The underlying probability space
For convenience of the reader, the example is repeated
at the bottom of this page.
Answer to Question 3.
We will show that the example is consistent with the
laws of probability. We will present a probability space
on which one can define an event A, random
variables X and Y and numbers
x1, y1 and y2 with the following properties.
- P(A) = 0.5
- X is independent of A
- P(Y=y1 | X=x1) + P(Y=y2 | X=x1) = 1
- P(A | Y=y1) = 0.7
- P(A | Y=y2) = 0.8
Consider i.i.d. random variables Z(j), for j = 1,2,...,10,
such that P(Z(j) = 1) = P(Z(j) = 0) = 0.5.
Let
- A = {Z(10) = 1},
- X be the sum of Z(j) from j = 1 to j = 9,
- Y be the sum of Z(j) from j = 1 to j = 10,
- x1 = 7
- y1 = 7
- y2 = 8
It is elementary to check that the properties (1)-(5) are satisfied.
One may wonder if the "paradox" is a result of
choosing a very unrealistic scenario. In other words,
is the probability that X takes the value x1
or larger exceedingly
small? It is equal to about 9%, which seems to be quite
reasonable.
Answer to Questions 1, 2 and 4.
The information obtained from Investor II
by Investor I does not change his assessment
of the probability of A. Note that
the event {X=x1 and P(A|Y) > 0.6}
is equal to {X=x1} so
P(A | X=x1 and P(A|Y) > 0.6) =
P(A | X=x1) = P(A) = 0.5.
Investor I should not participate in the investment.
Conclusion.
It turns out that it is logically consistent and
fully rational for
- Investor II to consider the investment profitable and
- Investor I to disregard the second investor's
assessment of the probability of A despite his own lack
of knowledge of any event or random variable that is not
independent from A.
This appears paradoxical in view of our everyday
experience. Consider any investment opportunity
depending on the outcome of some random
action A.
We all have access to great volumes of
"insider" information
(i.e., information which
is not available to the public)
and which is independent
from A. For example, you know what you had
for breakfast today, which books you read in the last two weeks, etc.
We instinctively disregard all the information that
falls into the "insider and independent" category
as totally useless. In such circumstances, a friendly advice based on some
information which is not independent of A
is normally regarded as worthy of following.
The "paradox" presented on this page shows that
various pieces of information can interplay in a way
that may contradict the intuition.
Insider trading paradox (example repeated for the convenience
of the reader)
Two investors, Investor I and
Investor II are friends and insider traders.
Each one of them has access to information not available
to the general public. Being friends, they share some
information and advice but they are not totally open
with each other. Investor I knows an
"insider" who can provide him with the value
of a quantity X, which to the general public
appears to be a random variable.
Likewise, Investor II can learn the value of a random variable
Y by calling a secret source.
Each investor knows what type of information the other one has
(for example, Investor I knows that
Investor II knows tha value of the random variable Y)
but they do not know the information itself
(for example, Investor II never tells Investor I
the value of Y before it becomes public).
A new investment opportunity appeared on the horizon.
If the Congress passes a certain law by the end of the year
(let us call this event A) then an investment
in a certain company would greatly increase in value. Otherwise,
all the money would be lost. The probability of A
is 0.5. The potential investment would be break-even
if the probability A were 0.6 or greater.
One could make a potentially very profitable
investment if one could know that the probability of
A is greater than 0.6. The value of this
probability can be affected by the insider information, of course.
Investor I checked the value of X - it happens
to be x1. He did it mostly out of habit, because
he knows that X is independent of A.
Hence, P(A|X=x1) = 0.5. The insider information
is of no help to Investor I.
Soon after that, his friend, Investor II, called him
and told him that, in view of his insider information,
the investment is definitely profitable, i.e.,
the conditional probability of A given the true value
of Y is greater than 0.6. However,
Investor II did not tell Investor I the value of
Y.
Since Investor II did not tell Investor I the value of
Y, the first investor decided to check if the optimism
of the second one is consistent with his own information.
It turned out that given {X = x1}, the random variable
Y can take only two values, y1 or y2.
The conditional probabilities happen to be
P(A|Y=y1) = 0.7 and P(A|Y=y2) = 0.8.
Hence, Investor II must be right - no matter what
the value of Y is, the investment seems to be profitable.
