[EM] average voting power and block votes

[Richard Moore]

When I posted on this topic a while ago, I was trying to argue that "voting
power" should not be calculated solely in terms of pivotal probabilities.

In the 3x3 committee voting blocks, if you are in block A and claim to be
casting the pivotal vote, couldn't the other voter who votes the same as you
make the same claim? Therefore, your voting power is shared with that voter.
Moreover, if your block claims to be pivotal, isn't there another block with
the same claim, with possibly two voters in that block claiming pivotal votes?

First, the voting power has to be shared by the two subcommittees, and then
it has to be shared within each subcommittee, so I think the power is
reduced by a factor of 4.

Also, if three voters in a subcommittee all agree, does that mean that each
voter gets a 1/3 share of the "pivotal" vote? This consideration adds back
some of the power that was reduced by sharing pivotal votes. The semantics
is getting strained here, since this is no longer a pivotal voting situation
where
one vote can change the outcome, but does voting power really disappear
when there's no single pivotal vote, or is it just distributed among the
voters?

I take the distributed view. Philosophically, I don't like the idea that your
vote counts only if it creates or breaks a tie or reverses the outcome. I'd
rather have a power of .0000001 every time I vote, than have a power of
0 except for once every 10 million times I vote.

Votes by the 3 subcommittees/power of each subcommittee:
AAA    1/3, 1/3, 1/3
AAB    1/2, 1/2, 0
ABA    1/2, 0, 1/2
ABB    0, 1/2, 1/2
BAA    0, 1/2, 1/2
BAB    1/2, 0, 1/2
BBA    1/2, 1/2, 0
BBB    1/3, 1/3, 1/3

Average power of each subcommittee:
( 1/3 + 1/2 + 1/2 + 1/2 + 1/2 + 1/3 ) / 8 = 1/3

Similarly each member of a subcommittee has an average power of
1/3 of the subcommittee's total power, or a power of 1/9.

Richard

PS -- Though I still haven't seen the Natapoff theorem, I speculate
that what he may have been on to is a similar analysis to Forest's,
but with unequal block sizes and with the smaller subcommittees
(or states) getting exaggerated voting power, as happens in the US
EC. Since the voters in the smaller blocks are more likely to be
pivotal within their blocks (as contrasted to pivotal in the overall
election), exaggerating the power of those blocks intensifies those
pivotal probabilities.


Forest Simmons wrote:

> 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