## Social Choice

Social Choice has to do with the aggregation of individual preferences to determine an overall social preference or social choice. Over 200 years ago a French enlightenment philosoper by the name of the Marquis de la Condorcet noticed that there is no straightforward way to do this and noticed what has come to be called the Paradox of Voting. The Paradox of Voting states that, given the distribution of voter's preferences, in certain cases there may be no solution which obeys certain common sense rules.

A number of people have worked on this problem over the years including the Rev. C. L. Dodgson better known as Lewis Carroll who wrote "Alice in Wonderland." In the 1950s, Kenneth Arrow wrote a book entitled "Social Choice and Individual Values" in which he formally and mathematically proved that Social Choice was impossible if a possible solution had to satisfy a set of rational and ethical criteria . His work was essentially a formalization of Condorcet's Paradox of Voting.

Various people have tried to water down Arrow's assumptions in an attempt to find Social Choice solutions that are possible according to the watered down set of criteria. There's also the possibility that Arrow's criteria, as originally conceived, are somehow wrong.

The following papers deal with various critiques of Arrow's theory and present various solutions to the social choice problem ranging from an analysis of tie solutions to an argument that Arrow's Independence of Irrelevent Alternatives criterion is ill-conceived.

Social Choice is an important field because, if in fact Social Choice is impossible, there is no solid theoretical ground for a satisfactory version of political democracy and also any possible version of economic democracy would likewise be invalidated.

Please note that the contact information on the title pages of the papers is out of date and invalid. Current contact information is provided on this website.

Optimal Selection of Alternatives with Applications to Voting and Consumers' Choice

Abstract

We devise a method for political and economic decision making that's applicable to choosing multiple alternatives from a larger set of possible alternatives. It is suitable for the selection of multiple members in a multi-member district. The method combines utilitarian voting with approval voting and sets an optimal threshold above which an individual voter's or chooser's sincere utility preference ratings are turned into approval style votes. We generalize utilitarian/approval hybrid voting which deals with a single member outcome to the case of multiple outcomes. The political case easily generalizes to the economic case in which a set of commodity bundles that will be made available by society is chosen from a larger possible set by the amalgamation of the individual choosers' inputs. As the set made available gets larger, the individual voter or chooser is more likely to gain greater utility or satisfaction.

Optimal Selection of Alternatives with Applications to Voting and Consumers' Choice (Microsoft Word 97-2003 .doc document)

Optimal Selection of Alternatives with Applications to Voting and Consumers' Choice (.pdf format)

Social Choice Models for Consumer Spending

Abstract

A companion paper to "POLITONOMICS: A Meta-Theory Encompassing Political and Economic Decision Making," this paper illustrates the nature of consumer choice among a number of consumption bundles. As the number of possible outcomes for each individual increases, the nature of the decision making process changes from political to economic.

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POLITONOMICS: A Meta-Theory Encompassing Political and Economic Decision Making

Abstract

In “Social Choice and Individual Values,” Kenneth Arrow said, “In a capitalist democracy there are essentially two methods by which social choices can be made: voting, typically used to make ‘political’ decisions, and the market mechanism, typically used to make ‘economic’ decisions.” This paper resolve s that dichotomy by developing a meta-theory from which can be derived methods for both political and economic decision making. This theory overcomes Arrow’s Impossibility Theorem in which he postulates that social choice is impossible and compensates for strategic voting, an undesirable aspect of decision making first pointed out by Gibbard and Satterthwaite. Thus the politonomics meta-theory spawns both political and economic systems which are indeed possible and which cannot be gamed. In a typical voting system the outcome of an election among several candidates results in one realized outcome – the winner of the election which applies to all voters. In a typical economic system, a consumer may choose among a variety of possible baskets of consumer items with the result that multiple realized outcomes are possible with a unique outcome for each consumer. We show that as the number of possible outcomes of a political-economic decision making process increases, the process becomes more economic and less political. We also show that as the number of possible outcomes increases, voter or consumer satisfaction or utility increases, and the need for strategy decreases.

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Algorithm for Employee Cooperative Shift Choice

Abstract

This paper deals with algorithms that allow employees to schedule their shift choices and pay levels in such a way that an overall budgetary constraint is achieved. The employee's input consists of several alternative shift choices and pay levels and the algorithm seeks to give each employee their highest preference consistent with meeting the budgetary constraint in a fair and impartial manner.

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Strategic Range and Approval Voting in the Abscence of A Priori Information

Abstract

This paper shows that in the abscence of a priori information the best strategy for voting in a range voting election, where each alternative is assigned a rating from zero to one, for example, is to first rate the alternatives sincerely and then compute the mean rating and use this as a threshold assigning every alternative with a sincere rating greater than the mean to a rating of one and every alternative with a sincere rating less than the mean to a rating of zero. This means that the voter would end up voting approval style. Approval Voting is a method in which each alternative is given a one or a zero by the voters, and the alternative with the most votes wins. It is possible to vote approval style within a range voting election. A voter voting this way would maximize his expected utility. We conjecture that, if every voter voted this way, expected social utility would be maximized as well. Also, the election would be stable in that no voter could gain an advantage or cause any other voter a disadvantage by voting any differently.

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Social Choice, Information Theory and the Borda Count © 1999 by John C. Lawrence

Abstract

This paper adds to the drumbeat of those who clamor for the Borda Count (BC) by eliminating the stigma of "arbitrariness" previously associated with the BC. It is proven that the BC is rational in the sense that, if a candidate dies, the expected value of the social profile is identical to the original with the dead candidate's name removed. Finally, the voting paradox is resolved by the BC when the most likely individual profiles are used. The fineness or coarseness of the grid on which individuals specify their preference profiles determines the amount of information conveyed. Since this grid is traditionally determined by the number of alternatives, there is no such thing as an irrelevant alternative. The problem of social choice, in general, can be viewed as the transmission of information from multiple sources (the individuals) to one receiver (society). Since there are finite information transmission constraints, there will be some probability of error, P(e), regarding the placement or ranking of alternatives both in individual and social profiles. As individual information is increased, P(e) in the social profile can be made to approach zero as closely as desired.

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Arrow's Consideration of Ties and Indifference" © 1999 by John C. Lawrence

Abstract

In "Social Choice and Individual Values" Kenneth Arrow (1951) asserts that, in his model of social and individual choice, ties are considered. However, as this paper makes clear only ties among alternatives and not ties among preference orderings are taken into account although Arrow claims to consider ties between binary orderings, at least, in Axiom 1 using R, the preference or indifference relation. Since he defines indifference as the logical ANDing of the elements of a tie, Arrow's treatment of ties is confined to the inclusion of indifference in a preference ordering. Moreover, Arrow clearly intends for the specification of individual and social preference orderings to be made in terms of P (the preference relation) and I (the indifference relation). Therefore, R should be defined in terms of them and not the other way around. An examination of Arrow's axioms, definitions and conditions reveals a number of inconsistencies and errors involving the use of R. There are at least three different interpretations of R used by Arrow at different times giving rise to at least two different models. These models are both examined and the axioms, definitions and conditions are clarified for each. If R (the preference or indifference operator) information is primary, then P (preference) information has been abstracted from and hence cannot be extracted from the data and the conditions must be stated without reference to P. If P and I (indifference) information is primary and R derivative, then the conditions as stated are not quite correct and must be modified. The implications of the inclusion of ties in both models are examined. If ties are included, Arrow's conditions must be rewritten in a more general manner. The inclusion of ties provides for the existence of the Social Welfare Function (SWF) and solutions are presented for both models for the case of three alternatives.

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The Possibility of Social Choice for 3 Alternatives © 1998 by John C. Lawrence

Abstract

In "Social Choice and Individual Values" Arrow (1951) discusses the possibility of ties for binary orderings. In particular, either xRy, yRx or both xRy and yRx are possible solutions when m (the number of alternatives) = 2 where R is the social "preference or indifference" operator. This is demanded by the axiom of completeness. Let us write "both xRy and yRx" as {xRy, yRx}. In the preceding sentence the word "and" is the English connective as distinguished from the logical and which we write "AND." If half the voters/consumers have xR

_{i}y and half have yR_{i}x (where R_{i}is the individual "preference or indifference" operator), it would be natural to assume (as one possibility) that the social ordering is {xRy, yRx} which we define as a tie. By extension, for three alternatives, if half the voters/consumers have xR_{i}yR_{i}z and half have yR_{i}xR_{i}z, it would be natural to assume (as one possibility) that the social ordering is the tie {xRyRz, yRxRz}. This reduces correctly to the binary solutions {xRy, yRx}, xRz and yRz when the appropriate alternative is removed both at the individual and the social levels. Arrow only considers ties among alternatives via his social choice function, C(S), and not ties among orderings. Since he demands orderings as the solutions for a Social Welfare Function (SWF), it would be more natural to consider ties among orderings which are also demanded by the axiom of completeness. Considerations of ties among orderings leads to the possibility of legitimate SWFs which are presented for m = 3 and which comply with the axioms of connectivity and transitivity and a strengthened version of Arrow's 5 criteria.

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Neutrality and the Possibility of Social Choice © 1998 by John C. Lawrence

Abstract

In "Social Choice and Individual Values," Kenneth Arrow (1951) postulates five criteria that a Social Welfare Function (SWF) must comply with, in his opinion, in order to be rational and ethical. One of these is Citizens' Sovereignty which states that the social decision must be a function of the individuals' preference information and nothing else. A strengthened version of this is the Principle of Neutrality which states that the SWF must yield the same social solution with regard to two alternatives x and y with x and y interchanged if x and y are interchanged in all the individual preference data. For the case of two alternatives, Arrow proves that social choice is possible only by virtue of violating the Principle of Neutrality (while complying with Citizens' Sovereignty) and treating x and y differently when half the voters prefer y to x and half prefer x to y. The way Arrow treats this tie case leads to the conclusion that social choice is possible in the case of m (the number of alternatives) = 2 whereas a treatment honoring the Principle of Neutrality and without the consideration of ties would lead to the conclusion that social choice is impossible for m = 2 also. If ties are considered legitimate for m = 2, then social choice which honors the Principle of Neutrality is possible for m = 2. This then opens the door to the possibility of social choice for m = 3, 4... .

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A New Approach to the Social Choice Function © 1998 by John C. Lawrence

Abstract

In "Social Choice and Individual Values" Arrow (1951) discusses two kinds of functions: the Social Choice Function, SCF, and the Social Welfare Function, SWF. The SCF is a function whose range is an alternative or set of alternatives. The SWF is a function whose range is an ordering of alternatives or sets of alternatives. Arrow's famous General Possibility Theorem, GPT, states that no SWF exists which meets certain criteria. However, the SCF may well exist even in cases in which the SWF doesn't because the production of an alternative or set of alternatives is less restrictive than the production of a set of orderings over the entire alternative set. The only role that the SCF plays in the GPT is in the specification of Condition 3, the Independence of Irrelevant Alternatives. In that condition Arrow specifies that the SCFs over a subset S of the set T (where T contains the total number of alternatives) produce the same social result in two different cases when the individual preference orderings within the set S are identical in both cases. The set T may have different orderings in the two cases, but the alternative or set of alternatives produced by the SCF must be the same. This is the weakest possible requirement for social orderings within the set S since it only requires the top alternative or set of alternatives to be the same. Stronger requirements would require that the top two or more alternatives within the set S be ordered in the same way, and the strongest requirement would be that the entire social orderings within the set S in the two cases be the same. The SWF produces orderings over T; a generalized SCF would produce orderings over S. Arrow provides for tie solutions via the SCF but only for ties among alternatives. Since the SWF requires orderings and not alternatives, it is natural to examine ties among orderings and not just ties among alternatives. When ties among orderings are allowed (as demanded by the completeness axiom), Condition 3 can be strengthened to the maximum and solutions which violate the GPT can still be found. We demonstrate this for m, the number of alternatives, = 3 and 4.

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Disproof of Arrow's Impossibility Theorem © 1998 by John C. Lawrence

Abstract

Arrow's Social Choice Impossibility Theorem is disproved by demonstrating that Arrow's treatment of tie situations was incorrect. Invalidating Arrow's proof in itself does not prove that Social Choice is possible. The possibility of Social Choice is proven by presenting an algorithm which represents a social welfare function that maps the domain of all possible combinations of individual choices into corresponding social choices. It is proven that the algorithm produces correct solutions for any number of alternatives and any number of voters which meet Arrow's criteria.

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A Social Choice Algorithm © 1997 by John C. Lawrence

Abstract

This paper presents an algorithm which represents a social welfare function that maps the domain of all possible combinations of individual choices into corresponding social choices. By means of a proper treatment of tie solutions, Condorcet's method of determining the outcome of an election is extended to cases that have previously produced a "paradox of voting." Our method is based on pairwise comparisons of candidates by voters and meets Arrow's five criteria and his axioms 1 and 2. Therefore, we have discovered a method which resolves the paradox of voting and extends to many other cases a generalized Condorcet solution. Solutions for all cases involving the R (preference or indifference) operator are worked out for m (number of alternatives) = 3. This paper lays the groundwork. A companion paper defines the algorithm in complete detail and proves that it provides viable solutions in every case.

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A General Theory of Social Choice © 1997 by John C. Lawrence

Abstract

In this paper we give a more elaborate development of our algorithm for social choice which generalizes Condorcet's method and satisfies Arrow's five criteria and two axioms as proven in the companion paper, A Social Choice Algorithm (1997). An example is worked out for the case of m, the number of alternatives, equals 5. The steps of the algorithm itself are precisely explicated. A proof is presented that the algorithm provides a social choice solution for all values of m when individual and social choices are expressed in terms of the R (preference or indifference) operator, and for all values of n, the number of voters. The demonstration of a Social Welfare Function (SWF) and the proof that it provides solutions in all cases proves that social choice is indeed possible. As was shown in the companion paper, A Social Choice Algorithm, the key to proving social choice possible is the admission of ties as possible solutions. In the cases for which there are tie solutions, an additional criterion can be used to winnow the solution set. We introduce the concept of "digital utility" to choose among the various ties the one or ones that have the best "goodness of fit" with the voters' preferences.

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A Refutation of Arrow's Impossibility Theorem © 1988 by John C. Lawrence

Introduction

In 1951 Kenneth J. Arrow published "Social Choice and Individual Values" in which he explored the question of whether or not individual preferences could be aggregated in some rational way in order to form a social choice. He postulated five rational and ethical criteria that such a social decision function should meet, and then proceeded to prove that no such social decision function existed. This theorem is known as Arrow's Impossibility Theorem, and an impressive literature concerning itself with what has come to be known as social choice theory has developed in the last nearly forty years. A sub-field of welfare economics has thereby been created. Some of the literature has been concerned with finding a way around Arrow's basic result that no rational social choice is possible by relaxing one or more of his criteria. Arrow's theorem has important political, economic and social implications since, if indeed no rational way to aggregate individual preferences is possible and Pareto optimality is the best we can do, then a populist democracy which closely reflects the will of the people becomes impossible and free market capitalism acquires a theoretically endorsed superiority over any kind of populist socialistic or democratic economy. This realization has produced pessimism and even nihilism among proponents of welfare economics. However, advocates of democratic voting systems should be equally concerned as Arrow's result tarnishes the validity of democratic elections as well.

Using information theory, the cardinal and ordinal components of individual preference rating information are computed and compared. With complete preference specification information, a modified and more rational "Independence of Irrelevant Alternatives" criterion and equal weighting of inputs to make interpersonal comparisons objective, it is shown that rational social choice, according to essentially similar postulates as Arrow used, is possible. This is first shown geometrically without recourse to a metric in order to demonstrate non-arbitrariness and motivate the formal proof after a metric is reintroduced. Criticisms of cardinal utilitarianism by Arrow and others are dealt with in detail. The exact nature of the social decision function (SDF) is treated from an ethical point of view and SDFs which are composites of Rawlsian and Benthamite SDFs are discussed. Political and economic implications are considered.

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Proving Social Choice Possible by John C. Lawrence

Abstract

In this paper we give a more elaborate development of our algorithm for social choice which generalizes Condorcet’s method and satisfies Arrow’s five criteria and two axioms. An example is worked out for the case of m, number of alternatives, equals 5. The steps of the algorithm itself are precisely explicated. A proof is presented that the algorithm provides a social choice solution for all values of m when individual and social choices are expressed in terms of the R (preference or indifference) operator, and for all values of n, the number of voters. The demonstration of a Social Welfare Function (SWF) and the proof that it provides solutions in all cases i.e.for all values of m and n, proves that social choice is indeed possible. As was shown in the companion paper, A Social Choice Algorithm, the key to proving social choice possible is the admission of ties as possible solutions. In the cases for which there are tie solutions, an additional criterion can be used to winnow the solution set. We introduce the concept of “digital utility” to choose among the various ties the one or ones that have the best “goodness of fit” with the voters’ preferences.

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The Possibility of Social Choice by John C. Lawrence

Abstract

A Social Welfare Function (SWF) maps the domain of voter preference orderings into the range of social orderings. Arrow's General Possibility Theorem states that no such function exists which complies with certain rationality conditions and 5 plausible criteria. However, while the range consists of orderings, Arrow only considers the possibility of ties among alternatives and doesn't consider the possibility of ties among orderings. When such ties are considered, it is possible to show that at least one SWF does exist which complies with an even strengthened version of Arrow's rationality conditions and other criteria. An algorithm is presented which represents a SWF that maps the domain of all possible combinations of individual orderings into corresponding social orderings. It is proven that the algorithm produces correct solutions for any number of alternatives and any number of voter/consumers.

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