Quote from: Torie (from e-mail, posted with his permission)

Assume that there is a sixth vote with no ward assignment elected at

large, who votes randomly. Is there some mathematical formula available the

one could use with respect to the relationship between the percentage of

votes the at large alderman has on the council, and the amount by which the

deviation in the number of votes the other councilmen,while still not

exceeding the 10% Limit? Could the number of council votes each of the five

aldermen have vary *at all* while still staying with the 10% range?

I don't believe that there is a mathematical formula.large, who votes randomly. Is there some mathematical formula available the

one could use with respect to the relationship between the percentage of

votes the at large alderman has on the council, and the amount by which the

deviation in the number of votes the other councilmen,while still not

exceeding the 10% Limit? Could the number of council votes each of the five

aldermen have vary *at all* while still staying with the 10% range?

Power share does not vary proportionately with voting weight. As we saw above, we could adjust the weights from (20, 20, 20, 20, 20) to (30, 20, 20, 20, 10) and see no change in voting power, and then suddenly at (31, 20, 20, 20, 9) there was a catastrophic change.

On the other hand, I've noted that by using simple population-based weights, but changing the threshold for success to something slightly different than 5%+1, that I could get just as good proportionality between population share and voting power, as by changing the voting weights. I sort of did this in my examples above. I initially had 100 total votes, so that a majority was 51 or 51%. In my last example, I used 101 total votes, and a majority of 51, or 50.5%. And in an unweighted 5-member body, a majority is 3/5 or 60%.

Simple-minded judges won't accept a redefinition of "majority" even if the new percentage is only slightly different and still represents the notion of "bare concurrence:.

But we can trick the judges by adding an extra weighted vote. Let's say that the at-large member has a 10% voting share. Then if he joins with district members who have votes representing more than 40% of the total vote, then the motion passes. A district member who is critical to getting past 40%, is also critical to the combination including the at-large member.

But we could boost the at-large member to 11%, and proportionately reduce the other shares. The district members now only produce more than 39% of the total. In the original version they needed 40/90 of the district vote (or 44.4%). In the second they needed 39/89 (or 43.8%).

We've subtly switched from "majority" which is fixed at 50%+, to "significant district concurrence" which can float.

This currently happens in Hudson, where the President's voting weight varies quite widely between votes for a simple majority, and the 2/3 and 3/4 super-majorities.

Use of an odd number of members, plus a president may have other problems. Consider an unweighted council. If the district members vote 2:3, the president can not change the result. Only if they vote 3:2 can he in effect veto their decision.

If there were an even number of members, he has a casting vote, whether formally or informally. If the district members vote 3:3 he breaks the tie. If the district members vote 4:2, then he has no effect.

Quote

You mentioned there is some kind of matrix chart that could be used

(presumably on an excel spreadsheet), that allows one to calculate these

critical vote percentages based on number of council votes assuming random

voting combinations of aldermanic votes? If so, may I have it?

(presumably on an excel spreadsheet), that allows one to calculate these

critical vote percentages based on number of council votes assuming random

voting combinations of aldermanic votes? If so, may I have it?

I use Computer Algorithms for Voting Power Analysis, in particular the program lpgenf.

If you set your voting weights in a column, you can copy and paste them directly in the "Weights" box, and be sure to set the correct "quota" or majority. You can then copy and paste the results from the web page into your spreadsheet using the "Match Destination Formatting" which will simply grab the numbers.

You are most interested in "Swings". The Normalized Banzhaf Index is the power share among the voting members. But in Hudson you want to use the power share among the district members, which is each member's swings divided by the total number of swings, excluding those of the president.

Add in a column with population, you can calculate population share and power share, and deviation (power-share / population-share - 1). All of these are expressed as percentages. I usually copy the results from the program over to the right side of my spreadsheet, and then copy the swings into a column used for computation.

The spreadsheet that I use is set up specifically for Hudson (11-members). It brute force generates the 2048 voting combinations, and then counts critical swings, etc.

In either case, using the program or the spreadsheet, iteration is doable, but not automatic. I could clean up the spreadsheet and send you a copy.

Quote from: Torie (from e-mail, posted with his permission)

In my view, as a legal matter, both (1) the discrepancy between the ward's

percentage of the population of the city and its percentage of the total

number of votes on the council, as compared to any other ward, cannot

exceed by more than 10% absolute number of council votes between the five

individual ward alderman cannot vary from the population in the wards by

more than 2% (so if there were two wards, one with 60% of the population,

and the other with 40% of the population, if there were 100 council votes

to allocate, the vote number for the 60% ward could go no higher than 62.4

votes, with the other no less than 37.6 votes ((2.4/60) +(2.4/40) = 10%),

and (2) the percentage number of critical votes cast given the number of

votes assigned, cannot exceed that 10% deviation amount from population

using the same mathematics as in (1). In the end, I am wondering just how

much ward populations can vary percentage wise from each other while still

fitting into both 10% tests. It seems to be with single alderman wards the

percentage population variance cannot be much with 5 wards, because the

critical vote percentage amount tends to go up exponentially vis a vis the

council vote percentage, until by the time you hit 50.1%, you are casting

100% of the critical votes, which is about a 100% deviation.

The 10% limit may not be applicable to local governments. See percentage of the population of the city and its percentage of the total

number of votes on the council, as compared to any other ward, cannot

exceed by more than 10% absolute number of council votes between the five

individual ward alderman cannot vary from the population in the wards by

more than 2% (so if there were two wards, one with 60% of the population,

and the other with 40% of the population, if there were 100 council votes

to allocate, the vote number for the 60% ward could go no higher than 62.4

votes, with the other no less than 37.6 votes ((2.4/60) +(2.4/40) = 10%),

and (2) the percentage number of critical votes cast given the number of

votes assigned, cannot exceed that 10% deviation amount from population

using the same mathematics as in (1). In the end, I am wondering just how

much ward populations can vary percentage wise from each other while still

fitting into both 10% tests. It seems to be with single alderman wards the

percentage population variance cannot be much with 5 wards, because the

critical vote percentage amount tends to go up exponentially vis a vis the

council vote percentage, until by the time you hit 50.1%, you are casting

100% of the critical votes, which is about a 100% deviation.

*Abate v Mundt*and

*Roxbury Taxpayers Alliance v Delaware County Board of Supervisors*In the latter case, the deviation range was 16.79%.

5% of 1280 is 26, which doesn't permit much flexibility when many individual blocks have more than 26 persons.

Nearly equal-population wards can be a problem with weighted voting. See Cortland County, which has a 17-member legislature where the districts are either combinations of towns, or divisions of the city of Cortland or the town of Cortland. Each member has a voting weight equivalent to the population they represent. But the districts are so similar in size, that any combination of 9 members represents a majority of the population (and therefore a majority of the legislative votes), and no combination of 8 members represents a majority.