D's payoff to be minimized is the expected sentence plus the disutility of trial. Reinganum finds that P’s plea offer (which signals the strength of his case) is accepted by a mixture of innocent and guilty defendants; thus, defendant types are not screened perfectly in equilibrium.
Baker and Mezzetti (2001) extend the Grossman and Katz plea bargaining model to include P’s costly evidence generation following failed plea negotiations; evidence can help filter out innocent defendants and increase the likelihood that a guilty defendant will be convicted. This modification results in an equilibrium in which all innocent defendants and some guilty defendants reject the plea bargain. Since the defendant types are not screened perfectly, it is optimal for P to try a fraction of the cases in which plea bargaining failed.
Bjerk (2007) also envisions a role for evidence generation following plea bargaining, but this evidence is not observed by P prior to trial, but rather by the jury during trial. The prosecutor has an initial observation on evidence, and this initial strength of P’s case against D is common knowledge to P and D. Furthermore, it is common knowledge that the evidence observed by the jury
at trial will be even stronger (respectively, weaker) if D is guilty (respectively, innocent). The jury observes neither the initial evidence nor the offer made by P to D. If the initial evidence against D is sufficiently strong, then in equilibrium P makes a plea offer that is rejected by all Ds. On the other hand, if the initial evidence against D is weaker, then in equilibrium P makes an offer that is accepted only by guilty Ds. When a case comes before them, jurors cannot know whether it involves an innocent defendant (who rejected a screening offer) or a defendant of either type (who rejected a harsh pooling offer); thus, jurors will convict (respectively, acquit) those Ds against whom the evidence realized at trial is sufficiently strong (respectively, weak).
Kobayashi (1992), Reinganum (1993), Miceli (1996), and Franzoni (1999) examine the effect of plea bargaining on the decision to commit a crime. Kobayashi considers conspiracies:
there are two defendants (for example, a price-fixing case) who face different (exogenously determined) initial probabilities of conviction based on the existing evidence. Each D can, however, provide information on the other D which increases that second D's likelihood of conviction.
Kobayashi assumes that the D with the higher initial conviction probability (the "ringleader") also has more information on the other D (the "subordinate"). Here P makes simultaneous offers to each D so as to maximize the sum of the expected penalties from the two Ds. Litigation costs are taken to be zero so as to focus on plea-bargaining as information gathering. He finds that “unfair” plea bargains, wherein the ringleader receives a smaller penalty than the subordinate, can improve deterrence. Reinganum's model takes all Ds as guilty and therefore takes P's payoff as expected sentence length minus resource costs. In this model the level of enforcement activities (as chosen by the police) and the expected sentence (as determined by plea bargaining) both influence D’s initial choice to engage in criminal behavior. Miceli considers two possible objectives for a prosecutor; he finds that a P who maximizes the expected sentence minus resource costs effectively
implements a legislature’s preferences over sentencing (i.e., a low probability of a long sentence), while a P who trades off the social losses from punishing the innocent versus freeing the guilty is unwilling to implement such a policy and instead offers a substantial sentence discount in plea bargaining. Finally, Franzoni assumes that P first bargains with D and then, if D rejects the plea offer, P decides how much to spend on an investigation. The investigation is assumed to (always) verify innocence, and to verify guilt with a probability that increases with the amount spent. Since innocent Ds always reject plea offers, P’s offer must induce enough guilty D’s to reject in order to make the subsequent pursuit of an investigation credible (recall Nalebuff, 1987, discussed in Section 16.1). This credibility requirement results in less-thorough investigations, lower sentences, and more crime than would occur if P was able to commit ex ante to the extent of investigation.