Reciprocal Score Voting: Difference between revisions
→Analysis
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== Analysis ==
Mathematically, the reciprocity adjustment step maps the space of score voting elections to the smaller subspace of score voting elections where all factions exactly match mutual ratings. It then proceeds as normal.
In the case of any asymmetry in support, the reciprocity ratio is <math>R(j,\phi) < 0</math> for the faction which did not cooperate, and <math>R(j,\phi) = 1</math> for the faction that did cooperate. Therefore, not cooperating penalizes the side which did not cooperate more. In this way, factions are encouraged to cooperate as much as possible to maximize mutual support, forcing them to strike a balance between supporting their favorite as well as supporting alternatives as much as they can. In the case of opposing factions, the mutual lack of cooperation has no effect.
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The <math>\min(\cdot, 1)</math> condition above is required so that support is never amplified by asymmetry. This is also necessary so that a smaller faction cannot parasite on the support of a larger faction, which will never rate the smaller faction above its own. A smaller faction artificially rating a larger faction too highly will only receive exactly as much support as the larger faction is willing to give to it.
=== Exaggerated approval-like voting ===
The optimal strategy under score voting is to vote approval-style: only using the minimum and maximum ratings. Under RSV this exaggerates faction overlaps, so that exaggerating voters will belong to multiple factions. This can benefit or damage them, depending on the behavior of other factions, but a precise analysis is difficult.
If every voter exaggerates, the reciprocity ratios will emulate honest score votes with the same outcomes, based on probability of a voter from a faction casting an approval. For example, suppose 30% of faction A voters approve of candidate B as well as A, casting A=10 B=10 ballots. Faction A's mean score towards B will be 3. In turn, 40% of faction B reciprocates, casting A=10 B=10 ballots.
The adjusted ballots for these voters will become A=10 B=3 and A=3 B=10 (B voters reciprocate more, so their support caps at 3), so that their strategical behavior reproduces, in a sense, the "mean belief" of their own faction: if 30% of voters exaggerate a score of 10, this becomes equivalent to the faction giving a score of 3, which produces identical results in a score voting election by linearity.
==== Bullet voting ====
[[Bullet voting]], supporting a single candidate, is a completely suicidal strategy under Reciprocal Score Voting. By giving zero support to other candidates you ensure your candidate will get very little support from others. If an entire faction bullet votes for their favorite they will receive support from nobody else. Unless they are a strong majority this will almost certainly backfire.
=== Chicken dilemma ===
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=== Passed/Failed criteria ===
Under complete ballot equilibrium (reciprocity ratios are all 1) RSV is identical to score voting. Therefore, RSV obeys whatever criterion is obeyed by this subspace of possible score voting elections. A basic analysis regarding [[voting system criterion]] reveals the following.
* [[Majority criterion]]: as the pre and post-adjustment ballots are both score ballots and operate in a similar fashion, strictly speaking RSV fails the majority criterion under more exotic circumstances. However, it does so only if the majority's and the winning candidate's factions highly rate one another, which is a less problematic situation. It is, however, very robust to strong majorities:
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* [[Mutual majority criterion]]: as in the chicken dilemma scenario mentioned previously and for the same reasons as in the majority criterion, strictly speaking it fails, but it is unlikely and failures likely less severe.
* [[Later-no-harm criterion]]: fails. However, it is more robust with respect to the top-rated candidates (the voter's factions), as per above analysis. The harm to a voter's faction can only happen if there is significant overlap, making these failures less severe in terms of satisfaction. Candidates which are not top rated will suffer more severely from LNH violations.
* [[Monotonicity criterion]]: as RSV ties together the support given to multiple candidates, it is monotonic in a very exotic way, especially with respect to the favorite candidates: a favorite candidate can have a lower chance of winning by lowering the scores of another candidate.
* [[Condorcet criterion]]: fails by virtue of not being a ranked system.
* [[Independence of clone alternatives|Independence of clones]]: passes, provided the clone is sufficiently similar, otherwise the clone itself may gain more support by inducing more reciprocation.
* [[Favorite betrayal criterion|No Favorite Betrayal criterion]]: passes, as rating your favorite highly can never hurt them.
* [[Summability criterion]]: fails. The procedure
== Variants ==
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