In methods with range [0,5] there are almost always min-max strategies that force the voters to use only ratings 0 and 5, as in approval.

In the case of methods that eliminate and normalize (STAR, etc) 1 is also used.

- If the intermediate candidates are set to 0 it is to favor as much as possible the most appreciated candidates, who have 5.
- If intermediate candidates are put at 5 is to disadvantage hated candidates at most, who have 0.

**S-TM Procedure**

You vote in a range with these values: {worst,1,2,3,4,best}.

For each pair of candidates the best is found, by adding up the points (win the highest sum), with these rules:

- the candidate with [best] receives 5 points, and 0 points to the other.
- the candidate with [worst] receives 0 points, and 5 points to the other.
- if both have intermediate values, so they are added as they are.

(- a vote in which the two candidates have both [worst] or [best], is not added).

The candidate who wins in the most pairwise matches wins.

**Tie**

- Only the 2 candidates (among those in tie) with the highest number of ratings [best] are considered, among which the one who won in the pairwise match wins.

or - Only the 2 candidates (among those in tie) with the lowest number of ratings [worst] are considered, among which the one who won in the pairwise match wins.

**Example (tie 1)**

Given an honest vote like this:

A [best] B [4] C [3] D [2] E [1] F [worst]

A's probability of winning does not change if the vote were like this:

A [best] BCDEF [worst] *(all at worst except A).*

while, the probability of victory of candidate F [worst] would increase.

There is no point in minimizing it in that way.

Respectively, if the vote were tactically like this:

ABCDE [best] F [worst]

the probability of victory for F would not decrease (apart from the rare cases of ties), while the probability of victory for A would decrease compared to honest vote.

Maximization is extremely disadvantaged (it can only serve in rare cases of tie, and it can disadvantage the victory of the true best candidate).

**Tie 2)** Maximization could be more disadvantaged (slightly favoring minimization) using procedure 2 in tie cases.

**Conclusion**

Limiting ourselves to the case of min-max strategies, the voter after assigning [best] to the candidates that he loves the most, and [worst] to those he hates most, will be able to feel free to assign intermediate scores to the other candidates.

Extreme case:

The intermediate ratings are maximized, making the honest starting vote become:

A [best] B [4] C [**4**] D [**1**] E [1] F [worst]

However, min-max intermediate candidates remains less favorable than min-max on the classic other systems with range.

P.S.

Due to the meaning of [worst] and [best] (not numeric values), it's not possible to uniquely convert a vote with range [0,5] into an S-TM vote.

**I think** that ratings with range [0,7] would probably be converted like this:

{7,6,5,4,3,2,1,0} --> {best, 4,3,2,2,1, worst, worst}

In methods with range [0,5] there are almost always min-max strategies that force the voters to use only ratings 0 and 5, as in approval.

In the case of methods that eliminate and normalize (STAR, etc) 1 is also used.

- If the intermediate candidates are set to 0 it is to favor as much as possible the most appreciated candidates, who have 5.
- If intermediate candidates are put at 5 is to disadvantage hated candidates at most, who have 0.

**S-TM Procedure**

You vote in a range with these values: {worst,1,2,3,4,best}.

For each pair of candidates the best is found, by adding up the points (win the highest sum), with these rules:

- the candidate with [best] receives 5 points, and 0 points to the other.
- the candidate with [worst] receives 0 points, and 5 points to the other.
- if both have intermediate values, so they are added as they are.

(- a vote in which the two candidates have both [worst] or [best], is not added).

The candidate who wins in the most pairwise matches wins.

**Tie**

- Only the 2 candidates (among those in tie) with the highest number of ratings [best] are considered, among which the one who won in the pairwise match wins.

or - Only the 2 candidates (among those in tie) with the lowest number of ratings [worst] are considered, among which the one who won in the pairwise match wins.

**Example (tie 1)**

Given an honest vote like this:

A [best] B [4] C [3] D [2] E [1] F [worst]

A's probability of winning does not change if the vote were like this:

A [best] BCDEF [worst] *(all at worst except A).*

while, the probability of victory of candidate F [worst] would increase.

There is no point in minimizing it in that way.

Respectively, if the vote were tactically like this:

ABCDE [best] F [worst]

the probability of victory for F would not decrease (apart from the rare cases of ties), while the probability of victory for A would decrease compared to honest vote.

Maximization is extremely disadvantaged (it can only serve in rare cases of tie, and it can disadvantage the victory of the true best candidate).

**Tie 2)** Maximization could be more disadvantaged (slightly favoring minimization) using procedure 2 in tie cases.

**Conclusion**

Limiting ourselves to the case of min-max strategies, the voter after assigning [best] to the candidates that he loves the most, and [worst] to those he hates most, will be able to feel free to assign intermediate scores to the other candidates.

Extreme case:

The intermediate ratings are maximized, making the honest starting vote become:

A [best] B [4] C [**4**] D [**1**] E [1] F [worst]

However, min-max intermediate candidates remains less favorable than min-max on the classic other systems with range.

P.S.

Due to the meaning of [worst] and [best] (not numeric values), it's not possible to uniquely convert a vote with range [0,5] into an S-TM vote.

**I think** that ratings with range [0,7] would probably be converted like this:

{7,6,5,4,3,2,1,0} --> {best, 4,3,2,2,1, worst, worst}

thanks, I was wrong, I corrected! ]]>

The candidate who wins in the most pairwise matches wins.

How are those counted?

]]>Considering 2 candidates, the points they receive in the various votes are added and the one with the highest sum wins.

The [worst] and [best] ratings have special rules that describe how to treat them in the sum.

The candidate with the most wins in pairwise match wins (all pairwise matches are made).

]]>Exactly like that.

I hypothesize that if a voter assigns [best] to a candidate, then he would definitely want the opponent to be at 0 points (respectively, if he assigns [worst] he wants the opponent to be at 5).

If he uses intermediate scores instead, he can agree to leave them as they are.

An STLR-style normalization could be done on the intermediate scores, eg: [1,2] becomes [2.5,5] but I think it would complicate the procedure too much.

]]>In methods with range [0,5] there are almost always min-max strategies that force the voters to use only ratings 0 and 5, as in approval.

Do such strategies work better than assigning intermediate scores to compromise candidates? Can you show an example?

]]>"Work better" in what sense?

If this is my honest vote in SV:

A [5] B [4] C [3] D [2] E [1] F [0]

and I want to maximize A and B's chances of winning (to the detriment of the others), then my vote will tend to go like this:

min: A [5] B [5] C [0] D [0] E [0] F [0]

If, on the other hand, I want to make sure as much as possible that E and F lose, then my vote will tend towards this:

max: A [5] B [5] C [5] D [5] E [0] F [0]

In methods such as STAR, on the other hand, it may make sense to also assign a rating of 1 (intermediate) to compromise candidates.

Better than SV, but the min-max problem is still there, also because the first step makes the two candidates win with the higher sum, so it's like SV (in fact, with clones, STAR becomes SV).

In S-TM (with tie procedure 1) maximizing makes no sense, while minimizing makes no sense except in the case of tie (very rarely).

]]>"Work better" in what sense?

In the sense of maximal support for the values and preferences of the individual voter or voting faction.

If this is my honest vote in SV:

A [5] B [4] C [3] D [2] E [1] F [0]

Honest valuations of the candidates are of course necessary for looking at strategy, but another required piece of information is how popular are the candidates with the voters at large. Voters will have some idea of this knowledge from polls, discussions, propaganda, considerations of conflicting interests, etc., and by the Gibbard theorem, they must take this knowledge into account when strategizing.

]]>Yes, but it seems to me that they often just observe who the 2 frontrunners are and then give max (5) to the best of 2, and min (0) to the worst.

More generally, the min-max problem for range methods is quite recognized as such, and it is the reason why maybe someone prefers a ranking with a single candidate for each position (because in that, you certainly can't min-max) .

If I want to maximize A (loved candidate), I vote:

A [5] BCDEF [0]

If I want to minimize F (hated candidate), I vote:

ABCDE [5] F [0]

If the voter wants to do both, then he won't know what scores to give BCDE candidates. The fact that I have to use result predictions to decide what scores to give BCDE candidates seems very wrong to me.

S-TM tries to solve this problem with [worst] and [best] ratings which allow to maximize/minimize candidates A/F without imposing conditions on B, C, D, E for which the voter will be much more encouraged to give honest (intermediate) evaluations.

]]>Yes, but it seems to me that they often

In what experience? Where has Score been in use, for matters of importance and contention, for long enough for voters to become accustomed to thinking about its broad effects, so that we can inquire what strategies they use under such conditions?

More generally, the min-max problem for range methods is quite recognized

Maybe it has only been recognized by thinkers who have held back their thinking from full consideration of the logic and evidence that apply.

]]>This is evidence:

if I want to maximize A (loved candidate), I vote:

A [5] BCDEF [0]

If I want to minimize F (hated candidate), I vote:

ABCDE [5] F [0]

If the voter wants to do both, then it is not clear what rating to give to the candidate BCDE.

This is a problem and it is not me who has to prove that in practice it does not show up (and possibly that it will not show up in subsequent elections).

However, I am collecting data on votes with ranges and what I notice for now is that it comes up quite often as a problem (at least, in online single winner polls). When I have enough data I will post about it.

I tell you that it also happens that some voters never give the score 0.

I have only stated that the S-TM is extremely resistant to a similar problem due to how the results are calculated (as well as the methods that use rank with one candidate per position).

I have made other small statements (even subjective), but the core of the speech does not change.

If the voter wants to do both, then it is not clear

I argue that the clarity will come when the rest of the background information of the election becomes available to the voter.

]]>Yes, in practice, whoever is best able to predict the results, the best will be able to exploit this knowledge to his advantage... not to mention that the political factions could spread false information precisely to condition votes.

I prefer a voting system in which the voter's vote depends as little as possible on the background information (and as much as possible on his interests alone).

]]>I prefer a voting system in which the voter's vote depends as little as possible on the background information (and as much as possible on his interests alone).

Yes, well, I can see that as a valid goal, if any movement in such a direction can be demonstrated, and if it can be done without worsening the power relations.

]]>(apart from the rare cases of ties)

It sounds to me as though ties would be common. Scores that could tie are from numbers of candidates, which could be three, rather than from numbers of voters, which can be in the millions, and rather than from scores added up from voters.

I believe Condorcet cycles are fairly heard of or expected, although I suppose they occur less than half the time, maybe less than a tenth of the time, I don't know. For example among four candidates, we could have that A beats B and D, B beats C and D, C beats A and D, and D beats no one. So the scores would be 2, 2, 2, 0, a three-way tie.

My first thought with respect to any newly proposed voting system is to examine the question of whether the system provides tit-for-tat balance.

I believe the first part of your system, the part up to but excluding the tie-breaking subprocedure, does provide balance. This first part considers the possible pairings of candidates. The obvious antivote to a given vote would invert the grades assigned to the candidates. Best would become Worst, Worst would become Best, 4 and 1 would be interchanged, and 2 and 3 likewise. Such an antivote would decide between any two candidates in the opposite way as the original vote. Since the outcome of the first part of the procedure depends wholly on which candidate wins each pairing, the procedure up to but excluding the tie-breaking aspect does provide tit-for-tat balance.

So, let's look at the tie-breaker. I think one of your tie-breaker variants is more IRV-like, and the other is more Coombs-like. I will just examine the latter, as I think they are enough alike that both do or do not provide balance.

Only the 2 candidates (among those in tie) with the lowest number of ratings [worst] are considered, among which the one who won in the pairwise match wins.

I will concentrate on

Only the 2 candidates (among those in tie) with the lowest number of ratings [worst] are considered,

This means that the antivotes I considered before, are not necessarily antivotes for this part of the procedure. The candidates who received Worst in a proposed antivote are those who received Best in the original vote, which has nothing to do with the determination of who is considered. So, the tie-breaking procedure and the system as a whole fails Tit-for-Tat. So, I suspect it smacks too much of the antidemocratic characteristics of FPtP. This is a consideration of power relations, so trumps the strategic help, I would say.

In devising KP Voting I did not start out by articulating to myself in so many words, that I was trying to provide a strategy that depended to a lesser degree on the background information of the election. However, in an indirect way, that was part of the real motivation, because I believe that IRV advocates believe that IRV does just that, and I was trying to incorporate IRV so that the system could attract acceptance from IRV advocates and Approval advocates. So, maybe you can judge KP as moving in the direction you seek, when it is compared to Approval.

]]>Assuming to use tie procedure 2, the worst that can happen in the case of cycles is that some of the few candidates belonging to the cycle receive [worst] instead of intermediate ratings.

This is if the voter can predict that there will be a cycle and which candidates will belong to it (it's not easy).

If they don't know which candidates belong in the cycle, then minimizing is very risky because it also remains true that in case of not tie, minimizing to [worst] is senseless.

**Tit-for-tat**

Given these votes:

A [best] B [3] C [2] D [worst]

A [worst] B [2] C [3] D [best]

When it comes to the management of the tie (procedure 1), the 2 candidates with the most [best] are held, so A and C are in tie.

With tie procedure 2, the 2 candidates with least [worst] are held which are B and C, which however result in tie (if there was only B, I would have to keep B and one between A and C in tie).

You always end up with a tie (it makes sense), even if there is still an imbalance towards certain candidates (those with intermediate ratings) rather than towards others.

However, tit-for-tat is not a criterion that I value very much (and I also point out that it only fails in the case of tie, otherwise it's satisfied, unlike FPTP which always fails).

**Score for tie**

If I use Score, I introduce the min and max strategies in case of tie, while with my procedure only one of the 2 strategies can be valid in case of tie.

I will inquire about KP voting.

]]>I have read KP and partly agree.

You look for a half way between IRV and AV.

S-TM instead is an half way to other methods:

- If everyone in S-TM voted with [worst] and [best], it would become
**AV**. - If everyone in S-TM voted with intermediate ratings, it would become
**Score**. **Condorcet**(pairwise comparisons) to mix AV and Score.- in case of tie, all candidates but 2 are eliminated on which the comparison is then made , similar to
**STAR**.

For instance for S-TM1 : when there are 20 candidates and 5 of them have the same sum of (adjusted )scores while 3 of them have also the same number of ratings [best] and 2 of them also have the same number of ratings [ worst] ]]>

- add something like: this is intended as a single winner method
- "The candidate who wins in the most pairwise matches wins." better: "The candidate with the highest sum of points wins"

Yes, there are unresolved tie cases as in any voting method. The important thing is that they are sufficiently rare.

If 1000 or more voters vote, it is extremely rare that multiple candidates have the same number of [best] or [worst] just as it is extremely rare in AV that multiple candidates have the same number of votes.

Frequent tie cases to be solved are those caused by condorcet cycles and for those the solution is proposed.

"The candidate with the highest sum of points wins" is not the method I am proposing. With a similar process, there could be many more min-max strategies.

@Keith

The benefits of STLR and STAR fall with the clones, becoming Score Voting. I don't think the added complexity of such methods is worth it in the long run (given that in the long run, political factions would understand this clone problem and exploit it to their advantage).