We can apply the analyses of social choice to political or economic problems, but in general they can be applied to a much larger set of problems including the judging of athletic competitions. Any situation which can be characterized by a finite number of distinct alternatives and a number of individuals, each of whom has a precise preference ordering among the alternatives, lends itself to a social choice analysis. The individual preferences are integrated by the social choice algorithm to determine the one preference ordering that represents the social choice. Arrow says in Social Choice and Individual Values, “We assume that there is a basic set of alternatives which could be conceivably be presented to the chooser. In the theory of consumer's choice, each alternative would be a commodity bundle; in the theory of the firm, each alternative would be a complete decision on all inputs and outputs; in welfare economics, each alternative would be a distribution of commodities and labor requirements.” He goes on to say, “In the present study the objects of choice [alternatives] 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 that accrue to his lot under each. it is simply assumed that the individual orders all social by whatever standards he deems relevant.”
This definition of a social state is perhaps over specified. It is hard to imagine that each individual would specify preferences over "the amount of each productive resource invested in each type of productive activity." Rather this would be subsumed under the total amount of commodities and labor specified. It's hard to imagine that individuals would specify things down to the level of diplomacy and the erection of statues. However, and this Arrow left out of his analysis, a social state does encompass the selection of political representatives such as a legislature. These officers then would be responsible for things like diplomacy and making day to day political decisions. Therefore, a social state would consist of economic and political components, and the individual could conceivably be involved in specifying his or her preferences over alternatives involving both.
1) An ice skating competition
We consider that there are 5 contestants and 5 judges. The goal is to come up with a ranking of the 10 contestants based on their performances. Let the contestants be A, B, C, D, E, and the judges be enumerated 1, 2, 3, 4 and 5. Each judge ranks the 5 contestants. For instance, judge 1 might have a ranking: C,A,D,E,B. Judge 2 might have a ranking B,E,A,D,C. etc. Let's say that all the judges’ rankings are fed into the social choice algorithm and the final result is E,C,A,B,D. Therefore, this represents the social choice.
2) Choosing a President
In this example, we consider n individual voters and m alternatives. The m alternatives would be all the candidates running for President. For instance they might be Jones, Smith, Johnson etc. etc. The n voters would consist of voter #1, voter #2, etc. up to voter #n. Each voter ranks the m candidates and submits a ballot with that ranking. The social choice algorithm than considers all the voters’ rankings and generates one candidate as the winner. Let us say it’s Smith. Notice that in this case the object is not to come up with a social ranking but only one candidate who represents the winner.
3) Choosing a Congress or Parliament
Let us say the Congress or Parliament is to consist of 300 people and there are 600 people who are running for these 300 seats. Therefore, each alternative would consist of each possible combination of the 600 candidates. Let us say there are n voters. Each voter then would rank all the alternatives from first to last. The social choice algorithm would generate the one alternative that would represent the 300 winners. Notice that the social choice would not need to be a ranking of all the alternatives similar to the Presidential case. In practice it would be a bit tedious for the voters to rank all alternatives so a method would have to be devised that would streamline this process. Computers should have no problem generating the solution. The practical problem is streamlining the process for the voters and then interpolating or “filling in the blanks” to get something approaching a complete ranking.
One way this could be done is the following: A voter could rank however many of the candidates he wished to express an opinion on and give each candidate a place ranking. For instance, a voter might rank Smith and Jones in a tie for first place. That would indicate that this voter very much wanted these two individuals included in the elected body. This particular voter might also give Martin and Johnson a rank of 600 or tied for last place. This would indicate that this voter wants very much for these two candidates to be excluded from the elected body. In addition, this voter might express preferences for a number of other candidates giving them rankings somewhere between 1 and 600 indicating different degrees of preference for or against certain candidates. Then it could be assumed that the voter is indifferent to candidates not included in her preference list. Based on these partial rankings, the computer could generate complete rankings for each voter. There would be a lot of ties between alternatives to which the voter would be indifferent.
In this way a truly national representative body could be elected without dividing the nation up into multi-member districts the way it's done, I believe, in Great Britain or single member districts the way it's done today in the US. This then would represent a truly national representative body each member of which would represent all the citizens and not just the citizens from his or her district.
4) The Handyman Problem
Here's a problem faced by numbers of people every day. How does one choose a plumber or an electrician or any other service person for that matter? Today most people would try and get a recommendation or referral from a friend, or consult the Yellow Pages of the phone book and then maybe the Better Business Bureau to see if a particular service person or company had any complaints against it. Then they would call inquiring about prices, expertise and availability. They might ask for the names of other customers to call as references to get their opinion as to the service provided by a particular company.
An alternative approach would be to set up a central clearinghouse for service people. When a plumber, for instance, was needed, the consumer would go online and fill out a short form which would list her preferences regarding price, availability, reputation, proficiency, customer satisfaction etc. etc. A computer algorithm would then match her up with a service person that best suits her needs as described by her preferences. Let's say the most important consideration for her is availability since she has a leaky pipe and needs someone to fix it right away. The next most important thing to her is price; the third most important thing is the plumber's qualifications and lastly she's concerned about customer satisfaction.
Meanwhile the central computer has accumulated data on each of the available plumbers including their availability which could include their exact real time schedule, their price for any particular time slot (they might charge more for coming over in the middle of the night or on weekends, for example), their ratings from other customers (customer satisfaction) and their proficiency rating (years of experience, competency level, training). Bear in mind that other similar queries from other customers would be coming into the central computer's web site simultaneously. Now the computer's job is first to generate a preference list for each customer, and then match up each customer with a plumber in the best possible way. Notice that there are two problems here. One is the social choice problem which is to generate any solution of matching plumbers to customers that is in compliance with certain basic rational and ethical assumptions. Secondly, we would want to come up with the best possible solution for matching plumbers to customers. Something like the “utilitarian” solution which would represent the “greatest satisfaction for the greatest number.” How would we measure this? Of all the possible solutions matching up customers with plumbers, one possible way of measuring satisfaction would be a Borda count over the customers. Let's say that there are n possible preferences that the computer generates for each individual. If Mrs. Jones got her first choice preference, this would count n points. If Mrs. Smith gets her third choice preference, this would count n-2 points and so on. If Mrs. Zimmer gets her last choice preference this would contribute 1 point to the total. We define total social satisfaction to be the sum over all customers of their satisfaction levels for each possible solution. The social solution which afforded the largest social satisfaction would be chosen. Of course, this is only one way of doing it. There may be other ways.
Now it's difficult to imagine a social choice problem as defined above not to have a solution. Therefore, it's difficult to imagine how Arrow's Impossibility Theorem would apply. Obviously, not all customers would be happy with their plumber, what time he arrived, his level of proficiency or what they had to pay him, but what else is new? There would obviously be a higher satisfaction level throughout the population of customers needing plumbers at any one time than if customers were matched to plumbers at random. Notice also that there would be certain trade-offs involved. A customer most concerned about price would probably not get a plumber in the shortest possible time or with the best possible credentials. Also it is implied that every customer would give feedback to the system regarding how well the plumber did his job, treated her, arrived at the agreed upon time etc. etc. Therefore, the system would have a built-in Better Business Bureau.
Now the system should of course include the plumbers' preferences over all possible alternatives so that not only is a high level of customer satisfaction attained, but also a high level of plumber satisfaction. A particular plumber might prefer not to get out of bed in the middle of the night to go fix Mrs. Smith's leaky pipe. He, therefore, might list his preferences in such a way that, if the system chose him for this job, he would be suitably recompensed for his effort. In other words he would charge a lot more to work in this time slot. A young eager beaver who wanted to increase his business and didn't mind getting out of bed in the middle of the night might charge less for this time slot and so would probably be selected by the system for Mrs. Smith's job. A particular plumber might only want to work Monday and Thursday mornings and Wednesday and Friday afternoons. He might want to go surfing the rest of the time. Therefore, the system would select him only for work during those time slots. The system would also make other choices in such a way that not only a high level of customer satisfaction was achieved but also a high level of plumber satisfaction.
Again the nature of the system would be to make compromises. The challenge to the system would be to make the best possible compromise. For Arrow's Impossibility Theorem to have any meaning, the system would have to break down under certain conditions. Before that would ever happen, I conjecture, the system would break down for more obvious reasons such as major simultaneous plumbing problems all over the city due to a flood, for example, and not enough plumbers to fix everything in an “Arrow Impossibility” impasse and can't decide if Mrs. Smith should get plumber A and Mrs. Jones should get plumber B or vice versa. Let the system “arbitrarily” decide to give Mrs. Smith plumber A and Mrs. Jones, plumber B. What’s the worst that could happen? In one particular instance Mrs. Jones would be unfairly treated compared with Mrs. Smith? So what? In the vast number of cases, the average customer would be better off than otherwise, and Mrs. Jones would eventually get her plumbing problem resolved. Big Deal!!
Another implication of this system is that people would, for the most part, be treated fairly. That means, for instance, that someone, by virtue of the fact that cost was not a consideration, could not command the services of any particular plumber at any particular time or, in other words, get preferential treatment. He certainly would be more likely to get better service within the system by virtue of the fact that he was willing to pay more to get his other wishes met. However, his satisfaction within the constraints of the system would not be valued more highly than any other person's satisfaction.
From this rather limited example, we can see how a whole economic system might work, how producers and consumers might interact, how prices could be set, how individually determined work schedules could be had, how high levels of quality and fairness could be achieved and how a high satisfaction level could be attained throughout an entire population.
5) The Employment Agency Problem
Social choice considerations can also be brought to bear in a centralized employment agency much as they have in Sweden where there is a computerized listing of job opportunities for the whole country. The use of this centralized job bank is free to the citizens. This is one way Sweden keeps unemployment down. However, this kind of service could be greatly magnified to bring a lot more varied choices for both workers and employers into play. Each employer, for example, could rank a number of alternatives each of which consists of preferred employee characteristics. His first choice could be for an engineer who will work 50 hours a week for $40,000 a year, is free to travel, will take work home on the weekend and will relocate to north of the Arctic Circle. His preference list could then go down from there to more realistic demands! A prospective employee, on the other hand, could list his preferences as follows: 1) engineer seeking $90,000 a year, location - Stockholm, no travel, housing allowance of $10,000 etc. 2) engineer seeking $80,000 a year, any large Swedish city, some travel etc. and then go down to his last preference which might be - engineer seeking $20,000 a year, location - the North Pole, will travel anywhere etc.
So each employer would submit a preference list for the kind of employee(s) he is seeking, and each prospective employee would submit a preference list for the kind of job he is seeking. Then the system's job is to match employers with employees in such a way as to maximize satisfaction within the entire system. A simple way, but not the only way, to rate overall satisfaction would be to do a Borda count of the satisfaction over all jobs where the satisfaction for each particular job would be the sum of both the employee's and employer's preference listing rank for that job. For instance, if a match-up between employee A and employer B resulted in the employee getting his second most highly listed preference out of 10 preferences and the employer's getting his fifth most highly listed preference out of 10 possible preferences, the satisfaction rating for this particular combination would be 9 plus 6 or 15. Considering all possible match-ups over the entire society, the one that produced the highest overall rating would be the social choice.
In this system there would be no need for individual employers to interview individual employees. They would be interviewed once by the employment agency. All their qualifications and preferences for a job including salary would be input to a central data bank. Each employer would submit his preference list for the type of employee he was seeking including qualifications, background and salary and the central algorithm would produce the match-ups in such a way as to maximize satisfaction over the whole society. Notice that those seeking employees and those seeking jobs do not demand a certain set of criteria but submit a preference list over a range of sets of criteria. In this way an employee may not get everything she wants in a job and an employer may not get everything it wants in an employee, but over the entire society and on average both employers and employees will be better off than they would be otherwise as long as the system seeks to maximize overall satisfaction which is similar, but not exactly, to the utilitarian concept of “the greatest good for the greatest number.”
This process would be dynamic and time varying. At any point in time there would be a certain number of job openings and a certain number of applicants. At a later point in time these job openings and applicants might be entirely different. It's hard to see how Arrow’s Impossibility Theorem applies to this problem. The worst that could happen is that certain job openings would go unfilled or that certain prospective employees would not be able to find jobs at any particular point in time, and this would have nothing to do with Arrow’s Impossibility Theorem. Employers would not be in the commanding position they’re in today with respect to employees in that their satisfaction in filling a certain job would be given no more preferential treatment than the employee's satisfaction in filling that job.
It is easy to see how the employment agency social choice problem can be generalized to the economic problem in which each worker-consumer specifies a “labor-commodity bundle.” In addition to specifying her preferences in a job, the worker-consumer specifies, instead of salary, her preferences over a commodity bundle of goods and services she wishes to receive in return for doing that job. The totality of specifications of preferences over the whole society of individual labor-commodity bundles would then place a constraint on the production and consumption for the entire society in such a way as to maximize satisfaction in the entire society. This constraint would ensure that there is no over production (more labor and produced goods than necessary) or underproduction (less labor and produced goods than necessary). Consequently, an optimal balance can be achieved in which supply equals demand and everyone is suitably employed without overworking to produce a surfeit of unnecessary goods. No underemployment and no overemployment. No scarcity of goods and no surfeit of goods.
Of course, reasonable interpolations of specified data would have to be made in order to come up with alternative preference lists. These would then feed into the margin of error for the entire system. Again the demand is not for perfection, but overall sub-maximal optimization over the entire society. Arrow’s Impossibility Theorem demands a perfect world. If you don't demand a perfect world, it's hard to see its relevance.
6) The Romantic Match-Up Problem
There are a lot of web sites like Match.com that help people find romantic partners. A man or woman peruses the pictures and profiles of the opposite sex and chooses whomever they wish to get acquainted with. One of the services offered is suggested match-ups. The site will email members suggesting certain people who they think are good matches. We can formalize the problem like this. Let each individual rank in order the individuals he or she thinks would be the best match-up for his or herself. Then the web site selects, based on an algorithm, the best one-to-one match-ups and informs the couples accordingly. Bear in mind that there would be no obligation for anyone to act on this information, and this doesn't mean that the suggested match-ups will necessarily result in happy or successful relationships. As formalized this becomes a social choice problem in that the system selects as inputs a set of individual rankings and then outputs the social choice consisting of a set of matched pairs. Here is one method of doing it. First, match up all the men and women who have selected each other as first choices. You can't do any better than that. Remove these matches from the pool. The remaining set of individuals will have first choices that do not correspond to someone who has selected them as a first choice. See if there are any individuals who have been selected as first choice by someone who has been selected by that individual as second choice. If so, match these couples up and remove them from the pool. You can't do better than matching a first choice with a second choice after all first choice-first choice matches have been removed.
Now we have to consider the philosophical problem of whether a first choice-third choice match-up is better or worse than a second choice-second choice match-up. It seems to me that the latter would be preferred. Therefore, select out of the pool all second choice-second choice matchups. In the same way continue with second choice-third choice matchups, third choice-third choice match-ups, third choice-fourth choice match-ups etc. Finally, we will get last choice-last choice match-ups. Needless to say these match-ups will probably be somewhat disgruntled. In fact somewhere down the list of match-ups, the couples will go from relatively pleased to relatively displeased with their match. However, that need not concern us here.
Our concern is whether or not we have matched everybody up, and the answer is no. There may be some people left unmatched. Let us ask another philosophical question: is a third choice-third choice match-up better than a first choice-third choice match-up or vice versa. It seems to me that the latter would be better. Therefore, we amend our algorithm as follows: first select all first place-first place choices, then all second place-first place choices, then second place-second place choices, then third place-first place choices, third place-second place choices, third place-third place choices etc. etc. So for a particular individual, let's call her Mary, we scan the men to see if Harry, Mary's first choice, has also chosen her first. If not, we look at Mary's second choice, Bill, and see, providing Bill’s first choice, Betty, has not chosen him as her first choice, if Bill has chosen Mary first or second. If not, we look at Mary’s third choice, Steve, to see, providing that Steve's first and second choices have not also chosen him first or second, if Steve has chosen Mary first, second, or third and so on.
By proceeding in this way, we are assured that Mary will eventually find a match, and everyone else will too. Therefore, Arrow’s Impossibility Theorem doesn't apply. If it did, someone would not get a match. Our value judgment is that a p place-s place match is better than an r place-s place match if p>r. So a first place-last place match, for example, would be better than a last place-last place match. If we arbitrarily assign a utility to each match-up equal to n (the number of men assumed to be equal to the number of women) minus the rank of the man's match plus n minus the rank of the woman's match and sum up the utilities over all matches, the result will be the social utility based on this particular algorithm. Therefore, if there are 100 men and 100 women, a first place-first place match-up would result in a utility of 99 + 99 = 198, and a last place-last place match-up would result in a utility of 0 + 0 = 0.
Now we ask is this algorithm the best possible i.e. does it produce the maximum social utility compared to any other algorithm? The answer is yes because at each stage and for each individual, we produced the best match-up we could. Therefore, we can proudly say that the Lawrence romantic match-up algorithm is an optimal algorithm.
There are other ramifications to this problem like what happens if the number of men is not equal to the number of women and what if not all men rank all women and vice versa. Also what if an ordering of selected matches is to be provided for each individual and not just one match-up. These issues will not be gone into here, but will be left as an exercise to the student. Ha. Ha.
7) Choosing the Members of a Team
Let us assume that a new league is forming in some professional sport. The league is to consist of m teams with n players on a team. There is a pool of p players to choose from where p > mn. How do you decide which players get assigned to which team? Let us assume that the owner of each team has a preferred roster of players. In fact let us assume that each team owner ranks all possible rosters of players in order of her preferences. Some interpolation will be required so as to make the owners’ job of listing alternatives practical. So we have a number of individuals with preference lists, a number of alternatives, and we must make not only a social decision, but the best possible social decision. Now we know that some social decision is always possible since we could arbitrarily number the players, assign the first n players to team 1, the second m players to team 2 etc. Therefore, Arrow’s Impossibility Theorem doesn't apply. So the question is what is the best possible match of teams to owners based on the owners' preferences? Again let us measure the utility of a team as R (the total number of alternatives) minus the order of the alternative that the team owner actually receives. For instance, let's say a team owner gets his first listed preference. His utility would be R-1. If he gets his last listed preference, his utility would be 0. The social utility would be the sum of the individual team owners' utilities.
We can proceed systematically as follows: List all the possible assignments of teams. No player can appear more than once in the collection of teams' rosters for any particular listing. For each possible listing, measure the social utility as the sum over the individual utilities. Choose the final assignments of players to teams which maximizes the social utility. Therefore, we have a method or algorithm for assigning players to teams and forming a whole new league based on the team owners' preferences and resulting in the “greatest happiness for the greatest number.” Utilitarians rejoice!!!
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