# Math PhD doesn't believe Score is immune to Arrow

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### Clay S

Nov 12, 2018, 6:35:56 PM11/12/18
to The Center for Election Science

it seems there's got to be a better way to make our argument if a math ph.d can't follow it.

### Steve Cobb

Nov 13, 2018, 8:27:32 AM11/13/18
to The Center for Election Science
I've heard it said that Arrow's theorem doesn't apply to Score if voters are rating candidates on an absolute scale, but does apply if they are giving the least preferred candidate the minimum score, the most preferred the maximum score, and everyone else in between. Ditto for IIA.

### Abd ul-Rahman Lomax

Nov 13, 2018, 10:57:47 AM11/13/18

He is not using math or math skills. He is imagining that, behind the votes, voters have a list of unexpressed preferences. The concept of preference strength is not considered by him. When a preference strength is low enough, the voter may choose to not express it, either because other considerations (hedging bets for strategic reasons) outweigh the preference strength, or the voter actually is undecided, can't tell which to prefer.

He is young and incautious, does not write like a mathematician. A mathematician would be extremely careful with definitions.

Arrow's Theorem is a mathematical analysis, with assumptions. It generates an impossibility proof, which is valid, given the assumptions.

The Wikipedia lede has:

In social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem.
As is common for Wikipedia, the lede is not precise. However, what is the input to the process that Arrow is describing?

Ranked preferences of individuals, or the same created by any criterion.

Arrow's theorem is an impossibility proof, with narrow application, to voting systems that provide lists of preferences, and that are deterministic based on those lists. Arrow's theorem is correct, nobody has found a flaw in it, as stated.

Approval Voting and Score Voting do not collect such lists. Nor do Plurality, and in real elections, nor does any voting system. We could claim that mandatory full-ranking STV in Australia does.

Arrow's theorem applies to an abstraction, that differs from real voting systems.

To apply Arrow's theorem to systems that are fed with incomplete rankings, the analyst must imagine that there is a "real ranking," even though there is no process for collecting it, because it is not disclosed. He can then imagine that the voter does not express this ranking -- by overvoting or undervoting. This can then create a violation of IIA.

But it's imaginary. We can then argue that it has some correspondence with reality, but, remember, this is an impossibility theorem. Not some evaluation-of-voting systems theorem. IIA is violated even if the shift actually has practically no effect on social utility.

This is the bottom line: We can think of systems that allow incomplete ranking (Notice: it must actually prohibit incomplete ranking, not merely allow it) as "violating" Arrow's Theorem, because it does not satisfy the stated requirements of the system. And that *sounds bad.* However, to be "bad," the requirement must be, by definition, necessary to sound outcome. And is an IIA violation an "unsound outcome"? How would we know?

Score Voting theory full addresses this, by creating a standard by which voting systems can be measured. Score Voting can suffer from the same problem as leads to IIA violation in the approval case. That is, voters may "fail" to express full ratings. The voting system may be crude and not allow full preference expression. Voters may be "dishonest," though that's poorly defined.

Mathematically, this is very, very simple: To rule that Arrow applies to Approval or Score, one must have

1. True internal rankings or ratings (ratings also rank, but allow equal ranking, and in this case, equal ranking would mean "no preference." For this reason, Warren suggested votes would be real numbers within the range of 0-1, as I recall. So there would be infinite resolution.

2. A voting process that translates internal ratings to votes on a ballot.

3. An election process that is deterministic based on the information on the ballots.

And, of course, top-two runoff would fail Arrow in the same way, most immediately by not being deterministic from the first ballot. Actual Robert's Rules preference voting is not deterministic. It will repeat an election indefinitely until a candidate is actually preferred by a majority of voters. (It does not do "runoffs" unless candidates withdraw or are not re-nominated).

Simple "majority vote" with unlimited runoffs may satisfy the intentions of Arrow's theorem, but not the specific conditions required. When an election fails in MV, under Robert's Rules, it is voided, the entire process, including nominations, must be repeated. Voters may then decide to change their preferences, because "preference" is not actually clearly defined. In a single list, it's clear. The list is an ordering, and it is strict, no equal-ranking. In the final election in repeated MV, a majority of voters have preferred the winner over all others. To be strict with this, a tie vote would fail. No coin toss.

Yet this still "fails" Arrow in not being deterministic. There is no limit to the number of polls that must be taken. In practice, however, humans start to value "getting it over with." That is a selection criterion, that, with changing conditions, shifts the internal preferences, and the voter will picks the top, all things considered. It's sincere, by any reasonable definition of sincerity.

Approval and Score can make this process more efficient, that's all. I prefer systems that require majority approval to declare a winner, and that are repeated when that fails. In a system that allows expression of preference strength, this can be so efficient that a need for repeated elections can be rare. We have lost the basic rule-making process of democracy when we allow elections without an explicit approval of a majority. Why do we do that? We pretend that the reason is efficiency, but a more real reason would be that it's always been done that way.

It is quite obvious that simply allowing overvotes will move voting systems toward efficiency in finding higher-social-utility winners. It's cheap, (actually possibly cheaper than standard rules) it is easily understood. It is dpne with multiple conflicting ballot questions. So why not with candidates, which are multiple conflicting ballot questions? Again, why not?

Again, it's obvious: some with power or influence care about some other value than democracy. And most people have never really considered the issues, and so go along with these "leaders."

Even when there are drastic and obviously unjust consequences to voting system defects, we do nothing but complain. And what the few who actually try to do something suggest or try to create, is little more than a band-aid on a broken leg. We have extremely poor collective social decision-making systems.

And nearly everyone, including voting systems activists, does not care. We just want the "right" decisions to be made. We don't want to work and think and reflect, we don't want to communicate and create collective intelligence. We would rather be comfortable in being "right."

And that includes mathematicians and scientists, very often.

We make artificial definitions in reasoning processes that then lead to the conclusions we like, too often forgetting that garbage in, garbage out.

That math PhD could follow this, if he would just let go of his preconceptions, his beliefs, and pay attention.

Some, a few, will do this. How to move that into actual social transformation is well worth studying, and that is what I did. Anyone who masters that process, who learns how to create major social transformation, and who documents it, makes it reproducible and practically usable, will deserve more than the Nobel Prize. They will deserve that their name be immortal, because the benefit of this would be incalculable.

I don't know how to do it. But I have ideas, and they seem possible. Something is missing, though, and what is that?

I do not think that the transformation requires better voting systems, though a transformed society will certainly use polling (and it's obvious that the best polling methods allow expression of preference strength). But polling would only be a piece of a transformed system. The best proposed voting system, in my opinion, is Asset Voting, which uses a vote-for-one ballot, in the simplest implementation, where election strategy suggests voting for the absolute favorite among all those willing to serve. The old-style ballot (a blank piece of paper) would work. This would then be used by the "electors," I call them, the people receiving votes in the primary ballot, to create seats in a fully-representative assembly, by transferring the vote (the "Asset") to them, a person receiving a quota would be elected.

I suggest the Hare quota. This is equivalent to requiring, not merely majority, to elect a representative, but *unanimity.* The Droop quota would imply a majority. Using the Hare quota will allow some seats to remain vacant, until the unrepresented votes are assigned. Definitely, not deterministic. Those seats could remain empty until the next election. This is actually basic democracy, only allowing representative decision-making. We are so close . . . yet so far.

The assembly then makes decisions *by majority vote.* No majority, no decision. Very simple. It is very close to existing systems, yet, my opinion, it would not harm political parties, but it would make them unnecessary. Hey, we can consider Warren as an inventor of it (along with a few others). Of course, to Arrow, this is not a voting system, because it is not deterministic from the ballot. That restriction is what makes Arrow's theory almost useless. Real, traditional, parliamentary systems do not, generally employ decision-making methods that violate "deterministic." Most critique of vote-for-one comes from usage where a mere plurality may win, so the real name is Plurality, which I distinguish from Majority. Plurality is heavily disallowed in Robert's Rules. No decision is made without a majority approval: The status quo remains until a majority agree otherwise.

Some Score Voting activists suggest that Score be used in parliamentary systems. I agree that it could be useful, but only if making a decision requires an absolute majority who "approve" the result. It's very simple to convert a Score ballot into an approval ballot. "Approval" means an explicit indication of "I approve of the election of this candidate, rather than that the election fail to be repeated." Because the concept of election failure and its highly useful function is mostly ignored, there has been little attention paid to this. The simplest way is to define midrange by this (on the ballot) This still allows refined expression of preference and preference strength, but sets nails the range to a range of approval preferences and a range of disapproval preferences, thus making the ballots of all voters *commensurable.*

Voters could still vote "insincerely," which must, in this system, mean the reversal of a sincere preference, but it would be dumb. I see no advantage gained by such a vote. If there is enough difference of opinion among the voters, if the electorate is badly polarized, the election would likely fail. And a majority of voters would have expressed a preference for that. So that's the "decision" made. None of the above.

But in a repeated election, there would be a new campaign. And, depending on conditions, the electorate would shift to those who still care about what choices remain possible. This, even with top two runoff, will push the election toward improving social utility (because those who don't care won't take the trouble to vote). People change their votes in runoff election, and new voters vote. (In studies, the result in TTR flips to the runner-up in the first ballot about a third of the time. With IRV, this is far more rare (i.e, that the first round leader loses). The loss of that fresh examination is one of the defects of "preferential voting"" as explained by Robert's Rules.) IRV proponents think this is "bad," because it is a knee-jerk opinion of voting that higher turnout is better than lower turnout. This all demonstrates a very poor understanding of how democracy actually works, in real practice. In fact, sometimes turnout in repeated elections is higher, *if the voters care strongly about the choice.* "The Lizard v. The Wizard."

IRV has steadily been killing the best voting system in actual practice. It's still flawed, but those flaws could be fixed. Simply Counting All the Votes (Approval) would improve it. The same with IRV, by the way. It makes a way that voters could fix the flaws in IRV. (Why are ballots, with "overvotes" at a rank, disregarded?) But there are better ways, the IRV ballot-counting is complex. Basically a bad idea.

### William Waugh

Nov 14, 2018, 5:19:43 AM11/14/18
to The Center for Election Science
+1

On Tuesday, November 13, 2018 at 10:57:47 AM UTC-5, Abd ul-Rahman Lomax wrote https://groups.google.com/d/msg/electionscience/d1cFakLTzkw/a3H0tOaDAAAJ

### Abd ul-Rahman Lomax

Nov 14, 2018, 9:33:38 AM11/14/18

Thanks. There are a number of typos in that piece. I will edit it if anyone requests that. I am, by the way, to be in the Bay Area for two weeks, arriving tonight at 10 PM (SFO). I will rent a car and drive out to my son's house in Marin County. If someone on the way would like to offer me a place to stop and spend the night, it would be nice, I will have been on a plane for many hours. I will be visiting scientists in the area, but I could include, voting systems activists and researchers in that.

Below, I correct a typo that actually alters meaning seriously.

On 11/14/2018 5:19 AM, William Waugh wrote:
+1

On Tuesday, November 13, 2018 at 10:57:47 AM UTC-5, Abd ul-Rahman Lomax wrote https://groups.google.com/d/msg/electionscience/d1cFakLTzkw/a3H0tOaDAAAJ
a correction:
Real, traditional, parliamentary systems [strike: do not,] generally employ decision-making methods that violate "deterministic." Most critique of vote-for-one comes from usage where a mere plurality may win, so the real name is Plurality, which I distinguish from Majority. Plurality is heavily disallowed in Robert's Rules. No decision is made without a majority approval: The status quo remains until a majority agree otherwise.

Part of what Arrow's theorem actually shows is that deterministic elections suck. Score would not be so vulnerable to the problem, but it still, as generally proposed, has it. I.e., there can be a situation where a majority of voters actually prefer that the election be repeated, but, instead, a winner is declared, because of what amounts to the highest expressed social preference (an equivalent of Plurality). As I wrote, this is almost trivial to fix, by using the rating system to express election approval/disapproval. All candidates not approved [rated above a defined level] are disapproved. In public systems, then, a procedure can be defined for repeated elections (as with top-two runoff, and I have suggested picking the top two candidates by an algorithm very likely to include the best social utility winner, *allowing, then, write-ins. (A Condorcet winner might be apparent from the results, and should then be included. Thus this method (the overall system including repeated elections) would satisfy Condorcet.

It is clear to me that society has, in the past, considered that choosing the best winner was important enough to justify the expense of a second election. What if a majority of voters in the election still fail to approve a winner? I suggest that (1) this would be rare, and (2) one of the consequences of democratic decision-making is that a majority can make a decision that we think is wrong, in some way. Election law can establish far more flexible procedures, including the use of deliberative process in a representative assembly. Heh! Asset Voting! But that's down the road.

The way forward, I suggested a long time ago, was for voting systems activists to employ advanced methods (such as Asset) in our own decision-making processes, and to encourage the use of advanced systems in NGOs. People need experience with transformed systems to be likely to accept them. They need to see if the systems actually work, and what problems might arise that we may not be able to anticipated. We, in the process of setting up the ESF, ran an Asset election, the only one that has ever been done, to my mind, so it was an historic event. How many people know that we did that. And what happened?

The result was essentially useless, even though it created a fully-representative (every member was approved by every voter, directly or indirectly) steering committee of three members. I could write about why this was useless, it's actually of high interest, (politics!) but ... we demonstrated how asset worked, and, as an election, it was spectacular. That result, which amounts to unanimous election, would probably have been impossible with any other system, other than repeated election in a group seeking unanimity (I've seen that happen in a situation that was highly polarized. They used Approval polling for a collection of alternatives, show of hands, then a high-social-utility winner, obvious from the approval votes was nominated -- not the plurality winner! -- and that election was then ratified with unanimity, including people who had declared "over my dead body" about what they now voted for.)

In an NGO, I would create Asset assemblies that are, functionally, primarily advisory in nature, but advising the Executive or Executive Committee, and also handling executive elections -- and removals ad interim. We need experience to design systems ready to actually assume power in governmental elections. Asset can be very light and inexpensive. The real work of an NGO, as to central organization, would be done by the Executive Committee (and staff or volunteers), under the authority of the EC. This is basic, traditional organization where there is available oversight by the representatives of all members; a smaller body than the full membership, able to coherently deliberate and advise if necessary.

Why Asset, with the Hare quota, is effectively the unanimous election of a representative by a set of voters, so it becomes true representation by uncontested choice, could be explained if anyone does not understand it. Using the Droop quota is a majority-election by the as-yet unrepresented votes. It falls short of true representation.