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
* 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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"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:
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The Decision Tree in The lenient System
Don't attempt
to rob
Attempt
to rob
Robber
0, 0
Don‘t give in
Give in
Victim
2, -10
Don‘t carry
out the thraet
Robber
-1, 0
Carry out the
threat
-5, -100
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:
Don‘t Attempt
to Rob
Attempt
Robber to Rob
0, 0
Don‘t give in
Give in
Victim
2, -10
Do not carry
out the threat
-1, 0
Robber
Carry out the
threat
-5, -100
However, in the severe system, the decision tree is as follows:
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The Decision Tree in The Severe System
Don't Attempt
to Rob
Robber
Attempt
to rob
0, 0
Give in
Do not give in
Victim
2, -10
Do not carry
out the threat
Robber
-7, 0
Carry out the
threat
-5, -100
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:
Don‘t attempt
to Rob
Attempt
Robber to Rob
0, 0
Give in
Don‘t give in
Victim
2, -10
Don‘t carry
out the threat
-7, 0
4
Robber
Carry out the
threat
-5, -100
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:
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The Decision Tree in The Model
Don’t attempt
to Rob
Robber
Attempt
to Rob
0, 0
Give in
Do not give in
Victim
M-Pr -M
Don’t carry out
the threat
Robber
-Pt 0
Carry out the
threat
M’-Pm -100
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'<Pt
and
2. M>Pr
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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
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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.
b. 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<aPr
.i.e., the robber will rob if and only if:
(Pm-M)/a<Pr<M
•
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
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•
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<Pt.
•
If Pm-Pr<Pt, then Pm-M<Pt.
•
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<Pm-M (the threat is incredible)
•
and
•
b. qPr+(1-q)Pt<Mq (this is rational to rob when there is no credible threat)
Pt< (q/1-q)(M-Pr)
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•
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)<Pt<Pm-M
•
or
•
2. M<Pr
•
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
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assumption. In the case of punishment for threats, there is an "Incentives
Reversal"
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