A problem for 200 years

Olsen (1972) clarified through a game-theoretical analysis how choices between alternative decision processes are made in organisations: in the ideal cases, either a process is chosen by which contrasts can emerge, through a voting procedure, or it is arranged that confrontation is avoided, unanimity on the issue concerned being achieved through successive soundings of opinion. Olsen stated: “The crudeness of the voting process itself should be expected to affect the frequency of its use” (Olsen, 1972, p. 271), i.e. in cases when more than two alternatives are presented. Olsen did not discuss this “crudeness” further, since it lay outside the scope of his analysis. Those who participate in social life, however, are soon confronted with such crudeness, mitigation of which has in fact been the object of reform proposals and analysis for 200 years.1 When there are three or more alternative proposals concerning the same issue among which a choice is to be made, the problem arises of deciding how these alternatives shall be presented for choice. In the Western world two main types of parliamentary voting methods are applied, namely the series method and the elimination method (these parliamentary procedures have since served as models for decision procedures in other organisations and societies in the countries concerned). A feature held in common by the series method and the elimination method is that the final conclusion of the voting can depend on the sequence in which the alternatives are put to the vote. In the case of the series method a person who votes as soon as the first proposal is put forward must adopt an absolute standpoint, either for or against the proposal. If a voter prefers the alternative presented secondarily, i.e., if it is neither the alternative he desires most nor that he desires least of all, it is not possible for him to give expression to this standpoint in the voting.2 It is necessary to choose absolutely between voting for the proposal — thus possibly missing the chance to bring about the alternative most desired — and voting in order to reject it, thereby risking the possibility that the least desired alternative wins — and indeed, perhaps backed by votes which could have been united to establish a majority for the subsidiary alternative.

In the case of the series method it may therefore happen that a proposal is rejected without having been taken to the vote. The only possible way to avoid this would be to present the proposals in an order such that none of the participants subsidiarily supported the proposal first in the sequence.

By the elimination method a proposal is rejected (eliminated) when it is put to the vote against another proposal and obtains a minority of the vote. The voting population tests all the proposals in this way, one proposal being eliminated at each vote. The outcome of the voting is the proposal that wins in the final vote.

The series method can in general be considered simple and easily comprehensible. In this respect it differs from the elimination method, which often creates confusion. However, the elimination method has the advantage that the subsidiary preferences of the participants are more likely to affect the conclusion — those who vote enjoy somewhat more freedom. On the other hand, the elimination method involves particular disadvantages with regard to deciding the sequence in which the proposals shall be submitted for voting. Generally, the voting problem has to a great extent been the question of the ranking of proposals in cases involving voting on many alternative proposals, i.e., of which proposal or proposals shall be put to the vote first. According to Ramstedt (1961) it is a general rule that the proposal or proposals voted on first should be that (those) considered to be the most extreme or that (those) which deviate(s) most from the “mean” point; for example, concerning taxes or expenses, the proposal(s) regarding the highest or lowest amount(s) should be put to the vote first. The proposal that comes next to the most extreme is then voted on, according to the voting method. Thus, an attempt must be made to rank the proposals on a falling or rising scale. Ramstedt (1961, p. 155) maintained that such a process must be based on the idea that the proposals between which a choice is made at a stage in the voting process (i.e., which exclude each other) have an inherent “logical” relation such that, for instance in the case of voting on taxes and expenditure, those who vote and whose proposals are rejected in the process are subsidiarily supporters of the “closest” alternative, i.e., the amounts of tax or expenditure adjacent on the scale of alternatives to the amounts rejected. Without discussing this idea in depth, it is only possible to agree with Ramstedt that such principles are practicable to only a very limited extent. Issues submitted for voting are not often of such a kind that the alternatives can be ranked on any meaningful scale. In addition, a scale may be meaningful to one person, but have a limited meaning for another. For instance, it is possible to conceive that those who submit proposals on expenditure wish to have either “reasonable” expenditure or no expenditure at all. As stated by Ramstedt, it seems that only with reference to this type of example is it justifiable to state that ranking of proposals in a sequence for voting can be based more on formal than on substantive considerations.

The following example shows that the final conclusion of voting in both the series and the elimination method can depend on the sequence in which the proposals are submitted. Suppose that there are three proposals A, B and C concerning an issue. The voters have grouped themselves around three different preference orders concerning these proposals, with 5, 4 and 4 in each group.3 The following table indicates how the three groups vote primarily and subsidiarily:

5

4

4

A

B

C

B

C

A

C

A

B

We first assume that the proposals are submitted for voting in the sequence A,B,C.

The series method:

First voting:

A submitted: 5 for and 8 against.

A is rejected.

Second voting:

B submitted: 9 for and 4 against.

B is adopted.

The elimination method:

First voting:

A and B submitted: 9 for A, 4 for B.

B is rejected.

Second voting:

A and C submitted: 5 for A, 8 for C.

C is adopted.

If the proposals are presented in the sequence C,B,A, A will win by both methods. If the sequence C,A,B is employed, A will be adopted using the series method and B using the elimination method.

Borda´s solution: Arrow´s and Black´s objections

Thus, problems of the above type have been under discussion for a long time. The many authors who in the last 200 years have considered the voting problem and put forward theories and propositions concerning voting methods have often tackled the question of the possibilities of following rational criteria for determining the voting arrangement. The problem has usually been stated in terms of the “voting paradox”. When there are three proposals A, B and C, the voting population may have a standpoint which through conventional voting gives A a majority over B, B a majority over C, and C a majority over A (Arrow, 1951, p. 2). The question is then that of how this cycle shall be broken. It was with reference to this paradox that in 1781 Borda put forward his proposition for a voting method. His solution was as follows. The alternative a participant wishes most to be adopted is given a numerical value equal to the number of alternatives; if there are five alternatives, the alternative which a participant prefers to all the others is given the numerical value 5. The alternative he next favours is given the same numerical value minus 1 (in this case, 4), and so on. Thus, the alternative that appeals least to the voter is given the numerical value 1. The alternative which overall obtains the highest numerical value for all the voters is adopted.

Borda presented his method in the following general form. A calculation of numerical scores is used such that the difference in the scores between two alternatives which are adjacent in the preference ranking of an individual participant is constant. If the alternative which is at the bottom of the ranking is given a points and the second lowest a+b points, the third lowest must be given a+2b points, and so on. The same voting result is obtained irrespective of the numerical values of a and b.4 This 200-year-old idea is still drawn upon heavily in the voting-theory literature. Gärdenfors (1970, pp. 25), for instance, presented strong arguments in favor of Borda´s method according to formal-logic considerations.

Gärdenfors, however, considered that Borda´s method fails to fulfill one of the already classical conditions put forward by Arrow in his presentation of the voting problem (Arrow, 1951). This analysis initiated a new epoch in this field. The choice problem was, according to Arrow, a question of aggregating individual preferences to yield a joint conclusion by means of a social welfare function. What was important and new in his contribution was his proposition that general criteria should be employed for evaluating the various methods of reaching a joint conclusion. He put forward the following conditions which such a method should fulfill if it was to be considered sensible.5

(1) if individual participants change their ranking of the alternatives, the joint order should preferably be changed correspondingly, or at least should not be allowed to change in the opposite direction;
(2) irrelevant alternatives must not affect the ranking of the other alternatives;6
(3) participants should have the right to choose freely between alternatives;
(4) the social welfare function is not to be directed by anyone.

 

Arrow concluded that it was impossible to fulfill these conditions simultaneously. As the conditions were considered reasonable, this was a discouraging conclusion. Since then little progress has been made. In the very extensive literature either based on or related to Arrow´s formulation of the problem and his conclusions, the second condition has given rise to the greatest speculation.7 Problems concerning the interpretation of this condition in particular, as well as the elaboration of choice and voting methods that satisfy the interpretation, have arisen because this is the condition which has proved to be the most difficult to fulfil. Hansson 8 (1970, p. 193) gave the following interpretation of the condition, which will suffice here. If there are two different choice situations, which, however, are such that the preferences between all the alternatives in a set of alternatives are the same for each individual participant in both situations, the joint order within that set shall be the same for both situations. How alternatives outside the set are arranged in order of preference shall not by any means affect the order within a joint set of alternatives: i.e., this shall be “irrelevant”. This is exactly the condition which Borda´s method does not satisfy.

There is also another class of objections to Borda´s method. Black (1958) considered Borda´s method to be excellent as an abstract rule, but to lack meaning in the sense that there is no explanation of why the alternative that obtains the most points in the point calculation should, in fact, be that preferred by the group.

Incompatibility between Arrow´s conditions and the economic theory of demand

Here we must confirm that those responsible for managerial arrangements in organisations, whether these are national assemblies, communal authorities, sports clubs or faculty councils, will not have found any valuable practical solutions to problems concerning voting methods in the accumulation of literature dealing with these problems over the last few decades. Persistent readers will have been able to deepen their insight and further their knowledge of the problem, but, as stated above, with meagre results insofar as practical reforms are concerned.9 In the voting literature there are many examples in which rules for the ranking of sports performances have been evaluated as potential voting rules. Indeed, in two different branches of sport a settlement is employed for deciding in a consistent way which is the best team or player, just as in the case of choosing the best alternative among many proposals. To our knowledge no rules have thus been found which have properties making them superior to Borda´s method. Of all voting rules Borda´s has enjoyed the highest esteem, even though it does not satisfy all of the conditions laid down by Arrow. Arrow´s contribution was to establish certain conditions such that a group voting choice could not be regarded as unreasonable on account of the method chosen for the voting. However, more is required when, like Black (1958), one desires a theory or a method which can provide deeper insight into the nature of the group choice.

The voting method to be discussed here is a method that is employed both in chess tournaments and ball tournaments. In the following we restrict discussion mainly to the case of chess tournaments. The method is that everybody plays against everybody.10 Calculations are made of how many times the individual players win; when there is a draw the win is divided between the players. Finally, the players are ranked according to their total number of wins. Among those concerned with chess it is a common interpretation that if for instance A has obtained the most points in a chess tournament, then A has given the best performance and is placed no. 1, even if A has, for instance, lost a game against B. The tournament has, in fact, demonstrated that B has overall been weaker than A in his games against the other players. Thus, it is not considered unreasonable nor does it have to be explained if the final winner happened to lose a game against no. 2 or another player who eventually ended further down on the ranking list. A corollary of this is that chess players agree to the fact that the order of preference between A and B can depend on whether or not C has participated. C could, for instance, be strong against B but weak against A, which would cause A to take precedence over B in a tournament with C participating, while B could take precedence over A if C did not participate. Thus, the chess tournament procedure produces merely a ranking of the actual participants in the particular tournament, based on their demonstrated skill. This is an order of preference which, in principle, need not necessarily apply to a tournament having a different group of participants. Thus it is the participation that defines the significance of the ranking.

Arrow´s condition regarding the independence of so-called irrelevant alternatives is therefore alien to those concerned with chess tournaments. They consider that an alternative (a player) that does not do so well as, for instance, the three best players, should be included, and that it should be taken into consideration how far it affects the adoption of the best alternative. Such a “mediocre” alternative may bring out new aspects of the other alternatives, aspects whose significance cannot be disclaimed a priori by outsiders and which cannot be rejected as irrelevant. The individual confrontations must be allowed to reveal how relevant such properties are for exposing different qualities or properties among the other alternatives.

As can be seen, a voting situation is highly analogous to the choice of a winner among a group of chess players. In chess there is no general measure of skill: the qualifications which are of importance in an individual tournament depend on the chess players taking part. Nor is there any general measure in a voting situation. A proposal about an issue is best understood in relation to other possibilities (alternatives) concerning the same issue, and the greatest insight into a person´s standpoint and evaluation both for that person and for others, will be obtained through such mutual comparison. In this way an individual participant forms a personal view of his own preferences, which then, depending on the decision process, becomes the basis for the group´s conclusion. Thus, we have followed the same argument as concerns the choice of the best player in a tournament in which everybody plays against everybody. In both cases a mediocre participant (a middle-good alternative) can be relevant in clarifying which other participant (alternative) is best. In both cases it is the alternatives that are decisive in the issue. Arrow´s condition entails that an outsider can dictate a priori the exclusion of certain possibilities in an evaluation of alternatives. This is not accepted according to present method. In this connection we cannot but refer to a basic idea in the economic theory of demand, namely that circumstances, i.e., the possibilities for choice (the alternatives), may reasonably affect an actor´s standpoint in a situation involving the choice of a best alternative. The actor is in the present case a group.

Arrow´s condition regarding the independence of so-called irrelevant alternatives has been the subject of very elaborate critical evaluation by many authors. However, from this literature it is not clear how an individual alternative can mould and be of importance to an issue (situation) on which a group is to decide, or that the concrete situation of the choice (the demand) is an elementary criterion in the theory of demand. In the voting theory literature this subject has not been handled in any way similar to that elucidated here through the analysis of the ranking procedure for competing chess players. We consider it improbable that Arrow would necessarily wish to retain this independence of irrelevant alternatives condition if his attention were drawn to how it appears in the context of a theory of demand; the same could probably be said of other eminent economists who have been concerned with the problem.11

Borda´s method: A meaning

Those who strive for democratic reforms therefore need not be impeded by Arrow´s conception of irrelevant alternatives (Arrow, 1951, 1963). We now consider more closely what can be learnt from the chess players´ way of thinking when choosing the best alternative. We have already seen how all of the real possibilities (alternatives, participants) can affect the characterisation of the individual possibilities in relation to the others. A meaning in terms of choice can now be perceived in Borda´s method in the light of how the best chess player is chosen in a tournament. It was precisely this type of meaning that Black sought for in Borda´s method which he, like many other voting theorists, valued highly in other respects. For the purpose of explanation, we now consider an issue involving the alternatives A, B, C, D, E, on which each individual participant of a group of three decides as follows:

1

2

3

A

D

C

B

C

E

D

B

A

C

E

D

E

A

B

Participant no. 1 indicates that A is better than four alternatives, that B is better than three alternatives, that D is better than two alternatives, that C is better than one alternative, and that E is the least favoured alternative. By analogy with the procedure in chess tournaments, i.e., by participant no. 1 making comparisons by pairs, A comes out better than the other alternatives four times, B three times, D two times, C once, and E not at all. If the conclusions of the three participants are to be equal and summed (aggregated) so that their joint conclusion can be established, it must be considered natural to sum for all participants the numbers of times each alternative, according to the ranking by each individual participant, is preferred to the other alternatives. The preference order of participant no. 1 gives the following grades: A, 4; B, 3; D, 2; C, 1; E, 0. The preference order for participant no. 2 gives D a better grade than the other alternatives four times, C three times, B two times, E once and A not at all. The preference order for participant no. 3 gives correspondingly for C, 4, for E, 3, for A, 2, for D, 1 and for B, 0. The alternative enjoying the greatest support is selected simply by associating numerical scores with the alternatives in their order of preference, with 0 at the bottom and then continuing upwards:

1

2

3

4 A

4 D

4 C

3 B

3 C

3 E

2 D

2 B

2 A

1 C

1 E

1 D

0 E

0 A

0 B

Then these scores are summed:

A    4 + 0 + 2 = 6

B    3 + 2 + 0 = 5

C    1 + 3 + 4 = 8

D    2 + 4 + 1 = 7

E    0 + 1 + 3 = 4

C comes out with the best joint grade. Above, an example was used to show how the final conclusion in the voting procedure can depend on the sequence in which proposals are submitted when using the series method or the elimination method. We illustrate the present method using the same examples:

5

4

4

2 A 

2 B 

2 C 

1 B 

1 C 

1 A 

0 C 

0 A 

0 B 

which yields:

A    5 x 2 + 4 x 0 + 4 x 1 = 14

B    5 x 1 + 4 x 2 + 4 x 0 = 13

C    5 x 0 + 4 x 1 + 4 x 2 = 12

Here A is rated best, and there is obviously no problem in determining the sequence in which the alternatives should be presented, since they are all submitted simultaneously.12 The conclusion is transitive, i.e., A is valued higher than B, B higher than C, and A higher than C. How could it be maintained that the group´s first preference here was other than A?

The importance of alternatives not being excluded from voting for logical and managerial reasons

Borda´s method is from a logical point of view an excellent method according to the formal-logic literature. Now we have seen that it is easily practicable, and that it has a meaning in terms of choice. We see a meaning in the fact that any of the alternatives presented can become relevant for the group´s evaluation of the alternatives mutually: no alternative can therefore be excluded a priori by means of a rule. Using Borda´s method there is no problem of determining how a set of proposals shall be submitted. That only proposals which exclude each other, i.e., proposals which are alternatives, should be put forward simultaneously is a consideration which remains. We here attempt to indicate how important it might be if it were possible, without managerial or logical problems arising, to deal with more than two alternatives by means of voting. Such an attempt must necessarily be rather speculative until a voting method has been tested that does not involve such problems, e.g., Borda´s method. Our analysis is based on Olsen´s (1972) analysis of alternative decision processes, where he makes use of two ideal types: the first where, in the organisation concerned, explicit voting is emphasised and little time spent in preparatory discussions—which Olsen terms the confrontation process — and the second where emphasis is placed on preparatory talks, on attempts to sound the opinions of a potential opponent—which he terms the sounding process. Thus an attempt is made to reach consensus through discussion, without employing explicit voting. If there were a practical method, such as Borda´s method, for handling all of the actual alternatives in a vote, this would presumably weaken the confrontational character of traditional voting, making it possible for a final vote to emerge from soundings of opinion, which would not involve confrontation with the risk of destroying mutual agreement reached during the soundings. In the following we explore this aspect further, by considering individual analytic points from Olsen (1972).

The nature of an issue will usually determine whether ease of dealing with many alternatives is important for the handling of that issue. Some issues are such that there is simply an “either-or” decision to be made. The question of traffic on the right or left of roads might seem to be such an example. However, even in this case matters are not so simple. Diversion of traffic from left to right will be associated with costs of variable size and form, depending on the way in which the diversion is handled and the many possible time scales for its implementation. Here there are, therefore, many real alternatives, and voting by Borda´s method will weaken the confrontational character of the issue. It will be possible to divide opinion in a more-or-less relation to the substantive alternatives under dispute. There will probably be a sliding transition from one alternative to another, which, therefore, will not cause participants to divide themselves into groups that dispute on the basis of quite opposite values and convictions. Using Borda´s method, Olsen´s following characterisation of the voting process will therefore not be so appropriate as previously: “Since the visibility of the process of confrontation is fairly high, and it is clear who supports and who fights the different proposals, the distribution of responsibility and thus the address for potential criticism or honor will be clear. The losing part also provides a basis for organizing future opposition against the winning part” (Olsen, 1972, p. 273).

Considering Borda´s method, the sounding process is of less value. Sounding out, according to Olsen, will be employed when the participants are willing to devote both time and energy to it, which should indicate that the participants consider the expenses of the conflict which would emerge in an explicit, traditional vote to be heavy. “Compared to a process of confrontation”, he says, “sounding out assumes that substantive rewards are discussed in terms of degrees, how much each part should get, and how much they have to give in. […] The sounding-out process produces a low visibility for those not participating and reduces their chances for raising criticism” (Olsen, 1972, p. 274). Voting according to Borda´s method will be characterised to a larger extent than current methods by views being expressed concerning the extent to which different groups shall receive different rewards, how much each of the parties shall obtain, and how much they must contribute. This will take place visibly, as opposed to the case for the sounding process.

The responsibility for the final conclusion will, by Borda´s method, be divided among those of the participants who did not have the final conclusion as the last alternative in their ordering of preferences on the issue concerned. Additionally, the method will probably provide a weaker basis for the formation of stable groups or the predominance of some individual line of conflict. An organisation employing Borda´s method seems likely to have many conflicts of a graded character about minor issues, and varying groupings of no great depth. Seemingly, this must strengthen the unity of the organisation as well as give rise to openness of viewpoints, in that this is implicit in the voting procedure.

Olsen realised that a permanent dilemma can remain for organisations in which different subgroups which control important resources (including expertise and external support) are not willing to be voted down. If an issue is presented with all of the real alternatives and decided on according to Borda´s method, it is not possible in the same way as for current arrangements to point out exactly the alternative which was voted down. In fact, it is most likely that many participants will, in their ordering of preferences, show a certain subsidiary support for the final decision. The risk of negative countermoves by a minority is therefore reduced and the responsibility for potential mistakes is divided among many. For these reasons less is to be gained by using the sounding process, which, according to Olsen, is the most reliable method of guarding against negative countermoves by a minority and of dividing responsibility for potential mistakes.

The relation between representative and voter also becomes somewhat different if Borda´s method is employed. Here a representative can express his preferences clearly in the voting, while, under existing processes, it has been impossible for a representative simultaneously to submit for voting his ultimate wishes and to give other alternatives subsidiary support. Voting procedures have not been able to reveal the graded preferences of the participants. The confrontation inherent in current voting procedures has indeed been evident, as considered by Olsen, but it is the problem of handling an entire scale of potential alternatives in individual cases which has often prevented the participants from presenting varied preferences and compelled them to overstress contrasts and adopt simplified viewpoints which have not indicated their real preferences. For example, elected representatives have not been able to document their graded preferences through the voting procedures so that they could refer to a scale of priorities: they have had to try to express their preferences, and their work in relation to them, differently. A still more obscure case arises when in an organisation certain parties, by means of soundings, decide that it is more advantageous to vote unanimously despite real disunity. These deficiencies are resolved if a standpoint is sought through a preference choice by means of individual point settlement by pairs (Borda´s method). In this case it is simple for participants simultaneously to express, by means of the submitted lists of preferences, their ultimate wishes and to strive for attainable objectives through their secondary preferences.

Only experience will indicate the validity of the assumptions we have made about the significance of Borda´s method. Many will probably hesitate over such an alteration in decision processes until they have seen the alteration tested. Some experience will be gained by implying Borda´s method to assess which of a set of alternatives receive considerable support, submitting all of the real alternatives for handling by Borda´s method. The results can then be applied to select a more restricted number of alternatives and to determine the sequence in which these should be submitted for voting, either by the series method or the elimination method. It will be interesting to see whether individual participants, when they vote according to the series or elimination methods, will indicate the same preferences as those already expressed using Borda´s method, or whether they take note of the test results and, by their voting, ensure that the decisive vote gives the same conclusion. If the participants entirely accept the choice according to Borda´s method, it will be easier for them, after the pilot study, to withdraw all the proposals other than the one which was evaluated as the highest by Borda´s method.

It can be expected that there will be less willingness to test Borda´s method in organisations in which the sounding process has been favoured; here some may fear that disunity will be manifested as a result of using the method. On the other hand, others may consider that the disunity emerging as a result of using Borda´s method is not so great as that which would result from traditional confrontation voting, and may find it profitable to escape from the time-consuming work of sounding. In certain organisations, those which have been organised into factions with clear divisions and where confrontation voting has been employed, some may fear that the fronts and factions will disintegrate when varied preferences are invited for presentation.

Can a participant profit by not declaring real preferences?

We have in the above presupposed that the participants consider it to their greatest advantage to declare their real preferences among the alternatives concerning an issue. There may be issues such that one of the participants can support his cause by declaring preferences other than his true ones. We take as an example a participant who actually prefers a set of alternatives in the order A,B,C. If he can take it for granted that the preferences of the other participants are such that either A or B would emerge at the top of the joint ranking list as indicated by the settlement, then in order to increase the likelihood of his most preferred alternative A emerging top he could downgrade B by adopting the preference ordering A,C,B (in this case he must expect that this will not upgrade C to the first place).

There is a possibility here for a participant to benefit by declaring his preferences insincerely in a concrete case, provided that he has sufficient knowledge of the relations between the alternatives for the other participants. How important is this? It is a general condition to be satisfied by any decision process that it should not give such possibilities. This condition was formulated absolutely by Ramstedt (1961, p. 111): the decision arrangement shall be of such a kind that there are no possibilities for tactical manoeuvring. Here, as elsewhere in discussing alternative decision procedures, it must, however, be comparison between methods that is relevant. How does Borda´s method stand here in relation to the conventional voting process? It was shown above that the conclusion of a vote using the elimination method or the series method can be manoeuvred according to the sequence in which the alternatives are submitted. In such a case the management of an assembly can manoeuvre the conclusion. Then the individual participants can affect the conclusion by tactical voting. If for instance the alternatives are submitted in the order C,B,A, a participant who actually has the preference A,B,C, can evaluate the potential advantages of voting for C in elimination voting if he expects that C does not have any chance of defeating A, but that B has a certain chance. However, probably more important in this context is the fact that the conclusion can be determined by whether or not an alternative is proposed or a candidate is put up. This can allow manipulation. It could happen that if two alternatives A and B were proposed it would result in victory for A, but if an alternative C were put forward in addition, the viewpoints could be distributed in such a way that those who earlier supported A were split between A and C, so that B would win.13

Johansen (1977) discussed the danger of someone being motivated under the conventional voting arrangement to declare false preferences when dealing with proposals concerning public goods. He maintained that the danger is slight. We agree with his assessment, but believe, as he does, that the possibilities nonetheless exist. Johansen´s arguments support our evaluation of similar possibilities in preference choice using individual point settlement by pairs (Borda´s method). As already pointed out, this method makes it relatively simple to put many alternatives to the vote. Therefore the number of alternatives presented can normally be expected to be greater than for traditional voting methods. With an increasing number of alternatives it becomes more difficult to calculate the conclusion which would result by a candidate not declaring his real preferences. Debates and argument before voting may provide some information about the preferences of other candidates, the completeness of such information probably depending on the total number of participants and on how divergent their preferences are. If the participants do not represent anyone, or are not chosen or commissioned by anyone to whom they owe explanation for their preferences, they are, in a way, more free to manipulate the results by declaring false preferences. If they are representatives for others, it is to be expected that the latter will wish to be informed about the preferences submitted; on the whole, it is to be expected that the voting settlement will be made public. Tactical preference lists may be accepted, possibly required on occasion, in such cases, but the representative in question would then be expected to be able to account for the reliability of his estimates of the preferences of others. However, in an assembly which in the present context can be assumed to consist of speculators, it is not easy for a participant to attain a reliable conception of the preferences of the other speculators. In such an assembly each individual member would presumably be wise not to make any assumptions concerning the preferences to be put forward by the others, and to declare his real preferences. Normally a morality of declaring real preferences can be expected to develop among the participants, encouraged by those they represent. How strong this morality becomes will presumably vary according to circumstances in the organisations concerned.

Experience from many discussions in academic circles about voting methods has shown that this very question of potential tricks and manoeuvring in elections and voting deserves great interest. As regards Borda´s method, we consider that speculative examples of how a party could benefit in a particular case are unsatisfactory. In fact, the question concerns not only tactical calculations in a concrete case, but also whether an individual´s cause is served in subsequent rounds of voting and in the long run by such conduct within the organisation concerned. In this light, other methods for decreasing the possibility of benefiting by tactical voting in a concrete case are simply irrelevant. There is no necessity to consider other than those for which a sound choice meaning has been found, as for Borda´s method and in the ranking of chess-players. In this connection the possibilities offered by Borda´s method for submitting all real alternatives to the vote are no less important, since this reduces the possibilities of manoeuvring alternatives out of consideration altogether before an issue is ready for voting.

Comparison between chess ranking and Borda´s method

We have suggested two voting methods which have the desired choice meaning and are more than calculation methods having certain logical qualifications; i.e., Borda´s method and the method applied in chess tournaments in order to choose the best player. The chess tournament method is handled in the choice theory literature as a calculation rule, a modification of the simple plurality rule (Copeland, 1951). Copeland presented the rule in the following form. Assume u(x) to be the number of alternatives defeated by x minus the number of alternatives which beat x according to the plurality rule. The alternatives are ranked according to the resulting values. We take the following profile as an example:

Participants

1

2

3

Preferences

x

y

w

 

w

z

z

 

y

x

x

 

z

w

y

This gives u(x)=1, u(y)=—1, u(z)=—1, u(w)=1. Voting-theory authors who have been concerned with Copeland´s contribution (Goodman, 1954; Luce and Raiffa, 1957; Gärdenfors, 1970, pp. 16-17; Fishburn, 1973) have not provided any choice-analytic basis for the method: for instance, why a defeat should downgrade an alternative and give a negative contribution. The method has been presented as a formal-logic contribution and as a calculation rule.

The method produces the same ranking as preference choice according to chess tournament procedure. This may be proved as follows. Suppose that in an example of a group choice there are n alternatives. The confrontations between pairs in the tournament procedure give superiority n1 times to A and n2 times to B. Suppose that n1 is larger than n2, and therefore A comes above B in the ranking list. By Copeland´s method A obtains n1 points minus n—1—n1 (n—1 being the number of confrontations each alternative participates in), i.e., n1—(n—1—n1)=2n1n+1 points. Similarly, B obtains 2n2n+1points. As n1 is larger than n2, 2n1n+1 will be larger than 2n2n+1. This, of course means that A emerges above B on Copeland´s index, in the same way as for the ranking according to chess tournament procedure. Thus, concerning the formal-logic properties of preference choice according to the latter procedure we can, equipped with this evidence, refer to the literature dealing with Copeland´s method. In short, on these criteria the method does well, while Borda´s method is somewhat better.

Fishburn (1973, pp. 171-2) has shown that although Borda´s method very often gives the same conclusions as Copeland´s method, sometimes the conclusions differ. We now consider an example to demonstrate the differences between the two methods. The example concerns a choice between three alternatives A, B and C, which are ranked on the lists of 20 participants as follows:

Lists

4

4

7

4

1

Preferences

A

A

B

C

 

B

C

C

B

A

 

C

B

A

A

B

By calculating points for individual preferences by pairs (Borda´s method, when a=0 and b=1), the conclusion is that A obtains 4x2+4x2+1=17, B obtains 4+14+4=22, and C obtains 4+7+8+2=21. The ranking will be B,C,A. By calculation of points for the group´s (the 20 lists´) preferences in pairs (chess tournament procedure), the settlement will be as follows:

A  against  B    9 against 11     superiority for B

A  against  C    8 against 12     superiority for C

B  against  C    11 against 9     superiority for B

Here there is superiority twice for B, superiority once for C, and never for A, so here the ranking is also B,C,A. However, by making a very slight change in the pattern of support (in the far-right column), the conclusions for the two methods are made to differ:

Lists

4

4

7

4

2

Preferences

A

A

B

C

 

B

C

C

B

A

 

C

B

A

A

B

Borda´s method gives: A, 8+8+2=18; B, 4+14+4=22; and C, 4+7+8+4=23. The ranking is C,B,A. According to the pattern from chess tournaments the result will be:

A  against  B    10 against 11     superiority for B

A  against  C      8 against 13     superiority for C

B  against  C    11 against 10     superiority for B

As before, this gives superiority twice for B, once for C and never for A, and so the ranking B.C,A. Thus according to Borda´s method the addition of a list of preferences C,A,B gives two extra points to C, so that C comes before B, whereas using the chess tournament procedure neither is a new win given to C nor is B deprived of its overall superiority in confrontations with C. Borda´s method takes each individual declaration about the preference between two alternatives into consideration, while using the chess tournament procedure (Copeland´s method) it is required that a new declaration (preference list) changes the group´s ranking of two alternatives in order that it shall count in the settlement. In this case also, Borda´s method is superior to Copeland´s; it is more sensitive to modifications in the preferences of the participants.

Practical questions regarding Borda´s method

What happens if a voter wishes to set aside two alternatives A and B from a set, i.e., to ignore them in the voting? It is then as if the voter concerned had produced two preference lists, one showing the ranking A,B and the other B,A, with each of these two lists carrying half the weight of a normal list. This may be illustrated by an example in which an assembly of 24 make up their minds on an issue involving six alternatives A, B, C, D, E and F. The participants have divided themselves into three groups of 7, 8 and 9, supporting three preference lists as follows:

Group

I

II

III

Support

7

8

9

 

A

D

B

Preferences 

B

F

D,E

 

C

B

A,C,F

 

D

A,C

 

 

E

E

 

 

F

 

 

It is simplest to carry out the calculations by giving the alternatives scores from the bottom upward, with 0 for the lowest alternative:

Group  

I

II

III

 

5  A

5  D

5  B

 

4  B

4  F

4  D,E

 

3  C

3  B

3  D,E

 

2  D

2  A,C

2  A,C,F

 

1  E

1  A,C

1  A,C,F

 

0  F

0  E

0  A,C,F

From group I, A obtains 5 points from 7 lists, or 35 points, from group II, (2/2+1/2)x8=12 points (A shares here with C places having the values 2 and 1, so the figures are halved), and from group III (2/3+1/3+0/3)x9=9 points, or in total35+12+9=56 points. Likewise, B obtains 97 points, C, 42 points, D, 85.5 points, E, 38.5 points, and F, 41 points. B scores the most points and is chosen. Now, when there are many alternative proposals and a participant is not interested in specifying his standpoint for more than the highest ones, it is possible simply to specify nothing concerning the remaining proposals. The preceding preference list to the right would, for instance, have emerged by group III having entered only the first two lines of its preference ranking, for B, D and E, i.e., even if nothing had been notified about A, C or F. Practical simplifications can also be arranged to take account of a voter who specially dissociates himself from one alternative and places it at bottom, while not being interested in marking differences on certain other alternatives preferred to the one definitely rejected. The preference list of group II above can serve as a moderate example of this, where the list produced could have assumed the following form:

D

F

B

E

(— denotes the position of alternatives not entered).

Often the number of alternatives increases as amendments to a main proposal are put forward. We are not now going to explain how such situations are dealt with under existing voting arrangements.14 As regards preference choice through individual settlement by pairs (Borda´s model), the procedure in such a situation is that whole alternatives are submitted for handling. For example, in a case involving two amendment proposals which do not exclude each other, there are five alternatives: the main proposal unchanged (A); the main proposal with the one of the amendments (B); the main proposal with the other amendment (C); and the main proposal with both amendments (D); the fifth alternative (E) is that the voter dissociates himself from all that is proposed. The participants then submit their preference lists for the alternatives A, B, C, D and E.

Election of persons

The basic concept of the decision process that we are introducing here is that the alternative receiving the best grade, so to speak, is adopted as the standpoint of the group. Suppose that two alternatives are to be chosen for the same purpose. This is most likely to happen when a group is to elect two representatives, for instance, for a managing board. Thus, a decision is to be taken on which two people shall sit on the board. If this proposal concerns three people, A, B and C, the alternatives are AB, AC and BC. The preference lists can assume the forms:

AB

AB

AC

AC

BC

BC

AC

BC

AB

BC

AB

AC

BC

AC

BC

AB

AC

AB

The procedure is then as described above. It is easy to see that the number of alternative combinations will increase rapidly if there are more people to be chosen, even if there are only 3, 4 or 5, and the number of different preference lists will increase still more rapidly and become more than it is practical to handle. Besides, it is clear that the method does not ensure proportional representation. The majority can have their alternative adopted in full, while the minority achieves none of its wishes.

It might be argued that the two candidates who obtained the highest grades in a preference choice between individuals should be selected. This would at least not be as complicated to handle. Whether this method is possible depends on the considerations that should be fulfilled by the voting arrangement. The procedure does not, in fact, produce proportional representation either. Those who, by means of their preference orderings, are in a position to give an alternative a sufficient number of points that it emerges first, are also likely to possess the strength to give their second preference a sufficient number of points to be second of all the alternatives. It the preference A,B obtains 5 votes, and the preference B,A 4 votes, A obtains 5 points and B obtains 4 points, and A is adopted if one out of A and B is to be chosen. If two are to be chosen out of two pairs of similar candidates, A1 and A2 and B1 and B2, the preference lists assume the forms A1,A2,B2,B1 and B1,B2,A2,A1, with 5 and 4 votes respectively. This gives A1 15 points, A2 10+4=14 points, B1 12 points and B2 5+8=13points. Here both places are determined by a majority of 5 out of 9. This form of settlement therefore does not exclude the majority dominance that is so familiar in existing procedures, at least without special countermeasures being taken to ensure proportionality. This is not discussed further here. This majority dominance probably applies not only in cases of electing representatives, but also in most ordinary issues.

Final discussion

It has been shown that one of Arrow´s classical criteria for group voting methods (the independence of so-called irrelevant alternatives) is not in keeping with basic concepts of choice and demand in the economic theory of demand which are, in addition, easy to understand. To Arrow´s criteria we would add the condition that in group choice all sides of an issue should be dealt with and all possibilities tested against each other. In addition, a method for group choice should have a meaning in terms of choice. Dealing with all sides of an issue and with all real possibilities can probably be achieved without formal handling; for instance, by sounding out of opinions, if this is desired. Preference choice through individual point settlement by pairs (Borda´s method) has, in this connection, the great advantage that it does not create any managerial or logical problems for the handling of an issue if many alternatives are submitted. It almost goes without saying that all actual possibilities should be handled, i.e., all of the possibilities which are preferred to some of the other proposals by any of those entitled to submit proposals.

It might be asked how important it is that all the sides of an issue are considered and that the participants are given an occasion to put them up against each other; for instance, what information this gives. Instead of trying to give a general answer to this question, we in a way reverse the problem. If all of the possibilities are not dealt with, some are excluded. There is hardly any reasonable way of doing this. The exclusion of real possibilities from being handled using existing decision processes is a feature on which a clear standpoint, factually or formally, has not been adopted. It has been in the nature of the handling of the issues considered that just one (by sounding) or very few real alternatives have been selected for formal handling.

By making a point of the fact that a group voting method should have a meaning in terms of choice, that it shall be more than an abstract calculation rule, we assume that the highest judges on the question of the meaning of a method are those who apply voting procedures to obtain decisions. In this connection we have described the way in which chess and ball players determine the best participant (alternative). In addition, we have sought for arguments in favour of preference choice through individual point settlement by pairs in the formal-logic choice-theory literature. Finally, we have discussed methods for group choice with regard to their social consequences, extending the issue beyond formal-logic choice theory. The problem is not to find an ideally correct method for group choice, but rather to choose a method which serves certain purposes and demands better than have done existing methods over the last 200 years.

References

Kenneth J. Arrow (1951). Social choice and individual values. New York: Wiley (2nd edn., 1963).

Duncan Black (1958). The theory of committees and elections. Cambridge: Cambridge University Press.

A. H. Copeland (1951): “A ‘reasonable’ social welfare function”, University of Michigan Seminar on Application of mathematics to the social sciences, 1951 (mimeo., not obtainable).

Robin Farquharson (1969). Theory of voting: New Haven: Yale University Press.

Peter C. Fishburn (1973). Theory of social choice. Princeton: Princeton University Press.

Einar Gerhardsen (1946). Tillitsmannen. Håndbok i praktisk organisasjonsarbeid. Oslo: Det norske arbeiderpartis forlag.

Frederik Glaven (1955). Om metoder for afstemninger og valg. Copenhagen: Dansk Videnskabs Forlag.

Leo A Goodman (1954). “On methods of amalgamation”, pp. 39-48 in R. M. Thrall, C. H. Coombs and R. L. Davis, eds.Decision processes. New York: Wiley.

Peter Gärdenfors (1970). Logiska villkor för demokratiska beslut. Philosophical institute of Lund University (mimeo.).

Peter Gärdenfors (1974). Group decision theory. Lund: Studentlitteratur.

Bengt Hansson (1970). “Valsystem och beslutsprocesser”, Minerva’s kvartalsskrift 14: 190-198.

Sigurd Heiestad (1958). Spillets ABC. Oslo: N. W. Damm.

Leif Johansen (1970). An examination of the relevance of Kenneth Arrow's general possibility theorem for economic planning. Oslo: University of Oslo, Institute of economics, reprint series no. 68 (From Optimation et simulation de macrodécisions. Namur: CERUNA).

Leif Johansen (1977). “The theory of public goods: misplaced emphasis?”, Journal of public economics 7: 147-152.

Jan Kobbernagel and Poul Sveistrup (1967). Afstemningsregler og afstemningsmetoder. Copenhagen: Einar Harcks forlag.

I. M. D. Little (1952): “Social choice and individual values”, Journal of political economy 60: 422-432.

R. Duncan Luce and Howard Raiffa (1957): Games and decisions. New York: Wiley.

Johan P. Olsen (1972). “Voting, ‘sounding out’, and the governance of modern organizations”, Acta Sociologica 15: 267-283.

Prasanta K. Pattanaik (1971). Voting and collective choice. Cambridge: Cambridge University Press.

Tolle Ramstedt (1961). “Parlamentarisk beslutsteknik”, in Organisationer—Beslutsteknik—Valsystem. Författningsutredningen: V, Statens offentliga utredningar.

Amartya K. Sen (1970). Collective choice and social welfare. San Fransisco: Holden-Day. University of Oslo (1974). Innstilling fra valgordningskomitéen.

 

Quality & Quantity 16 (1982) 433-54

  

1

See for example, Black (1958).

2

Our description of the series and elimination methods is based on that of Ramstedt (1961, pp. 112, 154).

3

The example is borrowed from Glaven (1955, pp. 25), with the difference that he has 5, 4 and 3 in each group. This does not make any difference here, but makes the example more suitable for subsequent illustration.

4

Concerning Borda´s propositions, the present description is based on the presentation of Black (1958, pp. 156).

5

In fact, Arrow set five conditions. The firsts condition is omitted here, because it is not of the same nature as the remaining four, being in fact a way of formulating the problem itself, as pointed out by Little (1952), footnote p. 422).

6

In the original version, p. 27: “CONDITION 3: Let R1,…,Rn and R1’,…,Rn’ be two sets of individual orderings and let C(S) and C’(S) be the corresponding social functions. If, for all individuals i and all x and y in a given environment S, xRiy if and only if xRi’y, then C(S) and C’(S) are the same (independence of irrelevant alternatives)”.

7

For background reading we refer to some essential works on the subject: Luce and Raiffa (1957), Black (1958), Farquharson (1969), Sen (1970), Pattanaik (1971), Fishburn (1973) and Gärdenfors (1974).

8

A few economists, e.g., Johansen (1970), have argued that it was not of the greatest significance to satisfy this condition, but have not claimed that the condition is unjustifiable.

9

This, of course, applies to voting problems. The case is quite different for problems arising when electing persons in assemblies. Election methods fulfil to a varying degree different considerations concerning the distribution of representatives. Elucidations of practical value concerning this question are available; e.g., University of Oslo (1974).

10

When there are many participants of if time is restricted, the number of games is reduced according to certain rules. Today it is quite common to reduce the number according to the Monrad system, which constitutes a reduction and a method of choosing the best participant in a shorter time with a satisfactory reliable conclusion; see for example, Heiestad (1958, p. 116). This question is, however, not in conflict with the conclusion that the most correct result is obtained if everybody plays against everybody.

11

Discussions with Trausti Jónsson on the ranking of chess and ball players and the analogy with choice between many alternatives, not least with regard to Arrow's conditions, have proved important for our understanding of the subject.

12

Glaven´s original examples with 5, 4 and 3 gave a dead heat between A and B, with 13 for both.

13

Hansson (1970, pp. 196-197) discussed a real example of such a problem.

14

Gerhardsen (1946, pp. 27) gave assembly leaders practical instructions for different voting situations; see also, Kobbernagel and Sveistrup (1967).