Plea Bargaining as a Mean to Maximum Sentence

Ⅰ. Introduction

A plea bargaining is a deal offered by a prosecutor as an incentive for a defendant or defendants to plead guilty. Plea bargaining agreement has been practiced in the United States since several hundred years. Although estimates vary, 95% of criminal cases were resolved by plea bargaining in United States.1 Yet, the wide use of plea bargaining is not without criticism. Skeptics of plea bargaining argue that the criminal justice system has been too soft on criminals by allowing for less sentence in exchange for a guilty plea.

Introduction of plea bargaining system aims to gain more information about the crime by allowing plea discounts to the defendant(s). So the main objective of plea bargaining is information-gathering effect, although the plea bargaining system is going to save trial cost borne by prosecutors. However, to induce the defendant(s) to plead guilty, a prosecutor shall offer lenient penalty in the plea offer, which tends to decrease the social welfare.2 Moreover, plea bargaining could be unfair when a plea bargain gives the most culpable defendant the lowest penalty.3 However, previous works have focused primarily on analyzing possibility of unfair settlement so far,4 and the first and main disadvantage of plea bargaining, leniency to the guilty was not thoroughly analyzed.

This paper tries to support plea bargaining system by providing method to minimize disadvantages in theoretical model. In section 2, a model of multiple defendants is introduced. Section 3 describes the effect of sequential offer by a prosecutor, and tries to characterize an equilibrium. Section 4 includes concluding remarks as well as points for discussion.

Ⅱ. Model

In this section, I consider plea bargaining model which was employed by Kim (2009). A prosecutor (P) has accused two codefendants (D_i,i = 1, 2), who jointly committed a crime. The sanctions need proof of their guilt at formal trial, while it is common knowledge that they jointly committed the crime.

If the defendants go to the formal trial, each defendant i is expected to be sentenced of s_i (>0) when he is convicted as guilty. Without loss of generality, let’s assume that s₁ ≥ s₂. The probability of conviction is set to q if there is no testimony from defendant(s) via plea bargaining. Here, q is strictly less than 1, as there is probability of acquittal after formal trial. Defendants may accept the plea offer with reduced sentences by the prosecutor. It is assumed that the defendant pleading guilty has to testify against the other defendant.

The sequence of movement in the model is as follows. First, P makes plea offers b_i∈ B ≡ [0, ∞] to each defendant D_i. It was assumed in previous literature that the prosecutor is able to make only simultaneous plea offers to both of the defendants.5 In the present paper, rather, I assume that the prosecutor can approach defendants to make plea offer one by one. Then, D_i decides whether to accept or reject the offer. If the first defendant D_i who P makes plea offer accepts the offer, he is sentenced to b_i at court with certainty. Also if D_i testifies against D_j via plea bargaining, D_j will be convicted with probability q'(> q) when D_j moves to the formal trial. For simplicity of analysis, let q' = 1. If he rejects the offer, then he goes to the formal trial. After knowing decision of the first defendant on plead, the prosecutor can choose to make plea offer to the second defendant, or she can just move the case to the formal trial without more plea bargaining.

It is assumed that D_i minimizes his expected sentence, and that P maximizes the penalties6 on the defendants. As it is common knowledge that both defendants committed a crime, the prosecutor who is representative of society tries to penalize the defendants by appropriate sentence. I assume that s_i is the appropriate level of penalty given D_i' s crime committal. So the objective of P is to maximize the probability of sentences while holding the level of sentences as high as possible.

The strategy of P is defined by σ_p = (b₁,b_2a,b_2r) ∈ B × B × B. Here b_2a is the plea offer to D₂ when D₁ accepted the offer b₁, and b_2r is the offer when D₁ rejected the offer. Note that b_2a and b_2r need be not necessarily different. Also P can choose not to make plea offer to D₂ if D₁ accepted the offer b₁. This can be the case where b_2a is set to be sufficiently high, or specifically be greater than q · s₂.

In my model, the strategy of D₁ is defined by σ₁ : B × B × B →Δ where Δ={d＼accept, reject}. The strategy of D₂ is defined by σ₂ : H × B →Δ where Δ={d＼accept, reject} and H is the history information set, which contains the decision of D₁. Lastly, in order to focus on strategic aspect of a plea bargaining, trial cost is set to be zero.

Ⅲ. Equilibrium with Sequential Plea Bargaining

The model with simultaneous offers has a unique equilibrium in which P offers (S₁, q, S₂) and both of the offers are accepted by defendants,7 under joint negotiations.8 The equilibrium is illustrated as E in [Figure 1].

[Figure 1]

Equilibrium with Sequential Offers

Now let’s characterize the equilibrium with sequential offers. First, plea offer to the second defendant is considered. There are two possible cases to the second defendant; one in which the first defendant accepts the offer with b₁, and the other in which the first one rejects. If D₁ already decided to accept the offer, D₂ will accept the offer if and only if b_2a ≤ s₂. If D₁ rejects the offer, then D₂ will accept the offer if and only if b_2a ≤ q · s₂. Thus, the decision of D₂ depends on the history information. Then the following result is easily derived.

Lemma 1 P chooses the offer to D₂ as (b_2a, b_2r) = (k, q · s₂), where k ≥ s₂.

Proof When D₁ accepts the offer and chooses to testify against D₂, in the formal trial D₂ will be convicted with s₂ for sure. Then, P has no incentive to offer b_2a less than s₂. Note that if b_2a > s₂, then it is virtually identical to the case where P does not make offer to D₂. When D₁ rejects the offer, P has two feasible options; to make D₂ accept b_2r, or to make D₂ reject b_2r. In the case where D₂ as well as D₁ rejects plea offers, then the expected penalties on the defendants are q · (s₁ + s₂). If P offers the maximum acceptable sentence q · s₂ to D₂, conditional on the testimony against D₁, then the expected penalties on the defendants will be (s₁ + q·s₂. Thus, she prefers offering b2r=q · s2. ■

Given the offer to D₂, the offer to D₁ is analyzed. If the first defendant accepts the offer, then the penalty on him is b₁. If he rejects the offer, the penalty becomes s₁ as D₂ is going to accept the offer of b_2r = q · s₂. So D₁ will accept the offer if and only if b₁ ≤ s₁. Then the following result is immediate.

Lemma 2 P chooses the offer to D₁ as b₁ ≤ s₁.

Proof From the above argument, it is trivial as P is maximizing penalty on the defendants. ■

<Table 1> shows the payoff matrix of defendants, given the prosecutor’s offers in Lemma 1 and Lemma 2. However, the normal form of the game is somewhat misleading. As D₁ moves first and D₂ moves after that, D₁ considers his payoff via forward looking movement of D₂. It appears that D₁’s (weakly) dominant strategy is ‘Reject’, but he is going to take ‘Accept’ as D₂ is going to take ‘Accept’ regardless of D₁’s choice.9 Now I have the main result in the following proposition.

<Table 1>

The Reduced Normal Form

Proposition 1 With sequential offers, there exist an equilibrium in which P offers (b₁, b_2a, b_2r) = (s₁, s₂, q, s₂) and both of defendants accept sequentially.

Proof From Lemma 1 and Lemma 2, it is shown that the defendants will accept the offers and have no profitable deviation. For the prosecutor, who has incentive to maximize penalties on the defendants, she has no profitable deviation, as the resulting penalties (s₁ + s₂) is the maximum possible sentence. ■

In proposition 1, I consider the equilibrium in which all the defendants accept the offers and there is no formal trial. The structure of game among the prosecutor and the defendants is illustrated in [Figure 2]. Unlike <Table 1> in the normal form, sequence of the model is explicitly described. However, another equilibrium in which not all of the defendants accept the offers is feasible, which is stated in the following corollary.

[Figure 2]

The Extensive Form (1)

Corollary 1 With sequential offers, there exist an equilibrium in which P offers (b₁, b_2a, b_2r) = (s₁, k, q, s₂) where k > s₂ and only D₁ accepts the offer.

Proof Similar argument in the proof of proposition 1 holds. And still the prosecutor, who has incentive to maximize penalties on the defendants, has no profitable deviation, as the resulting penalties (s₁ + s₂) is the maximum possible sentence. ■

The result of corollary is noteworthy. Note that the equilibrium outcomes from proposition 1 and corollary 1 is explicitly ‘fair’ in the sense that the defendants will be punished with deserved sentence. Moreover, fairness of this outcome, which provides virtually no discount to the defendants, is much stronger than those in Kim (2009).10 The equilibrium is illustrated by E' in [Figure 1]. This fairness is from the assumption that the defendants jointly committed the crime, and that possibility of innocent defendant(s) was ignored.

[Figure 3] illustrates the structure of model, in which the prosecutor makes the plea offer only to D₁ in the equilibrium. While P makes a kind of packaged offer (b₁, b_2a, b_2r) to the defendants at the first stage in the equilibrium in proposition 1, she can make separate offer to the defendants sequentially. The main result underscores that utilizing sequential offers can enhance bargaining power of the prosecutor, to increase penalties on the defendants.

[Figure 3]

The Extensive Form (2)

Ⅳ. Concluding Remarks

Recently, in Korea, it was announced that the plea bargaining system will be implemented in 2011.11 It was said that this introduction of plea bargaining system aims to gain more information about the crime by allowing plea discounts to the defendant(s). So the main object of this policy change is information-gathering effect, although the plea bargaining system is going to save trial cost borne by prosecutors. However, to induce the defendant(s) to plead guilty, a prosecutor shall offer lenient penalty in the plea offer, which tends to decrease the social welfare.

The present paper demonstrates that plea bargaining can be utilized to punish the guilty at the maximum with sequential offers. The prosecutor, by strategically timing and targeting her plea offers, can increase the level of expected penalties on the culprits. Considering cost saving motive, which was refrained from the present paper, along with information gathering effect is expected to enhance our understanding of plea bargaining system, I hope.