Mathematical criteria obeyed and disobeyed by score voting

This discussion is based on a French→English translation of this and this web page at the Vote de Valeur site, with additional remarks then added by us.

Criteria obeyed by score voting

1. Generically untied: In the limit of a large number of voters with some nonzero proportion voting randomly, the probability of a tied election (non-unique winner) goes to zero. Indeed with "continuum range voting" ties have probability=0 if there is even a single uniformly-random ballot.

2. No dictator (and no vetoer): "No elector shall be able to force a winner of his choosing (or force somebody of his choosing not to win) regardless of the preferences of everybody else."

3. Unanimity: "If a candidate is the unanimous favorite of every voter, it should be the winner." Also, "If every ballot scores A ahead of B, the voting method output shall rank A ahead of B, and in particular B cannot win."

4. Indifference to Irrelevant Options: "The voting system's output's relative ranking of two candidates A,B should depend only on their relative positions within ballots and not on those for other candidates C. So if/when we consider only a subset of candidates, the voting system should not change its output-ordering within this subset."

Arrow's impossibility theorem: Says that there is no voting system based on rank-orderings as ballots that meets criteria 2-4 and never is tied. But score voting (with some tiebreak rule added) does meet these criteria. It can, and does, evade Arrow's "proof of impossibility" because it does not use rank-orderings as ballots, but rather numerical candidate-ratings.

5. Monotonicity: Both

  1. If somebody increases their vote for candidate C (leaving the rest of their vote unchanged) that should not worsen C's chances of winning the election.
  2. If somebody decreases their vote for candidate B (leaving the rest of their vote unchanged) that should not improve B's chances of winning the election.

6. Loser-independence: "In an election where X is a winner: after adding a new candidate Y, the new winner can only be X or Y." Similarly, if a losing candidate is removed, that should not alter the winner.

7. Cloneproof: Definition: "A set of clones is a set of candidates ranked adjacent to each other on each ballot (in some order which may vary depending on the ballot), for voting systems based on rank-orderings as ballots. For voting systems based on numerical ratings, the 'clones' must all have ratings equal to within ±ε in the limit ε→0, on each ballot."

Cloneproofness: "If clones are added (or eliminated, so long as at least one remains), then the winner should not change except perhaps to a clone of the old winner."

8. Partitionable: "Partition the voters into two subsets, such the Northern and Southern ones. Suppose that the same candidate X is elected by the North and by the South. Then X must be elected countrywide."

9. Participation: "If a new ballot which scores the current winner W top, is added to the election, this should not alter the winner." (Actually, that was a special case of #8.) Also: "If a new ballot which scores some loser L bottom is added to the election, this should not cause L to win."

10. Reversal symmetry: "If a candidate is winning and then we 'reverse' every ballot, the same candidate will lose." (Applied to vote value, the operation is to negate the scale of values; for a rank-ordering ballot it is to reverse all the orders.)

11. Does not incentivize Favorite-Betrayal: A voter cannot achieve a better election result (in her view) by dishonestly scoring her true-favorite candidate, below topmost.

12. Neutrality: (aka symmetry under candidate renaming): Permuting the candidate names on each ballot permutes their winning probabilities in the same way.

Smith-Simmons impossibility theorem: No voting system based on rank-orderings as ballots can obey criteria 2, 7, 11, 12. But score voting accomplishes this "impossible" feat because its ballots are not rank-orderings of the candidates, but rather assignments of numerical scores to them.

Asterisk about "normalized" and "no opinion" score voting rule-variants: A score ballot is "normalized" if it employs the maximum allowed score and employs the minimum allowed score, at least once each. Some criteria above obeyed by score voting (e.g. #4, 6), can become disobeyed if we demand all ballots be normalized. Also, in "highest average wins" score voting rules where voters are allowed to express "no opinion" about candidates (instead of scoring them), #8, 9 can be disobeyed.

13. NESD: Assume all voters initially honest, and election initally untied. A single-winner voting system obeys the "NESD property" if, after every voter "exaggerates about A and B," i.e. changes her ballot to now rank A top and B bottom (or B top and A bottom; which depends on the voter and is "honest"), leaving it otherwise unaltered, then there exist untied elections in which somebody besides A or B wins. (The NESD criterion is not applicable to "multiround" voting systems, in which the voters vote a second time, knowing the result of the first election. Perhaps it could be made applicable by modifying the definition of NESD, but we do not attempt that now.) I speculate that 1-round voting systems failing the NESD criterion will yield 2-party domination.

Contrast with other voting systems: Plain plurality voting disobeys 4, 6, 7, 10, 11, 13. Instant (and non-instant) runoff voting both disobey 4, 5, 6, 8, 9, 10, 11, 13. The non-instant 2-round "plurality+top-two" runoff system (used in France 2013) and various IRV-like systems involving, e.g. ranking only the top three (but not all) candidates (used in San Francisco 2013), also disobey 7. The "Borda count" disobeys 4, 6, 7, 11. All Condorcet systems disobey 8, 9, 11, 13. Score-voting-like systems based on median instead of average score disobey 8, 9 (they would also disobey 1 & 13, but if certain tie-breaking schemes are added as in GMJ instead of just naively using the median score alone, then 1 & 13 come back to life).

Naively desirable-sounding criteria disobeyed by score voting

1. Secret preferences: (Also has been called "later no harm"): "If a candidate X wins and the preferences are changed among the candidates scored below X on some ballot, then X still should win."

(We think this criterion is undesirable: raising some rival R of the current winner W from last to second place on your W-top ballot, should instead be allowed sometimes to cause R now to win. Those in favor of this property have argued that without it, voters will fear hurting their favorite by honestly scoring their second-favorite high, thus will be incentivized to lie about their second-favorite. But with, say, a voting method like IRV complying with this property, voters still have incentive to lie about both their favorite and second favorite.)

2. Majority-top: "If a candidate X is the favorite of a majority of voters (i.e. is scored top on more than 50% of the ballots) then X must win.

(We think this criterion is undesirable: a candidate scored top by 51% and bottom by 49% of the voters, might really be a worse choice than one scored near-top by 100% of the voters.)

3. Mutual-Majority-top: "If every member of some subset X of candidates is scored above all others on more than 50% of the ballots, then some member of X must win.

(Undesirable for same reason.)

4. Condorcet winner: "If there is a Condorcet winner, i.e. a candidate C who is scored above R on more than 50% of the ballots (for every rival R, albeit the ballot-set is allowed to depend on R), then C must win."

But: Score voting does obey the closely related condition that "If there is a candidate C who would win an election versus R if all candidates besides C and R were erased from all ballots (and this is true for every R≠C), then C must win." (And this modified criterion might actually have been what Condorcet himself wanted!)

Also, in the presence of strategic voters voting methods which are guaranteed to elect Condorcet winners with honest voters, may no longer elect a candidate who would have been an honest-votes Condorcet winner – but approval and score voting under certain assumptions about strategic voter behavior will elect any candidate who would have been a Condorcet winner using honest votes! Paradoxically, this effect can cause approval and score voting actually to elect more Condorcet winners in real life, than methods obeying the Condorcet winner criterion!!

5. Smith-set winner: "If there is a set S of candidates such that for each candidate C∈S, more than 50% of the ballots score C above R (for every R∉S; the majority-forming ballot-set is allowed to depend on R and C), then some member of S must win."

But: Score voting does obey a closely related condition, similar modification to above.

Criteria, while interesting and useful for enhancing understanding, should not be your foundation

Bayesian Regret is what voting system comparisons should rest on, not logical criteria.

Problems with criteria: qualitative and "all or nothing" – each criterion is either obeyed or not by a voting system. No quantitative notion of "how often" and "how severely." Also, maybe "my criterion is important and yours is not"... except you disagree! That kind of argument never ends. Also, maybe there is some important criterion that has not yet been invented. Bayesian regret bypasses those problems: automatically assesses all possible criteria (whether anybody ever invented them, or not) correctly weighting according to frequency and severity of failures, quantitatively measures "how much social utility" a voting system "loses" compared to hypothetical and unachievable "optimum" system.

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