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When several people have to decide about a single yes/no issue*, the natural decision rule to use is the majority rule.

But when there are many issues to decide upon, the majority rule is "unfair" in the following sense: it is possible that the majority's opinion will be accepted on all topics and the minority's opinion will not be accepted on any topic. As an extreme example, it is possible that 51% of the population will decide about 100% of the issues.

I am looking for a decision rule which prevents this unfairness.

Formally, define a "uniform group" as a group of people who always vote in the same way. Define the "acceptance rate" of a uniform group as the percentage of issues on which the opinion of the uniform group got accepted.

Define a "fair decision rule" as a rule for which, for every uniform group containing X percent of the population, the acceptance rate tends to X when the number of issues tends to infinity.

MY QUESTION IS: Does there exist a fair division rule as defined above?

(* I restrict the question to yes/no issues, since when the issues are not binary the problems are much more complicated).

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I will object to your first-first sentence, because it ignores the intertemporal link between generations, and the fact that what is decided today will affect also the future, and those that are not here yet (or are not yet entitled to vote). Obviously this is philosophical, but it is a reality that human societies appear to try to take into account, when deciding what to put to vote or not. And in any case, it is not really needed as an opening to your question. –  Alecos Papadopoulos Dec 8 at 22:01
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I have provided an answer because I had an idea, but I wonder, why do you think this question is on-topic, or has good chances to get good answers in an Economics Q&A site? –  Alecos Papadopoulos Dec 8 at 22:31
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@AlecosPapadopoulos because in my university, voting procedures, and social choice in general, are taught in the economics department... meta.economics.stackexchange.com/questions/187/… –  Erel Segal Halevi Dec 9 at 6:20
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Social choice questions like this are, in my opinion, definitely on topic. Who would answer such questions if not economists? –  Ubiquitous Dec 9 at 8:02
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This is not so much political economy as it is social choice theory. Either way, it's definitely economics. "Social choice and public choice theory may overlap but are disjoint if narrowly construed. The Journal of Economic Literature classification codes place Social Choice under Microeconomics at JEL D71 (with Clubs, Committees, and Associations) whereas most Public Choice subcategories are in JEL D72 (Economic Models of Political Processes: Rent-Seeking, Elections, Legislatures, and Voting Behavior)." en.wikipedia.org/wiki/Social_choice_theory –  jmbejara Dec 9 at 17:02

2 Answers 2

up vote 6 down vote accepted

That's interesting: the flavor of the frequentist approach to probability used for a socio-political fairness criterion: if my measure as a population group is $0<p<1$, and known, then my opinion should be accepted by the whole at the same measure, as number of issues goes to infinity. In other words, current observed acceptance rate should be a consistent estimator of theoretical acceptance rate, and equal to my measure.

Then it is very easy to create such a decision rule, while saving public money: no need to hold one referendum after another, just construct a die, with as many sides as there are "uniform groups", with the die's weight distributed in such a way that the side representing uniform group $i$ will have probability of turning up equal to $p_i$. It won't be difficult to construct, and publicly and objectively test it for the desired properties.

Then, wherever an issue comes up for voting, just roll the die. And ok, spend some money for a suitable public ceremony.

Whenever there is a census, the relative size of each uniform group can be re-measured and a new die can be constructed.

Why do I have the feeling though that no uniform group is likely to ever accept such a scheme?

(This of course puts aside the importance of each issue, in general, for each uniform group, etc, but I took that from the OP which concentrates on number of issues, irrespective of what the issues are about, and to whom they matter and how much they matter, and how do we measure that etc).

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Or alternatively, just pick one voter at random and let that voter make the decision. –  Steven Landsburg Dec 9 at 4:10
    
@StevenLandsburg That's even better. Instead of focusing the news cameras on a die, we would focus them on a human face (kept secret till last minute so that the person's opinion is not revealed by association). Imagine the drama. –  Alecos Papadopoulos Dec 9 at 5:09
    
This is a good start-point, thanks! By the rule of large numbers, when there are many issues, the expected acceptance rate of each uniform rate is indeed proportional to its size. But, because the process is random, strange things may happen. For example, there is a chance that one person will always make the decisions although all other people disagree with him. So, the challenge now is: can your method be de-randomized so that it gives the same asymptotic behaviour deterministically? –  Erel Segal Halevi Dec 9 at 9:48
    
I just figured that the two procedures can be de-randomized by 'interleaving'. E.g., if group A is 60% and group B is 40%, then let group A decide on 3 issues, then let group B decide on 2 issues, then let A decide on 3 issues again, etc. Alternatively, put the voters in a circle in an arbitrary order and let each voter decide on a single issue in turn. This guarantees asymptotic fairness while avoiding the chance of having a single voter decide on all issues. –  Erel Segal Halevi Dec 9 at 17:13
    
This "deterministic approach" has the following characteristic: not everybody will have a chance to decide on some issue. –  Alecos Papadopoulos Dec 9 at 21:14

Rolling Dice?! Flipping a coin?! Randomly excluding voters?! To attain a fair vote?!

Here's a REAL set of answers that provide deterministic results and that start with the assumptions of the OP's conditions. Refer to the Addendum below if you need to understand how.

Some "fair" voting procedures that are deterministic:

You can skip to the section labeled with A,B,C etc...

In fairness, as a disenfranchised new member of the community (meaning that I can't vote), I would like to ask that those with the vote privilege please leave my vote count at zero if you likewise believe that my answer is of no value. Please leave a well-reasoned argument that is of value instead. I do edit my posts. Thank you...

INTRO. Much of the work by the philosopher, Alexis de Tocqueville, could be quoted and then paraphrased to sum the very problem you describe in your question: 'The rule of the 51 percent majority is the tyranny and oppression over the 49 percent minority.' This is especially true, in the case of YES or NO, all or nothing, when essentially the other half of voters are given neither equally nor pragmatically viable alternative benefits or consolations to substitute with which they may be indifferent (as in a basket of goods A or B for them) to be satiated. (In America, by the way, this could even be as bad as a split-haired 49.99% vs. 50.01% popular vote.) They, the lesser half, cannot be ignored as they do not disappear. By little stretch of the imagination this sets the stage for a very Pareto Inefficient outcome.

In your comment, you say "there is a chance that one person will always make the decisions although all other people disagree with him." As you originally alluded, the contrapositive is also applicable: 'Many people may make decisions although one person may disagree.' This is a challenge for new thinking when there already exists conventional thought.

BODY. What you're asking is 'For the most optimal outcome, how do you balance votes when there's a group within the total set of voters whose unchanging vote makes one majority outcome more probable than another (thereby even making the voting process itself superfluous.)'

There are several things that can be done. These solutions can be applied to remedying loaded coin/dice tosses mixed in with legitimate coin/die tosses or people who vote with bias:

A. IGNORE/REMOVE THE VOTES THAT NEVER CHANGE. If a subgroup always votes the same way, then the existence of them having a rationale is questionable. A vote is, in contrast to a random toss, assumed to be a discrimination between choices based on information. But voters can behave irrational in their choice. They may, without further consideration, always select one brand instead of another when faced between substitutes that have different labels but equal content. Perhaps they do this to minimize the risk of trying new. They operate on no, low, or old information. At any rate they skew the total vote by acting as a bias coefficient. 'Bias coefficient' means that the choice is completely inelastic. There are no other options or outcomes. This can mean that, because they neither question nor reason, they have no constructive voting input other than to skew the results. The solution: Simply ignore the never changing votes. Subtract the coefficient on the graph and bring the starting point back to zero. Conduct the real vote: Count the remaining votes as 100%, i.e., the votes which can sway either way depending on exogenous factors (as opposed to an inherent, endogenous bias.)

B1. WEIGHT THE VOTES AND DECIDE ON A DIFFERENT MAJORITY FRACTION. An inelastic vote bias gives the remaining voters who vote in line with the bias an unfair advantage over the other side of the YES/NO coin. This is a handicap for the other side. It takes less input for the former side to outvote the latter side - perhaps even when the majority of the latter votes are the actual thinking/rational voters who do constructively weigh YES vs. NO in decision-making. As you know, various sports employ handicaps to equate units of input, e.g., effort, on both sides. Q: How did David beat Goliath? A: By using an equalizer, i.e., a slingshot!

Also, pick a tie-breaker that is culturally tolerable. The US Congress uses the simplest fraction, 2/3 majority to represent a scenario where 2 out of 3 discrete/indivisible people would commit decisively one way versus another. In the 2/3 example, for the purpose of inclusion, redefine the inelastic subgroup as having 1/3 weight. The remaining voters can represent the other 2/3 of the vote. Multiply each vote of the remaining votes by some factor that makes their vote count numerically 2/3 the size of the first subgroup.

For example, the inelastic/biased group is made up of 90 voters or 40% of all voters. The remaining number of voters is therefore 90*60%/40% = 135 voters. Multiply the 135 elastic voters by a factor that gives them a 2/3 decision weight, i.e., 135*x=90*2 --> x = 180/(135) --> x = 4/3. In this example, the vote of each elastic voting person (who can be YES or NO) is equal to a 4/3 biased vote. This is actually a variation on A. The drawback is that the required majority might not be attained. The benefit is it makes the minority contingent smaller.

B2. Let's say that there is yet another subgroup within the elastic, changeable subgroup that does not have an equal probability of voting YES or NO. It may be partially biased. Let's say members in this variable subgroup may have a 2/3 probability of voting one way versus another. Again, find out the number of this special subgroup that has an uneven probability versus the number of those who have an even probability and multiply each side by factors that give each group, e.g., an equal 50/50 voting weight. For simplicity, one half has 2/3 probability of voting one way; and the second half has a 1/2 probability of voting either way. Multiply the first side votes by 3/2 and the second side votes by 2/1 to make the influential weight of both sides 1:1 again. If the numbers of either elastic subgroup are uneven then do the additional simple math for which you can refer above in B1.

C. INCREASE THE VOTING SAMPLE SIZE AND APPLY B. Imagine a room of only 2 voters: one stubborn, hard-headed person and one changeable,open-minded person. The outcome is either unanimous or 50/50 ambiguous. Increase your sample size! The problem is that neither of them and very likely at least one of them will not trust the newcomers especially after the winning/losing votes are tallied.

D. MAKE PEOPLE ACCOUNTABLE FOR VOTING CONSEQUENCES. [...this is my favorite one...] Hindsight is 20/20 but risk can truly sharpen one's focus. Involve risk management as part of the vote. Let the voters reap the fruit of their vote but also let them enjoy or suffer its taste. In this instance, voters would have to be registered/identifiable. Those voters who win get the benefits (and costs) of their vote (most fairly in proportion to the size of their vote.) If 67% of voters got to decide on how to use a budget, let them enjoy 67% of that budget toward their decision. Those voters who lose do not get to share in that benefit (or cost.) However, if the majority vote makes a bad decision, they must pay for it - not those who didn't vote for it. Most primates if not animals do not like to break even when budgeting input in exchange for output but the fear of loss is indeed greater than hope of gain. The perception of risks for voting vs. not voting can radically change voting behavior and motivate voters to acquire better information, not vote, or participate more actively thereby changing the voting sample (toward a more honest/constructive/informed turnout & vote) as well.

CONCLUSION. Fairness rules can be created and do exist (right here!) to balance a voting sample that contains biased subgroups in achieving a fair and deterministic outcome in both YES/NO votes or those that involve further complexity.

Hope these suggestions help, Erel!

BONUS. A lengthy list of quotes by de Tocqueville: http://www.goodreads.com/author/quotes/465.Alexis_de_Tocqueville

Enjoy!

ADDENDA. [Originally a response to a comment below asking for clarification. Important for inclusion but too long for intro.]

An election is a decision. A vote is a decision. The difference between either is the word used for "decision" & fraction criteria for finalization. A decision represents a probability. The probable decision is the sum of all probabilities divided by the total number of probabilities. Therefore, without complete/perfect info, a priori, an election decision is a probability; a single vote decision is a probability. Before casting a vote, a voter conducts an election with themselves. Each issue may be comprised of sub-issues, all representing probabilities, each with weight.

The probable voting decision that a voter makes is the sum of all sub-decision probabilities (each multiplied by their weight of importance - analogous to individuals voting) divided by the total weight of sub-decisions. Taking issues, sub-decisions, etc., to infinity, using the formula, gives us the probability of a vote when total issues are taken to infinity. The same applies to a voter having infinite elections with themselves or conducting infinite elections. Whether a group given probability=1 for their preference gets their way (at infinity) depends on the required majority.

If the probability of the group's decision at infinity is greater than the required majority vote, then the group will have their way at infinity. The answer to the question above takes this as a starting assumption, a mutually understood given, and then offers solutions for a "fair decision rule" that's interpreted to mean an 'equitable voting outcome that is balanced, that is "fair."' Sources of error it addresses are groupsizes/weights and that, even when taken to infinity, voters do not work with the same sets of info at the sub-decision levels to come to the 'same vote decisions'*.

*In the above, the "same vote decision" implies that the vote of one voter is identical in scope and thus equivalent to another voter's vote if it encompasses an identical number of issues/sub-issues, identical sets of information, identical cost/benefit analyses, and identical degrees of consideration with all other things being equal. Votes are not the same if all that which goes into each vote (for each voter) is different from one vote to the next which therefore creates a probable bias in the process leading up to the voting decision and thus the vote...

Case in point: Five people standing in line at a ballot box for an infinite number of elections to vote YES/NO that covers an infinite number of issues. They're living in an episode of Rod Serling's Twilight Zone where it just keeps repeating with more and more issues added to infinity. The first two people eternally read the newspaper everyday and do much research, consultation, & contemplation over weeks/years in order to decide YES/NO. The probability of the first group's vote either way, given variable tastes, is more variable. The other three people who like the same style and, who stubbornly refuse to change, base their decisions on comparing bumper stickers, 5-second slogans, and operate - with no exception - on the bias that marketing image & affiliation is everything and that the final decision is one about image & affiliation. The probability of the second group's vote given the same never changing preferences (and for the purpose of illustration) is 1 or very close to it. How to establish a fair vote? Refer to all of the above...

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If you're going to vote down, show some fortitude and comment please. –  SavedByZero Dec 9 at 16:38
    
"...in fairness, as a disenfranchised new member of the community (meaning that I can't vote), I would like to ask that those with the vote privilege please leave my vote count at zero if you likewise believe that my answer is of no value. And, in the said spirit, please leave a well-reasoned for/against argument of value instead. I do edit my posts. Thank you..." –  SavedByZero Dec 9 at 16:40
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I haven't voted on this answer, but I think that the problem is obvious. Question: find a rule for which, for every uniform group containing X percent of the population, the acceptance rate tends to X when the number of issues tends to infinity. Your answer never mentions likelihood of a given group getting its preferences. If you want upvotes, you are going to have to start answering the questions asked rather than give answers with minimal association to the question. –  Brythan Dec 10 at 3:29
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If you want more feedback, you'd probably be better off posting in meta or chat rather than comments. Downvoters are unlikely to come back to your question to read the comments. –  Brythan Dec 10 at 3:32
    
@Brythan Thank you. An election is a decision. A vote is a decision. The difference between either is the word used for "decision" & fraction criteria for finalization. A decision represents a probability. The probable decision is the sum of all probabilities divided by the total number of probabilities. Therefore, without complete/perfect info, a priori, an election decision is a probability; a single vote decision is a probability. Before casting a vote, a voter conducts an election with themselves. Each issue may be comprised of sub-issues, all representing probabilities, each with weight. –  SavedByZero Dec 10 at 15:55

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