[EM] average voting power and block votes

[Forest Simmons]

On Mon, 23 Jul 2001, Richard Moore wrote:

> Forest Simmons wrote:
> 
> > If my toothbrush is identical to yours, it doesn't necessarily follow that
> > each of us has only half of a toothbrush.
> 
> 
> But if there's only one toothbrush... (In a single-winner 
> election there is only one decision. You can't increase the 
> power of each and every one of 9 voters beyond 1/9, if 
> collectively they have a voting power of 1.)
> 

True, all voters share in one decision, sort of like sharing a toothbrush.

Banzhaf defined voting power of an individual or set of individuals to be
the probability that their vote or votes would be pivotal in a random
election conducted by the rules of the system being studied.

By that definition voting power is just a number, and if two people share
the same number, that number is not diminished in any way.

But I guess you are trying to find a better definition than Banzhaf's that
takes into account more completely the shared nature of voting power. 

I hope the example of three subcommittees of three members each can serve
as a test case for any new definition of voting power.

Here's an experiment that might help:

Simulate both the three blocks of three and the one group of nine systems. 
In both simulations keep track of how many times the winner agrees with
the choice of the first (or randomly chosen) voter. 

If the individual voters have any real power, the long range number of
agreements should exceed the long range number of disagreements.  If the
block system produces significantly more agreements than the non-block
system, then that would constitute evidence that the block system
increased individual voting power, and tell us that Banzhaf Power is
obsolete in this context. 

I suspect that the non-block voter has an advantage in this experiment,
but perhaps not as large as the 35 to 32 advantage that the Banzhaf Power
indices would have us believe, since the experiment counts agreement and
disagreement in all cases, pivotal or not.

Forest