Majority Judgement Study Group

To summarize in three lines.

It is widely known among researchers that majority rule, "a mechanism for decision-making by multiple people," is seriously flawed.

I brought the knowledge to the front to get a rough idea this time, and the mathematical and academic talk came later.

Introduce a situation in which "a majority of people do not like the decision" is reached by "choosing one of the candidates and voting by majority".

In that situation, the Majorty Judgement shows that the majority's preferred decision can be made

context setting

For a given agenda item P, 40% of people prefer option P1 and 60% prefer option P2

On the other hand, there is an agenda item Q, which is not as important as P. Both options Q1 and Q2 are supported by 50% each

There are three candidates A: (P1, Q1), B: (P2, Q1), C: (P2, Q2)

https://gyazo.com/d84911dd7a82029d0a57ea4dcc0fc5e0

Take a majority vote.

Mechanism: Each voter selects one person whose opinion is closest to his or her own and votes for that person.

https://gyazo.com/2e2915f11f376e2627853128931cf0f5

Candidates B and C each get 30% of the vote

Voters (P1Q2) who did not find a candidate that matches their preferences vote for candidate A who is closer to their opinion

https://gyazo.com/3b05c765cabd59547f5b0097ab8715bf

As a result, A gets 40% of the votes and takes first place

60% of people wanted P2 to be chosen, but P1 was chosen.

A majority vote can make a majority unhappy.

What's wrong.

Even if the person who chose B thought "I'd rather be C than A," there's no way to express that in a method that designates a single candidate.

A vote for B is equivalent to "if not B, then A or C".

Distortion on the candidate side

It's easier to get votes if you appeal to the options that the other candidates have not chosen, rather than the options that he or she truly believes are better.

It is a mechanism that encourages people to make untrue statements (to lie).

Distortion on the part of voters

A person from Faction B might say to a person from Faction C, "If we don't do this, the vote will be split and A will win, we have to vote for B, not C!" and solicit them to vote for B instead of C.

Behavior that encourages people to vote (lie) for someone who is not the candidate they truly believe is better.

strategicity

Majority rule can be strategically manipulated.

Strategic manipulation is to make the consequences more favorable to oneself by making statements that differ from the truth

I lie because it's more profitable to lie.

= Majority rule is not strategy resistant.

Is a strategy-resistant social choice mechanism (a mechanism that brings together everyone's opinions to choose one of several options) possible?

Make a Majorty Judgement

How it works: Everyone rates on a scale of 6, with the median being the best.

The meaning of the number 6 is discussed below.

For example, "6: Very good, 5: Good, 4: Somewhat good, 3: Somewhat bad, 2: Bad, 1: Very bad".

Voters rate all candidates on a 6-point scale

https://gyazo.com/c16a8d2a92c8e4cb31b1588a3f1c39fe

aggregation

https://gyazo.com/eaaa33fe506c256e32a2565fba348613

A is "6: very good" (20%), "5: good" (20%), and "2: bad" (30%), so the median is "2: bad

B is 30% "6: very good" and 30% "5: good", so the median is "5: good" (C is exactly the same)

Thus, B and C are tied for first place.

Majorty Judgement cannot lie and move his social reputation in this direction (see below).

change-over phenomenon

Related story.

For example, let's say you're talking about changing part of your company's office to be used for other purposes.

We had a discussion about how to change it and came up with a lot of ideas.

Each proposal was untenable, so we decided to put it to a vote of the employees.

I inadvertently put "no change" in that ballot item.

Composition: A: do not change, B: change, C: change

The vote to change it was split between B and C, and the vote resulted in a "no change" vote.

'Oh, no? I thought the majority wanted to change it?"

There are multiple ways to change it, but only one way not to change it, so when you take a poll like this, "don't change it" tends to be the choice

For example, unless you vote first on "change/no change" or have a "change is good" kind of corporate culture, you'll end up with a company that won't change.

Example

B6 A4 C2 40%

C6 A4 B2 40%

A6 B2 C2 20%

In aggregate, these are

Proposal A 6 20% 4 80

Proposal B 6 40% 2 60

A is selected as "4: Somewhat good

Consider real-world applications

6-level rating is not difficult to express in the WebUI and not difficult for users to operate

Doesn't that take a lot more time than voting for one person?"

The authors tested MJ on 12 actual candidates by the side of a polling station during the French presidential election.

Most subjects voted in less than a minute.

Less than 5 seconds per candidate

It is a 6-point scale for each candidate, so you can make a decision for each candidate. No need to think, "Which is better than the other candidate?"

Voter satisfaction was high with the ability to state, "Not this candidate."

It has to do with this, which I explained at the majority vote.

A vote for B is equivalent to "if not B, then A or C."

Voters often have the opinion "I don't care if it's B or C, just not A", but the "choose one of the candidates and vote by majority" system does not convey this

Meaning of 6 in "six steps."

As a mathematical model, it can be any number of things.

It can be "scored on a scale of 0 to 100."

In an experiment for the French presidential election, 12 candidates were rated on a 6-point scale, but 86% of the respondents used only 5 types

So, the authors believe that most people will only use a portion of it, even if it is more detailed than this.

For example, if you ask people to give a score out of 100, they will use a coarse scale, like "this one gets a 60, this one gets an 80.

What it means to be even

The authors believe that the lack of a neutral rating like "neither" makes it easier for differences to be expressed as "dare I say good or bad?"

This is not experimentally verified as good or bad, but merely the authors' interpretation.

It could be interpreted that the lack of a neutral option distorts the message by forcing people who really have neutral feelings to twist it into something good or bad, instead of accurately expressing their feelings.

I guess it doesn't matter either way, since the distortion in this case is not much compared to the severity of the "choose one of the candidates and vote by majority" (Nishio's interpretation).

In the "determine the initial value of the vote" pattern described below, it is more convenient to have a neutral value.

Increasing the number of steps can be interpreted as increasing expressiveness.

Functions as a lie-free questionnaire

A resigning employee said to me, "I always gave the highest score on the internal survey mechanically because I didn't want to be told anything by my supervisor. I've been doing it that way ever since I started thinking about quitting." The morning is about to end with an MTG where the survey supervisor who heard the message shuts down and no remedial action is taken.

This is one answer to the "why is it a problem that people are leaving when engagement is not declining from the survey results?" but it seems to be difficult to accept from those in charge.

While the survey design is anonymous and supervisors cannot directly identify respondents by name, a small team, such as five or fewer people, will be asked, "Did you answer negatively? What was the reason?" However, in small teams of less than 5 people, it is not clear whether everyone will answer honestly because the survey is anonymous.

Honest answers are damaging = gain by lying = not strategy resistant

Analyzing data collected by non-strategic mechanisms is difficult to interpret because the distribution of values is misaligned from the true distribution to begin with.

Majorty Judgement voting results can also be used as a 6-point rating questionnaire for each option

Good quality data with no lies

However, you can't hear everything you want to hear.

Q: So this example was an incentive not to vote honestly by leaking information about the vote to the supervisor, but wouldn't it be the same in MJ if the information is leaked?

A: Of course. In addition, even if no information is revealed, if there is no benefit to be gained by answering the question, it is "a loss of effort to take it seriously," so I don't take it seriously.

The basic premise is that there is an incentive to respond seriously, for reasons such as the distribution of resources within the organization depending on the results of the answers, etc. In addition, if a strategy-resistant method is used, the data gathered there can also be the results of a survey without distortion.

(a) summary

With respect to the mechanism for bringing together the opinions of several people to make a decision, the traditional majority rule tends to be used: "Vote for the candidate you think is the best and take the one with the most votes.

This has the problem of vote splitting.

It's also a mechanism that gives you an advantage when you lie.

The mechanism of the Majorty Judgement was introduced: "Rate each option on a scale of 6 and choose the one with the highest median value.

This is not a vote-splitting issue.

Lying is not advantageous (see below).

Data mixed with lies is useless.

MJ also functions as a questionnaire that does not contain lies.

strategicity

Math gradually from here

What does it mean that "lying is not advantageous" in MJ?

When your reputation is higher than your social reputation as determined by MJ, you can't fish your social reputation out of the water by lying.

https://gyazo.com/71033fdd8b9ce54b2e6f8f1d5acbc34e

The reverse is true as well, and you can't drag down your social reputation.

You can "lie and lower your social reputation when you value yourself more highly than society values you."

But I won't do it because it's a loss for me.

https://gyazo.com/9710b18fe00f83ffdfc348f95c6f65b9

This is a common feature of rank functions

Only rank functions?

If we let the evaluation vocabulary be an interval of real numbers and assume continuity of aggregate functions, then the rank function is the only function that satisfies strategy resistance, anonymity, congruence, and monotonicity

Why use the median function when there are many different rank functions?

When the majority evaluation is unanimously r, there is a property that the social evaluation is also r.

Only the median function has this.

If voter dissatisfaction is defined as "discrepancy between one's evaluation and social evaluation" and minimized, it is still a median function

Relation to [Boulder Rule

In Boulder Rule, voters rank each candidate.

Then, the higher the ranking, the higher the score, and the winner is determined by the sum of the scores.

https://gyazo.com/6419b00fbf2a3d4fde0af56138be74f7

Sometimes presented as an improvement over traditional voting, but weak in terms of strategy resistance

For example, let's say there are five candidates on the ballot, A through E, and only two of them are decent candidates, A and B, and the rest are bad guys who would massacre a certain ethnic group.

For simplicity's sake, let's assume there are three voters, two of whom are in the A camp and one of whom is in the B camp.

If you vote honestly, A wins.

https://gyazo.com/c52d6a383dce63a988a555f33cbac980

Here, if the B faction lies and says that A is even less preferable than the extremists, B wins.

B-ers lie because it's more advantageous to them.

https://gyazo.com/e5157d1f4bc7752bfcc027e9e5872cde

It is undesirable for group A to win by having only group B vote strategically.

So let's assume that group A did the same and strategically voted that "B is even worse than the extremists"...

https://gyazo.com/920c5f66b708aee0522f92f23b819ce4

What a bunch of decent candidates dragged their feet and an extremist that no one wanted won!

Comparing Boulder Rule to Majorty Judgement

N candidates rated in N steps with the same value prohibition corresponds to the Borda score

In Boulder Rule, take the sum of these, in MJ, take the median.

In short, a kind of [phenomenon where the mean is affected by outliers

The median of 5,5,1 is also 5.

https://gyazo.com/98b8b8fdeece4402f4b2335ed43c0c43

Try to make it with kintone

https://gyazo.com/76d0cbf4f097adfef0a617c56917b5c7

https://gyazo.com/a8dd4a46c198bd46de65b223f1e224f7

https://gyazo.com/b1f60667d7b764255a3afd26c57a5124

Aggregate by number of records and sort in ascending or descending order rather than by "aggregate value".

Then the 6 o'clock position on the pie chart would be the median.

I thought the trick was to put the number at the beginning of the radio button choices, but I was wrong.

Sorting is by order of choice, so 6 comes first in ascending order.

It doesn't matter that I put the number at the beginning.

I suppose we could aggregate for each of the multiple options.

It's done.

https://gyazo.com/1c753d70f5991245592b218e234f7508

It's weird that the order of the legend and the color order of the stacked graphs are reversed, but...

The color scheme might look better with a heat-map-like gradient.

Just look at the 50% line and make a decision.

In this case, since both 50% lines are "rather good", plan B wins with the highest "number of votes above the median".

https://gyazo.com/c62cce74b55261a79b82755551807737

Actually, it would be better if the choices were sorted based on this ranking, so it would be easier to see which one is in first place, but I couldn't figure out how to do it.

Is it indeed necessary to use JS customization?

Could not figure out how to limit one vote per person for multiple choices.

Especially what to do if there are "people who voted for Proposal A but not for Proposal B".

Proposal 1: Only counting all votes cast as valid votes.

I'm sorry to hear about the invalid votes due to an inadvertent oversight.

Proposal 2: Make it the median among those who voted.

I did so anyway in the above implementation, but wouldn't some kind of strategic manipulation be possible in this case?

Proposal 3: Make the choices odd-numbered, with no vote being neutral.

I think this is a sure thing.

Records for all voters are added in batches by the administrator, by neutral ballot.

Voters make edits whenever they like.

Just make sure you have admin rights to add records to the app.

The public can only browse and update

https://gyazo.com/380806f458ab5b045ebc85fb96bf72b2

Furthermore, if you only have access to the records for which you are the designated voter, you will not be able to view other people's votes.

Proposal 4: Put them all in one record and make them all required answers, so there will be no partially voted status.

Is it safe to publish the results during the process if it is strategically resistant?

If you're going to go to the trouble of using kintone to view aggregate data, it would be more interesting to visualize "how everyone is feeling at the moment" in real time.

Maybe that will further stimulate discussion?

Can I change my vote after hearing the discussion?

Something along the lines of, "At first I thought Plan A was very good, but I changed my mind when I heard the criticism."

The assumption is that all those eligible to vote are told that the vote is being taken.

Relationship to Arrow's impossibility theorem

A: The theorem is a model in which voters give their preferences as a message

The "candidate B is preferable to candidate C and candidate C is preferable to candidate A" kind of thing.

MJ does not. One that expresses the desirability of each candidate on an ordinal scale (e.g., a value on a 6-point scale or a real number from 0 to 100).

Assuming that the meaning of this scale is shared among the voters, I think a rating of about 6 is a roughly realistic assumption.

Candidate B is preferable to Candidate C, and Candidate C is preferable to Candidate A" has no information on whether C is so good that it just barely loses to B, or so bad that it is not good at all and just barely beats A.

MJ's method can gather more information.

Some argue that Arrow's "independence from unrelated candidates" is too strong a request.

To begin with, Arrow's impossibility theorem is not about strategy resistance.

This is another mechanism that uses preferences as a message.

Discussion after the presentation

A story about the many different patterns of majority rule.

I'm comparing various scoring rules.

p.58-59

Majority rule has the fatal flaw of "vote splitting," which means that it does not even reflect the majority opinion... Therefore, Toyotaka Sakai, author of "The Economics of 'How to Decide'," proposes the "Bordal rule" method as an alternative. This is because it is "the method closest to unanimity.

Majority Judgment (MJ), which we will cover in tomorrow's Auction Lab. MJ is amazing. I used to think that social choice theory was for "election systems" and "polling methods," but now I can use it for business.

I asked a question and got an answer.

nishio: I saw your 2015 "Doubting Majority Rule" 2016 "The Best Bordal Rule as an Alternative to Majority Rule" (Diamond Online) and 2020 "Winning by Mechanism Design".

Do you still think Boulder Rule is the best alternative to Majority Rule? Or do you think Majority Judgement is better?

From a strategic operations standpoint, MJ is much better than Borda. So if I had to choose between MJ and Borda, it would be MJ. The reason why I didn't appreciate MJ so much before is because MJ is very different and cannot be described within the traditional Arrovian Social Choice framework. It took me a long time to accept it.

The beauty of the Bordal rule, however, is that it can be conveyed in less than 10 seconds using only words: "The Bordal rule distributes 3 points for first place, 2 points for second place, and 1 point for third place. It can be communicated on the radio and in newspapers; MJ cannot be communicated on the radio or in newspapers; MJ can be communicated with a video, but it takes a minute of time that I can use to explain using a flip-flop.

nishio: thank you for your answer! i personally like the fact that MJ made the voter's message more informative than the traditional preference order, but in theory, not following the traditional model led to the difficulty of acceptance. the difficulty of explaining MJ is a matter of having more experienced people. I think that time may solve the problem if we increase the number of people with experience.

pair-loss criterion

Pair loser criterion: the property of not choosing a pair loser (a candidate who loses in any pair when two candidates are chosen)

In "Doubting Majority Rule," it is assumed that only Boulder Rule satisfies the paired loser criterion.

Q: Does MJ meet the pair loser criterion?

A: The candidate who loses in any pair in MJ is the candidate with the lowest median value, so of course he/she loses overall (= satisfies the pair loser criterion).

---Notes below

majority rule

discrepancy between votes

Naming the majority vote

Seems like the majority would be happy, but it's not.

5 to 7 steps

It doesn't matter how many steps you take.

A two-step process is called an endorsement vote.

Historically, endorsement voting was studied in 1970, and an improved version of it is MJ

tie break

Decide by the number of votes above the median.

Radical Opinion

Easier to get votes because there are no other candidates around with similar views.

median

Strong strategic operation

It's less susceptible to outliers.

Data containing lies is worthless.

No ideal voting system exists.

French presidential election

Recently, Balinski and Laraki have proposed a method called Majority Judgment to overcome the problems of conventional one-person-one-vote voting and social choice based on individual preference order. This paper introduces their discussion of its theoretical validity.

Keywords: one-person-one-vote, election, social choice, Majority Judgment, Arrow's theorem, rank function

non-limitability of the domain of definition

identity

non-dictatorship

Arrow's theorem

When there are more than three choices, there is no social welfare function that satisfies the above three

Evaluation by common vocabulary

As long as the order is in the common vocabulary and the meaning is shared.

Or any real number between 0 and 100.

Social evaluation function

Properties to have

neutrality

Independence from irrelevant objects

If this is satisfied, then an aggregate function exists.

aggregate function

Properties to have

anonymity

identity

monotonicity

Strategic Operability

Strategic Operations

strategic inoperability

The rank function satisfies all the requirements

Why the ranking function satisfies strategic inoperability

Are there other functions that satisfy the request?

If the evaluation vocabulary is an interval of real numbers and continuity of the aggregate function is assumed, then only the rank function satisfies the requirement

On the manipulation of valuations without motive

There is at most one person who can raise or lower the social rating

At most one person can manipulate the evaluation, and the only aggregate function that can be manipulated by at most one person is the rank function

Note: Continuity is assumed.

If the vocabulary is a finite set, there can be other aggregate functions that satisfy the condition

So what order is best?

The property that when the majority evaluation is unanimously r, the social evaluation is also r.

Only the median has this.

If we look for an aggregate function that minimizes dissatisfaction with the discrepancy between one's own evaluation and the social evaluation, we find the median value.

social experiment

An experiment with voters in the French presidential election

The candidates are the actual 12

Solicit to experiment on those who come out after voting.

74% of 2383 people participated in the experiment

1733 valid votes

I rated it on a scale of 6, but 86% used only 5, so that's good enough for me.

Most votes last less than a minute.

Satisfaction of being able to vote negative.

Balinski, M. and Laraki, R., 2007. A theory of measuring, electing, and ranking. Proceedings of the National Academy of Sciences, 104(21), pp.8720-8725.

Arrow's impossibility theorem

Satisfactory methods do not exist in the traditional voting model

More realistic model

Laplace and Galton Origin

New Ways to Avoid Impossibilities

Realistically feasible methods

Majority Judgememt

social choice

Collect multiple individual evaluations into a single evaluation

Two applications.

Voting: For example, choose one candidate from several, or choose one from several options.

Jury Desicion: for example, a jury decides the grade of wine or the score of a figure skating competitor

Social Decision Function SDF

Takes the voter's message as input and returns the ranking of the winner or candidate

There are many variations.

traditional model

Voter-ordered model

The meaning of the order is ambiguous.

There are three ways.

real world

The voter's message is simply a message, based on preference but not on itself

Based on the traditional model of voters expressing preferences

We have shown that there is no social rehabilitation function SWF that combines the three basic characteristics

Except in the case of two options

Model in which voters enter real values

Utility, not preference

Interesting but impractical, utility complex concept

Arrow's impossibility theorem reproduced

Unless the utilities are comparable.

This model was used to prove the [Gibert-Saththwaite theorem

There is no social choice function where the dominant strategy is for voters to express their true preferences.

If the output is rank ordering, shouldn't the input be a preference for OVER rank orderings?

Various suggestions

Lull's Proposal Condorcet

Cusanus Proposal Boulder

approval voting

claim

1: Arrow and other impossibility theorems indicate no solution in the traditional model.

2: Traditional models don't adequately model the message or purpose in the first place.

3: We need a new model.

The application of reality suggests another input.

Olympic figure skating, or wine ratings.

kelvin

Voting and Markets

Expression with money

lingua franca

Even if you personally prefer the wine, you may give it a lower grade.

In other words, input is not a preference but a rating by common language

Arrow's impossibility theorem is caused by the absence of a common language.

If hypothetically we were ranking on a 100 point scale and everyone was using 40-60 and Mr. A outputs 80-100, what would happen?

All of Mr. A's messages will be ranked first, and the only effect will be to pull up the median by one.

In other words, messages that are far removed from the common language are simply less informative.

Realistically, a common language is something like a 6-point scale, so there won't be too many serious differences.

In this paper, we point out a problem with the aggregate function that plays a central role in the Majority Judgment proposed by Balinski and Laraki, and show that it can be resolved by relaxing the condition of strong monotonicity. Furthermore, we show that the aggregate function is uniquely determined by the social evaluation when the evaluator's evaluation is widely divided.

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