University of Helsinki

1. Introduction

To base political research on the theory of games is a very recent idea. Pioneer work in this field has been carried out by William H. Riker, whose work The Theory of Political Coalitions from 1962 is the first coherent attempt to find a game-theoretical explanation of politically important coalition formation. The most significant result of this research is the 'size principle', which describes political decision-making as follows:

In social situations similar to «-person, zero-sum games with side-payments,
participants create coalitions just as large as they believe will ensure winning and
no larger.1

This basic principle means that the size of a coalition is exclusively determined by the player's estimate of whether his course of action is winning or not. If there is a situation of complete and perfect information (i.e. the players know the potential influence of each other from the point of view of the outcome and the choices of the other players), this leads to formation of minimum winning coalitions. The implied additional assumption in this argument is naturally that the players are rational - that they choose the alternative that leads to the outcome they prefer most.

Riker presents two additional principles: the 'strategic principle' and the disequilibrium principle'. The strategic principle is derived from the size principle, and the disequilibrium principle from the former two principles. Thus the basis of his coalition analysis forms a deductive system we can test by testing only the size principle.

Riker's theory has been criticized on the grounds that his simplified assumptions (particularly the zero-sum condition) have made its application to the real world difficult.² The intention of this paper is not, however, to examine the basis of Riker's theory but to extend its basic tenet, i.e. the size principle of a winning coalition and to apply this to the decision-making mechanism of a multi-party system and also

to test the reliability of the theory itself. In order to introduce exact operational
mathematical formulas, the construction of a set-theoretical description of a multiparty
system is necessary.

2. A Set-Theoretical Description of a Multi-Party System

Consider a given party Aj in a decision-making situation in parliament. Its MP's
have three decision alternatives (only MP's actually present are being observed):
i) to vote for the motion, ii) to vote against the motion, or iii) to abstain from voting.

In order to find a more sensible starting point for set-theoretical concepts, we can examine the situation in another way. For one thing we know that some of the party's MP's might be absent from the voting. Absent MP's of party Aj are assigned the symbol A_PJ. Therefore the number of its MP's in the voting situation is naturally A'_j=A_J —Apj. On the grounds of the MP's choice, A'j can further be divided into three exclusive sets:

a. The majority of the party: Aj (more than half its MP's),

b. The minority of the party: A^ ('inner complement') (less than half its MP's),
c. The abstainers of the party: ABj (those who were present but did not vote).

Definitions a and b correspond to decision alternatives i and ii, and definition c is
equivalent to decision alternative iii. Thus we get the following definitions:

Definition 1: A given party Aj can be divided in a given voting situation on the
basis of the choice of its MP's into four subsets as follows:

We can expand the definition to a case where there are r parties (Aj, . .., Ar).
We get the description shown in Table I.

In Table I the rows show how the parties divide into subsets, and columns the
corresponding subsets of the parties.

On the basis of definition 1 above we can state that

where

We can observe that the number of all possible subsets of the parties is 4 • r, where
r is the number of parties. For example, if there are eight parties, there are althogether
32 subsets to be considered in the analysis.

The concepts of winning and losing coalitions have a central role in Riker's theory. The set of all winning coalitions, W, can - in the case of a parliamentary roll-call analysis - be defined as a set of coalitions that determine for the whole representative body the policies in given issues. Similarly the set of losing coalitions, L, can be defined as those coalitions that cannot determine the policies of the representative body. Operationally they naturally are defined as the class of those coalitions that have won in some roll-call vote and the class of those coalitions that have lost in some roll-call vote.

In addition to W and L we find two other potential types of coalitions in analyzing
the roll-call vote. These are the coalitions of those who abstain and those who are
absent. We can symbolize the corresponding sets by B and P. From Table I we can

r

r

read that Bi=UAßm and P^UABn- Furthermore, we can see that the union

m=l

n = l

set of a given winning coalition, Wj, and a given losing coalition, Lt, consists of the

r

r

elements of the union set (U Ad U (U Ad- In a given roll-call situation this cor

i=l I=l

responds to the distribution of ayes and nays. We can now define the winning and
losing coalition in a multi-party system.

Definition 2: The winning coalition, W}, is a coalition that determines the policy of
the representative body and consists of the majorities of parties 1,..., k and the inner
complements of the parties k +1,..., r, when the number of the parties is r. Formulated
in symbols, that is

Definition 3: The losing coalition, Lis is a coalition that cannot determine the policies
of the representative body and consists of the inner complements of the parties
1,..., k and the majorities of the parties k +1,..., r. Or, in symbols:

Definitions 2 and 3 are easy to understand when we observe decision-making at the party level. According to definition 1, a faction of MP's in party A_, can oppose the majority choice. If, for example, A/s majority A-₃ belongs to the winning coalition this party's 'inner complement' A_} therefore belongs to the losing coalition, because A_} and A_} cannot take the same side in the voting. Taking this into consideration in the case of each party we arrive at the above definitions 2 and 3, according to which we can further conclude that W^ n I^=^.

According to the definitions of winning and losing coalitions we can now arrive
at a useful concept of coalition structure:

Definition 4: Coalition structure, CSi,4 is a union set formed from the winning and
the losing coalition.5 This can be formulated as

3. Riker's Size Principle in a Multi-Party System

If we can consider a roll-call vote an example of the type of situation where Riker's
basic assumption holds, we can expect that the coalitions formed there would be
minimum winning coalitions. Riker defines a minimum winning coalition as follows:

Let Sp be a set and Sp e Wmln (Wmin= a set of all minimum winning coalitions),
then (Sp-l)*Wmin.

According to this definition, subtracting even one member from a winning coalition means that it ceases to be winning. If we assign to a winning coalition the symbol W_t and to the corresponding minimum winning coalition the symbol W™¹", then the size of a winning coalition is defined by the parameter F=W! —W™ⁱⁿ, which shows how near the winning coalition is to a minimum one. r r

If we inspect a set of aye and nay votes-union set (U A^) U (u A^) and assume

i=l I=l

that the total number of votes is n, if n is an even number, the minimum winning
coalition, according to simple majority rule, is

(1)

If n in turn is an odd number the minimum winning coalition is

(2)

From these definitions we then get equivalent F values

(1)

(2)

(In calculating F we have taken the condition Wt +Li=n into consideration.)

According to Riker's hypothesis, F's value should be zero if the size of a winning
coalition is to be minimal. This is the case in (1) when Wi — Li=2 and in (2) when
Wi-L^l.«

It was implicitly assumed above that an individual MP is an independent decisionmaking unit. As we know from many studies concerning representative assemblies, this assumption is very unrealistic.⁷ It is more reasonable to take the party as the primary decision-making unit. Then we can take into consideration the party's 'cohesion' in a decision situation, that is, its propensity to act unanimously in rollcall votes. Before defining any measures for F it is necessary first to explain the set-theoretical basis of parliamentary voting behavior.

If we take a given CS_i5 we can distinguish between the theoretical and empirical values, W[^heor and Wr^P, and L?⁶⁰¹ and UT" of W_t and L^ Similarly we can define, for a given CS_i9 the values F^theor and F^emp (we take here only the case when n is even):

r _ r
W|heor and Lfeor can be determined in a given CSi when U A^fa U A^m=(f),
r 1 = 1 m=l
and U APn=^, i.e. when we do not consider those who either oppose the majority
n=l
of the party, abstain, or are absent. The set-theoretical conditions for determining
ptheor -n a gjven eg. are given jn Table 11.

A party's Fitheor in CSt is thus determined according to its number of seats in
parliament. We can thus calculate a priori, for any possible coalition structure,
a theoretical parameter expressing the size of a winning coalition. If the total number
of parties is r, the number of coalition structures formed from these parties is 2r~l.2r~1.
If we examine the determining of Femp in the same coalition structure CSi, we cannot
r_rr
be certain that U Al} U ABm., and U APn are necessarily empty sets, although it

I=l m=l n=l

is empirically possible. For example, in the Finnish Parliament there has rarely existed a roll-call vote where no MP of any party has either voted against the party majority, abstained from voting, or been absent. Thus we can give the set-theoretical conditions that determine F?^mp's value in the coalition structure CSj:

From this definition it can be seen immediately that if the sets above are empty

inCSl,thenF|heor=FrP.

Next we must find a relation that connects Wimp to Wjheor and similarly Ltmp to

Ltheor
i

Let us assume that parties in the winning coalition are 1,..., k, their absent MP's
union set is PWi, their abstaining MP's union set is BWI, and the union set of their
inner complements is Avi* so that WP^WJ11601'—PWi — BWI — A^{+AU.

This relation can be understood so that sets P_WI, B_Wi, and A_Wi reduce a theoretical winning coalition. From definition 2 above we know that A_LI increases Wf^mp's value. If the equivalent subsets of a losing coalition are assigned the symbols P_LI, B_Li, and A_Li, there exists analogously the relation

On the basis of the above formulations we can get the formula for calculating Fjmp:

From the formula we can easily see that Ffmp values can vary much within a given
coalition structure CSj (Fjheor being always constant), because Flmp is the function of

In order to test Riker's theory within a given coalition structure CSi? we can
define AFI=F{heor-F?mp. AF, can thus be calculated with the formula

The bigger PWi, BWi, and Ay/i are when compared with PLI, BLI, and ALi, the
smaller Fimp is when compared with Fjheor. AF—0 only when

If F|^heor>Ff^mp, then AF_t>o, and the size of winning coalition follows Riker's theory because AF_{ > 0 expresses a tendency toward a minimum winning coalition. If Fj^heor<Fi^mp, then AFi<o, and the empirical coalition is bigger than could be theoretically expected. The size of a winning coalition approaches the maximum winning coalition, and the result is contradictory to Riker's theory.

The elucidation of the set-theoretical analysis of parliamentary voting behavior is important in order to extend Riker's theory. The size principle can be extended within this frame of reference, as we have indicated above, to the analysis of the internal variations of different coalition structures, i.e. it is applicable to the analysis of multi-party systems. It is interesting to note that a two-party system is a special case of this analysis. There the sets P_{W i,} B_{W i,} A^i and P_LI, B_LI, /4_Li describe directly the subsets of majority and minority parties in a voting situation.

4. An Empirical Application

The above extension of Riker's analysis for the study of multi-party systems was tested with Finnish roll-call vote data. For the analysis 450 roll-call votes in the Finnish one-chamber Parliament were sampled from the period of 1964-1966. Because Finland had three different Cabinets during this time, the sample was stratified so that there were 150 roll-call votes from the period of each Cabinet. This stratification allows Riker's principle to be tested under different conditions, because the Cabinets differed considerably in their composition. Thus the first (Prime Minister Lehto) was a party politically neutral, so-called 'civil servant Cabinet', the second (Prime Minister Virolainen) a bourgeois coalition Cabinet, and the last (Prime Minister Paasio) a socialist majority Cabinet.

We have in Tables 111 and IV the coalition structures that appeared in roll-call votes during these Cabinets.⁸ As we mentioned above r political parties can be divided in 2r-12^r-1 ways into two opposing coalitions (the winning and the losing). As there are eight parties in the period studied, the number of potential coalition structures is 22^{B~l=l2B.8 ~ l}=l2B. From the period of Lehto's Cabinet we find 29 coalition structures (22.8 percent of the potential maximum), from Virolainen's period 23 (18.0 percent of the maximum), and from Paasio's period 31 (24.3 percent of the maximum). This indicates that the coalition structures (which coalitions 'can be

made') is normatively controlled. The same is reflected in the fact that only few coalition structures appear more frequently. The impact of the Cabinet coalition on the frequency of the roll-call coalitions is naturally significant. During Virolainen's Cabinet the correspondence was 13.3 percent (CS4) and during Paasio's Cabinet it was 40.0 percent (CS'3). We can see that the Cabinet coalition does not alone determine the roll-call coalitions.⁹ In order to test Riker's size principle, all AFl=Ftlheor-Fl^empfor the 450 roll-call votes were calculated. These are given in Table V where we can see that Fj^heor is greater than Ff^mp because AF, is most often positive. This is significant because it holds for the period of all three Cabinets. Negative AFa values (that contradict Riker's theory) can be found on only a few occasions in different coalition structures: eight during Lehto's Cabinet, seven during Virolainen's Cabinet, and five during Paasio's Cabinet. Generally these negative values appear in coalition structures which are uncommon. Thus in Lehto's period the range of the presence of these coalition structures was 1-4, in Virolainen's period 1-2, and during Paasio's period 1-11. Because the frequencies of these coalition structures are generally close to one, one cannot consider the appearance of negative AFi in them as strong evidence against Riker's theory. Only the frequency of CS\ (11) during Paasio's period and the negative AFj can be considered a significant deviation from Riker's size principle. Because AF{^heor in this coalition structure is only three, we can explain this exception by made') is normatively controlled. The same is reflected in the fact that only few coalition structures appear more frequently. The impact of the Cabinet coalition on the frequency of the roll-call coalitions is naturally significant. During Virolainen's Cabinet the correspondence was 13.3 percent (CS₄) and during Paasio's Cabinet it was 40.0 percent (CS'₃). We can see that the Cabinet coalition does not alone determine the roll-call coalitions.⁹

In order to test Riker's size principle, all AF_I=F|^heor-F₁ ^einpfor the 450 roll-call votes were calculated. These are given in Table V where we can see that F|^heor is greater than Ff^mp because AF_I is most often positive. This is significant because it holds for the period of all three Cabinets. Negative AF_a values (that contradict Riker's theory) can be found on only a few occasions in different coalition structures: eight during Lehto's Cabinet, seven during Virolainen's Cabinet, and five during Paasio's Cabinet. Generally these negative values appear in coalition structures which are uncommon. Thus in Lehto's period the range of the presence of these coalition structures was 1-4, in Virolainen's period 1-2, and during Paasio's period 1-11. Because the frequencies of these coalition structures are generally close to one, one cannot consider the appearance of negative AFj in them as strong evidence against Riker's theory. Only the frequency of CS^ (11) during Paasio's period and the negative AF_t can be considered a significant deviation from Riker's size principle. Because AFj^heor in this coalition structure is only three, we can explain this exception by

resorting to Riker's 'information principle'. The parties that belong to the winning coalition have minimized the risk of losing the vote by taking care that the Fj^mp stays persistently higher than its expected value. As we remember Riker assumes that Wf^mp tends toward minimum because the actors try to maximize their own benefits, but this rationality has greater predictive power in situations where information is more perfect than in situations of uncertainty. We can use the same 'information principle' to explain the negative AFj values in the cases CS_n, CS_I2, and CS₂₃ during Lehto's and Virolainen's Cabinets and CS'₉ during Paasio's Cabinet.

When AF, is positive this means that PWl+BWi+2 yiw, >Pu+BLl +2 ALi. The
decreasing of Fimp toward the minimum winning coalition increases the benefits
of the MP's that belong to Wfmp, according to Riker's theory. Furthermore MP's,

by belonging to P_Wi and B_Wi can avoid the responsibility of the consequences of the winning decision. The analogous interpretation can be made about the subsets of P_LI and B_Li. However, when the AFi is positive this trend is greater in the winning coalition. The subsets of A_WI and A_Li reflect the willingness of certain MP's to maximize their own benefits by opposing the majority of their parties. This may under certain conditions also be useful for their parties: the flexibility in the discipline in this respect may pay back in the following elections in terms of more votes.¹⁰ Because AFi>o, the maximization of the individual benefits of MP's takes place more often when they belong to the winning coalition.

To sum up, we can say that the results of our analysis within given coalition structures are very closely congruent with Riker's coalition theory, although the tendency to form minimum winning coalitions can be due to other factors than Riker assumes.

5. Conclusions

The set-theoretical description above aims at providing a logical foundation for the analysis of parliamentary voting behavior. We have also formulated it to give a conceptual frame within which one can extend Riker's theory of coalition behavior and test it reliably with roll-call data. The results support Riker's theory (its size principle) but also lead us to look for an underlying causal mechanism rather than to assume individual rationality, which Riker bases his theory upon.

The parameter AF_f used to test Riker's theory indicates the deviation of the observed empirical F-value from the theoretical F-value. We can still improve this measure by changing it from absolute to relative. This naturally will change the content of Riker's theory, which after this change could be stated as follows: In voting situations similar to n-person zero-sum games with side-payments in a given coalition structure CSj, the relative size of W'^mp to W}¹¹⁶⁰' is smaller than the relative size of the losing coalition L?¹"" to Lf^eor.

If we inspect the winning coalition Wj, the_deviation of Wf^mp from the W,^thcor is caused by the subsets of P_wt, B_Wi, A^i, and AU- In the same way the deviation of Lf^mp from L[^heor is determined by P_Li, B_Li, A_u, and J_{W i} • The greater the subsets P_Wi, B_Wi, A_wl are and the smaller the subset A_LI is, the closer Wf^mp is to W™¹".¹¹ In the relative measure one should not take A^ into consideration, because it boosts the size of the winning coalition irrespective of the will of its members.¹² If we divide the union set of (P_WI+B_WI +^_Wi) with Wj^heor and multiply the result by 100 we get the amount that these subsets decrease the winning coalition as a percentage. The same can be done in case of the losing coalition. Thus we will get the parameter AFi that measures the relative size of the winning coalition.

If we take into consideration the _equations W*mp=W}heor-Pwl-BWi-^wl+^Ll
and L*mp=Litlleor-PLi-BLI-JLI+JWI we will get

9 9 AFj has been derived so that when AFi >0, the results support the above improved' Riker's hypothesis. We have not calculated the parameter values according to this measure but it seems to be necessary to develop Riker's analysis in this direction in further studies that apply it to parliamentary decision-making.

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NOTES

1. W. H. Riker, The Theory of Political Coalitions, New Haven and London: Yale University Press, 1962, pp. 32-33.

2. Cf. C. Adrian and C. Press, 'Decision Costs in Coalition Formation', American Political Science Review 62 (1968), pp. 556-563; J. L. Bernd (ed.), Mathematical Applications in Political Science, 11, Dallas, Texas: Southern Methodist University Press, 1966, p. 165.

3. The symbols U, n, (f) are used in their customary meaning. See e.g. S. Miettinen, Logiikan perusteet I (Elements of Logic I), Helsinki: Yioppilastuki ry, 1971.

4. The set of all coalition structures can be symbolized by CS.

5. This formulation corresponds to the customary concept of 'group combination', of Tables 111 and IV.

6. If there are minority or majority rules in case of certain roll-call votes the calculation of F-values becomes more complicated; see M. Laakso, Poliittinen koalitioteoria politiikan tutkimuksessa (Political Theory of Coalition in Political Science Research), Unpublished Master's Thesis, University of Helsinki, Institute of Political Science, Helsinki, 1971, p. 47.

7. See e.g. P. Nyholm, Suomen eduskuntaryhmien koheesio vuosien 1948-51 vaatikaudella ja vuoden 1954 valtiopaivilla (The Cohesion of Finnish Parliamentary Groups during the Sessions 1948-51 and the Session 1954), Bidrag til kånnedom av Finlands natur och folk utgivna av Finska Vetenskaps-Societeten, H. 106, Helsinki, 1961.

8. The coalition structures of Paasio's period are symbolized by CS' because they were cornnosed of different narties than durine Lehto's and Virolainen's neriods.

9. For a more detailed analysis see Laakso, op.cit., pp. 72-74.

10. It is worthwhile to note that the parties that had relatively large 'inner complements' in our data for a Cabinet period turned out the greatest winners in the elections that followed this period. Thus the Social Democrats won in 1966 and the Conservative Party in 1970.

11. A weakness of the AF| parameter used here (and also the weakness of Riker's theory) is that if nearly all parties belong to the winning coalition then AFj>o because (P_Wi+B_Wi+2/4_Wi) j_s natural greater than (Pu+Bu+^Li)'

12. Similarly /<w, cannot be taken into consideration when calculating the losing coalition.