Welcome back to Math 103.
This is video number
14 on measuring power.
We have now measured the
distribution of actual power
in this weighted
voting systems, 6
colon 4, 3, 2 in
two different ways.
We have measured it according
to Banzhaf's approach
and according to the
Shapley-Shubik approach.
These two approaches differ in
the specific numerical details.
Each one comes to a slightly
different conclusion
about what fraction of the
power Northern Jersey has,
but they come to
them pretty much
the same overall conclusion
that Northern Jersey is getting
a much more than its
proportional share
that you would
have expected based
on what fraction of the
total weight it has.
And that, in turn,
was based on what
your share of the total
population it has.
So if you were intending
for Northern Jersey
to have 44 point something
percent of total power,
well, no matter how you measure
by these two approaches,
you end up concluding
that Northern Jersey gets
more, rather
substantially more power.
You also conclude,
by either method,
that Central Jersey gets rather
less than we had originally
intended, and Southern
Jersey gets a little bit less
than we had originally intended.
So they differ in the
details, but they come
to the same overall conclusion.
Now, there are examples you can
dream up, where they actually
come to very
different conclusions,
even on the general level.
But not here.
So why is that significant?
Because if you're
going to court,
making an argument that a
certain proportional voting
scheme is short changing one
of the regions of the state,
it's very helpful to have more
than one mathematical argument,
more than one piece of
mathematical evidence.
So if you rollout
the Banzhaf analysis,
and reach a certain conclusion
that Central Jersey is being
short changed, you don't want
the lawyer on the other side
to say, oh, you've got
your measurement of power,
I've got a different
measurement of power.
You want the various
measurements of power
to be reaching the
same conclusion.
And when they do, that
strengthens your argument.
Our next example pertains to
a very important real life
weighted voting system, namely
the US Electoral College.
Now, we're not going to do
a Shapley-Shubik analysis
of the entire US
Electoral College,
because there are 51 players
in it, all 50 states together
with DC.
And that's just out of control
in terms of the amount of work
that's involved.
In fact, we will,
in a few videos,
come back to the question of
just what a ludicrous amount of
work it would be,
and how it would not
be feasible to do it by hand.
It would not be feasible
to do it on a computer.
To convey the idea, but still
use some real world numbers,
let us suppose, for the
sake of a narrative,
that New York, Pennsylvania,
New Jersey, and Delaware secede
and form their own
four state country.
That would be an
insane thing to do,
but for our
mathematical purposes,
let's suppose
that's what happens.
In the actual US
Electoral College,
New York has 29 electoral
votes, Pennsylvania has 20,
New Jersey has 14,
Delaware has three.
So let's suppose that those real
life weights are the ones that
are used in the weighted voting
system that's being set up
in this new country.
And just for the
sake of example,
let us suppose that
36, of all numbers,
is selected as the quota.
I think we will come back to
the question of, well, what
if it's not exactly 36?
What happens if it's
only 34, for example?
For the moment, it's 36.
On the next slide, all
24 sequential coalitions
are written out.
24, wait a minute, 24
equals 4 factorial.
That's a number we could
have predicted here.
There were three factorial
sequential coalitions
with three players, and four
factorial sequential coalitions
with four players.
Hm, we might have to investigate
that further later on.
For the moment, let's
stick to the task
of finding pivotal players
in sequential coalitions,
counting up the number
of times each player'
is pivotal and finding
the Shapley-Shubik power
distribution of this
weighted voting system
here, based on our
electoral college example,
and then answering the
question, does any state
have a disproportionately
large or small share
of actual power compared to
its share of total weight?
You can see how
this is a real life
kind of a question when you
set up a system like the US
electoral college.
Before doing this,
you might find
it helpful to print out this
slide by going to resources
and finding the
week four folder.
Just to get us started, because
this is a little bit more
involved than previous examples,
let's do a few of these
together.
So when New York starts out
saying, yes, I want to join in,
I want to vote yes.
New York has 29 votes.
That does not meet the quota,
so New York is not pivotal.
Then Pennsylvania joins in.
Now we have New York and
Pennsylvania voting yes.
That's a total of 49 votes.
That does meet the quota,
and therefore Pennsylvania
is pivotal here.
So when New Jersey comes and
vote yes, the matter is really
already settled, the
motion is already passed.
When Delaware shows up, the
motion has already passed.
Another one I'd like
to call attention to
particularly is right here.
So Pennsylvania votes yes.
And Pennsylvania
starts things off.
So right now there
are 20 yes votes.
Then New Jersey joins in,
contributes 14 yes votes.
That makes a total of
34 yes votes right here.
So 20 for Pennsylvania,
14 for New Jersey.
Not enough yet to win.
And so it's only when Deleware
shows up and contributes
three more votes that
now the total is 37.
And that means that
Deleware is pivotal.
So Deleware is the
one who arrives
on the scene just in time to
cast the pivotal yes votes.
So Delaware, in
particular, is not a dummy.
And the non-dumminess
of Delaware
shows up in the
Shapley-Shubik analysis,
just as it would in
the Banzhaf analysis.
So I will leave the rest of
these to you to work out.
You can check your
work here, and see
who was pivotal in each
sequential coalition.
It takes a little bit of
doing working through all 24,
and finding a pivotal
player in each,
but it's really not so bad.
The bottom line
is that after you
found all of your
pivotal players
and count it up-- so
we know in advance
there are 24 instances
of being pivotal.
We know this in advance.
That's one of the great
things about Shapley-Shubik.
Since there's exactly
one pivotal player
in each sequential
coalition, the number
of instances being
pivotal is the same
as the number of
sequential coalitions.
So with four players,
that's got to be 24.
So after counting
up, it turns out
that New York has 10 instances
of being pivotal, Pennsylvania,
six, New Jersey, six,
and Delaware, two.
So we ask, how fair is
this, in the sense of how
does each player's, each state's
share of actual power according
to Shapley-Shubik compare
to their percentage
of total weight,
what they thought
they were going to get by
way of a share of power?
Well, New York is
slightly short changed,
but that's not so bad
in the scheme of things.
Pennsylvania,
somewhat shortchanged,
New Jersey is coming
out a little bit ahead,
and Delaware is
coming out ahead.
In proportional terms, Delaware
is coming out way ahead,
in the sense that
Delaware has almost twice
as much actual power as it
thought it was going to get.
Whether a player is a
dummy or does not depend
on how you measure power.
That is whether you're using
Banzhaf's approach or Shapley
and Shubik's approach.
You're either a
dummy or you're not,
just as you either
have veto power
or you don't, just as
you are either a dictator
or you're not.
So before going
onto the next video,
let's think about the following
question about the same example
based on the electoral college,
the same quota and weights
that we were looking at before.
Delaware is the state
with weight three,
so if anyone's a dummy,
it's going to be Delaware.
Now, since the quote
is 36 in this example,
here's your question.
What happens to be true about
the total weight of the players
to the left of
Delaware, in order
for Delaware to be
the pivotal player
in a sequential coalition?
In other words, if you've got
somebody, somebody, and then
Delaware, and then
somebody else,
what has to be true about
the sum of the weights
of the players
before Delaware that
would cause Delaware
to be pivotal?
For example, it might be useful
to look back at the coalition
Pennsylvania, New Jersey,
Delaware, New York.
In that order, the
sequential coalition,
here we concluded that
Delaware was pivotal.
What was true about
the sum of the weights
here that enabled Delaware to
be pivotal when it showed up?
