Various Flavor Variables of Elections:
Repitition:
1)Single round
2) Multiround
3) Continuous (portfolio)
4) Ad hoc (congress voting on bills)
5) Fixed time length
Number of Candidates
1) Single Candidate
2) Multi-Candidate
3) Variable Num Candidates
Hopefully I will come back to that subject, but for now I am assuming all Single Round Elections.
Voting System Criteria
Below I grouped the commonly listed “Voting System Criteria” as well as the criteria regarding Arrow’s Impossibility Theorem (and yes, a lot of it is taken from Wikipedia. It provided simplicity in cutting and pasting. I’d say that the quality of information is like that my proofs, good enough for now. Sorry, I would love to be able to say that everything is entirely rigorous, but I haven’t had the time to make that happen. Apologies. But I am confident enough in the mathematics to say that the conclusions should be true).
Majority criteria
A) Majority Domination – Single Candidate (aka Majority Criterion) —If there exists a majority that ranks (or rates) a single candidate at the top, higher than all other candidates, does that candidate always win?
B) Majority Domination – Multi-Candidate (aka Mutual Majority Criterion, MMC) —If there exists a majority that ranks (or rates) a group of candidates higher than all others, does one of those candidates always win?
C) Majority Domination – Single Loser (aka Majority Loser) —if a majority of voters prefers every other candidate over a given candidate, then does that candidate not win?
Order Criteria
A) Monotonicity criterion (Monotone)—Is it impossible to cause a winning candidate to lose by ranking him higher, or to cause a losing candidate to win by ranking him lower?
B) Condorcet criterion—If a candidate beats every other candidate in pairwise comparison, does that candidate always win?
C) Condorcet loser criterion (Cond. loser)—If a candidate loses to every other candidate in pairwise comparison, does that candidate always lose?
D) Reversal symmetry—If individual preferences of each voter are inverted, does the original winner never win?
Competition Criteria
A) Independence of irrelevant alternatives (IIA)—If a candidate is added or removed, do the relative rankings of the remaining candidates stay the same?[clarification needed]
B) Independence of clones criterion (Cloneproof)—Is the outcome the same if candidates identical to existing candidates are added?
C) Smith Criterion - the system always picks the winner from the Smith set, the smallest set of candidates such that every member of the set is pairwise preferred to every candidate not in the set
D) ISDA - the selection of the winner is independent of candidates who are not within the Smith set.
Tactical Criteria
A) Consistency criterion—If the electorate is divided in two and a choice wins in both parts, does it always win overall?
B) Participation criterion—Is voting honestly always better than not voting at all?
Computational Criteria
A) Polynomial time - Can the winner be calculated in a runtime that is polynomial in the number of candidates and the number of voters?
B) Resolvable - Are exact ties, in which the winner could be one of two or more candidates, vanishingly rare in large elections?
C) Summability (aka Summable)—How much information must be transmitted from each polling station to a central location in order to determine the winner? This is expressed as an order function of the number of candidates N. Slower-growing functions such as O(N) or O(N2) make for easier counting, while faster-growing functions such as O(N!) might make it harder to catch fraud by election administrators.
Robustness Criteria
A) Allows equal rankings—Can a voter choose whether to rank any two candidates equally at any position on the ballot? This can reduce the prevalence of spoiled ballots due to overvotes, and can give a less-dishonest alternative to some tactical voting strategies.
B) Allows later preferences (aka later prefs)—Can a voter indicate different levels of support through ranking or rating candidates?
C) Later-no-harm criterion and Later-no-help criterion—Can adding a later preference to a ballot harm/help any candidate already listed?
D) Continuity of Voting Signals – Is the domain of possible votes continuous?
Arrow’s Impossibility Theorem Criteria
A) Non-dictatorship - The social welfare function should account for the wishes of multiple voters. It cannot simply mimic the preferences of a single voter.
B) Unrestricted Domain (or universality) - For any set of individual voter preferences, the social welfare function should yield a unique and complete ranking of societal choices. Thus: It must do so in a manner that results in a complete ranking of preferences for society. It must deterministically provide the same ranking each time voters' preferences are presented the same way.
C) Independence of Irrelevant Alternatives (IIA) - The social preference between x and y should depend only on the individual preferences between x and y (Pairwise Independence). More generally, changes in individuals' rankings of irrelevant alternatives (ones outside a certain subset) should have no impact on the societal ranking of the subset.
D) Positive association of social and individual values (or monotonicity) - If any individual modifies his or her preference order by promoting a certain option, then the societal preference order should respond only by promoting that same option or not changing, never by placing it lower than before. An individual should not be able to hurt an option by ranking it higher.
E) Non-imposition (or citizen sovereignty) - Every possible societal preference order should be achievable by some set of individual preference orders. This means that the social welfare function is surjective: It has an unrestricted target space.
Note on Arrow’s Impossibility Theorem
Arrow's theorem says that if the decision-making body has at least two members and at least three options to decide among, then it is impossible to design a social welfare function that satisfies all these conditions at once.
A later (1963) version of Arrow's theorem can be obtained by replacing the monotonicity and non-imposition criteria with:
Pareto efficiency (or unanimity) - If every individual prefers a certain option to another, then so must the resulting societal preference order. This, again, is a demand that the social welfare function will be minimally sensitive to the preference profile.
Note on Discrete vs. Continuous Voting Systems
Lack of satisfying the Continuity of Voting Signals means that candidates can change positions, yet still receive the same rating. It also means that two candidates may be different yet still be expected to receive the same rating. Such systems can only incorporate a countably infinite amount of information. Therefore, in any “real-world” application (there cannot exist certainty about payout), there are scenarios where the incorrect candidate is elected. In general, the objective function for society is discontinuous, thus preventing any differential analysis. This means that choosing an optimal candidate is a zeroth order optimization algorithm.
Continuous Voting Theory Extension Theorem (unproven):
For every discrete voting system, a set of candidates, and a set of votes, there exists a continuous voting system that produces the same results.
Prescott
v.1.0
