The Robber Asks to be Punished



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The Robber Asks to be Punished

Uri Weiss*

Supervisors: Robert J. Aumann and Ehud Guttel
We have a strong intuition that increasing the punishment leads to less crime. Let's move our glance from the punishment of the crime itself to the punishment of the attempt to commit a crime. We will argue that the more severe the punishment of the attempt to rob, i.e. of the threat, “give me the money or else…”, the more robberies and the more attempts will take place. That is because the punishment of the attempt to commit a crime makes the withdrawal from it more expensive for the criminal, making the relative cost of committing the crime lower. Hence, the punishment of the attempt turns it into a commitment by the robber, and makes incredible threats credible. Therefore, the robber has a strong interest in increasing the punishment of the attempt.

2. Example

Imagine two different legal systems: lenient and severe. In the lenient system the expected punishment for attempted robbery (the threat) is 1 year in prison, while in the severe system it is 7 years in prison. In both, the expected punishment for robbery is 8 years in prison and for murder, it is 15 years in prison. Now the robber needs to decide whether or not to attempt a robbery, which for him is "worth" 10 years in prison. I.e., he is indifferent between doing nothing and a lottery that with probability 1/2 yields the expected benefit from the robbery, and with probability 1/2 yields the expected 10 years in prison.

The sequence of events is as follows:



  • First, the robber needs to decide whether to attempt to rob – i.e. to threaten – or not.

  • If he chooses not to make an attempt, then that’s the end of the game and the outcome is “no threat”. If he chooses to make an attempt, the victim decides whether to give in or not.

  • If the victim gives in, then that’s the end of the game and the outcome is “successful robbery”. If the victim does not give in, then the robber needs to decide whether or not to kill the victim and take the money.

  • If the robber decides to carry out his threat, then that’s the end of the game and the outcome is “murder and robbery”. If the robber decides not to carry out his threat, then that’s the end of the game and the outcome is “withdrawal from the threat”.

Hence, the decision tree in the lenient system is this:

We can see that in the lenient system, if the victim does not give in, the robber will withdraw (as -1 is greater than -5). Therefore, the victim will not give in when threatened. Therefore, the robber will not even make a threat (as 0 is greater than -1).

We can express this analysis as follows:

However, in the severe system, the decision tree is as follows:



In the severe system, if the victim does not give in, the robber will carry out the threat (as -5 is greater than -7). Therefore, the victim will give in. So, the robber will make the threat (as 2 is greater than 0).

We can express this analysis as follows:


The conclusion is that in our example lessening the punishment for an attempt will

prevent the robbery and even the threat!


3. The model

m - The money

qr – the probability that the robber will be caught in case of a robbery (and then will be punished and the money will be taken from him)

qt – the probability that the robber will be caught in case of an attempt (and then will be punished for the attempt)

qm – the probability that the robber will be caught in case of a murder (and then will be punished for murder and the money will be taken from him)

pr - The punishment for successful robbery

pt – The punishment for the threat (the attempt to rob)

pm – The punishment for murder

Pr = qrpr (the expected punishment for robbery)

Pt = qtpt (the expected punishment for the threat)

Pm = qmpm (the expected punishment for murder)

M = (1-qr)m (the expected benefit from a successful robbery)

M' = (1-qm)m (the expected benefit from taking the money by murder)

We will take the value to the victim to of being murdered to be -100.

The sequence of events is as described above.
Hence, the decision tree of the model is:

When will the Robber attempt a robbery?

He will do so if and only if:

1. The expected net cost of carrying out the threat (i.e., expected cost less expected benefit) is smaller than the expected cost of withdrawing,

and

2. The expected punishment for a successful robbery is lower than the expected benefit from it.


I.e., the robber will rob if and only if:

1. Pm-M'


t

and


2. M>Pr
The conclusion is that in the described game, the more severe the expected punishment for an attempt to rob (i.e. the threat, “give me the money or else…”), the more robberies and threats will take place.
3. Discussion

a. Intuitive Explanation

We have shown by an example and model that increasing punishment on the attempt to rob - .i.e., on the threat, will lead to more robberies and threats. Let us now explain it intuitively. Increasing punishment on threats leads to more robberies and threats, since this makes incredible threats credible ones. Increasing punishment may make incredible threats credible ones since this makes the cost of withdrawing form the threat more expensive, makes the cost of relative cost of caring out the threat cheaper. This is so because after the threat was made the punishment for the carrying out the threat becomes "sunk cost". The robber will bear the expected punishment of threat regardless to the question if it be carried out or not. Therefore, the expected punishment for murder the robber "sees", becomes now lower. Shortly, the punishment on the attempt to commit a crime makes the withdrawal from it more expensive for the criminal, making the relative cost of committing the crime lower.

Another conclusion from the model is that Robbery is also a situation of Litigotiation – negotiation in the shadow of the law. Hence, the theories of negotiation may be applicable in this situation.

How is it possible that is better for a person to be punished? In interactive situation it may be better for a person to limit his set if actions or to make some of his potential action expensive. It is, for example, the case in the "Chicken" game. One mechanism, to limit the set of options of a person is the law. If in the chicken game one player were subjected to a law that deviating leads to criminal punishment he would win the game; and therefore this limitation will promote his interest. In the "battle of sexes" it is also true that a player would be benefited if it was forbidden for him to go to the show the other player prefers. That is also the story regarding robberies. A robbery is a game of Litigotaion, .i.e. litigation under the shadow of the law. Giving in before the robber is like accepting a settlement proposal. The robber is equivalent to the plaintiff, sometimes to one with Negative Value Suit (NEV). If the law was punished the plaintiff for canceling a suit, his threat to continue suing, would become more credible, what promotes his interest.



  1. Modifications of Assumptions:

1. Adding limiting assumption

  • Let’s assume that in the given legal system the court is obliged to impose a punishment of aPr on attempt.

  • Hence, the robber will rob if and only if:




        1. M>Pr

        2. Pm-M

.i.e., the robber will rob if and only if:

(Pm-M)/a


  • Conclusion: we see that in this case we need to impose a lenient enough punishment or a severe enough punishment in order to prevent the robbery.

2. Repeat Robber

  • When the robber is a repeat player, he gains also reputation (r) form the committing of the threat. Therefore, this time his benefit from the committing of the threat will be M+r-Pm

  • Therefore he will try to rob, if and only if:

  • M+r-Pm>-Pt and M>Pr.

  • Therefore, our conclusion does not changed substantively.

3. When the victim does not know M

  • The one robbed can conclude that M>Pr from the visible preference of the robber to try to rob.

  • The one Robbed will give up if he believes that Pm-M

  • If Pm-Pr

  • Therefore, in a legal system in which Pt>Pm-Pr, for every information set the one robbed is in regarding M, the threat is credible.

  • Therefore, also when the one robbed does not know M, a severe enough punishment for the threat will lead to a result of “successful robbery”.

4. When some victims give up to any threat:

  • q of the victims give up to any threat and 1-q of the victims give up to only credible threats .

  • Hence, the robber will makes the threat iff:

  • 1. a. Pt

  • and

  • b. qPr+(1-q)Pt

Pt< (q/1-q)(M-Pr)

  • or

  • 2.a Pt>Pm-M (the threat is credible)

  • and

  • b. M>Pr (it is rational to rob when there is credible threat)

  • Therefore the robber will not try to rob iff:

  • 1. (q/1-q)(M-Pr)

  • or

  • 2. M

  • Let us now find M* - the most expensive asset the law may prevent form being robbed.

  • Pm-M*=(q/1-q)(M*-Pr)

  • M*[1+(q/1-q)]=Pm+(q/1-q)Pr

  • M*=(1-q)Pm+qPr

  • Let us now find Pt* - the punishment for the threat that may prevent the robbery of M*.

  • Pt*=Pm-M*=(q/1-q)(M*-Pr)

  • Pt*=Pm-[(1-q)Pm+qPr]=(q/1-q) [(1-q)Pm+qPr-Pr]

  • Pt*=q(Pm-Pr)=(q/1-q)[(1-q)(Pm-Pr)

  • Pt*=q(Pm-Pr).

Conclusion



The more severe the punishment on the attempt to rob, i.e. on the threat, “give me the money or…” will be, the more robberies and the more attempts will take place. That is because the punishment on the attempt to commit a crime makes the withdrawal from it more expensive for the criminal, making the relative cost of committing the crime lower. We have shown that this conclusion is valid also when we make the modification we did to the assumption. In the case of punishment for threats, there is an "Incentives Reversal"



* * PhD Candidate, the Center for The Study of Rationality. This paper is written under the supervision of Professor Robert J. Aumann and Professor Ehud Guttel. My gratitude is dedicated for them for their help that improved significantly this paper. I also wish to thank Ariel Porat, Eric Maskin, Alon Harel for our conversation about. I wish to thank the participants at The Israeli Law and Economics Conference, Stony Brook Game Theory Festival and Bonn University Law and Economics Workshop for their remarks.



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