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Social Choice and Beyond
Friday, February 19, 2010
Range-Approval Hybrid Voting and Economics
Mood:  d'oh
Now Playing: CNN
Topic: Social Choice

It has been shown previously that an optimal way to vote for one position such as President is to use Range-Approval  Hybrid Voting. This is a method in which each individual rates all the candidates on some scale, and the system normalizes all votes in such a way as to guarantee that each vote has maximum effectiveness. This makes it impossible for any voter to gain a strategical advantage by misrepresenting his or her sincere ratings of the candidates. I have stated that this model would apply to economic systems as well. However, an economic system is analagous to a political system in which at the simplest level each voter votes over the field of candidates and then two candidates are elected as Presidents. Then all citizens who prefer President A would be governed by President A and all citizens who prefer President B would be governed by President B. This could be extended to an arbitrary number of Presidents.


In an economic system there are many different outcomes with each citizen getting the most preferable outcome of those available. Ultimately, there would be as many outcomes as there are participants. This is analagous to a political system with many different Presidents and each voter governed by the President that he finds most preferable. This is not to say that voters or economic participants would get the outcome that is most preferred by them. But of all the possible outcomes, each individual’s satisfaction level should increase as the number of outcomes proliferates.


Considering the Range-Approval Hybrid model in which each participant’s ratings are divided into two groups by a threshold, this would only be applicable in the case where the outcome applied to all voters. In the case where there were two outcomes and each participant could have the one best for him, then it stands to reason that the threhold should be higher. The individual voter can afford to be more choosy. Quantitatively, this formula needs to be worked out. As the number of possible outcomes gets very large, the threshold will get very close to the participant’s first choice. Therefore, this model may or may not be realistic under these circumstances. Also there are constraints in the economic model that don’t necessarily apply to the political model. For instance, the outputs of the labor economy are inputs to the consumer economy so consumer preferences need to match up with workers’ preferences for the type of work that they do. For example, not everyone can work in the canned food industry if there is a limited demand for canned foods and a certain demand for frozen foods with no willing workers in that field. The demands for labor have to match up with the demands for consumption.


This non-linearity in the economic model mitigates against any coherent strategy on the part of the participants and so they will have more incentives to represent their preferences sincerely. In my Algorithm for Employee Shift Choice, employees state their preferences about shift choice and pay. Then those preferences are considered in such a way as to give each employee their highest preference insofar as possible. The strategies available have more to do with the actual algorithm  used than they do with the overall model.

Posted by jclawrence at 3:04 PM PST
Sunday, October 25, 2009
Why I Deserve the Nobel Prize
Mood:  a-ok
Topic: Social Choice

It may seem somewhat arrogant and egotistical to write a blog about why I deserve the Nobel Prize. However, I will make my case as follows. In the nineteenth century the English utilitarians, Jeremy Bentham and John Stuart Mill, posited that society should be organized according to utility theory, the essence of which was the greatest possible good for the greatest possible number. It was assumed that everyone could measure his or her own personal utility for various options, and those individual utilities could be amalgamated to come up with an overall social utility. This work has been carried on to the present day by those trying to find an alternative to GDP, namely the Greatest Happiness Principal or Sarkozy's recent commission to come up with a better way to measure societal well-being than current methods.

In the 1950s Kenneth Arrow attempted to carry the English utilitarian political and economic theory to the next stage by mathematizing utility theory. He changed the name to social choice theory. Social choice theory encompassed both voting theory and economic theory. However, Arrow only succeeded in proving, supposedly, that social choice was impossible. In other words no one could come up with a social theory based on the greatest possible good for the greatest possible number. This gave capitalists a huge sigh of relief because it was considered to be a theoretical endorsement of capitalism. Also it reinforced the idea that direct democracy was impossible giving encouragement to the idea that representative democracy was the best you could do. For this Arrow received the Nobel prize for essentially proving that socialism or economic democracy and direct political democracy were both impossible. Arrow's Impossibility Theorem became famous. Conservatives were overjoyed.

But some pointed out that Arrow's Impossibility Theorem meant that solutions were impossible only in certain cases. In many other cases it was actually possible to find a solution. Therefore, Arrow's Impossibility Theorem only meant that a soloution was not possible in every case. Others pointed out that the Borda voting method came up with a solution in every case, and also that Arrow's analysis only applied to rank order preferences. Preferences that were indicated by real numbers on the real number axis (a measure of utilities over various options) also were exempt from Arrow's Impossibility Theorem. So then additional work was necessary in order to discredit the possibility that direct democracy and economic democracy were less than desirable. That work culminated with Gibbard and Satterthwaite who proved that any social system that was not impossible according to Arrow's Theorem was subject to being manipulated or gamed. Some individuals could strategize their inputs to the system in order to gain an unfair advantage over other participants. This again delegitimized the notion that direct democracy or economic democracy constituted viable alternatives to capitalism and representative democracy.

Recent work by Warren Smith has shown a method for optimal strategizing or gaming the system whether or not you have knowledge of how others will vote. His work is for voting systems based on a voting method called range voting which is essentially a formalization of the English utilitarians' idea of personal utility. I came up with the idea of range voting independently of Warren but I will give him the credit for it. Warren and I have both pointed out that range voting escapes Arrow's Impossibility Theorem, and I have pointed out that, by an extension, it escapes Gibbard and Satterthaite's proof that every social choice system is manipulable. This is easy to see if we take Warren's proof for the optimal way for an individual to strategize and just let the system itself apply that to every individual. Since it is up to the system to amalgamate all individual inputs, part of that process would be to make every input an optimally strategized input thus denying any unfair advantage to any individual and maximizing the potential of every individual input whether that input be a political vote or an economic alternative. I call this method Range-Approval Hybrid voting and it can be applied to economic systems as well.

Therefore, both Arrow's Impossibility Theorem and the Gibbard-Satterthwaite Manipulability Theorem have been overcome, and each participating individual can be assured that he or she can express his or her true utilities whether political or economic and no other individual can game the system in such a way as to gain an unfair advantage. Any individual who tries to game the system will either come out worse off or at least will not come out better off. Then the resultant social choice or utility will represent the best possible solution for a system that is unmanipulable and can't be gamed. There is a price to pay, however, in that a better solution might be found if it were assumed that everyone were completely honest in their expression of personal utility, but that price is relatively small in order to guarantee a stable system.

So the age old quest for a social system based on individual and social uitility has been found despite the fact that Arrow proved such a quest impossible and Gibbard and Satterthwaite proved that any system that escaped Arrow's Impossibility Theorem could be gamed. This opens up new vistas (previously thought to be closed) for forms of societal organization based on cooperation and promises to reduce conflicts and limited choices available under present poltical and economic systems. If Arrow received the Nobel Prize for proving that human progress was impossible in this realm, I deserve one for proving that that avenue of human progress is indeed still open.

Here are the links to Range Voting and Social Choice and Beyond for additional background.

Posted by jclawrence at 6:52 AM PDT
Monday, January 19, 2009
Arrow's Example Doesn't Apply in Economic Case
Mood:  incredulous
Topic: Social Choice

Consider Arrow’s example on p. 27 of “Social Choice and Individual Values” where he shows a deficiency of the Borda count: “Let R1, ... Rn and R1', ... Rn ' be two sets of individual orderings and let C(S) and C'(S) be the corresponding social choice func­tions. If, for all individuals i and all x and y in a given environment S, x Ri Y if and only if x R/ y, then C(S) and C'(S) are the same (independ­ence of irrelevant alternatives). The reasonableness of this condition can be seen by consideration of the possible results in a method of choice which does not satisfy Condi­tion 3, the rank-order method of voting frequently used in clubs. With a finite number of candidates, let each individual rank all the candidates, i.e., designate his first-choice candidate, second-choice candidate, etc. Let preassigned weights be given to the first, second, etc., choices, the higher weight to the higher choice, and then let the candidate with the highest weighted sum of votes be elected. In particular, suppose that there are three voters and four candidates, x, y ,z, and w. Let the weights for the first, second, third, and fourth choices be 4, 3, 2, and 1, respectively. Suppose that individuals 1 and 2 rank the candidates in the order x, y, z and w, while individual 3 ranks them in the order z, w, x, and y. Under the given electoral system, x is chosen. Then, certainly, if y is deleted from the ranks of the candidates, the system applied to the remaining candidates should yield the same result, espe­cially since, in this case, y is inferior to x according to the tastes of every individual; but, if y is in fact deleted, the indicated. electoral system would yield a tie between x and z.”


However, if you use a corresponding economic example, there is no such problem.


Let x be 1 unit of work eg 1 hour at task X.

Let y be 1 unit of work eg 1 hour at task Y.

Let  z be 1 unit of work eg 1 hour at task Z.

Let w be 1 unit of work eg 1 hour at task W.


Let’s stipulate that each individual must work the same number of hours.


The general idea is to assign the tasks in such a way as to maximize social utility where utility is defined as sum over individual utilities and individual utility is defined as 4 x (time spent on most preferred task) + 3 x (time spent on second most preferred task) etc. ie a general Borda ranking. Let individuals 1 and 2 have the preference ranking xyzw and individual 3 have the ranking zwxy. Since 1 and 2 have the same preferences, we let them spend half their time on their first preference x and half their time on their second preference y. Let 3 do z since it’s his first choice. That leaves w. Let each individual do one third of w. Then the individual utility for 1 and 2 is (½)x4 + (½)x3 + (1/3)x(1) = 3 and (5/6). The individual utility for 3 is 4x1 + 3(1/3) = 5. Social utility is 12 and (2/3). The maximum possible utility is 16.


Now, if task y is deleted as in Arrow’s example, we have the preferences for 1 and 2 as xzw and 3 as zwx. Now let 1 and 2 each do (½)x and (½)w and 3 do all of z. Each person does 1 hour of work. Total possible utility = 12. For 1 and 2, utility = (1/2)x3 + (1/2)x1 = 2.Utility for 3 = 3. Total utility = 7. Alternatively, let 1 and 2 do (½)x and (1/4)w and (1/4)z. Utility = 3x(1/2) + 2x(1/4) + 1x(1/4) = 9/4 = 2 and (1/4). Let 3 do (1/2)z and (1/2)w. Utility = 3x(1/2) + 2x(1/2) = 5/2 = 2 and ½. Total utility = 7.


The  point is that eliminating y does not upset  the work schedules in the same way that it upsets political  rankings because the work can be divided among the workers in many possible ways. Hence Arrow’s Impossibility Theorem doesn’t apply.

Posted by jclawrence at 12:43 PM PST
Updated: Sunday, October 25, 2009 7:21 AM PDT
Monday, June 9, 2008
Social Choice Based Economic System Utilizing Range Voting
Mood:  happy
Topic: Social Choice

It has been shown that range voting offers a way out of Arrow's Impossibility Theorem. Arrow's Impossibility Theorem only applies to rank order voting methods and not point value methods. According to Arrow's book, "Social Choice and Individual Values," the field of social choice includes economic systems as well as voting systems. Since he thought social choice was impossible, no further discussion regarding economic systems was necessary. The assumed impossibility of social choice based economic systems was considered by some to be a theoretical endorsement of capitalism. However, Arrow had this to say about potential economic systems based on social choice, and since they are not as impossible as once was assumed, the topic is open for reconsideration:


In the present study the objects of choice are social states. The most precise definition of a social state would be a complete description of the amount of each type of commodity in the hands of each individual, the amount of labor to be supplied by each individual, the amount of each productive resource invested in each type of productive activity, and the amounts of various types of collective activity, such as municipal services, diplomacy and its continuation by other means, and the erection of statues to famous men. It is assumed that each individual in the community has a definite ordering of all conceivable social states, in terms of their desirability to him. It is not assumed here that an individual's attitude toward different social states is determined exclusively by the commodity bundles which accrue to his lot under each. It is simply assumed that the individual orders all social states by whatever standards he deems relevant. A member of Veblen's leisure class might order the states solely on the criterion of his relative income standing in each; a believer in the equality of man might order them in accordance with some measure of income equality.

We will consider a potential economic system which abstracts from the general social choice model which Arrow considers but is related to it. We submit that it is a form of economic democracy in that it's based on range voting. We consider only a very simplified, hypothetical system which is impractical without the many ramifications necessary in the real world. However, it is necessary for the sake of analysis to abstract from many real world ramifications in order to get at the basic structure. In particular we consider an individually based system in which an "individual's attitude toward different social states is determined exclusively by the commodity bundles which accrue to his lot under each." We also simplify each commodity bundle so that it contains only "a complete description of the amount of each type of commodity in the hands of each individual [and] the amount of labor to be supplied by each individual." "[T]he amount of each productive resource invested in each type of productive activity" is determined by consumer demand as specified by the aggregate commodity bundles off all individuals. Furthermore, each individual submits an input regarding only his or her own work-consumption schedules, and not those he or she desires for other individuals. Finally, collective activity is abstracted from so that each possible social state represents the aggregate of the individual inputs regarding only their own work/consumption.

Each individual rates his or her preferred individual state on a scale such as [0-9] or [0-99], for instance, in accordance with range voting procedures. The social state is then determined in such a way as to maximize social welfare or utility as measured by the summation of ratings over individual states such that the following condition is met. In each possible social state, the work to be performed shall be exactly what is necessary to produce the commodities to be consumed. In other words supply of commodities shall be equal to demand for those commodities as specified by the ratings of individuals over all possible work-commodity bundles. For instance, individual A might rate a work-commodity bundle in which he performed 20 hours per week of dentistry (assuming he's qualified as a dentist) in return for a copious amount of goods and services a 99. He might also specify a rating of 50 for a work-commodity bundle requiring 30 hours work per week and a less copious amount of goods and services. He then might assign a 1 to a bundle requiring 60 hours work per week in return for a meager amount of goods and services.


No real economic system could exist without money as a medium of exchange. It's just impractical to think that individuals would accept a system in which they were assigned a certain amount of work in return for a certain commodity bundle even if that maximized social utility. Therefore, work performed must be paid for in money and not by an "in kind" commodity basket. The commodity basket can be translated into monetary terms by pricing it such that the money received by each individual for his or her work exactly pays for it.  Pricing in such a system could be undertaken as follows. Pick some basic, simple and ubiquitous commodity and price it at 1 unit. (The units could be dollars, euros, pounds etc.) Then other consumer items could be priced in terms of that basic commodity considering the quantity and quality of labor and the quantity and quality of materials and other resources involved. For instance, a tube of toothpaste might be priced at 1 unit. Based on this, a particlular kind of automobile might then be priced at 20,000 units. Ideally, the aggregate amount of money dispensed by the system would be just sufficient to buy the aggregate amount of production as specified by aggregate consumer demand according to the the sum total of commodity bundles. Therefore, money supply would equal money demand, and there would be no inflation. Aggregate income could be computed in such a way that the amount of money in circulation would just be sufficient to buy all the consumer goods and services demanded according to the social state which maximizes social utility.

The social choice would then involve an assigment of work and income to each individual. It would be assumed that an individual's work preferences could be quite general involving different kinds of work and different hourly schedules. In general, an individual could do any type of work he or she was qualified for, and, at the lower end of the job spectrum, almost everyone would be qualified whereas at the upper end, only those with highly specialized training might be qualified. In general people would be qualified to do more than one type of work and would be free to submit more than one hourly schedule. Work weeks need not be standardized but could be individualized in accordance with worker demands.

Since individuals might not spend their income exactly in accordance with the commodity bundle they submitted with the corresponding work schedule, aggregate consumption would have to be tracked and adjusted so that there is little or no over or underproduction. Periodically, individual inputs regarding work-commodity baskets could be resubmitted, the social choice recomputed and adjustments made accordingly. Ideally, supply would equal demand both for work and commodities so that there would be no over or underemployment and no surplus or scarcity of commodities.


Fundamentally, the role of government would be to gather information from individuals, compute the social choice, disburse information to individuals informing them of their work-commodity schedules and monetary income (the one that maximized social utility), and oversee and track the production and consumption process making adjustments for the fact that actual consumer demand might not be the same as specified consumer demand. The production units could be either publicly or privately owned. Individual work schedules could be combined to construct a production unit so that production units might represent the collaborative work efforts of many individuals and production output for the enterprise might represent enough commodities to fill many consumer commodity baskets. Individual or enterprise inputs might include capital or other resources as well as labor.

The government would have to employ the services of massive supercomputers to do all the computation necessary. Information collection and work-commodity basket assigments would be centralized. Work schedule and consumer demands would be decentralized and individualized. Assigments would be flexible and subject to change both for work schedules and commodity bundle consumption schedules. The government could track changes in worker-consumer activity and make real time changes in production/consumption. The government might grant every citizen at least a minimum income for which could be purchased a minimum commodity bundle. A minimum amount of labor might also be required. Likewise, maximum work and/or consumer demands might be limited.


A social choice based economic system that utiilizes range voting has been described. Such a system would represent a highly simplified form of economic democracy. Individual work-commodity bundles would be preference rated in accordance with range voting and an amount of money associated with each. The system would then compute that social state which maximized social utility as the aggregate of individual utilites subject to the condition that production equal consumption. The system would also compute the pricing of consumption items and the amount of money to be distributed to each individual in return for the work and or capital resouces input by that individual. Money would just be a medium of exchange, and the total amount of money generated at any particular time would just be sufficient to buy the amount of production generated as specified by the social state which maximized social utility. Individual work-consumption assignments could be updated periodically or, perhaps, in real time in accordance with individual demands. Ideally, there would be no shortages or surpluses of commodities or labor and supply would equal demand. If priced correctly, supply would also equal demand in the money supply so there would be no inflation or deflation. There would be no unemployment by definition because exactly the amount of labor needed for production and no more would be required, and this would be distributed equitably by the maximizing of utility as a result of range voting.





Posted by jclawrence at 11:53 AM PDT
Updated: Monday, June 9, 2008 12:38 PM PDT
Friday, November 30, 2007
A Districtless Congress
Mood:  a-ok
Topic: Social Choice

The US is divided up into political districts. Each voter gets to vote for one member of the House of Representatives and two members of the Senate in this bicameral national assembly. There are 435 voting members of the House so there are 435 congressional districts. Cap1 The member from your district supposedly represents your interests, but none of the others do. In the Senate the political districts are the states. There are 100 senators, two from each state. The two senators from your state supposedly represent your interests but none of the other 98 do. So 1/435 or 0.22% of the members of the house represent your interests, and 2/100 or 2% of the members of the senate represent your interests.

This is a pathetic situation, and it's even worse if you did not vote for any of the congressman or senators who supposedly represent you. Say you're a Democrat and the congressman elected from your district (whom you didn't vote for) is a Republican. Then arguably you have no representation at all in the House. The same could be said of the Senate if you didn't vote for either of the senators who actually got elected. In other countries where they use other methods for making up the national assembly or congress like, for example, proportional representation, the percentage of the members representing each voter's interests is much higher. For example, if 28% of the electorate (including you) voted for the Green Party, then 28% of the seats in the national assembly would be Green Party members.

In a districtless congress each voter would vote for each representative, and each representaive would represent the interests of all voters. For example, if there are 300 seats in the congress and 500 candidates running for those seats there would be 500!/(300!)(200!) possible congresses or ways that this congress could be made up. In theory each voter could list each possible congress in order of his/her preferences, and then all the voters' specifications could be amalgamated to get the one congress that best represented the electorate as a whole. The problem is that it would be impractical for each voter to study the qualifications of each candidate and then come up with a list of all possible congresses. It would be too much work. However, there are ways to expedite this process as explained in more detail here. If each voter just listed the candidates (instead of the congresses) in order of preference, this list could be translated by software into an ordered list of congresses. Senate Furthermore, the list of candidates could be simplified by using the recommend- ations of the voter's political party or other trusted experts in part or in whole. A customized list could be generated by taking eclectic recommend- ations cafeteria style. Or there could be different lists available to the voter depending on the voter's profile related to his/her political objectives. A simple questionnaire given to the voter could generate a list according to the voter's predilections. There are a lot of different ways any particular voter's list could be generated with the voter having complete control and the final say.

One way of amalgamating the list information is by range voting. Using this method each possible congress would be given a numerical rating, and the ratings for each congress would be added up over all the individual voters to determine the winner - the one with the highest overall social rating. There is no need to rank the possible congresses in order over the entire electorate since only the top rated one would be chosen. Therefore, Kenneth Arrow's model for social choice and his impossibility results as presented in Social Choice and Individual Values are invalid.  In fact Arrow's model which calls for a full social ranking doesn't apply to most political as well as most economic situations. The only thing it seems to apply to is combining judges' rankings in Olympic figure skating where it is important to know not only first place but also second and third. In political and economic situations it's necessary only to know the top rated or first place result.

Posted by jclawrence at 2:48 PM PST
Thursday, June 7, 2007
A Voting Method Combining Range Voting with Maximizing Social Utility
Mood:  happy
Topic: Social Choice

With various voting methods, there is a method for individuals to vote, and then there is a method for combining the votes to determine the outcome of the election. For instance, with the Borda Count, all the candidates are ranked by the voters with the top ranked getting a number of points equal to the number of candidates and the bottom ranked getting 1 point. Then the votes are counted by counting the total points for each candidate, and the candidate with the most points wins. Range Voting is similar except the top ranked candidate can be assigned an arbitrary number of points usually determined by the ground rules of the election. Also 2 or more candidates can be assigned the same number of points. Then the points are counted for each candidate with the one getting the most points declared the winner. 

Computer simulations by the Center for Range Voting have shown that Range Voting is superior to other methods in that it minimizes Bayesian Regret. Bayesian Regret measures the difference between the social utility produced by the candidate whom, if he had won, would have maximized social utility, and the social utility produced by the winner as computed by the voting system used, in this case, Range Voting. The social utility is the sum over the individual utilities of all the voters. This presupposes that there is a meaningful measure of individual utility which is a foregone conclusion as far as the Center for Range Voting is concerned.

I would argue that, although there are many definitions of utility and the definition of utility used by the Range Voters is basically preference utility, it is, nevertheless, a meaningful form of utility. Each voter's utility is essentially revealed by his vote. In Range Voting with a range from 1 to 100, for example, if a voter rated some candidate a 100 and then that candidate won the election, that voter's individual utility would have been maximized. The number 100 may not have any meaning in itself, but just because it is the maximum point value that can be assigned in this example, the voter would be considered to have achieved maximum individual utility. On the other hand, if a voter rated the winner of an election as a 1, minimum utility would have been achieved by that individual.

My suggestion is this. Instead of summing point values over all individuals for all the candidates and then declaring the winner as the one with the highest point total, compute the social utility which would be the sum of the individual utilities for each candidate and declare as the winner the candidate who maximized social utility. Obviously, this system would minimize Bayesian Regret over all other systems! An individual's utility for any candidate would correspond to the point value assigned to that candidate. This system could be used for Borda Voting, Approval Voting, Plurality Voting or Range Voting. In fact, Range Voting is the generalization of Borda, Approval and Plurality. Any voter could submit his vote as a Borda, Approval, Plurality or Range Vote within the confines of Range Voting. For instance, with a range from 1 to 100, if a voter wished to be a plurality voter all he would have to do is vote 100 for some candidate and 1 for all others. For Borda, he would equally space his point values from 1 to 100 and then assign them in order of his preferences. For Approval Voting he would assign 100 points to all those candidates he approved of and 1 point to all others. Finally, for Range Voting, he would distribute point values among the candidates corresponding to his preference intensities.

Another consideration is strategic voting. Some feel that Approval or Plurality Voting within Range Voting is strategic, that really the voter has an "honest" distribution of point values over the candidates but then maximizes some and minimizes others. But how do you know that, or, more to the point, how can you assume that for the purposes of computer simulations? Maybe the maximin voter truly feels that this vote represents his true utility distribution. All the voting "system" knows is what the voter reveals by his submitted vote. You really can't tell if the vote is a strategic vote or not, so why worry about it, and why berate some method because the social utility is assumed to be lower than it would have been if all voters had voted "honestly."

The maximum social utility that can be achieved is a function of the distribution of utilities among the individual voters, the domain, if you will. Some distributions (or elements of the domain) will produce a greater social utility than others. How much social utility that can possibly be achieved depends on the distribution of utilities among the voters.

Another objection is that the computation of the maximum social utility for any election is much more complex than simply counting up the points. This is true, but it can be done and it was done in the computer simulations done by the Center for Range Voting. Otherwise, it wouldn't have been possible to calculate Bayesian Regret. In fact, these calculations can be precomputed and stored much in the way Google precomputes search results in order to speed up the search process. In addition shortcuts in the computation process may be discovered.

Posted by jclawrence at 6:32 PM PDT
Updated: Thursday, June 7, 2007 7:24 PM PDT
Monday, June 4, 2007
Arrow's Borda Count Example
Mood:  caffeinated
Now Playing: CNN
Topic: Social Choice


The following quotation is from "Social Choice and Individual Values," by Kenneth Arrow. We want to examine the "reasonableness" of Arrow's example of the Borda count which he maintains violates his Condition3: Independence of Irrelevant Alternatives.                                                                                                                                          



CONDITION 3: Let R1', ... , Rn' and R1', ... , Rn'  be two sets of individual orderings and let G(S) and G'(S) be the corresponding social choice func­tions. If, for all individuals i and all x and y in a given environment S,

x Ri Y if and only if x Ri'y, then G(S) and G'(S) are the same (independ­ence of irrelevant alternatives).

The reasonableness of this condition can be seen by consideration of the possible results in a method of choice which does not satisfy Condi­tion 3, the rank-order method of voting frequently used in clubs.2 With a finite number of candidates, let each individual rank all the candidates, i.e., designate his first-choice candidate, second-choice candidate, etc. Let preassigned weights be given to the first, second, etc., choices, the higher weight to the higher choice, and then let the candidate with the highest weighted sum of votes be elected. In particular, suppose that there are three voters and four candidates, x, y, z, and w. Let the weights for the first, second, third, and fourth choices be 4, 3, 2, and 1, respectively. Suppose that individuals 1 and 2 rank the candidates in the order x, y, z, and w, while individual 3 ranks them in the order z, w, x, and y. Under the given electoral system, x is chosen. Then, certainly, if y is deleted from the ranks of the candidates, the system applied to the remaining candidates should yield the same result, espe­cially since, in this case, y is inferior to x according to the tastes of every individual; but, if y is in fact deleted, the indicated electoral system would yield a tie between x and z.

A similar problem arises in ranking teams in a contest which is essen­tially individual, e.g., a foot race in which there are several runners from each college, and where it is desired to rank the institutions on the basis of the rankings of the individual runners. This problem has been studied by Professor E. V. Huntington,3 who showed by means of an example that the usual method of team scoring in those circum­stances, a method analogous to the rank-order method of voting, was inconsistent with a condition analogous to Condition 3, which Hunting­ton termed the postulate of relevancy.

The condition of the independence of irrelevant alternatives implies that in a generalized sense all methods of social choice are of the type of

2 This example was suggested by a discussion with G. E. Forsythe, National Bureau of Standards.

'E. V. Huntington, "A Paradox in the Scoring of Competing Teams," Science, Vol. 88, September 23, 1938, pp. 287-288. I am indebted for this reference to J. Marschak.


Now we examine this example in some detail.

Let's define social utility as the sum of the individual utilities where the individual utility for an alternative is equal to the point value of that alternative according to the individual's rating. For example, in the first case individuals 1 and 2 assign a point value of 4 to alternative x, 3 to alternative y, 2 to alternative z and 1 to alternative w. Individual 3 assigns a point value of 4 to alternative z, 3 to alternative w, 3 to alternative x and 1 to alternative y. The social utility then is 10 (4+4+2) for alternative x, 7 (3+3+1) for y, 8 (2+2+4) for z and 5 (1+1+3) for w. So the winner is x.

Now we consider the case in which y is removed from the election. With the Borda count, the social utility for x is 7 (3+3+1); for z: 7 (2+2+3); for w: 4 (1+1+2). As Arrow observes, there is a tie between alternatives x and z. Although Arrow thinks that the result should still be x, if you interpret the situation that z is now considered first by one individual and second by 2 while x is considered first by 2 and third by 1, a tie between x and z is not unreasonable!

However, let us consider the above example using Range Voting instead of the Borda count. With Range Voting and point assignments between 1 and 4, the point assignments for each individual remain the same. Then the maximum social utility is 12 in both cases (including y and excluding y). The result in the first case is the same: x wins with a social utility of 10. With y removed, the social utility remains the same since the individual point values remain the same. So x still wins with a social utility of 10! With Range Voting the election is truly independent of irrelevant alternatives at least in this example. This is explored more fully in my paper, "Social Choice, Information Theory and the Borda Count."

Posted by jclawrence at 1:19 PM PDT
Updated: Monday, June 4, 2007 1:39 PM PDT
Tuesday, April 18, 2006
Arrow's R Notation
Mood:  energetic
Now Playing: Cross Blogged with Will Blog for Food
Topic: Social Choice
In the arcane world of social choice, a man by the name of Kenneth Arrow looms large. In 1951 he published a book, "Social Choice and Individual Values," in which he supposedly proved that social choice is impossible. But what is social choice? Let us say we have a society composed of N individuals numbered 1,2,3, ... . Those individuals have to order a set of M alternatives with their most preferred alternative being their first choice etc. Let's indicate the alternatives as a, b, c, ... . Then a social welfare function accepts the individual orderings as inputs and produces as output the social ordering which is an ordering of the alternatives that applies to the whole society.

If individual 1 prefers a to b, we write aP1b. If society prefers a to b, we write aPb. So far so good. But we also want to provide for the case in which an individual is indifferent between a and b. We write this aI1b and aIb, respectively. Arrow's analysis then combines these two relationships into a relationship he denotes as R which means "prefers or is indifferent to" so aR1b means individual 1 prefers a to b or is indifferent between a and b. Arrow's rationale for this is the following: "Instead of working with two relations, it will be slightly more convenient to use a single relation, 'preferred or indifferent.'" (p. 12) (emphasis added)

Arrow then goes on to postulate two axioms. Axiom 1 states that either xRy or yRx and he notes that this does not exclude the possibility that both xRy AND yRx. Axiom 2 has to do with transitivity which will not concern us here. Again Arrow states (p. 13): "Axioms 1 and 2 do not exclude the possibility that for some distinct x and y, both xRy and yRx. A strong ordering on the other hand, [one with only preferences and without indifferences] is a ranking in which no ties are possible." This is blatant nonsense. One could have half the population with xPy and half with yPx [strong orderings] and that certainly would represent a tie so a tie is possible. What Arrow is implying without coming out and saying it directly is that in his world a tie between two alternatives is to be represented as a social indifference. This is completely arbitrary and limits his entire analysis.

One must assume that in Arrow's world each individual will submit his input in terms of R. That is individual 1 would submit aR1b, aR1c etc. until all pairwise comparisons have been made. For now we will go along with Arrow's demand that only pairwise comparisons need to be submitted. It can be assumed that individuals are not permitted to submit a comparison using the indifference relation since then what would be the purpose of introducing R to make the analysis "slightly more convenient." The whole idea of "slightly more convenient" is to reduce the number of relations from 2 (P and I) to 1 (R). However, Arrow proposes (without saying so) to use the I relation in the social choice to cover the case of a tie. Therefore, the social choice could be aRb, bRa or aIb.

Now the idea of the social welfare function (or of any function for that matter) is to connect each element of the domain (consisting of all possible combinations of individual choices) to an element of the range (consisting of all possible social choices). There are a great number of possible functions. Each function will hook up elements of the domain with elements of the range differently. The important thing is that each possible element of the domain is hooked up to one and only one element of the range. Arrow implies that any element of the domain that represents a tie (such as half the population having aRb and half having bRa) should be hooked up with the range element aIb. Respectfully, I disagree with this approach for the following reason: the half of the population that has aRb could actually prefer a to b (no one is indifferent), and the half of the population that has bRa could actually prefer b to a. That represents a tie to be sure, but society is hardly indifferent between the two alternatives. Arrow has confused a tie with an indifference! By so doing he has guaranteed that his analysis will yield the result that no social choice is possible.

Secondly, I would like to point out that individual information is lost when an individual submits his input as aR1b or "I prefer a to b or I'm indifferent between a and b." The system does not know which, and this introduces ambiguity at the outset. Not only that, but say an individual is indifferent between a and b. He has two ways to express it! He can submit either aR1b or bR1a. The resulting analysis becomes meaningless as the system knows not how many of the individual aRb's represent indifferences and how many of them represent preferences. Ditto for the individual bRa's! There can be no meaningful social welfare function given these kinds of inputs.

Therefore, I suggest that Arrow's approach is not acceptable and that his conclusion that social choice is impossible is invalid. A more rigorous approach is necessary involving the possibility of ties between orderings as elements of the range. One possibility of dealing with these ties is to randomly choose among them which I think my friend, Ben, at Oxford is considering as a doctoral these.

For more on this subject, please see my blog Will Blog for Food.

Posted by jclawrence at 1:44 PM PDT
Updated: Tuesday, April 18, 2006 1:50 PM PDT
Tuesday, February 28, 2006
Issues for a Social Choice Based Political-Economic System (3): Advertising
Mood:  a-ok
Topic: Social Choice
Would there be advertising in a social choice based economy? There wouldn't be any need for advertising since advertising is done only to entice you to buy a particular product. In a capitalist society, each company is privately owned and it's in its self-interest to get you to buy their products. This incentive would not hold in a social choice based economy or Preferensism. Instead of being enticed to buy, consumers would initiate the buying process by checking out the products available, the ratings of these products and the pricing. This can be conveniently done online where even today the entire range of products ratings, suppliers and pricing is available.

In Preferensism it is of no concern to society as a whole which products are chosen since whether or not certain business enterprises prosper or flounder is of no concern except for the fact that businesses that don't produce well or don't produce products that people want will be dismantled while other businesses will be expanded or started. Society is neutral and disinterested in which particular products are bought and sold. The workers, however, cannot lose their jobs since everyone has equal access to the job market. They will be reabsorbed in other businesses.

The motive for advertising today is to get you to buy a product or service so that the provider of that product or service can prosper - not because it is the best product or service for you. In Preferensism, the goal would be to assist the consumers in making the decisions that are best for them. Preferensist society is completely neutral as to which decisions are actually made by consumers because there are no vested interests. People don't have to fear losing their jobs. Enterprises don't need to fear going bankrupt. There would be no such thing as bankruptcy only disassembling and assembling of enterprises that would be done strictly in terms of the relevance of those particular businesses to society.

How to be an intelligent consumer should be taught in school since it is something people do almost every day of their lives. Resisting advertising and doing your own research into what is actually best for you is something that is possible today. The internet has websites that do ratings of products, price comparisons, ratings of suppliers etc. Advertising is no longer necessary in order to inform consumers as to what's available. A Google query on a particular generic term will bring up all the products that are available in that area.

Posted by jclawrence at 5:08 PM PST
Monday, February 20, 2006
Social Choice and Capitalism
Mood:  incredulous
Topic: Social Choice
In another post we compared a social choice based society or preferensism to socialism. The basic dissimilarity is in the ownership of private property. In socialism most of the property is socially owned. In preferensism there are less restrictions on property ownership. It can be both individually and socially owned. The basic similarity is that individuals are compensated for their work without the need for profit. So the work involved to produce goods and services consumed both individually and collectively redounds to the credit of both workers and property owners with each citizen having an equal say in how the work and compensation for it are distributed. In this way it is similar to socialism in the sense that socialism stands for "to each according to his work." Socialism does not imply equality of compensation since some people are capable of more and better work than others. The same can be said of preferensism. However, in preferensism property owners get a say in the outcome equal to every other citizen's say whereas in socialism there are no property owners, and everyone is a worker. In preferensism some might be strictly workers, some may be worker-property owners and some may be strictly property owners.

The similarity of preferensism with capitalism is the emphasis on innovation, invention and individual initiative. Innovators and inventors should definitely be rewarded. Intellectual property such as patents, music, artwork and literary works should be rewarded. The dissimilarity with capitalism is that owners of property would not have the sole determination of how that property would be priced in the marketplace. Prices are set according to an amalgamation of all citizens' preference lists.

In preferensism corporations would not be privately owned. They would be set up and dismantled according to the demand for their produced items. The realm of private property would include real estate, natural resources, intellectual property and anything else that can be accumulated. However corporations or businesses as entities in themselves would not be considered private property since only individual persons have a right to vote. In capitalism corporations are set up with the rights of individuals. In preferensism only individual persons would have the rights of individuals.

Therefore, there would be no stock market in preferensism since there would be no privately owned companies. In capitalism even publicly traded companies are privately owned in the sense that they are not owned by all of the citizens in general but only by that group of citizens who have bought stock in that company. A business enterprise would be set up in preferensism according to the demand by the citizens for the products which that enterprise would produce. Therefore, supply would equal demand. Several competing enterprises could be set up just to keep everyone honest and to distribute the work geographically. If only one enterprise were set up, there wouldn't be any checks or balances on that one business. Several competing businesses could keep things in check. But they wouldn't be competing price wise only in terms of efficiency and quality.

So relationships among producers, consumers, workers and property owners would all be regulated in preferensism so that no one citizen had any more power over the process than any other. In capitalism lone individuals are no match for large corporations.

In preferensism the law of supply and demand would be operative but would be contained or held in check due to the fact that aggregate consumer demand would be calculated from citizens' preference lists and work assignments and business start-ups and contracts would be such as to satisfy that demand. An excess beyond consumer demand would not be produced while production insofar as is possible would equal demand. Therefore, the aggregate amount of work and consumption of resources would be minimized since nothing in excess would be produced. No one would have an advantage in the economic process in the sense that they would be able to corner a market or dictate prices. Likewise, no business would have to fear bankruptcy since the businesses as entities in themselves would be assembled and disassembled according to societal needs.

The law of supply and demand would be in balance. There would be no unemployment since everyone would have equal access to the job market. The work week would be minimized and tailored to individual wants since work would tend to be spread around among as many people as possible. There would be no gluts in the marketplace since nothing would be produced that wasn't already earmarked for consumption. There would be no overproduction and no shortages. Natural resources would be conserved since there would be no wastage or wastage would be minimized.

Not everyone would have an equal outcome. Some would be richer than others, but the gross inequality that exists in capitalism would tend to be mitigated due to the fact that instant billionaires due to IPOs would not be produced.

Posted by jclawrence at 4:31 PM PST
Updated: Monday, February 20, 2006 8:09 PM PST

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