FORMING COALITIONS AND MEASURING VOTING POWER

Transcription

1 FORMING COALITIONS AND MEASURING VOTING POWER MANFRED J. HOLLER University of Munich IN this paper I will present various concepts of coalition formation and power measures and discuss them with respect to the view that situations where minor players possess greater potential for power are not anomalous, but occur rather frequently in real-world situations. We shall see that the analysis of the various concepts of coalition formation will teach us much about the implications of the so-called power indices and the character of a priori voting power. A new index will be introduced which considers the coalition value of a public good and takes into consideration the distinction between power and luck. The values of this index will be calculated for the parties of the Finnish Parliament (Eduskunta) for the period RIKER S SIZE PRINCIPLE Riker claimed that parties seek to increase votes only up to the size of a minimum Coalition., This follows from the well-known Size Prin~iple ~ which implies that, given a multi-member voting body u = (d; w1,w,,..., w,) a coalition so will be formed for which the sum of the corresponding voting weights of its members minimizes the difference A = so - d, given A > 0. Thereby, i = 1,..., n being the votes, w,, w,,..., w, the corresponding voting weights, and d the decision rule. The underlying idea of this solution is that payoffs for any winning coalition Sj are identical. If the coalition payoff is split between the members of the winning coalition according to their respective voting weights, each member s share will be maximized through the minimizing of the coalition partner(s) voting weight@). Hence, for a voting body u1 = (0; 0,, ) the minimum winning coalition (MWC) So = {, ) will be formed although S, = (0, ) and S, = (0, ) are also winning and minimal with respect to the number of members-they do not contain any dummy member, i.e., a member who is not necessary for the fulfilment of the decision rule d = 0. E. W. Packel and J. Deegan, Jr., An Axiomatic Family of Power Indices for Simple n-person Games, Public Choice, (1980) W. H. Riker, The Theory of Political Coalitions (New Haven, Conn., Yale University Press, 19), p Riker, Theory of Political Coalitions, pp. 8. Political Studies, Voi. XXX, No. (-71)

2 MANFRED J. HOLLER We can state that the voting power of the players who are not included in coalition So has to be zero in accordance with Riker s Size Principle.. THE DEEGAN-PACKEL INDEX A different concept of MWC was presented by Deegan and Pa~kel.~ They introduced an interesting paradox as regards the measuring of power. For a given game u on the player set N = { 1,,..., n) Deegan and Packel defined the set of MWC s M(u) as follows: M(u) = {S c NIu(S) = 1 and u(t) = 0 for all T s j SJ. From this definition it follows that T is a nonwinning coalition. Every strict subset of S is therefore nonwinning: v(s - {i}) = 0, for i E S. The set S - {i} is a strict subset of S, given that i E S. On assuming that (1) only minimal winning coalitions will form; () each such coalition has an equal probability of forming; () players in a (minimal) winning coalition divide the spoils equally, Deegan and Packel s measure of power is IM(u)l and IS1 are cardinalities of the corresponding sets. This measure expresses what player i can expect to get from a game u. If we apply this power index to a voting body uz = (1;, 0, 1, 1, 1) the power distribution is given by cuz = (18/0, 9/0, 11/0, 11/0, 11/0), This shows that a larger voting power is related to a player of voting weight 1 than to a player of voting weight 0. The Deegan-Packel index is not monotonic in votes. We may ask whether this index is a plausible way of attaching members to entities as implied in the term measurement. As long as we do not consider voting power to be monotonic in seats the Deegan- Packel index can be accepted. However, if we believe in a monotonic relation, and obviously this is a tenet to representational democracy, we must question the axioms fundamental to this index. Packel and Deegan contmented on this paradoxical result by referring to sociologists (such as Caplow ), who argue that situations where minor players possess greater potential for power are not anomalous, but occur rather frequently in real-world situations. J. Deegan, Jr. and E. W. Packel, A New Index of Power for Simple n-person Games, International Journal of Game Theory, 7 (1979), 11-. Packel and Deegan, An Axiomatic Family of Power Indices for Simple n-person Games, p. 1. H. Nurmi, Measuring Power, in M. J. Holler (ed.), Power, Voting and Voting Power (Wurzburg. Physica, 198). H. Nurmi, Power and Support: the Problem of Representational Democracy, Munich Social Science Review, (1978), -. Packel and Deegan, An Axiomatic Family of Power Indices for Simple n-person Games, p.. T. Caplow, Two Against One: Coalitions in Triads (Englewood Cliffs, N.J., Prentice-Hall, 198).

3 FORMING COALITIONS AND MEASURING VOTING POWER This might be so, but we can ask whether this argument is plausible for Packel and Deegan s conceptual framework. Firstly, we can recognize that Riker s Size Principle would clearly favour the coalition So = {,01 resulting from the voting body u. All other coalitions are either not winning or are related to a sum of voting weights which is larger than so =. Secondly, we must realize that the 1 per cent players in voting body u get the relatively higher power index, as one minor player does not suffice to form a winning coalition with the per cent player. That is how a 1 per cent player gets into three coalitions in accordance with M(u), whereas the 0 per cent player is member of only two coalitions. Thirdly, since, as assumed, players in a (minimal) winning coalition divide spoils equally, there is actually no reason why the per cent player should prefer a minimum winning coalition S, = (, 1, 1) to a non-minimum winning coalition S, = {,0, 1). Since the 1 per cent player is a dummy in S,, we could expect the major part (if not all) of the coalition s payoff to be allocated to the per cent player and 0 per cent player. Indeed, according to the above division rule, it is more profitable for the per cent player to form a coalition with the 0 per cent player and receive half of the coalition pie. Were he to co-operate with the two 1 per cent players, he would receive only a one-third share. Since this also holds under the given decision rule for the payoffs of the 0 per cent player, when comparing his share from coalitions {, 0) and (0, 1, 1, 1}, it seems doubtful that any coalition other than (, 0) is plausible under the conceptual framework of Packel and Deegan. By the division rule, as assumed by Packel and Deegan, Leiserson s bargaining concept, which favours the coalitions with small numbers of parties (players), gains plausibility and gives support to the coalition (, 0}. O One way out of this trap is introduced by Packel and Deegan themselves. They presented a second power index [{(u) which is also based on their concept of the minimum winning coalition and the above division rule but incorporates different occurrence probabilities of the coalitions. If the occurrence probability of the coalition {,0} is 1 and that of the others 0, [i(u) = 1/ and [[o(u) = 1/ for the voting body u = u,.. PROBABILITIES OF OCCURRENCE The underlying probability assumption might be considered rather crude, nevertheless it bears some theoretical justification. We can deduce from Leiserson s analysis that a two-member winning coalition dominates all coalitions which consist out of more than n = members., Kalisch, Milnor, Nash and Nering reported that coalitions of more than two players within experimental n-person games are seldom formed except by being built up by smaller c0a1itions.l~ This supports, what we would like to coin, the Principle lo M. A. Leiserson, Factions and Coalitions in One-Party Japan: An Interpretation Based on the Theory of Games, American Political Science Reoiew, (198), 77C87. I Packel and Deegan, An Axiomatic Family of Power Indices for Simple n-person Games, p. 1. I Leiserson, Factions and Coalitions in One-Party Japan. l G. K. Kalisch, J. W. Milnor, J. F. Nash and E. D. Nering, Some Experimental n-person Games, in R. M. Thrall, C. H. Coombs, and R. L. Davis (eds), Decision Processes (New York, Wiley, 19), pp

4 MANFRED J. HOLLER of MWC in Number'. This principle obviously cannot be in favour of members of small voting weights. It clearly says that the coalition (, 0) will be the only (stable) one for the voting body 0,. (Riker's Size Prin~iple'~ gives also support to coalition {, 0). This principle is, however, based on a different division rule.) Instead we can introduce a probability mix with respect to the coalitions as defined by M(u) for u = 0,. The use of the well-known Banzhaf power index actually implies a specific probability mix.i This has been shown by Straffin.I The Banzhaf power index bl of a player i is defined, as the number of swings for player i divided by the total number of coalitions containing player i. A swing occurs when the defection of player i changes a coalition from winning to losing. For reasons of comparability, the Banzhaf index is often standardized by the formula i = 1 Given the voting body ul, the power distribution as measured by this index is b, = (1/, 1/, 1/), For the voting body u we can calculate b, = (%, 0%, 1%, 1%, 1%). In the context of the Banzhaf index it is interesting to note that a fourth distinct concept of MWC is offered: 'for each minimal winning coalition there is at least one member whose removal would make the coalition nonwinning'.' ' The power of an actor i is thereby seen as his ability to threaten the other members of the minimal coalition, as implied by his ability to change the coalition from winning to losing. Hence, if only one player (and not all as in the Packel and Deegan concept) is critical to a coalition, it can be called minimum winning coalition. Thus, MWCs of the fourth type can include dummies. The Banzhaf index can be read as the measure of a player's probability to change the outcome." From this we can deduce that a 0 per cent player is more likely to effect the voting outcome than a 1 per cent player. This does not exclude the possibility of a specific 1 per cent player being, and a 0 per cent player not being, a member of a winning coalition, it just makes it less likely. Given the voting body u, there exists a MWC of the fourth type (as well as of the Packel and Deegan type) which does not contain the 0 per cent player but rather the 1 per cent players and the per cent player. This does not contradict the (positive) monotonic characteristic of the Banzhaf index with respect to voting weights, it only shows that the likelihood of occurrence is in general different to the realization. l Riker, Theory oj'polirica1 Coalitions, p.. Is J. F. Banzhaf, 'Weighted Voting Doesn't Work: A Mathematical Analysis', Rurgers Law Review, 19 (19), 17-. Also J. F. Banzhaf, 'One Man,.1 Votes: A Mathematical Analysis of the Electoral College', Villanova Law Review, 1 (198). 0. l P. D. Straffin, Jr., 'Homogeneity, Independence and Power Indices', Public Choice, 0 (1977) Nurmi, 'Power and Support', p. 18. Straffin, 'Homogeneity, Independence and Power Indices', p. 11.

5 FORMING COALITIONS AND MEASURING VOTING POWER. THE DIVISION RULE AND THE PUBLIC GOOD ASPECT Real-world situations where minor players possess greater power are compatible with all four concepts of the MWC and the deduced voting power measures. However, Riker s Size Principle and the Packel and Deegan concept might allocate larger a priori power to players with relatively smaller voting weights. This counter-intuitive outcome is due to the division rule which underlies these indices. It is consequent if the voting weights are positively related to a claim on coalition payoffs as assumed by the Size Principle. Yet it needs, however, further elaboration if the coalition payoff is divided by the number of members as implied in the Packel and Deegan concept. It cannot be advantageous for any player to form coalitions with minor players instead of with major players (measured in voting weights). However, when examined in conjunction with the division rule, it may be disadvantageous. All indices, as discussed so far, face the problem of distributing (or assigning) the value of a priori coalitions among their members. There might be no adequate solution to this problem, for the coalition value is a collective good. The private good approach, as implied in the discussed indices, is inappropriate if voting is not only a matter of allocating spoils. In a recent article Barry claimed that the concepts of dividing the value of a coalition violates the first principle of political analysis, which is that public policy is a public good (or bad). For illustration : If the death penalty is reintroduced, that pleases those who favour it and displeases those who do not. Similarly, a tax break is a good or bad for people according to their situation. The gains are not confined to those who voted on the winning side nor are the losses confined to those who are on the losing side.. POWER AND LUCK If we consider the value of a coalition (not the voting power as discussed by Brams to be a collective good, any member of the voting body whose preferences correspond with the outcome of the winning coalition can be considered as member of the specific coalition. A member who is essential for the specific coalition can exert power. A non-essential member (dummy voter) is merely lucky: the outcome will correspond to his preferences although he does nothing. Barry labels the difference between success (i.e., the coalition outcome corresponds to the member s preference) and luck decisiveness. Power is understood in Weberian tradition as the actor s ability to overcome resistance. According to this definition, an all-powerful actor, i.e., dictator, might not be decisive. If he is very lucky, all the outcomes he wants will occur even if he does nothing. According to Barry s definitions of power and decisiveness it follows that the more powerful an actor, the better his chance of being decisive. Power and decisiveness are closely related, however, as l9 B. Barry, Is It Better to be Powerful or Lucky? : Parts 1 and, Political Studies, 8 (1980), 18-9 and 8-. O S. J. Brams, Game Theory and Politics (New York, Free Press, 197), p Barry, 1s It Better to be Powerful or Lucky?, p. 8. Barry, Is It Better to be Powerful or Lucky?, p. 0.

6 MANFRED J. HOLLER 1 defined by Barry they cannot be clearly explained in terms of success, luck, and decisivenes~.~~ Yet if power is seen as an ability, a capacity or potential, to influence, bring about or preclude an outcome, it is identical to decisiveness as defined above. Power measures thereby become invariant to changes in the distribution of preferences. This seems adequate for measuring a priori voting power, when information concerning the decision rule and the voting weight distribution (but not concerning the preferences of the voters) is given.. THE STORY OF A NEW INDEX From this we obtain the following story for an appropriate power measure: (1) Any member of a minimum winning coalition is decisive for the coalition value. The (undivided) coalition value therefore expresses his power within the coalition. () An individual nonessential member does, by definition, not influence the winning of a coalition. He therefore has no power. It is sheer luck when an outcome corresponds with his preferences. () Since a nonessential member is not decisive for the winning of his preferred coalition, i.e., his preferred policy, he has no incentive to vote. () Because of (), only those winning coalitions will be purposefully formed ( not by sheer luck ), which win by means of the votes of their essential members. If, e.g., ABC is a winning coalition, then it will form if either all three members are essential, or if either AB, AC, BC, or any of the single coalitions A, B, and C is a winning coalition. ABC will not purposefully form, if, e.g., only A is essential but not sufficient for the formation of a winning coalition. If, however, coalition ABC forms, it is due to luck and not due to A s power (or decisiveness). We shall call the set of essential members a decisive set (of a coalition). Only those coalitions will form which have a winning coalition as decisive set, i.e., a winning decisive set. () Each winning decisive set corresponds with a specific coalition outcome (policy). The outcome of two coalitions differ if their decisive sets are not identical. If, e.g., AB forms the winning decisive set of the coalition ABC, the outcome of coalition ABC will be identical with the outcome of coalition AB. If AD is the winning decisive set of coalition ACD, then the outcome of coalition ACD will be identical with the outcome of coalition AD, but different to the outcome of coalitions AB and ABC. It follows that if we consider the coalition outcome (value) to be a public good, we must refer to the various winning decisive sets of the potential coalitions when measuring the a priori voting power within a specified voting body. Our definition of the winning decisive set is identical with the definition of the elements of the set of minimum winning coalitions M( u), which underlies the Deegan-Packel index. (Our story above does not imply that only these coalitions will form. It merely suggests that only these coalitions should be considered for measuring a priori voting power.) Barry, Is It Better to be Powerful or Lucky?, p. 0. See, for example, N. R. Miller, Power in Game Forms, in M. J. Holler (ed.), Power, Voting and Voting Power.

7 198 Party Seats (%) h-index b-index 191 Party Seats (%) h-index b-index 19 Party Seats (%) h-index b-index 198 Party Seats (%) h-index b-index 19 Party Seats (7;) h-index b-index TABLE 1 The distribution of seats and power in the Finnish parliament, I o z cn > Z CI m -0 0 s m?j

8 I Party 1 Seats (%) 7. h-index 1. b-index 1.7 Party 1 Seats (%).0 h-index 1. b-index 8.9 Party 1 Seats (%) 7. h-index 1. b-index 0.08 Party 1 Seats (%) 7.0 h-index 1. b-index.9 Party 1 Seats (%) I o = SDP. Suomen Sosialidemokraattinen Puolue, Social Democratic Party of Finland: = SKDL, Suomen Kansan Demokraatinen Liitto, Democratic League of the People of Finland; = Kok, Kansallinen Kokoomus, National Coalition Party; = Kepu, Keskustapuolue, Centre Party; = RKP, Ruotsalainen Kansanpuolue, Swedish People s Party in Finland; = LKP, Liberaalinen Kansanpuolue, Liberal Party; 7 = SMP, Suomen Maaseudun Puolue, Finnish Rural Party; S=SKL, Suomen Kristillinen Liitto, Christian League of Finland; 9 = SKYP, Suomen Kansan Yhtenaisyyden Puolue, Party of Finnish People s Unity: 10= SPK Suomen Perustuslaillinen Kansanpuolue, Finnish Constitutional People s Party; 1 1 = SPSL, Tyovaen ja Pienviljelijain Sosialidemokraattinen Liitto, Social Democratic Union of Workers and Small Farmers. z > z n P rn U L T 0 r m ; N 0.W

9 70 FORMING COALITIONS AND MEASURING VOTING POWER 7. THE PUBLIC GOOD INDEX When calculating the voting power, we should consider each minimum winning coalition only once, since by () of our story every winning decisive set corresponds with one specific outcome. Having abstracted from introducing specific preferences for the players of our voting game, we will give equal values u(s) to the various outcomes of the coalitions S E M(u). Due to the public good character of the coalition outcome, this value u(s) will be valid for each member of the specific coalition S. If we assume that every MWC being element of M(u) occurs with equal probability, the following measure for the (relative) a priori voting power of player i results: thereby n 1 hi(u) = 1. i= I If we standardize the value of the minimum winning coalition (S) with o(s) = 1, h,(u) measures the number of times a player i is a member of a minimum winning coalition S, divided by the number of times the n players of the set N are members of S E M(u). This index was used by Holler for measuring the voting power in the Finnish Parliament. The author gave no theoretical justification. He was doubtful about the validity of this measure, for it is not necessarily monotonic with respect to the voting weights. This can easily be verified by using the identical example for which the Deegan-Packel index has also shown nonmonotonicity. Given the voting body u = (1;, 0, 1, 1, 1), it follows the power distribution One could accept this result, hereby referring to the fact that for specific voting bodies (like the above) a player with a smaller voting weight can be member of more minimum winning coalitions and will therefore have a higher a priori voting power than a player with higher voting weight. Indeed, the larger the number of coalitions which player i can turn from winning into nonwinning by changing his vote, the more likely it is that the resulting policy corresponds with player i s preferences. One could alternatively question the equiprobability of minimum winning coalitions. For instance, since Leiserson s bargaining concept favours coalitions with small numbers of players, the coalition (, 0) becomes more likely than others. We may thus choose probability weights for the considered minimum coalitions which increase the power measures for both the per cent player and the 0 per cent player. The monotonicity of apriori voting power and voting weights could then be re-established. It is, however, not the concern of this paper to discuss this alternative in detail. * M. J. Holler, A Priori Party Power and Government Formation, Munich Social Science Reoirw, (1978), pp. ff. Leiserson. Factions and Coalitions in One-Party Japan.

10 MANFRED J. HOLLER 8. SOME EMPIRICAL RESULTS Table 1 shows some empirical results from calculating a priori voting power for the parties in the Finnish Parliament in the period of In the first row you will find figures attached to the eleven parties which have been represented in the Finnish Parliament since the election of 198. In the second row the voting weights (Lee, the relative numbers of seats) are listed in decreasing magnitude. In the third row you will find the values of the index h, and in the fourth row there are the (standardized) values of the Banzhaf index. For smaller parties (party to 11 ), with exception of party (LKP) in the years 198 and 1970, the values of the index h are larger than the values of the Banzhaf index and the relative seat shares. For larger parties (party 1 to ) the values of the index h tend to be smaller than both of these measures. For the periods 191, 19, 19,1970, 197, and 197 the power weights as measured by the index h are not monotonic with the seat shares. Thus, by this empirical result we cannot say that the paradox of nonmonotonicity is an exception. It seems to occur rather frequently. However, from the discussion above we can see that there is nothing paradoxical about this phenomenon if we accept the notion of power and the hypothesis of coalition formation underlying the index h. Holler, A Priori Party Power and Government Formation, pp. 8 ff. 7 I