[EM] average voting power and block votes

[Forest Simmons]

In a recent posting someone reported reading of a claim that the Electoral
College system elevated the voting power of the average voter because of
its block voting feature.

Various replies showed that indeed the voting system affects the average
voting power of the voters. 

One example was random candidate: average voting power is zero.

Another example was random ballot: average voting power is the reciprocal
of the number of voters.

Another example was ordinary plurality: average voting power is
approximately (Pi*n/2)^(-.5), where n is the number of voters.

Other examples were given, but as far as I can tell the fundamental
question was left unanswered: does block voting tend to increase or
decrease average voting power? 

Here's a simple example that tips the scale away from block voting:

Suppose that a committee with nine members is divided into three
subcommittees of three members each.

A question has come before the committee for a yes/no vote.

Each subcommittee is a block in their voting system, with two out of three
members determining the position of a subcommittee, and two out of three
subcommittees determining the position of the committee.

Under this system each member has a 25 percent chance of casting a pivotal
vote.

One way to see this is as follows:

If you are a member of one of those subcommittees, then you have a 50%
chance of being pivotal within your subcommittee, and your subcommittee's
vote has a fifty percent chance of being pivotal in the final outcome,
etc.

Now suppose that the subcommittees were done away with, and the nine
committee members voted individually on a question. Under this system each
member has a 27.34375 percent chance of casting a pivotal vote.

One way to see this is as follows:

If you are on this committee, your vote is pivotal if and only if the
other eight members of the committee are evenly divided on the question. 
The probability of that is "eight choose four" divided by two to the
eighth, in other words  ((8!/4!)/4!)/2^8 which reduces to 35/128, which is
exactly 27.34375 percent.

In summary, this simple example illustrates the fact that block voting
reduces average voting power.

As we mentioned before, in the case of the US Electoral College, it turns
out that the members of the larger states have greater voting power than
those of the smaller states. Now we have another reason for abandoning the
EC: it gives the average voter less power. 

Here's another example on a larger scale:

Suppose that a certain country has N provinces with M members in each
province.

In the case of no EC the average voter's power would be approximately
  (Pi*M*N/2)^(-.5)

If each province voted as a block, then the average voter's power would be
approximately
  ((Pi*M/2)*(Pi*N/2))^(-.5)

The respective ratio of these two average voting powers is  (Pi/2)^(.5) ,
which is a number greater than one (in favor of the non-EC system).

Note that our first example is the case where both N and M are equal to
three. The exact ratio in that case was 35/32, which agrees well with the
(Pi/2)^.5 approximation. The approximation improves drastically as N and M
get larger.

Forest