Smith//Score
Smith//Score is a rated voting method which elects the Score voting winner in the Smith set (i.e. the Score voting winner when all candidates outside of the Smith set have been eliminated). It is a Smith-efficient Condorcet method.
Condorcet//Score elects the Condorcet winner if there is one, otherwise it elects the Score voting winner.
The Smith set or Condorcet winner can be found off the rated ballots by doing pairwise comparisons between the candidates based on the scores each voter submitted i.e. if a voter scored Candidate A a 5/5 and Candidate B a 4/5, then their ballot is treated as giving A one vote in a pairwise matchup against B.
Example
Example for both:
17 A:10 B:9 C:8
17 B:10 C:9 A:8
18 C:10 A:9 B:8
49 D:10 E:10
Scores are (A 468, B 467, C 469), and (D and E 490).
52 voters prefer any of (A, B, C) over 49 for all other candidates (D or E), but these 3 candidates are in a Condorcet cycle, so they are the Smith set. Smith//Score would elect C for having the highest score in the Smith set. However, Condorcet//Score would elect D or E, since there is no Condorcet winner, and D and E are tied for being the Score winner. Note that Score and STAR voting would elect either D or E in this example.
Strategic voting in Smith//Score greatly resembles that of Score voting, with the added need for burial strategy. Example:
| Number | Ballots |
|---|---|
| 35 | A:5 B:4 C:0 |
| 25 | A:4 B:5 C:0 |
| 40 | A:0 B:0 C:5 |
A is the CW (beats B 35 to 25 and C 60 to 40) But with burial:.
| Number | Ballots |
|---|---|
| 35 | A:5 B:4 C:0 |
| 25 | A:0 B:5 C:1 |
| 40 | A:0 B:0 C:5 |
There is now an A>B>C>A cycle, with B having the most points (265 to A's 175 and C's 200) and thus winning. This is an example of the chicken dilemma. [1]
Notes
Smith//Score can be thought of as electing from the smallest group of candidates who voters gave more points to than all others in head-to-head matchups, subject to those candidates being in the smallest group of candidates preferred by more voters than all others in head-to-head matchups.
The Smith//Score ranking of the candidates (i.e. who came in 1st place, 2nd, etc.) can be found by first finding the generalized Smith set ranking and Score voting-based rankings (this is found by ordering the candidates from most points to lowest), and then "filling out" the Smith ranking by ordering the candidates in each consecutive Smith set based on their Score ranking. For example, if for seven candidates A, B, C, D, E, F, and G, the generalized Smith ranking is A=B>C=D=E>F>G, and the Score ranking is G>F>E>D>C>B>A, then the Smith//Score ranking is B>A>E>D>C>F>G (of the two candidates tied for 1st, A and B, B was higher in the Score ranking than A. Of the 3 candidates tied for 3rd, C, D, and E, the Score ranking was E>D>C, and for the two final candidates, F and G, are each in their own Smith sets and thus need not be ordered any differently).
It is also possible to do Smith//STAR or any other Smith// version of a cardinal method. When the rated ballots only allow voters to approve or disapprove of each candidate (for example, score them a 0 or a 1), then Smith//Score and Condorcet//Score become variations of Smith//Approval and Condorcet//Approval. It is also possible to implement Condorcet and Smith//Score by using ranked or rated ballots with approval thresholds.
(It's also possible to allow voters to decide who wins i.e. if a majority of voters indicate on their ballots which method they prefer, or how strict a condition they want the Score winner to pass to win instead of these methods' winners.)
One potential issue in Condorcet-cardinal hybrid methods is that if a majority subfaction prefers someone over the candidate the majority prefers, this can help the minority's preference. Example:
26 A:10 B:8
25 B:10
49 C:10
There is an A>B>C>A cycle in which C wins with 490 points to A's 260 and B's 458. To avoid this situation, it may be better to do a variant of Smith//Score where you repeatedly eliminate everyone not in the Smith set and eliminate the candidate with the fewest points, similar to a Tideman's Alternative method. Alternatively, the ballots can be checked to see if there are enough A>C voters who also prefer B>C to justify electing B.
One way to visualize Smith//Score is to start by looking at the Score winner. Then, assume the voters who prefer someone to the Score winner are placed in a head-to-head matchup between the two candidates. Whoever pairwise wins wins the matchup, and a new candidate is introduced for another matchup to take place. In this matchup, the winner is again the pairwise winner. Repeat until no new candidates can be placed in the matchups. If at some point a candidate loses a matchup, and later reappears and wins a matchup, then that is a sign of a cycle, in which case the highest-scoring of the candidates in the cycle is ranked higher than them, so long as all matchups have been processed. When looking at each matchup, one way to further visualize it is to imagine that the voters give the two candidates the same scores they gave them in the election, and then they polarize these scores based on their preference i.e. someone who scored the Score winner a 7/10 and the other candidate in the matchup an 8/10 is now treated as giving a 0 and 10 respectively. This solidifies the understanding that the use of Score in the cycle resolution is because, in part, it allows voters to "resimulate" head-to-head matchups to make their preferred candidate win when a strategic burial attempt has occurred.
Previous discussion of Smith//Score: [2]
Variants
There are several additional or alternative ways to combine the pairwise and cardinal data.
Electing Score winner with significantly more points
It is possible to modify these two methods by checking if the Score voting winner (whenever they differ from the winner of either method) satisfies certain condition(s), such as having (say) 10% more points than the winner of these methods, and if they do, electing them instead. The same modification can be done to favor a STAR Voting, Majority Judgment, or any other cardinal method winner as well.
Example:
51 A:5 B:4 49 A:0 B:4
Scores are A 255, B 449.
A is the Condorcet winner (51 voters prefer A>B), but if the special winning requirement for the Score winner to beat the Condorcet winner or Smith//Score winner is that they must have at least 30% more points, then the Score winner B wins.
Eliminating candidates with significantly fewer points than the Score winner
It is also possible to do something similar by first eliminating any candidate who has less than, say, 50% as many points as the Score winner, and then doing Condorcet or Smith//Score among the remaining candidates.
Example:
49 A:5 C:1 3 C:5 48 B:5 C:1
Scores are A 245, B 240, C 112.
C is the Condorcet winner, yet is scored only a 1 out of 5 by almost all voters. Since C has such low utility and thus can't get 50% as many points as the Score winner, C is eliminated. Between the two remaining candidates A and B, A is now the Condorcet winner and wins under either Condorcet or Smith//Score.
Adjusting pairwise threshold of victory
A more continuous way to implement this idea would be to increase the ratio of votes received in pairwise matchups by candidates with higher utility than the Condorcet or Smith//Score winner (or just in general, increase the pairwise victory margin ratio required for the lower-utility candidate to beat the higher-utility candidate in the matchup) in some proportion to how much higher the higher-utility candidate's utility is (which can also be thought of as what proportion of the two candidate's combined utility does the higher candidate's utility constitute). So for example, if the Condorcet winner has 30% approval and the Score winner has 40% approval with only a small pairwise loss to the Condorcet winner, then the 10% difference in approval could be used to justify changing the margins of that pairwise loss to make it a small pairwise victory for the Score winner, such that the Score winner becomes the Condorcet winner. This approach can be modified by a constant i.e. if you want to make it harder for the Score winner to win in this way, reduce the amount of additional support they receive in their pairwise matchups. Also, a threshold can be added such that, for example, a candidate with 52% of the points in a matchup doesn't end up pairwise beating someone they had a 51% pairwise loss to.[3]
Example:
51 A:10>B:9 49 B:10
Scores are A 510, B 949.
Because B has 65% of the points between A and B, one could potentially justify overturning their 51% pairwise loss to A, thus making B the Condorcet winner.
If B had had less points though:
51 A:10>B:1 49 B:10
Scores are A 510 B 541.
B would have slightly less than 52% of the points between A and B, and thus one could arguably either not apply any changes, leaving A's pairwise victory intact and making A the Condorcet winner, or apply changes to the pairwise matchup such that B is treated as having up to a slightly less than 52% pairwise victory against A, making B the winner.