In this video, I'm going to go over
the nitty gritty details
of this model.
This involves a little bit of math,
so it's an optional video
and if you feel comfortable with math
go ahead and watch it,
but if you don't, then you can skip it.
without really losing too much insight
into how the model works.
Here's the details.
We're going to let N be the number of
strategies each person has,
And M be the number of weeks for which
the attendance is known.
Let t be the current time which is,
that is, the current Thursday
that we're predicting the
attendance for,
and the previous weeks,
therefore are:
t minus one, t minus two, etc.
And we're going to denote the
attendance at time t by capital A of t.
Each strategy is going to look like this.
So a strategy that's going to predict
the attendance for week t,
is going to be a hundred times a sum,
where the sum is some weight w,
which is some constant,
times the attendance at the previous week
plus some other constant times
the attendance at the week before that,
etc., all the way up to the memory limit,
plus some other constant.
So this is a very general,
sort of linear, combination
of previous weeks, times a hundred.
And these weights, w sub i, are
in the interval minus one to one.
Each person has N such strategies,
and what makes the strategies different
from one another are the weights,
and some of these weights can be zero,
which means we ignore
the data from that particular week,
and the strategies are different
from person to person.
Now one of these strategies is determined
to be the current best one.
We'll denote that by S star,
and the decision that each person makes
is as follows:
If S star of t, that is the best
strategy that we've determined,
I haven't told you yet how that's
been determined,
but I'll tell you in a minute,
so if S of t is greater than the
overcrowding threshold,
which is some value like 60,
then don't go, otherwise go.
This S star of t is different
for each person,
but everybody uses this rule,
with their own S star of t
to decide whether or not to go.
OK
Moving on,
To start off with,
everybody's N strategies
are initialized with random weights.
Everybody is given an initial history
so that they can actually
start making decisions.
The previous M time steps
is initialized at random,
with some value between 0 and 99.
So you can make some predictions
on the first M time steps.
And, here's how the best current
strategy is determined.
So at each time step,
after each person makes a decision,
they then learn the current
attendance at that time step,
then they determine which strategy
would have been the best predictor.
This strategy will be used by
that person on the next round.
OK so how they do that
is as follows:
Each person's going to determine,
for each time step between now
and M previous time steps,
what current strategy would
have had the least error.
The error is the difference
at each time step,
between the prediction made by a strategy,
and the actual attendance.
So this strategy S,
has a difference between
what it predicted this time and
the actual attendance
and what it predicted last week
minus the actual attendance
and so on,
all the way to the memory limit.
The best current strategy is
the strategy S star
that has this lowest error
over all the strategies.
So that's how the best strategy
is determined.
And that's it.
In the next video,
we'll look at a NetLogo implementation
of this model.