Per-option quorum

By Jochen Voss, last updated 2011-08-31

This page discusses a possible feature of the Debian voting system. Here I want to examine the following voting system:

Let V(a,b) be the number of votes which prefer option a over option b. Let R be some positive number (the quorum).

step 1:: remove each non-default option x, where V(x,default) < R
step 2:: Use Condorcet voting with Clone-proof Schwartz Sequential Dropping on the remaining options.
step 3:: In case of a tie after CpSSD the elector with a casting vote chooses the winner from the Schwartz set

Which of the criteria from http://electionmethods.org/evaluation.htm does this voting system preserve? In the following examples D denotes the default option and a vote like ABD means A is prefered over both B and D, and B is prefered over D.

Monotonicity Criterion (MC):: Still holds.
An option option which was previously removed in step 1 can be added if a vote ranks it higher. Everything else is as in Condorcet voting.
Condorcet Criterion (CC),
Generalised Condorcet Criterion (GCC),
Strategy-Free Criterion (SFC),
Generalised Strategy-Free Criterion (GSFC):: These do not hold.
Example: R=1, only vote: AD
A is the "Ideal Democratic Winner" and the only member of the Smith set. It is also prefered by a majority to D. But the default option D wins.
Strong Defensive Strategy Criterion (SDSC),
Weak Defensive Strategy Criterion (WDSC):: These do not hold.
If there are not enough voters, the majority has no way to prevent the default option from winning.

Result: with the introduction of an per-option quorum we loose CC, GCC, SFC, GSFC, SDSC and WDSC. We still have MC.

Note that the circumstances for these criteria failures only arise when there is an option which only a small number (e.g. less then 44 in the 2003 DPL election) of voters prefers over the default option.