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Common Pool Resource (again)
This week’s experiment is virtually identical to the last experiment, with one
small tweak: If the resource is allocated for your group, I will give your group’s
messenger a medal for each group member, who must wear the medal around
their necks for the remainder of the experiment. You do not get to keep the
medals, I will collect them at the end of the experiment. The medals give you
a visual indication of who in the class has had sucess in the game. In case you
need a reminder about the experiment...
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Our Experiment
Our experiment will consist of four rounds of the same scenario. Groups of n
members share a common resource pool where the size, z, is not known. Each
group member j ∈ {1...n} requests rj units from the random resource pool. If
(r1 + r2 + ... + rn < z), each member j is granted his/her request; otherwise, all
group members get nothing.
Your first task is to form groups: anywhere between 2 and 5 people per group
is acceptable. Note that the expected size of the resource (per person) E[Z]
n = 10
is held constant regardless of the size of your group. Specifically, for all group
sizes the resource level is a random variable drawn from a uniform distribution
over support [0, 20n]. For example, if n = 2 then z is distributed uniformly over
support [0, 40], E[Z] = 20 and the expected resource per person is E[Z]
= 10.
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At the other end of the spectrum, if n = 5 then z is distributed uniformly over
support [0, 100], E[Z] = 50 and, once again, the expected resource per person
is E[Z]
= 10. See Figure 1 for a graphical depiction of the densities for various
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group sizes. Expected profits for individual j are equal to their request size rj
times the probability that the resource is allocated P (r1 + r2 + ... + rn < z)
E[Πj ] = rj P (r1 + r2 + ... + rn < z) = rj
20n − r1 − r2 − ... − rn
20n
(1)
Once you have formed a group, each member of your group privately fills
in their information slip with their student id on one side and their
request on the other: Resource requests are to be made independently
and anonymously. A messenger for the group collects all the information slips
for their group (student id side up) and brings them to me. I will enter the
requests into a computer program that determines if the (randomly generated)
resource is large enough for the requests to be allocated. I will then tell the
messenger whether or not the requests were allocated, who in turn tells the other
members of the group. The messenger also returns the slips to the members of
their group, who in turn record whether the requests were allocated, and their
resulting profit. Each round is independent, and you must change groups after
the completion of each round.
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density
20
40
60
80
100
0
20
40
60
80
100
Z
density of Z when n=4
density of Z when n=5
density
0.000
0.010
0.000
0.010
0.020
Z
0.020
0
density
0.010
0.000
0.010
0.000
density
0.020
density of Z when n=3
0.020
density of Z when n=2
0
20
40
60
80
100
0
Z
20
40
60
80
100
Z
Figure 1: Densities of z for n ∈ {2, 3, 4, 5}. Dashed line is the expected value.
After the experiment is complete I will collect your information slips as you
leave the class. It would be helpful if you could hand me your slips in order and
I will staple them. Also don’t forget to record your total profit (for all rounds)
on the top slip.
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Warm up
Suppose you join a group of 4 other people. For your group:
• Suppose that you request 10, and the other requests are 10,10,10,10. What
is the probability that the realization of the resource exceeds the sum of
all the requests?
• What is your expected payoff if you request 10, taking as given the requests
of the other people?
• What would be your expected payoff if you requested 30, taking as given
the requests of the other people?
Suppose you join a group of 1 other person. For your group:
• Suppose that you request 10, and the other request is 10. What is the
probability that the realization of the resource exceeds the sum of all the
requests?
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• What is your expected payoff if you request 10, taking as given the requests
of the other person?
• What would be your expected payoff if you requested 15, taking as given
the requests of the other person?
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