Bjerk Plea Bargaining limits

Question 1

What are the timing of events in the Bjerk model?

Answer 1

1. The type of defendant is determined. The probability the person is guilty upon arrest is λ. The defendant knows his own type.
2. An initial signal of guilt is determined θ(a). This signal is drawn from different distributions depending on whether or not the defendant is innocent or guilty. This signal can be interpreted as the strength of the case. This higher this signal, the more likely it is to have come from a guilty individual. Both the prosecutor and defendant observe this signal.
3. After the prosecutor observes the initial signal of guilt θ(a), they offer the defendant a plea bargain.
4. The offer is accepted or rejected. If rejected the defendant goes to trial where some new evidence is revealed. This new evidence changes the initial guilt signal to a new guilt signal θ(t). For the innocent, this new guilt signal is lower than the original, for the guilty it is higher. The jury decides whether they are guilty or not. If guilty they receive a sentence of Z_bar. Anyone with a guilt signal greater than some θ* will be convicted at trial, as such defendants are convicted if θ(t)>θ(*).

Question 2

What are the differences between the Bjerk and Reinganum models?

Answer 2

1. At the point of the plea bargain offer the defendant knows the strength of the case. Through a discovery process as when the plea offer is made the prosecutor must reveal evidence to the defendant.
2. The defendant still has information about the probability of conviction.
3. Sentences are in years not utility terms and the prosecutor and defendant have risk preferences.
4. There are no trial costs for either the prosecutor or the defendant - results come from risk preferences as opposed to trial costs.
5. Prosecutor and defendant are joint senders and the jury is the receiver (as opposed to the prosecutor sending message s to the defendant in the Reinganum case).

Key difference: The plea offer has to be separating because the defendant knows the strength of the case, unlike the Reinganum model where the plea offer could be separating or pooling.

Question 3

What does theorem 1 say about the prosecutors optimal use of plea bargaining?

Answer 3

Because prosecutors believe the jury will convict any defendants for which the probability of guilt at trial θ(t) is greater than the threshold guilt level θ*, we get:

1. Defendants who emit signals below the threshold, such that θ(a)<θ*, it is optimal for the prosecutor to use plea bargaining to sort the innocent from the guilty by offering a plea sentence given by the function Z_g(θ(a)).
2. For defendants who emit initial guilt signals greater than the threshold θ* is is optimal for the prosecutor to not offer a plea bargain that is acceptable to either innocent or guilty such that they go to trial.

Question 4

What are the equilibria in the model?

Answer 4

A jury finds it optimal to not convict any defendant who emits a guilt signal at trial less than θ. There are a continuum of equilibria categorized by a different level of the threshold θ. Plea bargains in these equilibria are only taken by a fraction of the guilty and the remaining fraction of the guilty as well as all the innocent go to trial. This degree in separation between these groups, however, differs within different equilibria.

When θ* is very high, defendants of either type are unlikely to emit a guilt signal at trial that is greater than this θ* threshold, thus to get a guilty person to accept a plea deal the prosecutor has to offer a very short sentence.

If θ* is low, optimal plea bargaining leads to less sorting between the guilty and innocent, but the prosecutor will be able to impose longer sentences on those who plead guilty.

Question 5

What is the result if prosecutors believe juries will be relatively likely to convict any given defendant?

Answer 5

[θ* is low].
Is is optimal for prosecutors to offer plea bargains (that only the guilty would accept) to only a small fraction of defendants. This means a large fraction of those who go to trial are guilty, making it optimal for juries to set a low standard of evidence required for conviction, thus confirming the belief that convictions at trial are likely.

Question 6

What is the result if prosecutors believe juries will be relatively unlikely to convict any given defendant?

Answer 6

[θ* is high].

It is optimal for prosecutors to offer plea bargains (that are acceptable only to the guilty) to a relatively large fraction of defendants. This means a large fraction of those who go to trial are innocent making it optimal for juries to set a high standard for conviction at trial.