Score DSV

Score DSV (Declared Strategy Voting) is a hybrid of Condorcet and Score voting that is closely related to Smith//Score. You take score ballots, find the Smith set, then "renormalize" each ballot so 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.

Using the essential (bipartisan) set instead of the Smith set would be even more strategy-resistant (as the Nash equilibrium strategy only involves the bipartisan set), but it's much harder to explain and rarely different in practice.

If there is a pairwise champion, i.e. only one member of the Smith set, you use that candidate and the candidate who comes closest to beating them. 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.

Motivation

Why is this a form of Declared Strategy Voting? If all voters provide their honest utilities, the renormalized score ballots represent the average score they would assign, if they were following the ideal score-voting strategy and knew the Smith set. 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 (but using the average avoids making the system nondeterministic per se).

If there is a Condorcet winner, all voters' ideal strategy will be to vote approval-style with a , and the Condorcet winner will win (satisfying the Condorcet criterion).

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, Smith criterion

Obviously non-compliant: participation criterion (conflicts with Condorcet), 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, i.e. only in relatively-contrived cases when a highest-utility (after renormalization) candidate is not part of the Smith set: local independence from irrelevant alternatives, Generalized Strategy-Free criterion.

The method is clone-independent 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). A variation of Score DSV which uses the bipartisan set instead of the Smith set for renormalizing is fully independent of 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 for Smith sets with up to 3 candidates (if using the Smith variant) or 4 candidates (if using the bipartisan 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. Suppose that:

1. Candidate A is the original winner.
2. Now, some voters rate candidate A higher without changing the order in which they rank the other candidates relatively to each other.
3. Then it is possible that some other candidate B is kicked out of the Smith set.
4. 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.

This method does not meet the participation criterion, but it does meet a weaker version User:Homunq calls 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. To his knowledge, it is the only Condorcet method to meet this criterion.