Social Choice

Robert M. Hayes

Overview

Individual Choice Social Choice Preference Contexts Arrow’s Voting Conditions Social Welfare Functions Data Envelopment Analysis

Individual Choice

This presentation will be focused on social choice. That is, on the means for combining the individual choices of a group of individuals into a group choice.

Having said that, it should be recognized that everything to be said regarding social choice is fully applicable to individual choices as well.

To illustrate, consider the simple decision about whether I want to have Meat, Fish, or Chicken for dinner. Obviously a simple choice, one that a diner makes virtually every day (among various options, of course). But, in making that decision, one may want to weigh three quite separate sets of criteria: (1) Cost, (2) Taste, (3) Health.

Relative rankings might well be: Cost: Chicken is preferred to Fish is preferred to Meat Health: Fish is preferred to Meat is preferred to Chicken Taste: Meat is preferred to Chicken is preferred to Fish

In a very real sense, these represent three different voters whose choices must be combined when the individual makes the decision.

Utility Transitivity

In von Neumann and Morgenstern, Theory of Games and Economic Behavior, the starting point for considering the utilities, or criteria for choice for the individual, is as follows:

(1) For an individual the system of preferences is complete. That is, for any two alternative events, the individual is able to tell which is preferred. 

(2) The individual can compare not only events but combinations of events. 

From this starting point, they move to a set of axioms: 

Utility Axioms (A1) Completeness: for every u, v either u = v, u > v, or v > u and only

one of those three applies (A2) Transitivity: if u > v and v > w then u > w (A3) Combining 1: if u > v then u > au + (1-a)v and if u < v then u < au +

(1-a)v (A4) Combining 2: if u > w > v then there is an a such that au + (1-a)v > w (A5) Algebra of combining: if c = ab then a(bu +

(1-b)v) + (1-a)v = cu + (1-c)v  Each of these axioms, rational though they are, can be questioned. In

particular, it is by no means self-evident that an individual can assess his preferences among all possible alternative events. He simply may not know whether he prefers u to v or vice versa, and that does not mean that he is indifferent (i.e, that the utilities are equal).

But it is the axiom of transitivity that I want to examine here.

Multiple Dimensions To do so, I want to suggest that the choice among alternative events may

involve consideration of several dimensions or aspects of those events. Indeed, in any real situation it usually will do so.

To illustrate, suppose I am at a restaurant and, as previously described, have the options of choosing a filet mignon steak, a salmon steak, or a chicken for my dinner. Which is preferable for me at that time? Clearly, I will make a choice, if only because I am hungry and the waitress is impatiently waiting for my decision. But I am torn in weighing several factors—what I will call the dimensions of the utility.

Later we will be discussing Arrow's "voting paradox" theorem. If I treat the three dimensions as the voting members of the society of my mind, I face the paradox.

Now, obviously it can be argued that an internal debate within my own mind is hardly comparable to an external one between political alternatives. But nevertheless, the internal debate is real and the paradox holds.

Transitivity in Individual Preferences

Transitivity in value judgments is a crucial concept. A value judgment, such as preference or choice among alternatives, is said to be “transitive” if, when x is preferred to y and y is preferred to z, then x preferred to z.

In the example of the evaluation of alternative options for dinner, transitivity would means that, if chicken is preferred to fish and fish is preferred to meat then chicken must be preferred to meat.

But as the example shown earlier demonstrates, meat might well be preferred to chicken! Hence, while many individual value decisions may well be transitive, there may also be cases when they are not. Indeed, when decisions are multi-dimensional, the likelihood is that they may well be non-transitive.

Societal Decisions In fact, I will suggest that societal decisions imply such individual internal

debates. The dimensions of the utility function apply within each individual and not just among the members of the society. Specifically, each individual must and does weigh the relative importance to himself of at least three dimensions:

(1) What does something cost? (2) What is the balance of benefits and losses to me personally? (3) What is the balance of benefits and losses to larger groups (to family, to

society, to the world)?

Thus, assessment of the utility function involves at least two stages:

(A) Assessment of the relative position of alternative choices on each dimension.

(B) Assessment of the relative importance of the dimensions.

Social Choice

Different options will be preferred in different orders of ranking by different individuals. And in fact they may well differ in the extent to which their rankings are transitive.

How can individual preferences be “aggregated” into a social choice over all possible sets of preferences?

Preference Contexts In what follows, I am going to refer to preference contexts. These

are the array of possible preference evaluations. If the preference evaluations are transitive and there are N

options to choose among, there are N! possible preference evaluations (i.e., there are N ways of selecting the most preferred, then (N – 1) for the next most preferred, etc.).

However, if preference evaluations are not necessarily transitive and there are N options to choose among, there are N*(N – 1)/2 pairs of options. For each pair, say x and y, either x > y or y > x so there are 2N*(N-1)/2 ways of making those choices between pairs.

For N = 3 the following are the potential not necessarily transitive preference evaluations:

x > y, y > z, x > z x > y, y > z, z > x x > y, z > y, x > z x > y, z > y, z > x y > x, y > z, x > z y > x, y > z, z > x y > x, z > y, x > z

y > x, z > y, z > x

Voting Contexts

Next, let’s look at “voting contexts”. This is the array of possible selections of preference evaluations from the preference set by the voters.

If there are N options and P voters, then the number of voting contexts for transitive evaluations is given by

V1 = (P + N! – 1)!/(P!*(N! – 1)!) The number of voting contexts for potentially non-transitive evaluations is

given byV2 = (P + 2N*(N-1)/2 – 1)!/(P!* (2N*(N-1)/2 – 1)!)

For even small values of N and P, these numbers, especially V2, get large. For N =3 and P = 3, V1 = 56 and V2 = 120. For N = 4 and P = 4, V1 = 17,550 and V2 = 766,480

Simplest Case

The simplest case is surely a social group consisting of just two persons faced with a choice between two options.

Thus there are three possible preference contexts: 2 prefer x to y or, symbolically, x > y 2 prefer y to x (y > x) 1 prefers x to y (x > y) and 1 prefers y to x (y > x)

Note that transitivity is not an issue, since there are only two alternatives being considered.

Of course, if the voters both agree on the relative ranking of the two options, the social choice is clear.

The conflict arises when they do not agree, and the conflict can only be resolved if one person dictates, or perhaps persuades, and in either event determines what the social choice should be.

Three Persons & Options, Non-Transitive

Consider 3 persons and 3 options, with non-transitive evaluations. There are 8 ways in which individual preferences may occur, and the number of voting contexts is 120.

For purposes of illustration, let’s focus on one of the patterns for individual preference, say x > y, y > z, and x > z (so it is a transitive pattern), and suppose that at least one of the three persons conforms to it. There are then just 36 voting contexts, since (2 – 8 – 1)!/(2!7!) = 36. They are distributed as follows:

14 are x > y, y > z, x > z. which is transitive 5 are x > y, y > z, z > x, which it non-transitive 5 are x > y, z > y, x > z, which is transitive 2 are x > y, z > y, z > x, which is transitive 5 are y > x, y > z, x > z, which is transitive 2 are y > x, y > z, z > x, which is transitive 2 are y > x, z > y, x > z, which is non-transitive 1 are y > x, z > y, z > x, which is transitive

The General Case The general case, then, arises when there are more than two persons and/or

more than two options involved in the social choice. Let’s look at the case of two options but more than two persons. Clearly, if there is an odd number of persons, one of the options must be

preferred by a majority, even if it is only a majority of 1. Thus, the Supreme Court, with 9 persons, ought generally to be able to arrive

at a decision, even if it is 5 to 4. But if there is an even number of persons, there is the potential for an evenly

split vote, with decision unresolvable except by fiat or dictatorship or, potentially, persuasion or compromise.

Of course, the larger the number of persons, the smaller the likelihood of an even split.

The means for dealing with the general case would appear to be either some form of voting, with the social choice determined by the majority view, or dictatorship, in which one person or one group of persons determines the social choice.

Condorcet

Turning now to the alternatives for voting, Condorcet proposed the following

Compare each pair of alternatives If more voters strictly prefer A to B, declare A is socially

preferred to B Condorcet Principle: If one alternative is preferred to all other

candidates then it should be selected

Example

3 options A, B, C

10 voters with the preferences 4 vote A > B > C 4 vote B > C > A 2 vote C > B > A

Result: B wins. (6/10 prefer B to A, 8/10 prefer B to C)

Vulnerability to Minority Alternatives

The Condercet rule is vulnerable to irrelevant or minority alternatives

Twenty voters: 8: X > Z > Y 7: Y > X > Z 5: Z > Y > X

37 X , 42 Z, 41 Y so X wins Remove Z and 32 X, 28 Y so Y wins

Borda Count

Each voter lists alternatives in order of preference On each ballot compute the rank of each alternative Rank order alternatives based on increasing sum of ranks across

all voters

This rule makes eminent sense and is at the basis of all approaches to proportional voting, which is intended to avoid the effects of minority options.

Problems with the Borda Count

There are problems with the Borda count, rational though it may seem to be.

First, it does not always choose the Condorcet winner! For example, consider 3 voters

2: B>A>C>D 1: A>C>D>B

Borda scores: A 5, B 6, C 8, D 11 But B is the Condorcet winner, since B is strictly preferred to A, to

C, and to D, in each case by a majority of one voter. Admittedly, A is preferred to C and to D by a majority of 3, but

that does not countermand the fact that, overall, by the Condorcet rule, B is preferred to each of the others.

Inverted-order paradox Consider a Borda rule for 4 alternatives

1. X > C > B > A 2. A > X > C > B 3. B > A > X > C 4. X > C > B > A 5. A > X > C > B 6. B > A > X > C 7. X > C > B > A

X=13, A=18, B=19, C=20 Remove X: C=13, B=14, A=15

Manipulating Preferences

As well, most voting schemes are manipulable. That is, an individual can cast an “untruthful” vote to improve the

social outcome for himself. Again consider rank-order voting.

Manipulating Preferences

P1 P2 P3

x(1) y(1) z(1)

y(2) z(2) x(2)

z(3) x(3) y(3)

Suppose these are truthfulpreferences. Note that thereis no consensus

Manipulating Preferences

P1 P2 P3

x(1) y(1) z(1)

y(2) z(2) x(2)

z(3) A(3) y(3)

A(4) x(4) A(4)

Suppose P3 introduces anew alternative, and these are still truthful preferences.

Rank-order voteresults.

x-score = 7y-score = 6z-score = 6A-score=11

Manipulating Preferences

P1 P2 P3

x(1) y(1) z(1)

y(2) z(2) A(2)

z(3) A(3) x(3)

A(4) x(4) y(4)

Suppose P3 now lies aboutPreferences.

Rank-order voteresults.

x-score = 8y-score = 7z-score = 6A-score= 9

z wins!!

Arrow’s Voting Conditions

Kenneth Arrow identifies five conditions that voting should satisfy.

Collective rationality Pareto criterion Transitivity Non-dictatorial choice Independence of irrelevant alternatives

What is collective rationality?

This is the assumption that a social choice has the same structure as that for the choice by individuals.

While such an assumption, on the surface anyway, makes sense, there are good reasons for thinking that a social group might have options beyond those available to the individuals within it and that the criteria for determining the preferences among options might be quite different for the group from those of individuals.

What is the Pareto Criterion?

If alternative X is preferred to alternative Y by every individual, then the social ordering should also prefer X to Y.

Note the importance of the assumption of collective rationality, so that we can indeed talk about social orderings in the same way that we talk about individual orderings.

On the surface of it, the assumption of Pareto efficiency seems to make sense, and it seems difficult to argue otherwise.

What is Transitivity?

When a relation R is that xRy and yRz implies xRz, the relation is said to be transitive. For example, “less than” is transitive among all real numbers. Thus, if 2 is less than P, and the square root of 3 is less than 2, we can be certain that the square root of 3 is less than P.

Equality also is transitive: if a = b and b = c, then a = c. In everyday life such relations as “earlier than,” “heavier than,”

“taller than,” “inside of,” and hundreds of others are transitive.

If the relationship R is “preference”, transitivity states that, if X is preferred to Y and Y is preferred to Z, then X is preferred to Z.

On the surface, this appears to make eminent sense, but we have seen several examples, including that of individual choices, in which it may not apply. Indeed, of all of the desirable voting conditions, this one appears to be most likely NOT to apply.

Non-Transitive Relations

It is easy to think of relations that are not transitive. If A is the father of B and B is the father of C, it is never true that A is the father of C. If A loves B and B loves C, it does not follow that A loves C.

Familiar games abound in transitive rules (if poker hand A beats B and B beats C, then A heats C), but some games have non-transitive (or intransitive) rules. Consider the children’s game in which, on the count of three, one either makes a fist to symbolize “rock,” extends two fingers for “scissors,” or all fingers for “paper.” Rock breaks scissors, scissors cut paper, and paper covers rock. In this game the winning relation is not transitive. 

What is non-dictatorial?

This states simply that there is no one individual or group of individuals whose preferences automatically are those of society, independent of the preferences of all other individuals.

Certainly, in a democracy, we are committed to the view that dictatorship is anathema. But the facts are quite otherwise. Indeed, in most companies, as social structures, the decisions are likely to be made by a virtual dictatorship.

Having said that, it must also be said that the fact that the social choice turns out to be the preferences of one individual or group does not necessarily mean that the social choice was dictated by that person or group. It may simply mean that it turned out that way.

What is independence of irrelevancies?

If the alternatives include options that really are not available, they should not affect the decisions. Specifically, in assessing the preference between two alternative options, the decision should not be based on consideration of any other alternative options.

We have shown the example of a spurious alternative being used to determine the outcome of voting, and this condition has, at least in part, the objective of avoiding such malefaction.

Having said that, there is good reason to think that decisions about the relative value of two alternative options might well need to consider their relationships to other options.

Arrow’s impossibility theorem

Arrow’s impossibility theorem states that, if there are 3 or more alternatives and a finite number of voters then there is no protocol that satisfies these properties, desirable though they may be.

Proof of Arrow’s theorem - 1

Assume R is Pareto efficient and independent of irrelevant alternatives Let A be the set of all voters We will refer to the preference of a voter with respect to two alternatives, x

and y, as “x > y” meaning x is preferred to y by that voter. And we will refer to the group choice as “x is chosen over y” if the group, as a whole, chooses x over y.

Step 1. Suppose that for two options, x and y, x is chosen over y. Then a sub-set S of the voters in A is called “decisive” for alternative x over y if x > y for each voter in S. At the maximum, S might include all of the voters for whom x > y, but it might not. The key point is that the view of the group of voters S agrees with the group decision. S will be called “faction” of the voters and those voters not in S the opposing faction.

Step 2. Suppose S is decisive for x over y, and consider any other pair of alternatives, say a and b, for which all voters are in agreement concerning their preferences with respect to x and y. Specifically, let’s suppose that all voters agree that a > x and that y > b. That means that for voters in S, a > x > y > b and thus, by transitivity, a > b. For voters in the opposing faction, though, there may be disagreement concerning the preferences between a and b.

Proof of Arrow’s theorem - 2 Step 3. Since S is decisive for x over y, we have x is chosen over y

Since a > x and y > b for every voter, by the Pareto principle we have a is chosen over x and y is chosen over b.

Then, by transitivity, a is chosen over b Hence, S is decisive for a over b

Step 4. Consider now a third option, z, not already resolved (those in Steps 2 and 3 having been resolved). Divide S into two sub-groups, S1 and S2 such that for all voters in S1, x > y and y > z (and, by transitivity, x > z), and for all voters in S2, z > x and x > y (and, by transitivity, z > y). Then, for all voters not in S1 or S2, y > x and x > z. (and by transitivity, y > z).

Step 5. Since S is decisive for x over y, x is chosen over y. Now, either z is chosen over y or y is chosen over z. If z is chosen over y, S1 is then decisive for y over z and by Step 2 for all pairs on which there is total agreement in comparison with y and z. On the other hand, if y is chosen over z, then by transitivity, x is chosen over z. In any event, either S1 or S2 is decisive.

Step 6. The process of Steps 4 and 5 (using either S1 or S2, depending on which is decisive) is now repeated until all options have been included. At that point, one sub-group is decisive for all the social choice among all options. That single group becomes the de facto dictator of the social choices.

Step 7. This contradicts the assumption that there is not a dictator.

Non-Transitive Paradoxes

In his book The Colossal Book of Mathematics (Chapters 22 and 23) Martin Gardner identified what he calls “non-transitive paradoxes” by which he means situations in which one’s intuition is that a relationship must be transitive but it is in fact not so.

Martin Gardner says, “The oldest and best-known paradox of this type is a voting paradox sometimes called the Arrow paradox after Kenneth J. Arrow because of Arrow’s “impossibility theorem,” for which he shared a Nobel prize in economics in 1972. In Social Choice and Individual Values, Arrow specified five conditions that almost everyone agrees are essential for any democracy in which social decisions are based on individual preferences expressed by voting. Arrow proved that the five conditions are logically inconsistent. It is not possible to devise a voting system that will not, in certain instances, violate at least one of the five essential conditions. In short, a perfect democratic voting system is in principle impossible.” 

As Paul A. Samuelson has put it: “The search of the great minds of recorded history for the perfect democracy, it turns out, is the search for a chimera, for a logical self-contradiction.. . Now scholars all over the world—in mathematics, politics, philosophy, and economics—are trying to salvage what can be salvaged from Arrow’s devastating discovery that is to mathematical politics what Kurt Gödel’s 1931 impossibility of proving consistency theorem is to mathematical logic.” 

Reality

Having said that, it must also be said that Arrow’s proof of impossibility relates not to the reality of voting but to the theoretical objective of having a process that is guaranteed, under all circumstances, to conform with the identified conditions.

In reality, voting does take place and, more or less, does reflect the democratic objectives. Social choices are made, and to varying degrees, they do reflect the objectives of the society.

The key point about Arrow’s theorem is that it clearly identifies the effects of the conditions and removes the chimera of an “ideal voting system” from consideration.

Having identified the conditions and their effects, one can now consider how those conditions might themselves be modified.

Social Welfare Functions

We turn now to examining a number of alternative social welfare measures as potential basis for social choice decisions, each of which attempts to represent mathematically an alternative for social choice:

Pareto Optimum Utilitarian Optimum Weighted Sum Optimum Mix-min Optimum

Pareto Optimum

“Pareto Optimum” arises from a social welfare measure that determines the optimum allocation of the resources of a society as those make at least one individual better off in his own estimation while keeping others as well off as before, again in their own estimation. Mathematically this can be expressed as:

Maximize W = u(i), subject to u(i) v(i)

where u(i) is the intended benefit for each individual and v(i) is the current benefit for each individual.

Note that the effect of Pareto optimum is to preserve the status quo for those who already have.

Utilitarian Optimum “Utilitarian optimum” arises from a social welfare function that

determines the optimum allocation of the resources of a society as those make the sum of all individual assessments of well-being the maximum.

Mathematically this can be expressed as:

Maximize W = u(i)

where u(i) are the intended benefits for each individual.

Note that Utilitarian optimum might well involve reducing the benefits for those who have if, by doing so, the total benefits for everyone as a whole are increased. The effect might be “taking from the rich in order to give to the poor”. Progressive income tax laws are probably intended to be Utilitarian optimum.

Weighted Sum Optimum “Weighted-sum Optimum” arises from a social welfare function that

determines the optimum allocation of the resources of a society as those make the sum of all individual assessments of well-being the maximum but weighting those individual assessments by some criterion of importance.

Mathematically this can be expressed as:

Maximize W = a(i)*u(i)

where u(i) are the intended benefits for each individual and a(i) is the relative importance of each individual.

Obviously such a social welfare function is undemocratic, but not every social choice context is necessarily democratic.

Max-Min Optimum A “Max-min Optimum” arises from a social welfare function that

determines the optimum allocation of the resources of a society as those make the person with the least well-being as well off as possible.

Mathematically this can be expressed as:

Maximize W = min(u(i))

where u(i) are the intended benefits for each individual.

Thus, Max-min optimum tries to improve the conditions for those who are least well off in the society. Recognizing that is the case, it is important also to recognize that Max-min is a crucial criterion in making game-theoretic decisions. Thus, to be successful in economic decision-making, one must maximize one’s minimal gain.

Data Envelopment Analysis

In many cases, the choices of individuals are reflected in what they in fact do, not in voting but in their behavior, and the question at hand is whether there are means by which those behaviors can be objectively combined to reflect a consensus.

There is a methodology, called “Data Envelopment Analysis” which attempts to do precisely that.

The nature and application of that methodology is the subject of another presentation.

THE END