http://www2.warwick.ac.uk/services/exampapers/ec/2012/ec2020.pdf
Q4) part d)
We are looking for a δ ∗ such that for δ < δ ∗ each agent is better o deviating,
for δ > δ ∗ the agent is better o cooperating. Let's consider agent i, then
10
5δ
j
∗
+ Pr δ > δ
EU Cooperation|δ = δ = P r δ < δ
3+
1−δ
1−δ
Since δ j follows a uniform distribution, then P r δ j < δ ∗ = δ ∗ , while P r δ j > δ ∗ =
1 − δ ∗ . Hence
10
5δ
∗
i
∗
+ (1 − δ )
EU Cooperation|δ = δ = δ 3 +
1−δ
1−δ
i
∗
j
and
5
5δ
j
∗
EU Deviation|δ = δ = P r δ < δ
+ Pr δ > δ
15 +
1−δ
1−δ
5δ
5
+ (1 − δ ∗ ) 15 +
= δ∗
1−δ
1−δ
If δ i = δ ∗ , then EU Cooperation|δ i = δ ∗ = EU Deviation|δ i = δ ∗ . Hence
to nd δ ∗ we need to solve the equation
5δ
10
5δ
5
∗
∗
∗
∗
δ 3+
+ (1 − δ )
=δ
+ (1 − δ ) 15 +
1−δ
1−δ
1−δ
1−δ
i
j
∗
which implies
δ∗ =
5
8
Therefore, introducing incomplete information about the discount factor of the
other agent makes cooperation more dicult 58 > 21 . As the number of agents
increase uncertainty about the discount factor of the other agents makes cooperation more and more dicult.
1
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Q4) part d) We are looking for a δ