ECON 3208 Game Theory
Topic 5. Mixed Strategies (Supplement)
Reporting a Crime Example
Note: This document corrects what I said in the lecture today. You do not need to study this
material unless interested; it is provided in response to your classmate’s question.
Suppose p equals the probability of each person calls, unlike how we defined in the class. Then, the
return from not reporting equals
0 × P r(no one calls) + v × P r(someone calls)
= v × P r(1 − no one calls)
n−1
= v(1 − p)
since the two events–no one calls and some one calls–are exhaustive. Setting it equal to the return
from reporting, v − c, we get
1
c n−1
.
(1)
p=1−
v
from the Reporting a Crime example. When n increases, the exponent 1/(n − 1) decreases. Then,
the following increases (not decrease) since 1 − vc is between 0 and 1.
1−
1
c n−1
v
(2)
Finally, p (the probability of any person calling) decreases. That is, the larger the group, the less
likely the crime gets reported. We reached the same conclusion in the class.
1