University of Lund

1. Contents

Contents - General Objectives

The objectives of this paper are to establish the strategic potentials and other
properties of two common voting procedures and to discuss the potential effects of
these properties on parliamentary decision-making.

Research Methods

A basic problem for a democratic society is the choice of decision rules which will satisfy essential democratic requirements. The nature of such requirements has been discussed in the literature along with the degree to which common decision mechanisms meet these criteria. Here we create a criterion adapted to our simulation approach (cf. below page 40).

Simulation has been used especially in modeling dynamic political processes and has been used along with mathematical evaluation in dealing with the paradox of voting.¹ Evidently simulation is essential in situations where the probability models and/or the probability calculations are so complex that mathematical evaluation is impossible or extraordinarily tedious. In my opinion this is the situation when we try to establish strategic potentials of most voting procedures.

Model Design

The formal model on which our simulations are based is for obvious reasons rather crude. Viewing the real world, however, we observe that any collective choice situation consists of a set of decision-participants, a set of alternatives and a set of preference statements.

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* This is a condensed version of my article with the same title published in Statsvetenskaplig Tidskrift l (1973). Here I am only reporting the results of my analysis of the elimination and the quality procedures. In Statsvetenskaplig Tidskrift I deal with the acclamation, plurality, successive, vote-count, and point-count procedures as well.

* This is a condensed version of my article with the same title published in Statsvetenskaplig Tidskrift l (1973). Here I am only reporting the results of my analysis of the elimination and the quality procedures. In Statsvetenskaplig Tidskrift I deal with the acclamation, plurality, successive, vote-count, and point-count procedures as well.

* This is a condensed version of my article with the same title published in Statsvetenskaplig Tidskrift l (1973). Here I am only reporting the results of my analysis of the elimination and the quality procedures. In Statsvetenskaplig Tidskrift I deal with the acclamation, plurality, successive, vote-count, and point-count procedures as well.

The set of decision-participants or 'voters' (the 'who cluster) can be structured into seven different systems of voting blocs if the number of blocs is assumed to be less than six. The set of alternatives can be structured according to the number of dimensions that exist in a set. Finally we can restrict the set of possible preference statements in a number of different ways.

A situation is defined by a specific system of voting blocs, a certain number of dimensions and certain restrictions on the set of possible preference statements. In each situation we randomly create 10,000 preference patterns.³ In each pattern certain properties of certain voting procedures are tested.

We must stress that this simulation approach is the only realistic way to establish such complicating features as the strategic potential of voting blocs under specific voting procedures. This can be done in the simulation by making all possible changes in a specific type of preference statement from a voting bloc.

The concepts used in this section are quite familiar to any political scientist working in the fields of probability modeling, theories of voting, and collective choice. They might not, however, be familiar to readers who have had only limited experience in these research areas. Hence, in the following section we discuss our interpretation of concepts like probability modeling, preference statements, voting blocs, strategic potentials, simulated changes, etc.

2. Concepts Used

Probability Modeling

The simulation analysis presented below is one of many possible applications of formal probability models to collective decision-making processes. The formal model within which our simulations take place was first published as one of sixteen papers⁴ showing in detail the many and varied uses of probability theory in the development of formal models of collective decision-making. They also reveal the considerable potentiality and utility of probabilistic models and - like all models - illustrate the hazards and liabilities that attend this approach. The pros and cons of probability modeling are extensively discussed by Niemi and Weisberg in their introduction and conclusion.

What might be stressed as the main advantage of this investigation is the ability to study a vast number of different preference situations which otherwise would not be amenable to treatment. My investigation of Scandinavian roll-calls shows the impossibility of investigating the strategic aspects of the elimination procedure on the basis of empirical data.⁵ The main disadvantage is the inability to justify particular probability assumptions.

Preference Statements and Voting Blocs

A preference statement is regarded here primarily as a representation of a decision
participant's evaluation of a set of alternatives. Obviously a set of preference statements

cannot be transformed into a collective choice unless they are comparable. The effect of this demand is to give relevant preference statements a very crude character. In parliamentary decision-making, collective choices made through rollcalls are usually based on pair-wise ordinal preferences. That is, the outcome is based on a crude 'better-worse' evaluation of the alternatives which in many cases would be deemed not to represent the preferences of individual participants in an adequate way.

The nature of our study, however, is such that the problem of whether the preferences of decision-participants are adequately represented by, say, a rank ordering is not a relevant problem here. We simply note the fact that collective choices are made by representing the preferences in a specific way. Our problem is to analyze the properties of different voting procedures when individual or bloc preferences are represented by a specific type of preference statement. We will, however, assume that every participant could, on demand, produce a complete rank ordering of the alternatives.

The decision-participants of this study are voting blocs, where a bloc consists
of voters who have agreed to evaluate the alternatives under consideration in the
same way. A voting bloc could, of course, be a single individual.

Decision Principles

The procedures to be analyzed in this study are based on the majority principle
as defined below. We also define an equality principle, which will be a useful
tool for comparing the various voting procedures.

Our definition of the majority principle is based on two concepts: the majority alternative and the cyclical set. We define a majority alternative as one that obtains a majority of the votes in pair-wise comparison with all other alternatives on the agenda. That is, if and only if a particular alternative obtains a majority when paired against all other alternatives, each in turn, then it is a majority alternative. An example illustrates this definition.

Suppose we have an electorate of three voters (I, 11, III) which must decide
upon a collective policy from among three alternatives (als a2, a3). The voters'
preference orderings are:

That is, reading down the table, voter I prefers a^ most, prefers a₂ next, and a₃ last. Consider a^ For the dyad (a_ls a₂), the preference orderings indicate that voters I and 111 choose aj. For the dyad (a_ls a₃), voters I and II choose a^ Thus, 3i obtains a majority when paired against each of the remaining alternatives and, according to our definition, is the majority alternative. Note that aj is not most

preferred by a majority of the voters. Only voter I most prefers a^ Of course, if a majority of voters, like I, ranked a_± highest in their preference orderings, then it would satisfy our definition of majority alternative. We simply observe that this is not necessary, as the example indicates.

Some sets of preference orderings do not, however, possess a majority alternative.
Consider the following arrangement:

Alternative ai is preferred by a majority to a₂ (I, III), but not to a₃. Voters II and 111 prefer a₃ to BJ. Thus, aj is not a majority alternative. Neither is a₂ since, as we have already seen, a_r is preferred by a majority to it. That leaves a₃, which is preferred to ai (11, III) but voters I and II prefer a₂ to a₃. We have then an unusual result: is preferred to a₂, a₂ to a₃, and a₃ to &i. The alternatives form a cycle with regard to majorities. A set of rank orderings where no majority alternative exists will be denoted a cyclical set.

By our definition the majority principle states that if a majority alternative exists, then this alternative should be chosen as the outcome. When the set of rank orderings is circular, however, majorities are frustrated in their preferences for a collective choice and no alternative is the obvious selection by the majority principle.

Finally the equality principle states that the probability that a bloc's most preferred alternative wins should be proportional to the weight of the bloc. This equality principle will be a useful tool in an attempt to test the 'fairness' of some of the procedures which are introduced in the next section.

Voting Procedures

The voting procedures to be analyzed in this study are the elimination and the quality procedure as defined below. In almost any case a tie for first place can occur, and some tie-breaking mechanism is needed (such as a chairman who only votes in case of a tie or a rule that the status quo wins in case of a tie). Since some such mechanism always exists, we do not deal further with ties.

The elimination procedure. The initial poll is taken between the first two alternatives in a predetermined voting order. A second poll is taken between the winner of the first poll and the next alternative in the voting order. This process continues through the entire list of alternatives. In the end one of the alternatives emerges as the winner. In some 'amendment' variants of the elimination procedure the set of alternatives is not defined when the first vote is taken - that is, amendments can be put forward during the voting process. This does not, however, change the basic structure of the procedure. Nevertheless the distinction between a welldefined

set of alternatives coupled with a predetermined voting order and this
amendment variant should be kept in mind.

The quality procedure. The voters register a complete rank ordering of the alternatives under consideration, and the outcome is selected on the basis of these rank orderings. If a majority alternative exists, this alternative is the winner. If a cycle exists a voting order is determined at random and the elimination procedure is used to determine an outcome. The main distinction between the elimination procedure and the quality procedure is that when the latter procedure is used the voters do not know the voting order when they record their preferences.

Strategic Potentials, Moves, and Countermoves

We noted above that cyclical sets have a potentially devastating effect on majority rule. Empirical investigations show, however, that cyclical sets seem to occur less frequently than is expected from theoretical calculations.⁶ This is probably due, first, to the fact that cyclical majorities can only occur when alternatives are viewed multidimensionally and, second, that these sets themselves tend to be open to compromise and logrolling. Thus when potential cycles exist, they are 'solved' by the decision processes that precede the actual voting.

What must be explored, however, is the possibility that the cycles can be used to promote the interests of individual voters or blocs. This possibility depends, in the case of the elimination procedure, on the fact that if the set is cyclical then the outcome will to a large extent be determined by the voting order. In fact, in the case of three alternatives the last alternative will always win.⁷ With this in mind consider the example on p. 39 where aj is the majority alternative. A simple analysis shows that voter II is the only one who can change the set of rank orderings into a cyclical one. This is done by changing his rank ordering to a₂, a₃, a_lt This strategic potential can be utilized only when there is a favorable voting order, here a_ls a₃, a₂ (or a₃, a_1? a₂). With this voting order, in the first poll, between aj and a₃, voter II should vote for a₃ contrary to his 'true' rank ordering. In the last poll, between a₂ and a₃, voter lis forced to vote for a₂ since otherwise the outcome will be a₃, I's least preferred alternative. Note that voter I has no alternative but to vote for aj in the first poll and for a₂ in the second. Voter 111, however, can make the only possible strategic countermove by voting in the first poll for a! against his own primary alternative a_s.

A specific voting order might thus give a voter or a bloc of voters a strategic possibility. The most favorable voting orders are those in which the voter's primary preference is the last alternative in the voting order (cf. below p. 46).⁸ To try to take advantage of the paradox by voting contrary to one's 'true' preferences is here denoted a strategic move. All strategic moves are primarily directed against the majority alternative. Hence that majority who prefer the majority alternative to the 'pushed' alternative can always make a completely effective strategic countermove by changing the majority alternative into an alternative most preferred by

a majority of the voters. This will, however, imply that one (some) bloc(s) must
vote against its (their) 'true' first, and sometimes second, third, etc., preferences.

An empirical investigation of roll-calls in the Swedish and Finnish Parliaments, where the elimination procedure is used,⁹ shows that the strategic move is a tool used from time to time with the desired effect. For obvious reasons, however, the frequency of strategic counter-moves is impossible to establish.

In this paper the strategic potentials of a specific procedure will be established by analyzing if and to what extent changes in the preference statement can change the winning probability of different blocs in different situations. The simulation analysis of strategic potentials of the elimination procedure and the quality procedure must, however, be based upon a thorough knowledge of the cyclical sets. Accordingly we must classify the cyclical sets on the basis of the internal majority relations. We must, moreover, establish in each type of cyclical set three different probabilities or sets of probabilities, as follows:

1. The probability, if the quality procedure is used, that each alternative will be
the winner.

2. The probability, if the elimination procedure is used, that the last alternative

in the voting order will be the winner, the probability for the next to last, etc.
3. The probability, if the elimination procedure is used, that a specific alternative

will be the winner when it is the last alternative in the voting order.

As an example, consider our cyclical situation above on p. 40.

1. Each alternative has a probability of one in three of being the winner if the

voting order is determined according to the quality procedure, i.e. at random.
2. The last alternative in the voting order will always be the winner.
3. Each alternative will win when it is the last alternative in the voting order.

The first set of probabilities is computed since it represents the random variant of the elimination procedure, here denoted the quality procedure. The second set of probabilities is computed since it enables us to show the general effect of the voting order in cyclical situations. Finally the third set is computed since it represents the important variant of the elimination procedure in which one bloc of voters has the power to make its own alternative the last one in the voting order.

In our simulations we will use the first and the last set of probabilities to establish upper limits on the whining probability of a bloc's primary preference in three types of situations. The situations are a) no strategic moves are made, b) strategic moves from one bloc are made whenever possible but there are no countermoves, and c) when a successful strategic move is made, certain countermoves are carried out.

These simulations will thus enable us to establish the extent to which the elimination
procedure and the quality procedure produce outcomes in accordance
with the equality principle with or without strategic moves and countermoves.10

These simulations will also give us an idea of the strategic potentials of each of these procedures. In addition, the frequency with which favorable changes can be made by a voting bloc in sets where a majority alternative exists will give us an idea of the extent to which the two procedures are in accord with the majority principle.

The classification of cyclical sets and the establishment of the three sets of probabilities
are carried out in the next section.

3. A Classification of Cyclical Sets

A classification of cyclical sets on the basis of the internal majority relations is a necessary first step in any attempt to analyze their effects. The classification is limited to sets of alternatives with up to five alternatives and is carried out in two steps. The first step is based on the distribution of majority relations, while the second step utilizes indirect majority relations.

Consider the first step. The distribution of majority relations in any set of preference orderings can be evaluated as follows. Mark each alternative with the total number of its 'positive' majority relations. We will then have a cyclical set if and only if each alternative is marked n-2, where n equals the number of alternatives.

Proof: Since no alternative can have a majority relation with itself, no alternative can ever be marked > n - 1. If one alternative is marked n - 1 it is a majority alternative since the figure n - 1 implies that the alternative obtains a majority when paired against every other alternative. If all alternatives are marked

< n - 2, this implies a cyclical set since each alternative has at least one majority
against it.

One possible distribution in a set consisting of five alternatives is: a! = 3, a₂ =2,a3 =2,a4 =2, and a₅ = 1. A closer study of different representatives of this distribution shows that the position of alternatives with the same number of direct majority relations may be entirely different due to the indirect relations.

Each of these three cases is a representative of the given distribution. The positions of the alternatives are, however, different in each case. Consider, for example, a₅. In case I this alternative has a majority over the strongest alternative, aj. In case 11, it has a majority over a_s, which in turn has majorities over the two other 'two-majority' alternatives, a₂ and a₄. In case 111, finally, a₅ has a majority over a₂, which has majorities over one 'three-majority' alternative (aj) and one 'two-majority' alternative (a₄).

The second step in the classification is based on these differences in the indirect majority relations. The importance of these relations is best illustrated by analyzing how the voting result would differ in the three cases under the elimination procedure. An analysis of the 5! = 120 possible voting orders in each of the three cases results in the following three distributions of winning alternatives:

In other words, if a set of preference orderings has a structure as set forth in case I, and if the elimination procedure is used, then aj will be the winner in 54 of the 120 possible voting orders, a₂ will be the winner in 14, etc. Such an analysis of all possible voting orders will produce the three sets of probabilities defined in section 2.5.

Hence we first make a systematic classification of cyclical sets (for three, four, and five alternatives) and then analyze each voting order in each case in order to establish the winner and its position in the voting order. The results of the second step are shown in Table I below.

When a set of three alternatives is cyclical there is only one case, viz. the symmetrical
'circle.'

That is, each alternative is marked 1, and any circle can be transformed into
the one above by a re-labeling of the alternatives.

When a set of four alternatives is cyclical we have two distributions, since no
alternative may be marked > n - 2 = 2. Six majority relations can only be distributed
among four alternatives in the following two ways:11

The first distribution is analogous to the case of three alternatives. That is, we
have a symmetrical 'circle' among three alternatives and a fourth alternative with
no 'positive' majority.

The second distribution is slightly more complicated. The uniqueness of this
case can be shown12 by the following study of the majority relations.

Of the two alternatives, a_A and a₄, which have only one majority, one alternative necessarily has a majority over the other, e.g. a₄ over aj.. Then aj must have a majority over a₂ or a₃, e.g. a₂. This will establish a unique structure: a₂ has a majority over a₃ and a₄, and a₃ has a majority over a₄ and a_l. Hence only one case exists.

Due to lack of space we do not, here, report our analysis of sets of five alternatives
13

In Table Iwe present the results of an analysis of each14 voting order in each
case to establish the winner and its position in the voting order.

On the basis of this information the probabilities defined on p. 42 are worked out. The first probability, i.e. the random probability, equals the number of times the alternative would win divided by the total number of voting orders. The third probability requires some simple summations and divisions. Expressed in words, the probability of winning when an alternative is last in the voting order equals the number of voting orders when it is the last alternative, and it wins, divided by the total number of voting orders when it is last. Notice that the structure of the elimination procedure implies that an alternative that is first or second

in the voting order can never be the winner in a cycle. This occurs since those alternatives
by definition have at least one majority against them. Hence these two
probabilities always equal zero and can be excluded from the table.

5. A Presentation of the Elimination and the Quality Procedures

The Elimination Procedure

The native country of the elimination procedure is Great Britain. Thus in addition
to Sweden and Finland, the procedure is used in the English-speaking countries
and in previous English, possessions (def. see p. 40).

The elimination procedure yields varied ordinal information about the preferences of the voters. It is usually possible to reconstruct some complete rank orderings. The important trait is, however, that it is never possible to establish the existence of a cyclical set since less than all pairwise votes are taken.¹⁵ An empirical analysis¹⁶ of roll-calls from the Swedish and Finnish Parliaments I made has, however, made the existence of 'real' and 'strategic' cyclical sets very likely.

Despite their empirical likelihood, empirical data do not furnish sufficient information to establish fully the properties of this procedure. Hence we are forced to use hypothetical situations. On the basis of these situations we can conclude that the existence of cyclical sets is what causes problems under the elimination procedure. A majority alternative will always be chosen when the elimination procedure is used.¹⁷ A cyclical set, however, is troublesome both when it occurs naturally and when it is used as a tool to promote the interests of individual voters.

This fact led us to classify cyclical sets on the basis of the internal majority relations and to establish three different probability distributions in each type of cyclical set (section 3). The equality aspect and the strategic potentials of the elimination procedure will be thoroughly analyzed in our simulation section.

We must stress the experimental character of our analysis. In actual parliamentary decision-making, negative properties of decision procedures will be to some extent neutralized by countermoves by parliamentary members. That is, 'the rules of the game' are one determining factor when the members decide upon their actions. The reluctance on the part of parliamentarians in Denmark and Norway to allow more than two alternatives in the final voting process is one highly plausible effect of the properties of the successive¹⁸ procedure.

When the elimination procedure is used, one factor that tends to reduce the number of alternatives is the use of strategic countermoves as defined above on p. 41. In situations where a 'real' cycle would tend to materialize despite the inhibiting factors mentioned on p. 41, we find another type of move that will tend to diminish the number of alternatives that reaches the voting phase. Consider once more our cyclical situation on p. 40. Assuming sufficient information, all three voters will know that if a_x is given last place in the voting order, this alternative will be the winner unless some move is made. In such a situation voter II can and should vote for a₃ in the first roll-call - that is, against his own primary preference. Otherwise a_ls which is considered worst by voter 11, will be the winner. There is, however, a definite reluctance on the part of parliamentary members to vote against their primary preference. This fact will ensure that this type of move will usually be taken care of in the decision process preceding the voting phase. The implicit disadvantage, contrary to what was generally found above, for those alternatives in a cycle having positions late in the voting order should, however, be kept in mind.

The Quality Procedure

This procedure is probably first considered by parliamentarians accustomed to the
elimination method. The quality procedure has been discussed above on p. 41
and will be thoroughly analyzed in our simulation section.

However, it should be stressed here that this procedure sharply diminishes strategic
considerations but does not eliminate them. A simple example will help clarify
this point.

Ex. Suppose there are three voters, I, 11, and 111, and three alternatives, al} a2

and aa

This is the same situation as on p. 39. We concluded on p. 41 that voter II could change the set of rank orderings into a cyclical one (by changing his rank ordering to a₂, a_n, aj). We also concluded that this strategic potential could best be utilized with a favorable voting order.

If the voting order is not known in advance, II can still make the strategic move that changes the set into a cyclical one. Then, however, he will run a considerable risk, since after such a move a_l} a₂, and a₃ will each have the same chance of winning (assuming no counter-moves by III).

Seemingly II would run the risk of a3 winning only if he favored a2 very much
and/or if he did not dislike a3 that much. Under either of these conditions, however,
a strategic move might be made.

Two conclusions can be drawn from this example:

a. The quality procedure is not completely satisfactory if we want a procedure in
which there are no strategic possibilities.

b. The intensity of preferences plays a significant role in collective decisionmaking, and the problem of constructing acceptable procedures based on more differentiated preference statements would seem to be highly relevant. An introductory discussion and analysis of this very complicated problem has been carried out.¹⁹

6. The Theoretical Model

A Classification of Systems of Voting Blocs

In this section we present an exhaustive and mutually exclusive classification of systems of voting blocs, where each bloc consists of voters who have agreed to rank the alternatives in the same way. The classification will be carried out for assemblies with up to five voting blocs. Throughout the analysis blocs will be considered cohesive. Since our interest lies primarily in situations governed by majority rule, the classification is based on the ability of the blocs to form majority coalitions. A motivating assumption is that, ceteris paribus, it is easier to form a two-bloc coalition than a three-bloc coalition, a three-bloc coalition than a fourbloc one, and so on.

A simple example helps clarify our approach. Listed below are four different
systems, each consisting of four or five blocs.

The weight attached to each bloc is simply the proportion of assembly members
who belong to that bloc. In each system, of course, the weights add up to one.

Note that these systems differ in the ability of various blocs to form majority coalitions. In the first system, blocs A, B, and C can form two-bloc majority coalitions freely among themselves. Bloc D, on the other hand, is unable to transform any coalition into a winning one. That is, no coalition rises from a minority to a majority position when D enters. The second system is exactly the same as the first with regard to the ability of the blocs to form majority coalitions. In the third system bloc A has a uniquely advantageous position in that it can form two-bloc majority coalitions with any of the other blocs in the system. The other blocs can form two-bloc majority coalitions only with A. The fourth system is the same as the third except that the fifth bloc (E) is unable to create any majority coalition.²⁰

The following classification is thus based on the ability of the blocs to form majorities alone or in coalition with one or more of the other blocs in the system. The proof of the exhaustive and mutually exclusive nature of the classification is found in Bjurulf.²¹

1. Two blocs in the voting process

Except when both blocs have equal weights, one bloc will hold a clear majority with a weight > 0.5. Cases iri which both blocs (or coalition of blocs) have a weight of 0.5 will be regarded as a special case here and below. Since tie-breaking mechanisms are usually available, these cases will not be treated further.

2. Three blocs in the voting process

a. One bloc has a clear majority. (Ex. A = 0.6, B = 0.3, and C = 0.1.)
b. All two-bloc coalitions have a clear majority. (We will call this the three

bloc case.) (Ex. A = 0.4, B = 0.4, and C = 0.2.)

3. Four blocs in the voting process

a. One bloc has a clear majority. (Ex. A = 0.6, B = 0.2, C = 0.1, and
D = 0.1.)

b. The three-bloc case: All two-bloc coalitions among the three largest blocs have a clear majority. The fourth bloc cannot contribute to a majority coalition (i.e. by changing a minority coalition to a majority one). (Ex. A = 0.4, B = 0.28, C = 0.28, and D = 0.04.)

c. The largest bloc is the only one that can create a two-bloc majority. (Ex.

A = 0.4, B = 0.2, C = 0.2, and D = 0.2.)

4. Five blocs in the voting process

a. One bloc has a clear majority. (Ex. A = 0.60, B = 0.15, C = 0.10
D = 0.10, and E = 0.05.)

b. The three-bloc case: All two-bloc coalitions among the three largest blocs have a clear majority. Blocs D and E cannot contribute to a majority coalition. That is, no three-bloc majority can be created in which two of the three blocs do not belong to the three largest, and thus already have a clear majority position. (Ex. A = 0.30, B = 0.30, C = 0.30, D = 0.05, and E = 0.05.)

c. All two-bloc coalitions that include A have a clear majority. No three-bloc coalition without A has a clear majority position because a two-bloc coalition containing A can always form in opposition to this. (Ex. A = 0.45, B = 0.15, C = 0.15, D = 0.15, and E = 0.10.)

d. The smallest bloc cannot contribute to a majority coalition. Otherwise, all
two-bloc coalitions that include A have a clear majority. (Ex. A = 0.40,
B = 0.20, C = 0.20, D = 0.15, and E = 0.05.)

e. Ais necessary for the formation of a two-bloc majority. This can, however, only be done with B or C. Either A or (B + C) is thus necessary and sufficient for the existence of a three-bloc majority. (Ex. A = 0.40, B = 0.24, C = 0.24, D = 0.07, and E = 0.05.)

f. A and B are necessary for the formation of a two-bloc majority. All threebloc
coalitions containing A or B form a majority. (Ex. A = 0.40, B = 0.40,
C = 0.08, D = 0.07, and E = 0.05.)

g. No two-bloc majority can be created. All three-bloc coalitions form a majority. (Ex. A = 0.21, B = 0.20, C = 0.20, D = 0.20, and E = 0.19.) A special case in this category is when all blocs have equal weights. This is equivalent to the case of five individuals as considered by previous investigators.

The Dimensionality of the Preference Orderings

A group of preference orderings can always be characterized by a set of dimensions on which individuals and alternatives are placed. Occasionally all of the preference orderings will 'fit' a single dimension. This case is important, of course, because of Black's (1958, Chap. IV) well-known finding²² that the voting paradox cannot occur when preference orderings are single-peaked (which is equivalent to unidimensionality).²³ Most often, however, two or more dimensions are needed to fully characterize the preference orderings. While it is notoriously difficult to work with more than one dimension, some aspects of the multidimensional situation have recently succumbed to the analysis of Davis, Hinich, and Ordeshook ²⁴ In probabilistic analyses, however, little attention has so far been paid to the dimensionality of the preference orderings. One exception to this is Niemi,²⁵ who worked with the proportion of individuals having single-peaked preference

functions. Here we utilize a different approach, which we hope will be useful both
in this and other applications.

In our variant, bloc preferences are locked into certain positions on the dimensions by assuming that each bloc has one specific primary preference. For example, suppose the single dimensions are a_ls a₂, a₃, a₄, a₅ and a_ls a₃, a₅, a₂, a₄. Each bloc is assigned one alternative as the primary preference of that bloc, e.g. A= a_1? B= a₂, etc. Then when each bloc is assigned a preference ordering, the set from which the selection is made must satisfy two requirements: 1) singlepeakedness, and 2) one specific alternative as the primary preference, Thus, in the example, the preference ordering of bloc A must be a_ls a₂, a₃, a₄, a₅ or a_ls a₃, a₅ > a_2> a_4- The preference ordering of bloc B will be a₂, a_ls a₃, a₄, a₅ or a_2« a_s» a_i> a_4> a₅ or a_2> a_s> a_4, a_i> a_s oi a₂, a₃, a₄, a₅, a_x or a₂, a₄, a₅, a₃, aj or

a2> as> a4> a3> al or a2> as> a35 a4> al or a2? asj a3s al> a44-

In this way we can analyze the effects of having the blocs situated more or less uniformly along a left to right scale. The 'dimensional' analysis will start with all blocs positioned along one scale. This situation is analyzed by studying the degenerate case in which the two dimensions are identical. Having analyzed this case we will turn to some of the other cases, viewing them as greater or lesser deviations from the left to right scale.

Assumptions about Conflict and Tied Ranks

In addition to the systems of voting blocs and the dimensional character of the preference orderings, two other factors will be varied in the simulation. First, we will make several assumptions about the possibility that two or more blocs have the same first choice. Theories of decision-making are most concerned with situations of conflict. Hence cases in which two large blocs (especially the two largest) have the same first preference are of somewhat less interest. In parts of the simulation, therefore, we will require that some or all of the voting blocs have different first choices.

Secondly, for a part of the simulation we wish to introduce the possibility Of ties in the preference orderings. We will do this in a limited way here by assuming either that preferences below the first choice may be tied or that preferences other than the first and last choices may be tied. These assumptions strike us as a fairly realistic initial way of bringing in tied ranks, since we presume that best and worst alternatives are among the easiest to distinguish. The probability of different types of ties under these assumptions is given in footnote.²⁶

7. A Computer Simulation of Strategic and Equality Aspects of the Elimination Procedures

The probability distributions we are seeking can be closely approximated by a
simulation in which sets of individual (or, in this case, bloc) preference orderings
are chosen at random and then analyzed. An excellent description of the detailed

procedure is given by Klahr,²⁷ although in his case the analytical objective was cyclical social orderings. As in other probabilistic analyses, we assume throughout that all individual (bloc) preference orderings are transitive and that in voting on a pair of alternatives each bloc votes for the alternative that is higher in its preference ordering. We also assume that all preference orderings are equally likely - subject to the constraints discussed above.

A Computer Analysis of the Elimination Procedure

In our analysis of the strategic and equality aspects of two variants of the elimination procedure we have in each case computed four distributions of the winning probability of each bloc. The first distribution is the winning probability of the different blocs in 10,000²⁸ randomly generated sets of preference orderings when the voting order is decided at random after the registration of the preference statements and assuming no strategic moves. In the first situation to be analyzed, where all blocs have different first choices (Table II), this distribution should add up to 1.00 (except for rounding error). The second distribution is the same as the first except that the entry for each bloc represents the probability of winning when that bloc in non-winning positions has tried all²⁹ possible strategic moves and no other strategic moves or countermoves have been made. Obviously this distribution usually adds to more than 1.00. The same is true of the third distribution where the winning probability for each bloc is calculated as if that bloc had the power to make its own first alternative the last one in the voting order. The last distribution is the same as the third except that each probability presumes that the bloc under consideration has tried all possible strategic moves (when it is in a non-winning position) and no other strategic moves or countermoves have been made. We should emphasize that we have used the probabilities given in Table I when computing the winning probabilities shown in Table 11. That is, we have calculated the strategic potentials given a bloc with emphasis on the success of the first choice. Whether a bloc facing a Vs probability of succeeding by the strategic move, a Vs probability of no change, and a Vs probability of a worse alternative will in fact make the strategic move is a problem dependent on intensity of preferences and on psychological factors. Here we simply add ¥3 to the bloc's winning probability in an attempt to calculate the magnitude of the potential strategic moves.

Our first results are derived from the situation in which no dimensional assumptions are made, where the five voting blocs have different first choices, and where no tied ranks are permitted. The results for each system of voting blocs are given in Table 111.

The first³⁰ case in Table II is the three-bloc case (cf. p. 50). Recall that since no ties are permitted, any distribution within the case will yield the same result.³¹ Accordingly, in the first case the three largest blocs have the same winning probability (within the sampling error of ± 0.01).³²

Comparing the four distributions, note that the two 'dummy blocs' D and E have no strategic potentials. Their winning probabilities relate solely to the positions of their primary preferences in the preference statements of blocs A-C. Turning to the result for the three main blocs we note that when the voting order is decided at random, even with strategic moves, these blocs cannot reach the winning probability obtained when they control the voting order.³³ The strategic gain is about 10 percent. The gain when controlling the voting order, however, is about 20 percent. Furthermore, when a bloc is in control of the voting order and employs strategic voting, the gain is especially large.

For cases c-g we find a moderate increasing strategic gain when the voting
order is decided at random. The gain is, however, always below the 10 percent
gain found in the three-bloc case b.

For cases c and d (closely resembling the Scandinavian situation) we should
note that when the largest bloc votes strategically and has control of the voting
order it will almost always win.

Case f resembles the case of three balancing blocs, case b. Here there are two large balancing blocs. These two blocs and the three blocs in the b case have almost identical winning probabilities in all four distributions. The results of the remaining three blocs in the f case, however, do not resemble the results for the dummy blocs in the b case. In case f the smallest blocs have considerable strategic potentials.

For cases b, e, f, and g we should note that the first strategic distribution is
fairly close to the actual distribution of weights. Such is not true for cases c and d
(even though the last two distributions differ even more).

This situation with five alternatives in the final voting process is important in an attempt to uncover some of the problems facing the blocs if each one wants to bring its own solution to the voting process. The fact that in the Swedish and Finnish Parliaments five alternatives frequently remain in the phase just preceding the voting emphasizes this.

Obviously, however, the situation with three alternatives is also of vital theoretical
and empirical interest. In Table 111 sets of three alternatives are first analyzed
in a type of situation closely resembling the preceding one.

In Table 111 we should remember (e.g. in case b₂) that only the three largest blocs have conflicting first choices and that each of the dummy blocs has a first choice identical to the first choice of one of the big blocs. Accordingly a winning probability of 0.33 for D and E is to be expected. Obviously the first three probabilities in the first distribution should add up to 1.00.

In general we find that the first probability distribution is somewhat higher and
that the last is lower but that all the tendencies found in the five-alternative situation
are also found in the three-alternative situation.

The next change in the parameters of our voting system is to permit indifference between alternatives - i.e. ties are allowed in the bloc's preference statements. This adds two complicating factors. First, we cannot eliminate tied ranks in the social ordering simply by seeing that no pairs of blocs have weights that add up to exactly 0.50. This is most easily seen in the case of equal weights, in which each bloc in the five-bloc system has a weight of 0.20. With no indifference permitted, a unique majority alternative always exists. With ties, however, we may have a weight of 0.40 for one alternative, 0.40 for another, and 0.20 indifferent. We will see, however, that this problem usually does not change the results very much. A second complication is that our classification of systems of voting blocs would have to be considerably expanded to take care of all the 'subcases' that can arise when ties are permitted. For example, the following two examples are both e systems:

However, if bloc A is indifferent between alternatives l and 2, D and E can contribute to a majority coalition in ej but not in e₂. Rather than creating an elaborate classification we have used two subcases for each of our previously defined cases.

The results of the first simulation with ties permitted35 and three alternatives
are given in Tables IV and V.

The main difference between Table 111 and Table IV is found in the first probability distribution. We find that the tied preferences tend to increase the winning probability of the largest bloc. As in Table 111, the sum of the probabilities for the three largest blocs should be 1.00.

Turning to the strategic gains, however, and comparing them with the gains found in Table 111, we find that the increases in distributions 1 through 4 are very similar. The differences can mostly be attributed to the generally high values of the probabilities in Table IV.

In Table V other subcases represent our previously defined cases. The effect of this is that the sharply reduced weight for the largest party diminishes the first winning probability. The increases from the same basic winning probability were found to be almost identical. In general the basic tendencies found in Table II are still with us. That is, neither the decrease in number of alternatives nor the introduction of indifference changes the tendencies.

Our last simulation in the analysis of the elimination variants was meant to incorporate the dimensional character of the preference orderings as outlined above. A thorough investigation would, however, consume a tremendous amount of computer time. Hence we will have to postpone a complete analysis until the next computer generation is installed.

We have, however, made a thorough investigation of the unidimensional situation. Since it is highly probable that the two-dimensional situation would yield results 'in between' the unidimensional case and the multidimensional case analyzed above, we feel that this constraint is acceptable.

In Table VI we try to clarify how the analysis of the unidimensional situation is presented. Since no ties are permitted, any subcase within our seven main cases will yeild the same result. In Table VI we have picked out two subcases in the g case. In the first subcase (g₂) the blocs have the positions ABCDE on a left to right scale. Since all blocs have single-peaked preference statements³⁶ the A bloc can only have the preference statement a_ls a₂, a₃, a₄, a₅ and the E bloc only a_g, a₄, a₃, a₂, aj. For the other blocs we make a random selection among the possible ones. That is, for bloc B one of a₂, a_1? a₃, a₄, a_g or a₂, a;!, a_ls a₄, a₅ or a₂, a;;, a₄, a_ls a_r, or a₂, a;;, a₄, a₅, a^ is randomly selected. When computing the strategic potentials, however, all 24 possible preference statements were investigated to see which one produces the best result assuming that the blocs primarily concentrate on getting their first preference as the outcome. The same four distributions were computed as above.

The result for case g2 is a verification of Black's theorem that 'if the member's
curves are single-peaked OmC(1 will be able to get a simple majority over any other
motions als a2, ...am put forward.'37 Accordingly the first preference of the C
bloc will always emerge as the majority alternative since bloc C holds the median
position on the scale. It automatically follows that no strategic moves can improve
the position of bloc C.

The strategic potentials are concentrated around the median power center. We find again that the strategic potential when the voting order is selected at random after the voters have registered their preferences is limited when compared with the strategic potential for a bloc in control of the voting order.

As predicted, the results for the gi case are identical (within the margin of error) to those for g₂. Notice, however, that in this case the smallest bloc (E) holds the median position. That is, the blocs are ordered ABEDC. Since all of the g subcases are identical from the point of view of creating majorities, the positions and not the distributions of weight determine the winning probabilities.

In Table VII we analyze one and the same ordering in our remaining fiv cases. The ordering analyzed is ABCDE. In all cases Black's theorem is veri fied. In addition, note that the strategic potential is concentrated around the media] position. In all cases we found a sharp increase in strategic potential when a bio is in control of the votine order.

Turning to the specific probabilities we should note the identical values for th C bloc for cases b₂ through e₂. This is due to the fact that in all these cases the < bloc has the opportunity to defeat the majority alternative (a₂) by voting with th A bloc (i.e. using a^. In case f₂, however, the C bloc does not hold a majorit in coalition with A. The large strategic potentials of the A bloc, especially whe

in control of the voting order, are clearly shown in the Table. Remember here that
all these strategic potentials imply that the majority alternative (a2) would not be
chosen.

The two cases in Table VIII are especially interesting from a Swedish point of
view since they closely resemble the political structure in contemporary Swedish
Parliament.

The second case is a good representation of the present (1972) Swedish political context in cases where one dimension predominates, although many people would argue that the B and the C blocs should change positions. In any case, when acting separately the B through D blocs are dummy blocs in this unidimensional structure. Empirical investigations of Swedish roll-calls seem to indicate that this is a situation frequently occurring, although multidimensional situations also occur quite often.³⁸

If there is a change in the present Swedish situation, the first case in Table VIII is the most probable new organization. This case will be further discussed below. Here it is sufficient to emphasize the high strategic potential of the largest bloc when it is in control of the voting order.

8. Summary

A significant potential for strategic moves inherent in the elimination procedure would be sharply reduced (but not eliminated) when using the variant known as the quality procedure. This is the main conclusion of analyzing the results of our simulation study.

Turning to these results we will first summarize the general effects. Then we
will report the specific effects of voting blocs and otfier features on the strategic
and equality aspects.

Two basic variations (in my opinion the most important variants)³⁹ of the elimination procedure were analyzed. In the first type one bloc has the power to make its own first alternative the last one in the voting order. In the second type the voting order is randomly selected either before or after all voters have registered their preference statements.

In the first variant there was considerable potential for strategic manipulation of the results in all situations in which no bloc controlled more than 50 percent of the first choices. The opportunities for strategic voting were found to be in rough proportion to the numerical strength of the blocs. This variation of the elimination procedure bears a strong and unexpected resemblance to the plurality procedure, in which only the first preferences determine the outcome. The plurality procedure is, however, cruder since it might, for example, have a result that is considered the worst one by more than half of the voters.⁴⁰ This is never possible when using the elimination procedure since the last vote will ensure that the worst alternative is always eliminated.

The resemblance to the plurality procedure is most pronounced in cases in
which we have one dominating bloc (the c, d, and e cases). In these cases we

find that the strategic winning probability for the largest bloc is generally higher
than 0.90.

For the c case the winning probability might even reach 0.99. However, in the cases with two (f), three (b), or five (g) balancing blocs the less crude character of the first variant of the elimination procedure will, in spite of the frequent potential for strategic voting, insure a better adherence to the equality principle.

The strategic potentialities, especially when the largest bloc is in control⁴¹ of the voting order, will tend to limit sharply the number of alternatives brought into the voting process. Hence issues are mostly decided in processes taking place prior to the final voting, and it is likely that strategic potentials will be most significant in this phase of the decision process.

Parliamentarians, since they often determine how the voting order should be selected, might view strategic voting as a natural and indispensable part of political life - even if they recognize the negative implications of strict majoritarian or egalitarian viewpoints. Such parliamentarians might reasonably prefer the variant of the elimination procedure in which the voting order is decided at random but known to the voters when they register their preferences. This variant would probably drastically increase strategic maneuvers in a multiparty system. This conclusion is based on the fact that under a random selection of the voting order, all blocs holding a certain 'majority-coalition potential' would have an opportunity to promote their primary preferences through strategic voting. Of course in each specific situation the voting order that happened to be randomly selected would play a crucial role. In the long run, however, this variant would produce outcomes more in accordance with the equality principle than does the first variant.

A vast change in the properties of the elimination procedure occurs when the voting order is selected at random but not known to the voters when they register their preferences (the quality procedure). We found that such a procedure results in very limited strategic possibilities, even if we assume that blocs take advantage of strategic voting whenever possible. That is, we regard it as very doubtful that a bloc confronted with a Vs probability of a better result, a Vs probability of no change, and a Vs probability of a worse result would make a strategic move. Nonetheless, in our calculations we have assumed that a strategic move would be made, and we have counted the positive probability (e.g. Vz) in calculating probabilities of strategic voting. We have not, however, made any reductions for probabilities of a worse result. Thus we have calculated the upper limit of strategic probabilities. Despite this fact, we find that strategic voting in this case leads to only moderate improvements in the winning probability.

These results suggest that all blocs (when the quality procedure is used) have a fair chance (in relation to their weight and majority position) if they bring their first choices into the voting process. This will tend to increase the number of alternatives in the voting process and will make the blocs more 'equal' in the decisions preceding the voting itself. It is obvious, however, that these properties will also make the voting more uncertain. Parliamentarians, in general, tend to regard this uncertainty as a very negative property and will usually prefer the first variant

of the elimination procedure despite its negative properties from strategic and
equality aspects.

Turning now to the more detailed results, we find that neither a change in number of alternatives nor the entry of tied preferences changes the general tendencies reported above. When relaxing our basic assumption of conflicting first choices among blocs, we found as expected a heavy increase in the winning probabilities when no strategies were employed. This is due to patterns in which one alternative received more than 50 percent of the primary preferences. These patterns are trivial and serve no analytical purpose and only complicate the analysis. In regard to the effects of different systems of voting blocs, we reported above the general difference between the 'balancing' cases and the 'dominating' cases. Here it is sufficient to note the different character of the smaller blocs. The dummy character of blocs D and E in case b and the balancing positions of blocs C through E in case f are obvious in our results. We should also mention that case g is slightly distinct from the other two balancing cases.

From our analysis of the unidimensional situation we should note our corroboration
of Black's theorem that 'if the members' curves are single-peaked Oraed will
be able to get a simple majority over any other motions als a2,... am put forward.
42 More importantly, we also found that the strategic potentials are concentrated
around the median power center. The limited strategic potential when
the voting order is selected at random after the voters have registered their preferences
as compared to the strategic potential for a bloc in control of the voting
order was found here as well. Finally we illustrated the present Swedish situation
and the most probable new organization (if there is a change) in unidimensional
contexts. These cases illustrate the strongly dominant position of the largest bloc
in the present situation and the hazardous position of a one-bloc (e.g. B) minority
government in the new organization. The strategic potential of the largest bloc
if in control of the voting order is very large in the new situation (0.92). Moreover
the strategic potentials of blocs C and D if in control of the voting order are considerable
(0.51 and 0.11, respectively). The 'random' strategic potentials are on
a much lower level, and we should also remember that the strategic moves in
that situation mostly imply a considerable risk.

In this article I have analyzed certain properties of two voting procedures in certain specified situations. Our analysis has been concentrated on properties and effects of the final voting process. The next step in the research process is to analyze other aspects of this as well as other legislative stages in an attempt to build a dynamic model of the parliamentary voting process. The predictive value of the debate preceding the voting and the predictive value of the committee report are examples of specific objects in such an attempt.

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NOTES

1. R. G. Niemi and H. F. Weisberg (eds.), Probability Models of Collective Decision Columbus, Ohio: 1972, p. 393.

2. A set of preference orderings is single-peaked if there is an ordering of the alternatives on the abscissa such that when utility or degree of preference is indicated by the ordinate, each preference ordering can be represented by a curve which changes its direction at most once, from up to down (i.e. has at most one peak). A dimension is said to exist in a set of alternatives if a set of voters has a complete agreement on a standard of judgment in such a way that their preference orderings will be single-peaked.

3. A sample size of 10,000 implies a 0.95 confidence level that our estimated probability is within 0.01 of the true probability (W. J. Dixon and F. J. Massey, Jr., Introduction to Statistical Analysis, 1957, pp. 84-85). In all simulations that follow we have sampled 10,000 sets of preference orderings.

4. B. H. Bjurulf, 'A Probabilistic Analysis of Voting Blocs and the Occurrence of the Paradox of Voting,' in Niemi and Weisberg, op.cit.

5. B. H. Bjurulf, Parlamentariska voteringmetoder, SOU 1972:15 Bilaga 4 (3), pp. 364-365.

6. Ibid., and R. G. Niemi, 'Majority Decision-Making with Partial Unidimensionality,' American Political Science Review 63 (1969), p. 494, and Niemi and Weisberg, op.cit., Chapters 9-10.

7. R. D. Luce and H. Raiffa, Games and Decisions, New York: 1957, p. 369.

8. D. Black, The Theory of Committees and Elections, Cambridge: 1958.

9. See note 5.

10. Test-runs involving countermoves indicated that a vast amount of computer time would be needed. This fact and our conclusion above that a completely effective strategic countermove always exists were the main reasons behind a decision not to analyze the effects of strategic countermoves when using the elimination procedure. The amount of computer time involved is the main argument for not analyzing appropriate countermoves when the vote-count procedure is used.

11. The labels of the alternatives are assumed to be arbitrary. That is, the distributions (ax =2, a₂ = 2, a₃ =2, and a₄ = 0), (a_A =2, a₂ =2, a₃ = 0, and a₄ = 2), (a_t =2, a₂ = 0, a₃ =2, and a₄ = 2) or (aj =0, a₂ =2, a₃ = 2, and a₄ =2) can by a re-labeling of the alternatives be transformed into one another. It is generally assumed here and below that a re-labeling of the alternatives is always permitted.

12. This and the other proofs in this context have not been carried to their logical extensions. However, since they have been tested by a simulation method, I consider them adequate for our purposes.

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12. This and the other proofs in this context have not been carried to their logical extensions. However, since they have been tested by a simulation method, I consider them adequate for our purposes.

13. B. H. Bjurulf, 'A Simulation Analysis of Selected Voting Procedures,' accepted for publication in the Swedish Journal of Political Science.

14. n alternatives =n! voting orders, i.e., 3 alternatives = 31 = 3x2xl = 6 voting orders, 4 » =4! =4x3x2xl = 24 voting orders, and 5 » = 5! = 5x4x3 x2x 1 = 120 voting orders.

15. Niemi and Weisberg, op.cit., Chapters 10 and 11.

16. Bjurulf, Parlamentariska voteringsmetoder.

17. Black, op.cit., p. 43.

18. Hence, when analyzing procedures that are embodiments of the majority principle, we use the terms 'strategic move,' 'strategic countermove,' and 'anticipating action.' The anticipating actions are, of course, countermoves. They are not, however, strategic countermoves since they are not reactions against strategic moves.

19. See note 13.

20. Duverger pointed out that 'three kinds of party can be distinguished on the basis of strength; parties with a majority bent, major parties and minor parties' (Political Parties, London: 1965, p. 283). I have tried to indicate with these examples that Duverger's classification can be expanded and stated more precisely.

21. Bjurulf in Niemi and Weisberg, op.cit.

22. Black, op.cit., Chapter 4.

23. See note 2.

24. O. A. Davis, M. J. Hinich, and P. C. Ordeshook, 'An Expository Development of a Mathematical Model of the Electoral Process,' American Political Science Review 64 (1970).

25. Niemi, op.cit.

26. A preference ordering with ties can be created as follows. First, if there are to be no ties for the first choice, choose a number 1-5 (for the five alternative case) randomly. Suppose it is 2. Then alternative 2 is said to be the first preference. Then randomly select four numbers with replacement from the numbers 2-5, inclusive. Suppose the numbers are 4, 3, 3, 2. We then assign alternative 1 to the fourth preference, alternatives 3 and 4 to the third preference and alternative 5 to the second preference. The resulting preference order is thus Alternative 2 5 » 3, 4 » 1 A similar method is used when ties are not allowed for the first or last choices. With this procedure for creating ties, we have the following probabilities: No ties for first choice No ties for first or last choice P (no ties) = 1(_im)/44= 24/256 P (no ties) =1( 3 )/3» = 6/27 P (two tied) -12( 21⁴1Q )/4⁴ = 144/256 P (two tied =6( * )/3³ = 18/27 ££T tled> " 6( 22⁴00 >/41 - 36/256 "W~ <««> -5< 3^o W- '/»• P (three tied) =12( QQ )/4« = 48/256 P (four tied) =4( 4^*QQ )/4* = 4/256 In these fractions, the multinominal coefficient represents a way of distributing scores to three or four alternatives. The preceding coefficient is the number of ways of obtaining this distribution. For example, in P (four tied), QQQQ) means that the same random number was drawn each time and the preceding coefficient denotes the fact that this number could be any one of the numbers 2, 3, 4, or 5.

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27. D. Klahr, 'A Computer Simulation of the Paradox of Voting,' American Political Science Review 60 (1966).

28. See note 3.

29. That is, the complete set of strong rank orderings was used.

30. The case with a weight > 0.5 is the first case in our classification. However, this case was not analyzed since the result is trivial.

31. Any weighting system within each category of the classification will give the same result (except for sampling errors). The specific systems we used are listed in note 32.

32. The specific systems used are as follows: Case Voting bloc b_t A - 0.47 B - 0.46 C = 0.045 D = 0.014 E-0.011 G! 0.41 0.20 0.15 0.14 0.10 dj 0.40 0.22 0.17 0.14 0.07 B! 0.41 0.24 0.22 0.07 0.06 f_t 0.40 0.39 0.09 0.07 0.05 gl 0.24 0.23 0.21 0.18 0.14 bo 0.40 0.31 0.25 0.03 0.01 Co 0.47 0.21 0.15 0.10 0.07 do 0.46 0.30 0.15 0.07 0.02 eo 0.46 0.30 0.19 0.03 0.02 fo 0.40 0.36 0.09 0.08 0.07 go 0.2 0.2 0.2 0.2 0.2 b"₃ 0.29 0.27 0.25 0.10 0.09 C₃ 0.37 0.18 0.16 0.15 0.14 d₃ 0.31 0.23 0.21 0.20 0.05 e₃ 0.31 0.24 0.23 0.12 0.10 f₃ 0.31 0.20 0.18 0.16 0.15 In systems eo, f₃, and go, ties in the social ordering can occur. In case go this possibility is greatest, and the random chance of a tie in the social ordering is 0.30. Note that in Tables 11—VII the terms bloc and party have been used synonymously.

33. Control is here defined as the situation in which the bloc has the power to make its own first alternative the last one in the voting order.

34. See note 32.

35. All results, when ties are permitted, assume that preferences other than the first choice may be tied. Results based on the assumption that preferences other than the first and last choices may be tied show only expected distinctions.

36. See note 2.

37. Black, op.cit., p. 18.

38. N. Stjernquist and B. Bjurulf, 'Partisammanhållning och partisamarbete. En studie av voteringarna i riksdagens andra kammare 1964 och 1966,' Statsvetenskaplig Tidskrift 4 (1968).

39. Our two variants of the elimination procedure are in my opinion the basic types. In realworld parliaments these types are divided into many subvariants as indicated by my discussion above and in Parlamentariska voteringsmetoder, pp. 370 ff.

40. See note 13.

41. See note 33.

42. Black, op.cit., p. 18.

Acknowledgements

In writing this article I have benefited from discussions with many people. My
teacher and advisor was Professor Nils Stjernquist. Many discussions at each stage

of the research process have been of vital importance in producing the final
product.

Rune Dahlgren was my link to the computer and has also served as statistical
advisor. Our discussions during the program-writing process have helped clarify
my analysis as well as adding new features to it.

Professor Richard G. Niemi of the University of Rochester translated this paper
into readable English. In this translation process his comments and criticism have
added precision and clarity to the paper.

Professors Arthur Goldberg and William Riker of the University of Rochester, Professor H. F. Petersson, Assistant Professors Bengt Hansson, Gunnar Sjoblom, and Torbjorn Vallinder, Lecturer Staffan Zetterhohn and my wife Carin Bjurulf have all read and criticized some version(s) of the paper. Mrs Inga-Marta Lindgren and Ulla Petersson served as patient and skillful typists through several versions of the article.

My work has been financially supported by grants from the Swedish Council
for Social Science Research which I gratefully acknowledge.