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We are considering a set of people making decisions sequentially
about whether or not to purchase a product.
Without further information, we assume that the expected value of the product
is 0.
So users will want to buy the product, if additional information
suggests that product is more likely to be good,
and don't want to buy if the additional information suggests
the product is more likely to be bad.
In the previous segment, we've already set for a single decision maker.
We concluded that if she does her research and gets a high signal,
she'll buy the product.
And if she gets a low signal, she will not buy the product.
Now we're considering the second decision maker.
But what this means is that observing the first person's action
allows other decision makers to infer what her signal must have been.
If she buys, she must have had a high signal.
If she doesn't buy, she must have had a low signal.
So now consider the second person's decision.
She does her own research and can infer the signal
the first person had from the first person's action.
Assuming they both get a high signal, or they both get the same signal,
the decision is easy.
With two high signals, it's even more likely that the product is good.
So the second person would also want to buy.
And similarly with two low signals, it's more likely the product is bad.
And therefore neither of them should buy.
The interesting case is if they don't get the same signal.
So again, the first person's action contains
the information about her signal.
But let's assume the first person buys, and the second person's
research results in a low signal.
What should this person do now?
To figure out what her decision should be,
we again want to work out the base rule.
What is the probability the product is good,
given the sequence of signals we have seen so far?
So we're interested in the probability that the product is good,
conditioned on that the first two signals came out
high followed by a low.
So using base rule, this is the probability
that the product is good times the probability
that we see these signals conditioned that the product is good,
divided by the probability that we see these two signals in any case.
The first of this term is the probability that the product is good.
And that's just p.
And that's easy.
The second term, the probability that the product--
we get the signals high and low in order given the probability is good
is a bit more complicated.
High signal has probability q.
A low signal has probability 1 minus q.
And these two signals are independent.
So we need to take the product to get the probability that they
both happened.
So that gives us the numerator, p times q times 1 minus q.
And obviously, we have to work out the denominator.
What is the probability that we see the high-low pair of signals?
So as usual, this pair of signals can happen in one of ways,
either because the product is good, and we got the signals.
Or because the product is bad.
The first term-- product is good times the probability
that we got a high-low pair of signals, given that the product is good.
We just worked that.
It's p times q times 1 minus q.
And similarly we can work out the other part.
The probability that the product is bad is 1 minus p.
And now the probability that we get a high-low pair of signals
is 1 minus q times q.
This time the probability of a high signal has probability 1 minus q.
And the low signal has probability q.
So taking this and substituting back into the base formula we worked on,
we get this very complicated-looking expression-- p times q times 1
minus q in the numerator, and the probability
we just worked out in the denominator.
This looks rather complicated, but actually can be simplified
because there's a q times 1 minus q term in all of the parts.
So simplifying with that term, we end up with p divided by p plus 1
minus p, which is exactly p.
So what we learned is that the sequence of high and low signals,
this second person is now back to having a probability p
that the product is good and hence has an expected value of 0.
We're going to again assume that this person trusts her own signal
a little bit more than trusting the signal she
heard from the previous person.
And given her bad signal, she will decide not to buy.
Under this assumption, again, for both of the first two people,
we can infer a signal from their decision.
They buy if their signal is high.
And they don't buy if their signal is low.
Now we're up to thinking about the third person.
The third person now will know about the sequence of three signals--
the first two people that she can infer the signal from
and her own signal, which is the third signal.
And this is where information cascade will start.
If the first two people get the same signal,
say for example-- they both got high signals-- they both buy.
We can infer they must have had high signals.
The third person we're going to notice will ignore her own signal.
Even if her own signal is low, at this point
she can infer that the first thing consisted
of two highs followed by her low.
The majority of the signals were high.
And this is more a suggestion that the product is good.
The probability that the product is good is not again
going to be above the original probability p.
And therefore has a positive expected value.
And the rational decision for her to make
is to go with the majority of the signals and buy.
And similarly, if the first two people don't buy,
which must have happened because they had low signals,
the third person should ignore her possibly high signal
and don't buy even on the high signal.
That's because at this point, the majority of the signals
are low, which is a suggestion that the product is more likely to be bad.
Expected value is now less than what it used to be.
It's now negative.
And the rational decision is to not buy the product,
despite her personal high signal.
And again, if the first two signals turn out to be the same,
then the third person in our model is rational decision making.
And information cascade starts.
And the person ignores her own signal and goes with the first two decision
makers.
So what we learn is that the information cascade can occur,
even where all people are rational.
Rationally, they can ignore, or its rational for them
to ignore, their own signal and follow the crowd.
Because from the previous decision makers, at least the first few decision
makers, they can infer the signals that those people had.
And if they veer off some number of signals, the majority of which
is possibly say low, they should follow the majority signal,
even if their own signal is different.
Of course, it's important to notice that an information cascade can easily
be wrong.
With a good product, the first two signals
are more likely to be high than low.
But it's possible that they both come out low.
And a wrong information cascade starts.
So let's step back a second and think about what
one can do to avoid incorrect information cascades.
And one important aspect of what caused the information cascade is
that we didn't actually see people's information, only saw the action.
So sharing information can break the information cascade.
A bad information cascade gets started if the first signals were maybe high,
despite the fact that the product is actually bad.
If the later people not only made the decision
based on the previous information but also shared the information,
then that will break the information cascade pretty fast.
A few low signals shared will break that everyone's
making the decision based on the first two high signals.
And with many independent signals, people
will almost surely make the correct determination about the product.