# Score DSV

Score DSV (Declared Strategy Voting) is a hybrid of Condorcet and Score (Range) Voting. You take score ballots, and find the Smith set, then "renormalize" each ballot so that the worst member of the smith set is at 0 and the best is at 100. Say Violet the voter rated the smith set members A, B, and C as 40, 38, and 30, and the non-smith members D and E as 100 and 0; that ballot will be counted as 100, 80, 0, 100, 0. The ballots are added, and the highest score wins.

Actually, I'd prefer to use the Dutta set instead of the Smith set, but that would require explaining it.

If there's a pairwise champion, ie, only one member of the Smith set, you use that candidate and the candidate who comes closest to beating them. If all the members of the Smith set are tied on a ballot, then you set them at 100 if they're tied-for-best, at 0 if they're tied-for-worst, and at 50 otherwise. All candidates better than all members of the Smith set on a ballot get 100, and all candidates worse than all members of the Smith set get 0. Thus, the Condorcet winner wins, almost always with an average score of over 50 (you can fix the tied-in-the-middle score to take into account number of candidates above and below to make sure the average is over 50, but it's not worth it).

## Contents

## Motivation

Why is this a form of Declared Strategy Voting? Assuming all the voters vote their honest utilities, the renormalized score ballots represent the (average of the) ideal score-voting strategy if the voters knew the Smith set (but didn't have enough information to know which candidates were likely to come out ahead within that set). If there's a Condorcet winner, all voters' ideal strategy will be to vote approval-style, and the Condorcet winner will win, thus this method satisfies the Condorcet criterion. If there are several members in the Smith set, the ideal strategy is actually a mixed strategy; the example voter Violet from above should have an 80% probability of strategically voting 100, 100, 0 and a 20% probability of voting 100, 0, 0. If there are more than a few hundred voters, these probabilities will inevitably average out, and so the mixed strategy is equivalent to the average vote of 100, 80, 0.

## Less strategy than other systems

Because the system is working out your strategy for you, you would have to be very clever indeed to work out an even better strategy. You need to know not only who is likely to be in the Smith set, you also need to know the chirality (is it A>B>C>A or A>C>B>A?) and the relative position of the members of the set. In real-world situations, even the best of polls would only indicate the possibility of a Condorcet tie (Smith set larger than 1), they would never be able to give you this extra information.

Moreover, even if every voter has absolutely perfect information about all other voters' true preferences, for every group of G voters who has a dishonest strategy available to gain an advantage, there is a larger and/or more motivated** group H of voters with a defensive counterstrategy available - one that would make G's strategy backfire to their own harm, if G dares to use it. In the real world, that means that nobody has a motive to start the massive coordination that a successful strategy would require, because the word will get out and others will use a defensive strategy.

Also, consider what kinds of strategies exist in other systems. In Condorcet methods, a typical strategy to consider is a "favorite betrayal" strategy. An A>B>C voter who thinks C could win an A>B>C>A Condorcet cycle can "betray" A to vote B>A>C. If this breaks the cycle, B wins. In effect, the voter was saying that they care more about the possible advantage of getting B over C, than about the possible disadvantage of getting B over A. In Score DSV, this voter's honest A>B>>C vote (such as 100, 90, 0) will probably have the same effect. In other words, this system, by allowing a more expressive ballot than most Condorcet methods, allows honesty to replace the many cases of strategy, especially those with the strongest voter motivation behind them.

Finally, unlike score voting, this system is not only resistant to strategy, it is forgiving of the lack of strategy. A voter who, through imperfect understanding or laziness, simply votes approval-style, ranks the candidates, or truncates their vote will still have almost the same power as a voter who carefully figures out a precise utility for every candidate. There will be essentially no systematic bias in favor of candidates whose supporters are more skilled and/or willing strategists.

**"More motivated" relative to the range of utility of the Smith set. A voter who voted the Smith set 20, 19, 10 is "more motivated" for B>C than one who voted 60, 50, 10 (90% vs. 80%), even though the un-renormalized gap is less. Still, individual differences of this kind should average out, so in general they will be "more motivated" in an absolute sense as well.

## Example

There are 4 candidates, and 3 groups of voters. The true utilities are:

- 10 voters: 100, 90, 80, 0
- 9 voters: 80, 100, 90, 0
- 8 voters: 82, 80, 100, 100

### Score DSV results

The Smith set is A, B, C, so the strategic ballots are:

- 100, 50, 0, 0
- 0, 100, 50, 0
- 10, 0, 100, 100

The totals are 1080, 1400, 1250, 800, and candidate B wins. Effectively, the smallest group of voters has acted as a "tie breaker" between the favorites of the larger two groups. (If their utility for A had been over 89, A would win)

### Under other systems

On an honest score ballot, the last group of voters has the advantage of "caring less" about making sure candidate D doesn't win; they can honestly vote 10, 0, 100, 100, exaggerating their utilities beyond what the others do. So they get their favorite member of the Smith set, C, even though they're fewer than the other two groups.

Most Condorcet methods would elect A, as that is the candidate with the weakest defeat (versus C).

IRV would eliminate C and elect B.

## Criteria compliance

Obvious compliances: Pareto criterion, Plurality criterion, Strategy-Free criterion, Blank Ballot Criterion. Obviously non-compliant: Consistency criterion, invulnerability to compromising, invulnerability to burying, Favorite Betrayal criterion, Later-no-harm criterion.

This method complies with the Condorcet criterion but not the Smith criterion. The latter non-compliance is seen as a feature; for instance, if there are 4 candidates and 3 voters with the votes:

- 100, 99, 0, 98
- 99, 0, 100, 98
- 0, 100, 99, 98

then candidate D is the highest-utility compromise, but is not part of the Smith set. However, if the third voter's true preference were 0, 99, 100, 98, the system should NOT elect D even though they're still the utility winner and the score winner, because to do so would give voters 2 and 3 motivation to strategically bullet-vote for C.

There are some other criteria which Score DSV fails for similar "good" reasons, ie, only in relatively-contrived cases when a highest-utility (after renormalization) candidate is not part of the Smith set: Mutual majority criterion, local independence from irrelevant alternatives, Schwartz criterion, Generalized Strategy-Free criterion,

The method is not universally clone independent under the standard definition of clones (that they must simply rank the same as the originals). However, if the clones have scores within the range of the Smith set for every voter (for instance, if they have identical scores to the originals), then the method is clone independent. A variation of Score DSV which uses the Dutta set instead of the Smith set for renormalizing is actually fully independent of inferior clones.

The method complies with the Strong and Weak Defensive Strategy criteria. As noted above, it does even better: defensive strategies are always available to larger and/or more-motivated groups (where motivation is counted relative to the Smith set) than the corresponding offensive strategies.

Score DSV is monotonic with up to 3 (serious) candidates, or 4 if you use the Dutta set (which can't have 4 members) instead of the Smith set. However, if the Smith set is larger than 3, there is a case where it could be non-monotonic for a minority group of voters, as pointed out by Marcus Schulze: "Candidate A is the original winner. Suppose some voters rank candidate A higher without changing the order in which they rank the other candidates relatively to each other. Then it is possible that some other candidate B is kicked out of the Smith set and that, after renormalizing the ballots, candidate A is worse off." In real life, a Smith set of over 3 would require a truly extreme degree of symmetry.

This method does not meet the Summability criterion, although, unlike IRV, it only requires a maximum of two summable counting rounds. A minor modification of this method, in which average scores for every 3-member subset of the Smith set are used, does meet this criterion (at summability of order 3, not 2 like other Condorcet systems); this variation also happens to meet the Smith criterion and some of the related criteria noted above (all of them, if the Schwartz set variation above is also used), but the extra complexity is not worth it.

This method does not meet the Participation criterion, but it does meet a weaker version I call the "defensive participation criterion": *If a given election chooses X, and new voters are added who prefer X over Y, then the set of voters who prefer X over Y have some way of voting to make sure Y does not win without reversing any preferences or falsely voting any candidates equally.* As far as I know, it is the only Condorcet method to meet this criterion.