I. Introduction

The article describes a method to bring about a mathematical aggregation of preferences. Problems which economists have been dealing with are emphasized. These matters are mainly under discussion in public economics, but often other organizations than public authorities have to deal with the same kinds of problems, e.g. a joint-stock company or a cooperative society or organizations of firms. Of course, political scientists have tackled the problem, but as already mentioned it is not restricted to public organizations. It is, in fact, right to consider the research on the aggregation of preferences on the whole as a sociological problem, but still it is natural enough that economists attend to the task, when it is concerned with the distribution of economic goods, as is done here.

This is no opportunity to discuss the nature of social conflict or methods used in the aggregation of preferences or known proposals about such methods except for pointing out briefly what is most important for what follows. It is familiar that the majority of votes is most often decisive. If we look at a group of decisions, which are made in that way, there is no knowing, whether wins or losses will be divided somewhat fairly among the participants, neither regarding the number nor the importance of issues. Also, it is often difficult to vote on more than two alternatives on an issue and therefore attempts are made to avoid that. Voting on two alternatives is mainly done in such a way that votes are cast for or against a proposal.

At established assemblies like national assemblies it is common that the participants coalize so that those who form a coalition compromise before voting. By the coalition form it may be controlled to a certain extent, how wins and losses are distributed among those who are coalized. By that, for instance, the standpoint of the coalition may depend on how much emphasis each of the members places on individual issues. Whether the outcome is reasonable depends to a certain extent on the position of the participants in the coalitions they form and also how influential those coalitions are. At any rate, the distribution of the influence cannot be measured objectively.

Here there will just be made mention of three proposals concerning reforms of methods to make joint decisions, i.e., the plurality rule and the point system (cf. e.g., Musgrave1 (p. 129-130) and Coleman´s political money.2 The plurality rule determines definite proportions between the alternatives for the preferences for each, while the point system should make the arrangement of preferences with free difference possible. A striking defect of the system is that a proposal about a small appropriation will be late to win among proposals about big appropriations, if the size of all the appropriations together is predetermined. Coleman (1970) compared the monetary system and the arrangement of voting. As a result of that analysis he put forward his proposals about a new arrangement of voting. It may be noticed that there exists a particular similarity between his proposals and the arrangement, which I am describing here, although we have not directly influenced each other. At first he mentions that every member of an assembly receives at the beginning of the session a definite number of votes for him to use on whatever bills he likes. Coleman does not account for what is to be done, if there are more than two alternatives on an issue, or whether it should be permissible to put forward the same proposal again and again, or whether the voting is to be secret, until it is over. These are all important points that require a satisfactory solution. It is, of course, not right, as Coleman does, not to regard a rejected proposal as a point of view and a decision.

II. Description of the arrangement

Those who are entitled to vote receive votes, e.g., one vote per day, 100 votes per month or 1 000 votes per passing year, and mutually in the proportion of influence which has been decided that they should have, e.g., in such a way that he who enjoys twice as much support from voters or stockholders as someone else, will receive twice as broad a stream of votes. The stream of votes can be used or kept at will. When the votes are used the participants shall be able to distribute them among proposals according to how much emphasis they place on working for or against the passing of proposals, where they, for instance, can take into account how many votes they think profitable to keep in reserve because of the issues that may subsequently come to the vote. This is, in fact, similar to the case, when someone receives a continuous stream of money, and is either entitled to spend it immediately or collect it in reserve for later use, and spends it on what is offered in such a way that the last five-pounder in one deal produces as great satisfaction as the last five-pounder in any other deal. When spending money people also think of the security the enjoy by saving money. You seldom know, how well the stream of money will continue to flow to you nor how much you may be in need of money, because your position or your purchasing power may change. In mathematical aggregation of preferences it should be fairly certain, how the stream of votes will last, whereas it is worse to ascertain, how important issues may turn up. A discussion is needed on what conditions influence that uncertainty and how the arrangement may be made in order to decrease the uncertainty.

In the monetary system it is a rule that money retains its value, irrespective of when it has been acquired. It has to be specially estimated and decided for how long votes shall be valid.

A. A choice between two alternatives

A proposal comes to the vote. Then two alternatives will be voted on, that of passing the proposal or that of ignoring it. The alternatives are called A and B. Participants arrange the alternatives, at first the alternative they want less, and then the other one, placing by it the number of votes they want to give in its favour. Then the votes for each alternative are summed up. The alternative, which has received the majority of votes, is considered as approved, because the participants have as a whole been willing to sacrifice for it more of their right to influence decisions. When results are obtained, votes shall only be taken from those who won. In all, just a little bit more shall be taken from them than was contributed to the other alternative. This is fairly similar to the case that in business, where tenders are sought, you have to offer a little bit more than the previous offers did, if you want to get the deal, where only he who makes the highest offer and gets the deal has to pay. Those who offered less do not have to pay anything. Similarly, it is also the case here that votes are taken from those who won, while those who supported the alternative which lost keep the votes they offered to sacrifice for it.

All the votes, which those who won offered to give, shall not be taken from them. This is done to prevent an alternative, which is supported by almost everybody and which received relatively few counter-votes, from being dearly bought. Therefore the number of votes of the counter-proposals plus an infinitely small number of votes shall be taken from those who win as a whole but in practice it turns out that the number of votes of the counter-proposal will be taken from those who gain the victory. If the alternative A received n votes and wins and the alternative B m votes, m votes shall be taken from those who voted for A. That figure shall then be divided among all participants by multiplying his number of votes for A by m/n. The voting shall be secret, until it is finished in order to prevent tactical voting. The idea behind this pricing and other ways of pricing and their effect require further examination and discussion.

B. Three or more alternatives on an issue

If there are three alternatives that shall be voted on the voting takes place at the same time. The method will be described by an example. The same method is used, although the alternatives are more than three. The alternatives may have become three, when an amendment to A was put forward. The third alternative is to support none of the proposals. One of the participants (X) arranges the alternatives in order of preference as follows: C is the worst alternative. He offers to give 10 votes in order that A is accepted rather than C. He offers one vote in order that B is chosen rather than A. Another participant, Y, arranges in order of preference as follows: A is the worst alternative. He offers four votes more for B than A. He offers 20 votes more for C than B. Z finds B the worst one. He estimates C 3 votes better than B and A 12 votes better than C. According to this, the participants shall first arrange the alternatives from the worst one up to the best one stating the difference of votes at each interval. Thereupon the votes are summed up. (The calculation of the difference of votes over two or more intervals shall be done by an accountant and not by him who votes). A receives 10 votes from X, nothing from Y and 15 from Z. B receives 11 votes from X, 4 from Y, and nothing from Z. C does not receive any votes from X, 24 from Y and 3 from Z. If there are no more votes, the conclusion will be as follows: A receives 25 votes, B 15 votes and C 27 votes. Those who have the right to vote on the issue offer most of their influence for C. As a result, it is considered as chosen and accepted.

Then we have to calculate, how many votes shall be taken from those who supported the victorious alternative C. It is the companions Y and Z. A received 25 votes altogether. To find out, how many votes Y and Z have to give away in order to make the outcome equal for A, we first have to reduce the votes for A and C to such a degree that the same participant possesses votes for both alternatives. In this case it is only Z that has votes on both sides, 15 or A and 3 for C. Therefore the outcome for both alternatives is reduced by 3. Then there are 22 votes left for A (10 from X and 12 from Z) and 24 votes for C, all of them from Y. 24 votes times 22/24 = 22 shall be taken from Y.

The next thing to do is to calculate, how much shall be taken from those who voted for C in order to make the votes equal for B. B received 11 from X and 4 from Y. In this case Y has votes for both alternatives. When both sides have been reduced by 4 Y-votes 11 votes are left for B, compared with 23 votes for C, 20 of them from Y and 3 from Z. 20 times 11/23 = 9 13/23 votes shall be taken from Y to make the votes equal for A and 9 13/23 votes for his part in making the votes equal for B. The voting is now regarded as a repeated voting: A against B, B against C and C against A. The votes which Y has offered for C in one voting keep their value, although more than one alternative competed with C. Accordingly, he who has had a share in the victory has sided with one alternative (but has not, in itself, taken sides against so and so many alternatives) and has to pay for that the price which is set up, as already described.3 Thus it does not matter, whether more proposals on the issue have been composed and put forward, which have been estimated as a whole and by individual participants as worse than the alternatives, which according to vote settlement cost individual participants most. Y, therefore, loses only 22 votes for contributing to C´s victory. Z loses 1 10/23 votes because of vote settlement between C and B.

The settlement of the voting can be summed up as follows:

22 votes are needed for C to make votes equal for A; 22/24 votes per each vote for C after a parallel reduction to 0 on the one side.

22 C-votes distributed among the participants Y: 24 x 22/24 = 22

11 votes are needed for C to make votes equal for B; 11/23 votes per each vote for C after a parallel reduction to 0 on the one side.

11 C-votes distributed among the participants

Y: 20 x 11/23 = 9 13/23

Z: 3 x 11/23 = 1 10/23

The conclusion of the vote settlement:

C is chosen. 22 votes are taken from Y because of votes for A, but nothing because of votes for B. 1 10/23 votes are taken from Z because of votes for B.

Two principles are followed in the vote settlement: Before the pricing, the votes, which are compared, are reduced equally much for both alternatives and down to 0 on the one side for those who have votes for both the alternatives, which are being calculated. Votes for the victorious alternative are compared with those for each of the other alternatives, as if we had a repeated voting, where the victorious alternative and the votes for it recur.

As one can see, the arrangement can be used, although the alternatives become many. It is easy to indicate in order of preferences, if participants have taken their bearings. A participant does not have to keep in mind at the same time more than the two alternatives, which stand next to each other for him. Amendments to a bill can become practically unlimited in its procedure. If a participant is only interested in a small part of the issue he is entitled to declare his point of view by saying that those alternatives will receive n votes, which contain a given article, but else nothing, or he will offer m votes for the bill, if the given article is lacking there, but no votes for the alternatives that contain the article.

C. How control over an issue can be preserved

The customary arrangement of voting is built up in such a way that it does not formally reduce one´s right of voting to vote again and again on the same issue. At first a decision is made. A short time later a proposal is put forward to reject the decision. Those who previously supported the decision usually have the same number of votes to be able to turn down the counter-proposal as they had before to support the decision. This is the formal side of the arrangement. It is a different thing, which often turns out to be decisive, that a minority, which has got a decision through by taking part in a coalition, has no security tied up with regulations, how long the decision will last. The coalition may de dissolved or change, and it depends on the position of the minority in the game what happens to the decisions, which have been made at their request.

In mathematical aggregation of preferences it is to be expected that a decision will most often cost votes. If counter-arrangements are not made, those who want a decision to keep its value may have to waste votes repeatedly on what actually is the same issue, if proposals to change the decision are constantly being put forward and voted on. It is not fair that one can be deprived of votes in such a way. It even invites the making of proposals which will above all aim at winning votes from an opponent, who has a superior issue.

Decisions can be very different in the sense that some will, in fact, not be recalled, while others are such that new decisions may be made independently of former decisions. If laws are made on sheep bathing in the year 1971 one can therefore hardly be forced by circumstances to enact laws on bathing in the following year, or later. If it has been decided to harness a river in the year 1971, and it was actually done, it is to be expected that a proposal, which is put forward in 1972—the year after the harnessing—that the river shall flow as it did before the harnessing, will receive far fewer votes than may have been cast against the harnessing when it was approved.

These examples indicate that it can be different, how circumstances can practically remove an issue from the agenda after a decision, and this becomes still clearer from the following example, which is used to show, how people may be given an opportunity to avoid having to waste votes repeatedly on what actually is the same issue, whereas the opponents on the issue keep votes, as they are defeated. The example is that a proposal is put forward about harnessing a river for water power. Automatically a counter-proposal is put forward that the power plant, as described in the proposal, shall not be made. If the proposal about the power plant is turned down the same number of votes will be taken from the opponents of the power plant as were contributed by the supporters of the power plant. If no counter-arrangements are made the same kind of proposal could be put forward, as soon as the proposal has been turned down. It is to be expected that the supporters of the power plant would then contribute about the same number of votes as they did the first time. If the opponents stop the harnessing of the river again they will then again lose about the same number of votes as they did in the first voting. This is obviously not fair. It might be advisable to examine, if it was better to prohibit the taking up of the same proposal, until after a certain period of time. It could either be a general rule or a decision connected with the counter-proposal that no power plant be erected. Such a solution to the problem can easily be unreasonable. The premises for the standpoint, which is taken in a voting, change either quickly or slowly, and it is difficult to foresee, how quickly the premises may change. In addition, there may appear new participants, who did not have any right to vote before. Even though it was decided that the same proposal could not be voted on, until after a certain interval of time, it cannot be completely avoided that proposals, which serve the same purpose and arouse the same kind of opposition, will be put forward and brought to the vote. A plan about a power plant can await a later date or the design can change more or less. It may become difficult to set such a limit to new proposals that it is unmistakably distinct what is a new proposal and what is really the old proposal revived. It seems possible to solve this problem in such a way that those who have contributed to a decision and sacrificed votes for it will be able to acquire a special right to influence the issue for a certain period of time, or that they will be capable of rendering such a right to others by a special decision in connection with the issue. Then there will be no question of locking up the decision, but giving specially appointed parties (e.g., representatives, ordinary voters, authorities or organizations) special votes, which they shall be able to spend on the decision, no matter whether they are in due course willing to defend the former decision, take part in altering it or even oppose it.

This can take place in various ways. The proposal about the harnessing of a river can be used further as an example. As before, it will be an automatic counter-proposal that the power plant, according to the proposal, shall not be constructed. Another proposal can be made with the following contents: The river shall remain unmoved, and a certain number of special votes shall for a certain period of time be disposed of by a specially appointed party on proposals, which concern the flow of the river. Many proposals can, of course, be made on this point, where the number of votes, time and party can be variable and the number of votes can change in the course of time. It is to be expected that those who are against the power plant would be willing to offer more votes for the proposals on the issue that will give themselves or those, whom they expect to share their views on the power plant, a right to influence operations at the river for a long period of time rather than for a short time, but probably they will not offer many votes to keep such a right in the distant future. Those who want the power plant to be erected sooner or later would probably cast more votes against proposals, which give the opponents of the power plant the right to control special votes concerning the harnessing of the river for a long period of time rather than for a short time, and, of course, they cast most votes for the power plant proposal itself. Here we have a new kind of decisions. One is invited to obtain the right to influence issues by means of special votes without one´s standpoint on the issue having to remain unchanged. It is familiar that in purchase and hire contracts regulations can be found which allow for revision. Similarly, it is well-known that special parties acquire a permanent right to control special issues or categories of issues. National assemblies, for instance, entrust communal authorities and institutions with particular issues—it needs to be discussed, whether and how such decisions about the conditional transfer of votes may be altered, and if such votes shall be consumed, when they are used. That will. however, not be done in this article, but it seems as if that has to be determined in the general decisions of the organizations concerned.

D. Dividing a fixed size

Here we have a problem with which organizations of a financial character are often confronted. I will describe the problem and how it is dealt with by taking an example of the making of a budget in a communal district or a town. It is, in fact, not the case that the size of the community´s budget is determined formally by others, but people tend to think it is so. Then they say that the sources of income are fully utilized and prefer to have more for distribution. When the income, which is considered somewhat fixed, shall be distributed among expenditure items the task is carried out by stages, taking first the unavoidable items and then others. Then two or three items are compared at a time, stage by stage, in order to find out if one item may be increased at the expense of another. Thereupon the budget is put forward, hardly ever more than one alternative, and little chance is offered, when the budget is formally passed, to make amendments to it, where one amendment would be succeeded by just as big amendments to other items within the frame of income. In the mathematical aggregation of preferences it will be no problem of procedure to vote at the same time on many different budget proposals with the same amount of expenditures and income. The number of possible alternatives will, however, become easily too great for one to gain a complete view over all of them.

This will be illustrated by an example. It does not seem complicated to divide a particular size into 5 equal parts. Yet there are many possible combinations. Take as an example, if 5 boats are to be granted a licence to fish. If there are 6 applicants 5 boats may be arranged in 6 different ways. The number is not so great that it will be easy to arrange all the alternatives and vote on them in the way that has already been described, giving first place to the least attractive combination, then the worst but one, together with the number of votes which are given in order that that alternative be preferred to the first one, and so on, to the best alternative and the number of votes, which are offered in order that it be accepted rather than the best but one. If there are 8 applicants for the licences, the alternatives are 56 (8!/5!3! = 56); if there are 10 applicants 252 combinations are possible, in the case of 12 applicants 792 alternative combinations, and even though there were only 15 applicants 3003 different combinations would be possible. Here it has already long ago become almost impossible to arrange all the alternatives. All this would, however, have got even more complicated, if the size of the licences could have varied, for instance, if a definite quantity should have to be divided among 5 boats with free variance of quantity per boat (number of days, catch).

This is intended to show, how easily one can find many more alternatives than are dealt with formally and consciously by means of the methods that are used for division. Probably it is also often the case, when one shall divide fixed sizes or sizes that are determined by others, as, for instance, bank loans or the right to fish on special fishing grounds, that one tends to think most or only of securing for oneself and one´s clients a share. It is, however, easy to find examples where some person wishes to prevent another from being favoured with something, although it does not necessarily increase his share. It is not to be expected that everybody will acquaint himself and take sides on all issues. In mathematical aggregation of preferences it is possible to ignore the alternatives that come to the vote, if one feels that he is unable to judge about them, but if one takes sides, one indicates it, as already explained.

The following example explains the matter further. It concerns four applications for licences to fish on special fishing grounds. Only three licences shall be granted. This time there are only four alternative combinations, so that they can all be easily arranged at the same time. The number in the example is limited in order to gain a better view over the matter. The applicants are A, B, C and D, and the alternative combinations become ABC, ACD, ABD and BCD. There are 6 participants in the voting.

No. 1 wants to give A a licence, but wishes to prevent B from getting to the fishing grounds, as he will cause damage there. He does not know the circumstances of the others and does not form a special view of their applications. He gives A 11 votes, offers 9 votes against B, and gives C and D no votes.

No. 2 wants to give B a licence, provided that C will get a licence too (B may in some way or other be dependent on a companionship with C), but he does not care, whether A or D participate or not. No. 2 therefore gives B 13 votes, if C is in the combinations, i.e., 13 votes for BC, or else nothing.

No. 3 wants to give C and D separate licences and not both of them together (he may consider it will suit those concerned badly to have two boats, C or D). He does not care, what will happen to A or B. Therefore he places 7 votes on alternatives with C and not D and also 7 votes on alternatives with D, but without C.

No. 4 wishes to grant A and B separate licences, but thinks it is, however, even more important for them to get a place at the same time - they may support each other - and does not care, what happens to C or D. (In this case the participants evaluate the premises of the applicants in a contradictory way, when no. 1 wants A and B to be kept separate, whereas no. 4 wants them to be joined. It is not the intention that this explanatory example necessarily interprets the consistent evaluation of the participants; it is only intended to show, how one can use votes in accordance with one´s own evaluation). Alternatives with A receive 15 votes from no. 4, those with B also receive 15 votes, and alternatives with AB receive 5 votes in addition.

No. 5 wants to offer something for C and D separately. He thinks that they will compete with each other for a crew and equipment, although it would, however, be more advantageous to have them joined rather than either of them separate. No. 5 offers 10 votes for an alternative with C without D, 10 for D without C, but 12 for combinations with C and D.

No. 6 wants to give a licence to C and D together and thinks it will cause trouble, if A is joined with them. He, therefore, places 24 votes on an alternative with C and without A, but only 16 votes, if A is included.

The alternatives receive votes according to the following summary (usually the calculations must be carried out mechanically—they will become too extensive, if done manually):

ACD receives 9 votes from no. 1, because he offered 9 votes against B, which is the same as if 9 votes were added to all the alternatives without B. As may be seen there are votes from no. 4 and no. 5 on all the alternatives. This may easily happen, when one does not take sides in connection with the whole alternative, but only in connection with the individual features of the alternative and when no alternatives can be combined without containing at least one of the features, which the participant wanted to be brought forward. Such a conclusion would not, for instance, be unnatural in a voting on a budget, which is conditioned by a certain size of expenditure and income. Although this is the outcome for 4 and 5, it will not cost them more than if they had arranged all the alternatives and placed the worst alternative down to 0, keeping the same numerical distance between the alternatives as is shown in the summary. This is, as a matter of fact, done in the vote settlement, when the number of votes of the participant concerned is in the comparison reduced in a parallel way and down to 0 on the one side.

ABC is chosen by 76 votes. To calculate, how much shall be subtracted from those who have votes for that alternative the votes for ABC are compared with those for the other alternatives, as has been already described.

The purpose of this example is, of course, not to maintain that this is the method for the distribution of values that will lead to the most economical utilization of the fishing grounds, and that this method is superior to other administrative means, such as granting of licences against payment or by inviting tenders. In fact, it may become a matter of voting to take sides on methods of distribution.

III. Further points on how the method can be used

a. Transaction of budgets

1) Proposals about appropriations, which determine directly the size of a source of income

Here it is proposed that a certain sum of money is appropriated for special purposes and that the money is taken from a source of income, which is strengthened correspondingly, while the appropriation has no direct influence on other appropriations. Of course there can be other proposals on the same issue. The sum can vary as well as the purpose (then it is stated in the proposal that it will replace another proposal) and the source of income can be variable. The proposals may be about an appropriation for the building of a road between two places with three different road beds, where each road bed can have three different forms (breadth and wearing course), but the money shall be taken from a general sales tax, income tax or a car charge. Here there are 27 proposals on the same issue. The 28th alternative is that no money shall be appropriated in the way indicated in the proposals. This alternative emerges automatically as a proposal according to general decisions. Therefore 28 proposals are voted on. Thus an opportunity is offered to take side as to whether the advantages that result from the appropriation exceed the difficulties caused by the taxation. The charges, e.g., a car charge or sales tax, will then be determined for each period according to the regulations that are valid for the source of income, when all the appropriations from the source of income concerned have been decided. By this the advantage and the disadvantage of each individual appropriation are directly connected, as one may be expected to do in one´s own private economy, but no direct sides are taken as to how high the charges for the individual sources of income shall be. Of course an attempt may be made to estimate, how high the charges will be. when it is known, how big appropriations have been decided or proposed.

2) Appropriations from sources of income of a definite size

Then there will be voted on all the proposals about an appropriation from each source of income for each period. The method used is the same as already described when a fixed size is divided.

3) The whole budget decided in one voting

This is done in the same way as in the division of a fixed size, with the difference that at the same time sides are taken as to how big the size of expenditure and income shall be by arranging proposals about income and mark up in order of preference. It is natural for those who make proposals about appropriations to refer to a source of income in the proposal, for instance, by referring to a corresponding increase of purchase tax or equally much income from a 10% purchase tax. Naturally, it often calls for alternative proposals. In this way the same procedure can be continued in the way the participants are accustomed to in the transaction of a budget, where participants are generally opposed to the increase of taxes and charges at the same time as they support new appropriations without referring to a corresponding reduction of other appropriations. In the vote settlement such conflicting points of view will converge.

4) Measures taken because of economic fluctuations

These may to a certain extent collide with other economic objectives. The solutions, which have already been described, offer an opportunity to evaluate the opposing objectives. Proposals can be easily made about a different big deficit or surplus in the budget, and a budget can be passed, where special appropriations are, for instance, dependent on unemployment in the whole country or in certain parts of the country. The difference from the present state of affairs is that many proposals on the same issue can be voted on as a whole.

5) Appropriation proposals with votes that follow them year by year

It has been shown, how one can secure for oneself a relatively permanent control over an issue, if one wishes. This may be arranged in the voting on the budget. Then the items which are thus secured, follow their votes into the alternative combinations, if there is no mention of anything else, or that those who have votes behind such items are informed that their guarantee of votes is involved. They can also transfer the secured votes according to their preferences, for instance, in order to work at the acceptance of some of their alternatives. The votes follow the issue, although it may change its form during the period that has been decided.

6) Permanent counter-votes

Just as votes can follow appropriation proposals for many years, there can follow votes against certain appropriations. This does not seem to add anything to what has been said about methods to keep control over issues.

7) Prohibitions of a vote on particular appropriations

Instead of letting counter-votes follow particular appropriations proposals for many years, decisions can be made with more or less permanent votes that a proposal about an appropriation for a particular issue cannot be voted on. Probably there would be no reason for making such a decision, until an appropriation proposal has been put forward. There may be some reason for such a reaction, for instance, if a proposal is made about an appropriation for a building and the sum is far from being sufficient to finish the building. If a small appropriation has been decided and the construction of the building has started, one will feel oneself forced to complete it. To prevent people from being able to force issues upon others in this manner there must exist the possibility of deciding that such proposals must not be brought to a voting on budget combinations. The same thing can be reached by deciding, as a condition to an appropriation, that a payment must not be made, until the appropriation has reached a particular size. Such a condition can be secured with permanent votes.

In connection with the idea that decisions can be made, which prohibit a voting on particular proposals, we must mention that a general examination is needed of what conditions must be fulfilled, so that a proposal can be voted on. It is a known fact that in the general decisions of organizations there are contained different strict conditions that must be fulfilled, before a proposal can be voted on.

B. Distribution of other goods

Here I shall only point out the possibilities of a mathematical aggregation of preferences, when it is the question of distributing the right to utilize natural resources, in regional planning (which can often be regarded as different from the utilization of natural resources), in the distribution of bank loans and when the production shall be distributed within the organizations of firms. Regional planning is most often a public business, while the other tasks can be either public or not in this country, although the public authorities are dominant, most of the banks being public, for instance. Whichever the case may be, it does not seem to have great influence on the logical sides of the aggregation, and actually there do not seem to appear any considerably different sides than those already discussed, whether the resources are distributed by means of payment, prohibitions and instructions or by the granting of licences. Sometimes the granting of licences is restricted by a certain frame and then becomes similar to the division of fixed sizes or sizes, which are determined by others (e.g., bank loans, regions for planning, and sometimes fishing licences), while the alternative combinations within the size can become numerous. Here it will often be natural to distribute goods for short or long periods of time, perhaps with the opportunity of revising the issue at any time, and then, it may become advantageous to have, by a decision, secured for oneself special votes in order to influence the revision.

C. Tactical voting or misuse of votes

It has become clear that it is the purpose of the method that the participants shall be able to distribute votes among issues according to the importance they attach to them. It is necessary to realize, although it will not be done thoroughly here, whether the participants may be able to distribute their votes among the issues, not only in view of the relative importance they attach to them, but also for the purpose of depriving others of votes or saving their own votes, irrespective of their evaluation of the issue in question. A participant can try to save votes, if he believes that the alternative he prefers will win, although he places fewer votes on it than it deserves according to his judgement. Similarly, it can also be attempted to deprive others of votes, if one believes that the alternative one prefers will not win, even though one gives it more votes than one thinks it deserves, whereby the price of the victorious alternative has been increased. We can call this tactical voting or misuse of votes, if we want.

It is a bit complicated to estimate, how extensive this may become. It must be assumed that it will be possible to misuse votes to a certain extent, and the advantages of the method must be evaluated in view of that. By experiment one can, of course, get a clearer picture of to what extent votes can be successfully misused. One can imagine those experiments as simulation conducted by researchers or the method is taken up in a consultative or determinative form. This may, however, be clarified to some extent by making up situations and examining the possibilities of tactical voting.

Apparently tactical voting cannot succeed completely, unless one knows how the other participants will distribute their votes among the alternatives. Therefore the voting must above all be secret, until it is over. The condition for a successful tactical voting is based on general knowledge of the position of votes, which seems to decrease with the increasing number of participants and the alternatives that may win. In the ordinary arrangement of voting there are generally just a few alternatives, often just two at a time. Generally there has been a greater number of real alternatives that the participants might have wished to consider and possibly taken sides on. Mathematical aggregation of preferences, as already shown, offers an opportunity to take sides concerning many alternatives in one voting. In that case it will become more difficult than ever to predict the distribution of the votes.

The number of participants can be changed in two ways, by increasing or decreasing the number of what can be called parallel participants, and by making it possible that representatives be controlled. They can be controlled by making the exercise of votes public after the counting of votes, which will presumably make the representatives exert their influence according to the nature of the issues without any misuse of votes so that they have to account for the exercise of their votes to those they represent. Similarly, the latter can be offered an opportunity to control the representatives´ exercise of votes by their own participation in the voting, when they wish to and think it safer. The method permits it logically and technically, but it cannot be discussed here further.

There are more aspects to the method as regards the possibilities of tactical voting. Issues with permanent votes seem to be in less danger. If one issue is big in comparison with other issues in question, the conditions for misuse are more favourable than in the case of relatively small issues. For the participants, the relative size of issues becomes to some extent a question of the period during which their votes are valid, but how quickly votes lose their value is also a question of whether one can take participants by surprise by making proposals and bringing them to the vote. That, on the other hand, reminds us of the problem of determining the access to get issues voted on. As known, there are various regulations concerning that, and in this connection it is natural enough to think of the possibility of holding a special voting (by means of mathematical aggregation of preferences, possibly by means of an independent vote budget) on whether or not issues can come to the vote. 4

IV. Mathematical aggregation of preferences and the monetary system

The comparison made must be short this time. By the monetary system an over-arching principle of evaluation and a form for an order for values, which replace the barter system, were introduced for the exercise of economic power. In modern times the prices of goods have, as a rule, been announced by those who offer them for sale. The buyer must keep himself within a limited budget, so that he feels that one deal immediately reduces his chances to buy other goods.

Compared with the ordinary form of voting, where all participants have the same formal right to influence each decision independently of the result they have already gained by the exercise of their votes, the mathematical aggregation produces an over-arching principle of evaluations for the formal possibilities to be able to influence a group of decisions, bringing in the budget constraint, so that supporting a decision most often reduces one´s possibilities of having influence on other decisions. Coalitions as a means of compromising are a form of exchange without a common denominator for a formal influence. They are replaced in the mathematical aggregation by a formal common standard of measure and a regulator of influence on decisions.

In the mathematical aggregation those who want a decision state the maximal price, which they are willing to offer for it, whereas the final pricing depends on the demand for other alternatives on the issue.

Money usually keeps its formal value without any limitations on time, which is, however, often confined to organizations (preferably states). Its power, however, changes in accordance with the trend of prices. Votes can be allowed to keep their value for as long as desired and within the organizations they choose, and their power will become dependent on the total quantity of votes available and the demand for decisions.

The monetary system as a method to exert economic power was a new abstraction that replaced the barter system. Although this is a new way of thinking and a new abstraction, which has to be learned by those who are not accustomed to it, it is the generally accepted view that it has made it easier for most people to use their economic power to influence the distribution of goods. Although the monetary system has thus benefited people great manpower is, however, bound up with the use of money. The mathematical aggregation of preferences is a new abstraction to regulate the distribution of influence on decisions. It is no small task to analyse, how many people are occupied in expressing views and aggregating preferences for decisions, different as the methods are. This must be analysed for a better understanding of what the mathematical aggregation of preferences has to offer.

Quality & Quantity 8 (1974): 121-138