Plea Bargaining as a Mean to Maximum Sentence

This paper outlines the findings of a model of plea bargaining with multiple defendants, in which a prosecutor makes plea offer sequentially. It is shown that plea discount can be minimized with sequential offers and that not all of defendants shall be induced to plead guilty. By allowing sequential offer, a prosecutor has more power in the plea bargaining, which may increase social welfare by giving appropriate level of punishment to the guilty.


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.
1 See US Department of Justice (2000). 2 I followed Adelstein and Miceli (2001) in that punishing truly guilty defendants is directly linked to social benefits. So less severe punishment imposed on the guilty implies less social welfare, if trial cost is ignored. 3 Kobayashi (1992) tried to explain unfair plea bargains. 4 For example, Kobayashi (1992) and Kim (2009).

Ⅱ. Model
In this section, I consider plea bargaining model which was employed by Kim (2009). A prosecutor (�) has accused two codefendants (� � , � = 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 � is expected to be sentenced of s � (>0) when he is convicted as guilty. Without loss of generality, let's assume that � 1 � � 2 . The probability of conviction is set to � if there is no testimony from defendant(s) via plea bargaining. Here, � 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, � makes plea offers � � � � � �0, ∞� to each defendant � � . 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, � � decides whether to accept or reject the offer. If the first defendant � � who � makes plea offer accepts the offer, he is sentenced to i b at court with certainty. Also if � � testifies against � � via plea bargaining, � � will be convicted with probability ���� �� when � � moves to the formal trial. For simplicity of analysis, let �� = 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 � � minimizes his expected sentence, and that � maximizes the penalties 6 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 � is the appropriate level of penalty given � � �s crime committal. So the objective of � is to maximize the probability of sentences while holding the level of sentences as high as possible.
The strategy of � is defined by the plea offer to � 2 when � rejected the offer. Note that can choose not to make plea offer to the case where � 2� is set to be sufficiently high, or specifically be greater than � · � 2 .
In my model, the strategy of ∆= �� � accept, reject�. The strategy of ∆= �� � accept, reject� and decision of � 1 . Lastly, in order trial cost is set to be zero.

Ⅲ. Equilibrium with Sequential Plea Bargaining
The model with simultaneous offers has a unique equilibrium in which �� 1 , �, � 2 ) and both of the offers are accepted by negotiations. 8 The equilibrium is illustrated as [ Figure 1] Equilibrium with 7 It is assumed that a tie in payoffs is resolved in favor of acceptance of plea offer. 8 In fact, this is true only when � and both of the offers are accepted. See Kim (2009) for details.
Plea Bargaining as a Mean to Maximum Sentence probability of sentences while holding the level of sentences as high as possible. is defined by σ � = �� 1 , � 2� , � 2� � � � � � � �. Here, � 1 accepted the offer � 1 , and � 2� is the offer when rejected the offer. Note that � 2� and � 2� need be not necessarily different. Also can choose not to make plea offer to � 2 if � 1 accepted the offer � 1 . This can be is set to be sufficiently high, or specifically be greater than In my model, the strategy of � 1 is defined by σ 1 : � � � � � � ∆ . The strategy of � 2 is defined by σ 2 : � � � � ∆ and � is the history information set, which contains the in order to focus on strategic aspect of a plea bargaining, Equilibrium with Sequential Plea Bargaining ultaneous offers has a unique equilibrium in which � of the offers are accepted by defendants, 7 under joint The equilibrium is illustrated as � in [ Figure 1].

Equilibrium with Sequential Offers
It is assumed that a tie in payoffs is resolved in favor of acceptance of plea offer.
. Here, � 2� is is the offer when � 1 need be not necessarily different. Also � . This can be is set to be sufficiently high, or specifically be greater than where where is the history information set, which contains the to focus on strategic aspect of a plea bargaining, � offers under joint ) are equilibrium 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 � 1 , and the other in which the first one rejects. If � 1 already decided to accept the offer, � 2 will accept the offer if and only if � 2� � � 2 . If � 1 rejects the offer, then � 2 will accept the offer if and only if � 2� � � · � 2 . Thus, the decision of � 2 depends on the history information. Then the following result is easily derived.
Proof When � 1 accepts the offer and chooses to testify against � 2 , in the formal trial � 2 will be convicted with � 2 for sure. Then, � has no incentive to offer � 2� less than � 2 . Note that if � 2� � � 2 , then it is virtually identical to the case where � does not make offer to � 2 . When � 1 rejects the offer, � has two feasible options; to make � 2 accept � 2� , or to make � 2 reject � 2� . In the case where � 2 as well as � 1 rejects plea offers, then the expected penalties on the defendants are � · �� 1 � � 2 �. If � offers the maximum acceptable sentence � · � 2 to � 2 , conditional on the testimony against � 1 , then the expected penalties on the defendants will be �� 1 � �·�2. Thus, she prefers offering �2�=�·�2. ■ Given the offer to � 2 , the offer to � 1 is analyzed. If the first defendant accepts the offer, then the penalty on him is � 1 . If he rejects the offer, the penalty becomes � 1 as � 2 is going to accept the offer of � 2� = � · � 2 . So � 1 will accept the offer if and only if � 1 � � 1 . Then the following result is immediate.
Proof From the above argument, it is trivial as � 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 � 1 moves first and � 2 moves after that, � 1 considers his payoff via forward looking movement of � 2 . It appears that � 1 's (weakly) dominant strategy is 'Reject', but he is going to take 'Accept' as � 2 is going to take 'Accept' regardless of � 1 's choice. 9 Now I have the main result in the following proposition. 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 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 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) With sequential offers, there exist an equilibrium in which � � and both of defendants accept sequentially.
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 �� 1 � � 2 � is the maximum possible 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 Bargaining as a Mean to Maximum Sentence 143 ) � offers 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 is the maximum possible 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 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 � 1 , ��� 2 ) Corollary 1 With sequential offers, �� 1 , � 2� , � 2� � = �� 1 , �, �, � 2 � 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 maximum possible sentence.
The result of corollary is noteworthy. Note that the equilibrium outcomes from proposition 1 and corollary 1 is explicitly 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 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 � 1 in the equilibrium. While �� 1 , � 2� , � 2� � to the defendants at the first stage in the equ [Fig 10 Fairness in Kobayashi (1992) or in Kim (2009) is relative among defendants, while that in the present paper can be seen as societal justice.
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 �� 1 � � 2 � 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 irtually no discount to the defendants, is much stronger than those in Kim The equilibrium is illustrated by � � in [ Figure 1]. This fairness is from the assumption that the defendants jointly committed the crime, and that possibility defendant(s) was ignored.
illustrates the structure of model, in which the prosecutor makes the in the equilibrium. While � makes a kind of packaged offer to the defendants at the first stage in the equilibrium in proposition 1, Figure 3] The Extensive Form (2) Fairness in Kobayashi (1992) or in Kim (2009)  � offers argument in the proof of proposition 1 holds. And still the prosecutor, who has incentive to maximize penalties on the defendants, has � is the The result of corollary is noteworthy. Note that the equilibrium outcomes from in the sense that the defendants will be punished with deserved sentence. Moreover, fairness of this outcome, which irtually no discount to the defendants, is much stronger than those in Kim . This fairness is from the assumption that the defendants jointly committed the crime, and that possibility illustrates the structure of model, in which the prosecutor makes the makes a kind of packaged offer ilibrium in proposition 1, Fairness in Kobayashi (1992) or in Kim (2009) is relative among defendants, while that in the 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.

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.