Multi-district paradox

Fiorenzo Franceschini1 · Domenico A. Maisano

Analysing paradoxes in design decisions: the case of “multiple-district”

paradox

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the paradox is about Instant-Runoff Voting.

it is about choice of three options given to two separate groups

then given to the groups after they are combined in to one group.

the winners are different.

the reason is partly that when they are two groups, the loser is determined just on first round results, which are (deliberately) unbalanced in the set up.

As a choice is eliminated just by virtue of having fewer first preferences in one arbitrary districts, and no one is eliminated in the other arbitrary district, the back-up preferences come into play (or do not come into play) in deliberately unbalanced way,

Also districts are different sizes. not a lot but more than ten percent different.

Thus in two-district situation, the winners are not same as the winner in the "at--large " situation.

This structure is not transparently applicable to elections because when the two groups were combined, there would be six candidates, not just three.

And when there are two districts, the same choice could not run in both districts.

In the O1, O2, and O3 contest

District A 17 votes

IRV result O2 winner -- if O1 eliminated, O2 = 10; O3 = 7

if O2 eliminated O1 = 5 ; O3 = 12

if O3 eliminated O1 = 10 ; O2 = 7

District B 15 votes

IRV result -- first round results O1 = 6; O2 = 8 O3 = 1 O2 wins on first count. no transfer

if O1 eliminated, O2 = ; O3 =

if O2 eliminated O1 = ; O3 =

if O3 eliminated O1 = 7; O2 = 8

At-large contest 32 votes

first round results O1 = 10; O2= 14; O3 = 8

if O1 eliminated, O2 = ; O3 =

if O2 eliminated O1 = ; O3 =

IRV result O1 winner -- if O3 eliminated O1 = 17; O2 = 15

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if districts had been evenly drawn with 16 votes in each district in such a way that each district (C and D) contained the same first preferences votes -- O1 = 5; O2 = 7; O3 = 4, the district results are much closer to the at-large winner.

O3 votes Wilbert transferred so we look at the second preferences:

the eight O3 voters marked back-up references as this 7 O3-O1; O3-O2 = 1

so say district C contained 5 O1; 7 O2; 4 O3-O1

District D contained 5 O1; 7 O2; 3 O3-O1 , and 1 O3-O2

result in district C would have been:

if O3 eliminated O1 = 9; O2 = 7

O1 winning by having more votes

result in district D would have been:

if O3 eliminated O1 = 8; O2 = 8

O2 winning by by having more first pref votes

so at least in one district, winner is same as at-large winner,

and in the other the at-large winner lost by just a tie-breaker.

Proves that with only 32 voters, it is easy to artificially create situation where unexpected result is certain and where two-district structure produces result different form at large situation.

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And anyway it is possible to have two districts, of 15 and 17 voters each, that each produce same winner as the at-large winner:

District E = district C contained 4 O1; 7 O2; 4 O3-O1

winner is O1 on 2nd round

District F district C contained 6 O1; 7 O2; 3 O3-O1 1 O3-O2

winner is O1 on second round.

so yes, multiple districts can produce results contrary to overall winner but can also produce exactly the same winner.

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