1) Schulze Method – what I guess is considered the most mathematically sound method of discrete voting (the votes voters are able to caste are from a discrete sets (like the integers))
2) Fixed P-Norm – an example of a continuous voting system, utilizing one of the most common mathematical functions
3) Variable P-Norm – an example of a dynamic continuous voting system (the voting “accounting system” (as you may call it) changes based on the votes (but in an algorithmic way) that is able to overcome the shortfalls of it’s static cousin, the P-Norm Voting System
P-Norm Voting Systems and Dynamic P-Norm Voting Systems
Under a P-Norm Voting System, each voter assigns a rating to a subset of the candidates. There is a restriction that the sum of the ratings raised to the pth power is less than or equal to the weight of that voter. If all the weights of the voters are identical, any arbitrary real number chosen to be the weight of a voter will produce a congruent voting system.
Schulze Method scorecard:
The passes for “always good”:
Monotonicity
Majority Domination – Single Candidate
Condorcet
Condorcet loser
Reversal Symmetry
Polynomial Time
Resolvability
Smith Criterion
ISDA
Fails for “always good”:
Consistency
Participation
Continuity of Voting Signals
Majority Tyranny (forgot what I meant by this, woops! More on this later)
Passes of “sometimes good”
Majority Loser
Majority Domination – Multi-Candidate
Passes of “sometimes bad”
Majority Domination – Single Loser
Majority Domination – Multi-Candidate
Fixed P-Norm Voting System scorecard:
Always Good Criteria Satisfied:
Monotonicity
Majority
Majority Loser
Mutual Majority
Condorcet
Condorcet loser
Consistency
Participation
Resolvability (should be trivial)
Continuity of Voting Signals (trivial)
Smith Criterion (97% chance)
ISDA (98% chance)
Always Good Criteria Failed:
None
“sometimes good” Criteron passes:
None:
“sometimes good” Criteron Fails:
Mutual Majority
Majority Loser
“sometimes bad” Criteron passes:
None
Dynamic P-Norm Voting System scorecard:
“always good” criteria satisfied:
Monotonicity
Majority Domination – Single Candidate
Majority Domination – Single Loser**
Majority Domination – Multi-Candidate **
Condorcet
Condorcet loser
Consistency
Participation
Majority Tyranny*
Reversal Symmetry
Polynomial Time (trivial)
Resolvability (should be trivial)
Continuity of Voting Signals (trivial)
Smith Criterion (99% chance)
ISDA (99% chance)
*there may exist a vote such that a p cannot be chosen that would simultaneously satisfy all the *’d criteria.
**the satisfaction of this criteria is p dependent, thus it can be made to pass when good, and not pass when bad (for the most part)
Note: Dynamic P-Norm Voting Systems unilaterally fail no standard criterion.
Theorem: no static voting system can satisfy Mutual Majority Criterion and Majority Tyranny (I forget how I defined Majority Tyranny here. Sorry, I think that the definition is semi-superficial: more to prove a point about how passing Mutual Majority Criterion is not always a good thing (i.e. it’s not a good criterion. It is not always good or always bad. There are conditions on whether or not it is a good thing or not. Those often depend on the vote. The concept may be of import, but in its current incarnation, I don’t think this criterion is legitimate (i.e. the amount of time spent discussing the issue far outweighs its value. If you are going to have an in-depth discussion on Voting Theory, or any subject, you need to make sure that all the terms are well-defined first. While the TERM may be well-defined, the criterion aspect of it is not. Thus, I think the amount of confusion of regarding actual criteria will negate any extra time spent investigating this property further. And that’s my reasoning for leaving the vagueness there, sorry))) or the Majority Loser criterion and Majority Tyranny criterion (as they are mutually exclusive).
Bottom Line:
It is a Chipotle vs. Q’doba type situation. You can walk in to both, and they look identical. You can eat at Q’doba (if you haven’t eaten at Chipotle first) and say “hey this is some good shit”. However, once you go to Chipotle, you begin to repress your Q’Doba experience. If at a later date, you have to do a rigorous comparison of Chipotle to Q’Doba, you are at a loss for words because after having Chipotle, you realize that talking about Q’Doba doesn’t do anybody any good. And that’s my excuse for not being more rigorous or more cogent :)
Prescott
v.1.0
