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On the real voting power index
European Journal of Political Economy, Vol. 4, No. 2 (1988), 285--291 © Transfer Verlag, Regensburg, FRG
On the Real Voting Power Index BjSrn Erik Rasch*
1. Introduction In a recent article, Stenlund, Lane and Bjurulf(1985) (hereafter, SLB) have distinguished between formal and real voting power. In models of formal voting power, attention is focused on the ability of actors to affect outcomes in constitutional settings (e.g., the Shapley-Shubik and Banzhaf indices). Real voting power, on the other hand, can be seen as "the actual amount of voting power as determined by political circumstances, in particular the pattern of coalition building" (SLB, p.59-60). In this note I will argue that there is no easy route from formal to real voting power, and hence, that there are severe problems inherent in the type of measurement advocated by SLB. The main reason for this conclusion is that reai-world collective decisions are regularly reached through rather complex voting procedures, and the effect of voting procedures on coalition formation and power is not captured in the model of real voting power (or in fact, in any power index at all). ~. Voting Power
Formal and real power indices are based on quite different probability assumptions. It is central to the (formal) Shapley-Shubik model that the different permutations of voting sequences are equally likely. In the (formal) Banzhaf indices, it is assumed that all coalitions or all winmng coalitions are equally likely. In contrast, the essence of empirical or real voting power analysis is that estimated probabilities of certain coalitions are utilized. To be more precise, in the framework of SLB, the real voting power of a voter i is: RVPi= ~IiP(S), where I equals 1 when voter i can swing coalition S from winning to losing, and 0 otherwise. P(S) is the estimated probability of coalition S. In their analysis, SLB estimated P(S) by "using data from the actual coalitions which occurred in the Swedish Riksdag 1971-1979" (p. 63).
286
Real Voting Power Index
3. Amendment Voting Method Issues faced by the Swedish Riksdag quite often involve more than two alternatives (Bjurulf and Niemi, 1978). To reach a decision on these issues, the Pdksdag uses a variant of the amendment procedure. This is a method of palrwise comparisons. First, a voting order is determined. Then, the first two alternatives (of the chosen voting order) are paired using a majority vote. The defeated alternative is eliminated, while the winner of the first poll meets the next alternative in the voting order. Again one alternative is eliminated, and the surviving alternative paired with a new one; and so forth. The alternative that wins the final comparison is adopted as the decision. The set of actors voting in favor of the adopted alternative at the last voting stage, is called the winning coalition (see below). When there exist more than two alternatives and none of them commands a majority of first preference votes, it is immediately clear that:
Proposition I: The winning coalition on an issue depends (more or less) on (i) the order of voting, and (ii) the strategic attitudes of the actors (parliamentarians or parties). A simple example may be illustrative. Suppose that three actors or groups, A,B and C, of about equal size have preferences over three alternatives a, b, and c. Actors' ranking are: A: abc (ais best; cis worst) B: bac C: cba. Alternative b then is the Condorcet winner (i.e., it beats both a and c in pairwise comparisons). Three voting orders are possible, all of them leading to b as the outcome if voting is sincere. With voting order abc i.e., the alternatives a and b are voted upon first - the winning coalition will consist of group A and group B (coalition AB). Note that alternative a is eliminated at the first stage, and b next beats c. Is it reasonable to call AB the winning coalition in this example? I think so. Both A and B have voted in favour of the alternative adopted; both actors bear responsibility for the final decision. This conclusion is, however, not without problems. For example, it totally ignores the first vote of the decision process. We could argue that because C votes in favour of alternative b in the first comparison, this group has also been necessary in making b the outcome. At the first voting stage, group C has the power to get b eliminated. (The price of voting against b in this specific axample would, however, be high. It would lead to a, i.e., C's worst alternative, as the outcome). Consequently, each of the groups has helped b on its way,
-
Real Voting Power Index
287
and they may all be regarded as winners. This conclusion is, I think, not very appealing or helpful. Intuitively, there are differences between the groups with regard to power and success, and no differences at all are grasped by calling A, B, and C winners. On the other hand, one could argue that as long as both A and C have had their first preferences voted down, they are both losers. Then, group B is regarded as the sole winner of the voting process. But, in institutions governed by majority rule, it would be strange to view minorities as winners. Thus, our definition of winning coalition is clearly more appealing than the feasible alternatives. (The remaining voting orders acb and bca both lead to B and C as sole winning groups and b as the selected alternative.) So far, strategic voting has been disregarded. In general, with perfect information, strategic behaviour always leads to the adoption of existing Condorcet winners under the amendment procedure (see, e.g., McKelvey and Niemi, 1978). However, the situation may be different if information is incomplete or biased, leaving only a subset of the voters with possibilities of successful strategic voting. For example, let the voting order be bca. Only group A is sufficiently informed. When c and b are paired, actor A can vote strategically in favour of c (the worst alternative), thereby making the next poll a comparison between a and c. Thus, the winning alternative will be a, and AB the winning coalition. In order to relate the above remarks to voting power measurement, a clarification of how voting orders are formed in parliamentary settings becomes crucial. A central possibility is that the voting order is determined by application of more or less general rules. Examples of rules re/~ulating the choice of voting orders are: "vote upon status quo alternatives last", and "begin the voting process with the two most extreme alternatives, and introduce successively less extreme alternatives". To highlight some of the problems voting rules create in power analysis, we again turn to the example. Suppose now that a is a status quo proposal. Alternative b is a governmental bill, and c is an amended bill. The rules of the American Congress, where an amendment procedure is used, would invariably lead to BC as the winning coalition. There is no admissible voting order but eba. In the Swedish Riksda$, it is not required that status quo alternatives are voted upon last (Bjurulf and Niemi, 1978). Consequently, AB becomes a possibleor even likely - winning coalition. Hence, differences in winning coalitions may be due to differences in voting order rules (and not to differences in voting strength or preferences~. Voting rules clearly influence the likelihood of the occurrence of the possible winning coalitions in parliamentary voting. Consequently,
288
Real Voting Power Index
the real voting power of the actors is also affected. It seems strange and counter-intuitive to assign real voting power to an actor whose membership in a winning coalition simply rests on the mechanical functioning of a specific rule. To be powerful it is not sufficient just to be member of winning coalitions. Thus, the differences between formal and real voting power in the Swedish Riksdag may not solely or sufficently reflect "how political parties in their coalition behavior restrict the numbers and types of coalition possibilities in accordance with ideological preferences" (SLB, p.73). What happens if voting order formation is not restricted by rules, and parties a~'e free to select whatever voting order they want? This requires that the parties have preferences over voting orders or winning coalitions in addition to or in combination with preferences over alternatives. (Otherwise, the situation would be indeterminate.) Again, we can turn to the above example. For simplicity, strategic voting is excluded, and it is assumed that the actors are indifferent between (i) winning coalition BC as a result of voting order acb, and (ii) winning coalition BC as a result of voting order bca. (Remember that alternative b is adopted regardless of the voting order chosen or the winning coalition finally realized.) The possible strict preferences in this (over)simplified setting is shown in Figure 1. From Figure 1 we see that in two cases (1 and 8) the actors unanimously prefer one winning coalition to the other. In two other cases, as long as majority rule guides the choice of voting orders, the voting order is determined by the two actors who will end up as members of the winning coalition. Case 2 (coalition AB) and Case 4 (coaltion BC) are of this type. So far, then, winning coalitions also seem to be "ruling" coalitions or powerful actors. The real voting power index will probably function quite well in these cases. The remaining cases, however, are more problematical. Consider first cases 5 and 7. The voting order is determined by B, who is a member of the winning coalition, a~d by C, who is outside the winning coalition - a loser at the voting stage. In the two final cases, 3 and 6, actor B opposes the choice of voting order and yet is part of the resulting winning coalition. Thus actor B's first preference is adopted, but the actor's powerhflness can indeed be questioned. Here we are dealing with power relations too complex and ambiguous to be grasped in the real voting power model, and I believe, too common in real-world legislatures to be overlooked by the analyst (cf., Rasch, 1984). It must at least be shown in every application of the real voting power index that these difficulties represent no problem (and this is not done by SLB in their study).
Real Voting Power Index
289
It might be argued that it is not surprising that strange results occur if we allow party A to prefer the winning of coalition BC to the winning of AB.I Existing power indices are based on the assumption that actors strive to be members of winning coalitions. However, because we are dealing with real voting power measurement, it is crucial that the assumptions of the model utilized are realistic. The assumption that actors always waist membership in the winning coalition oversimplifies the matter, and may, accordingly, give rise to misleading index measures. The conclusion in the wake of the discussion in this section can now be formulated, assuming, that the actual voting body at least from time to time votes upon issues where more than two alternatives exist: Proposition $: The measurement of real voting power (i) is possibly distorted if the voting order is formed by application of general voting order rules, or (ii) may not grasp the real power relations, at work if the formation of voting orders is "free" (all possible voting orders are equally admissible).
4. Conclusion Although this article has been mainly critical or negative to the real voting power index, it is thereby not implied that the models of formal or a pr/or/voting power need to be equally dubious. It all depends on what problems the analyst wants to illuminate, and on how the results of calculations are interpreted. Thus, formal power indices may be suitable analytical instruments. It does, however, at least require that the actors exclusively strive for membership in winning coalitions. Moreover, it is probably necessary that there exist strong bargaining mechanJ~sms in the voting bodies under study. Such mechanisms may work effectively to reduce the alternative set; it becomes likely that the final vote on any issue will contain no more than two alternatives.
290
Real Voting Power Index
Figure I:
A
Alternative preferences of actors A, B, and C over winning coalitions AB and BC.
B
C
Winning coalition
Case
AB p BC
AB
Case I
BC p AB
AB
Case 2
AB p BC
AB
Case 3
BC p AB
BC
Case
AB p BC
AB
Case 5
BC p AB
BC
Case 6
AB p BC
BC
Case 7
BC p AB
BC
Case 8
AB p BC AB p BC a BC p AB
AB p BC BC p AB BC p AB
a: "AB p BC" means that winning coalition AB is (strictly) preferred to winning coalition BC.
No~es: * University of Oelo, Institute of Political Science, P.O.Box 1097, Blindem, 0317 Oslo 3, Norway. I am grategul to Richard Niemi, Manfred Holler, 35rgen Hermansson and an anonymous referee for most helpful suggestions and comments. 1 The reason for such a "perverse" preference may be that the decision is assumed to be unpopular. A party can free itself from any responsibility for the outcome by acting on the perverse preference when the voting order is formed.
Real Voting Power Index
291
References Bjurulf, B. and Niemi, R. (1978). "Strategic voting in Scandinavian parliaments", Scandinavian Political Studies, 1, pp. 5-22. McKelvey, R. and Niemi, R. (1978). "A multistage game representation of sophisticated voting for binary procedures", Journal of Economic Theory, 18, pp. 1-22. Rasch, B.E. (1984). "Et spillperspektiv p~ voteringsstyrke (A Game---Theoretical Perspective on Voting Strength)", Tidsskri~ for samfunnsforskning, 25, pp. 563--582. Stenlund, H., Lane, J.E. and Bjurulf, B. (1985). "Formal and real voating power", European Journal of Political Economy, 1, pp. 59-75.
Abstract: Both formal and real voting power indices completely disregard the significance of voting methods to processes of coalition formation. This may result in misleading figures of voting strength if estimated probabilities of certain coalitions are utilized in the indices. It is shown that application of the real voting power index of Stenlund, Lane and Bjurulf (a measure closely related to the Banzhaf index) to parliaments using the amendment voting procedure, is problematical. If a parliament votes upon three or more alternatives, the winning coalition on the issue depends more or less on the order of voting. The measurement of real voting power is distorted both if the formation of the voting order is based on application of general voting order rules, or if it is totally unregulated.
On the Real Voting Power Index BjSrn Erik Rasch*
1. Introduction In a recent article, Stenlund, Lane and Bjurulf(1985) (hereafter, SLB) have distinguished between formal and real voting power. In models of formal voting power, attention is focused on the ability of actors to affect outcomes in constitutional settings (e.g., the Shapley-Shubik and Banzhaf indices). Real voting power, on the other hand, can be seen as "the actual amount of voting power as determined by political circumstances, in particular the pattern of coalition building" (SLB, p.59-60). In this note I will argue that there is no easy route from formal to real voting power, and hence, that there are severe problems inherent in the type of measurement advocated by SLB. The main reason for this conclusion is that reai-world collective decisions are regularly reached through rather complex voting procedures, and the effect of voting procedures on coalition formation and power is not captured in the model of real voting power (or in fact, in any power index at all). ~. Voting Power
Formal and real power indices are based on quite different probability assumptions. It is central to the (formal) Shapley-Shubik model that the different permutations of voting sequences are equally likely. In the (formal) Banzhaf indices, it is assumed that all coalitions or all winmng coalitions are equally likely. In contrast, the essence of empirical or real voting power analysis is that estimated probabilities of certain coalitions are utilized. To be more precise, in the framework of SLB, the real voting power of a voter i is: RVPi= ~IiP(S), where I equals 1 when voter i can swing coalition S from winning to losing, and 0 otherwise. P(S) is the estimated probability of coalition S. In their analysis, SLB estimated P(S) by "using data from the actual coalitions which occurred in the Swedish Riksdag 1971-1979" (p. 63).
286
Real Voting Power Index
3. Amendment Voting Method Issues faced by the Swedish Riksdag quite often involve more than two alternatives (Bjurulf and Niemi, 1978). To reach a decision on these issues, the Pdksdag uses a variant of the amendment procedure. This is a method of palrwise comparisons. First, a voting order is determined. Then, the first two alternatives (of the chosen voting order) are paired using a majority vote. The defeated alternative is eliminated, while the winner of the first poll meets the next alternative in the voting order. Again one alternative is eliminated, and the surviving alternative paired with a new one; and so forth. The alternative that wins the final comparison is adopted as the decision. The set of actors voting in favor of the adopted alternative at the last voting stage, is called the winning coalition (see below). When there exist more than two alternatives and none of them commands a majority of first preference votes, it is immediately clear that:
Proposition I: The winning coalition on an issue depends (more or less) on (i) the order of voting, and (ii) the strategic attitudes of the actors (parliamentarians or parties). A simple example may be illustrative. Suppose that three actors or groups, A,B and C, of about equal size have preferences over three alternatives a, b, and c. Actors' ranking are: A: abc (ais best; cis worst) B: bac C: cba. Alternative b then is the Condorcet winner (i.e., it beats both a and c in pairwise comparisons). Three voting orders are possible, all of them leading to b as the outcome if voting is sincere. With voting order abc i.e., the alternatives a and b are voted upon first - the winning coalition will consist of group A and group B (coalition AB). Note that alternative a is eliminated at the first stage, and b next beats c. Is it reasonable to call AB the winning coalition in this example? I think so. Both A and B have voted in favour of the alternative adopted; both actors bear responsibility for the final decision. This conclusion is, however, not without problems. For example, it totally ignores the first vote of the decision process. We could argue that because C votes in favour of alternative b in the first comparison, this group has also been necessary in making b the outcome. At the first voting stage, group C has the power to get b eliminated. (The price of voting against b in this specific axample would, however, be high. It would lead to a, i.e., C's worst alternative, as the outcome). Consequently, each of the groups has helped b on its way,
-
Real Voting Power Index
287
and they may all be regarded as winners. This conclusion is, I think, not very appealing or helpful. Intuitively, there are differences between the groups with regard to power and success, and no differences at all are grasped by calling A, B, and C winners. On the other hand, one could argue that as long as both A and C have had their first preferences voted down, they are both losers. Then, group B is regarded as the sole winner of the voting process. But, in institutions governed by majority rule, it would be strange to view minorities as winners. Thus, our definition of winning coalition is clearly more appealing than the feasible alternatives. (The remaining voting orders acb and bca both lead to B and C as sole winning groups and b as the selected alternative.) So far, strategic voting has been disregarded. In general, with perfect information, strategic behaviour always leads to the adoption of existing Condorcet winners under the amendment procedure (see, e.g., McKelvey and Niemi, 1978). However, the situation may be different if information is incomplete or biased, leaving only a subset of the voters with possibilities of successful strategic voting. For example, let the voting order be bca. Only group A is sufficiently informed. When c and b are paired, actor A can vote strategically in favour of c (the worst alternative), thereby making the next poll a comparison between a and c. Thus, the winning alternative will be a, and AB the winning coalition. In order to relate the above remarks to voting power measurement, a clarification of how voting orders are formed in parliamentary settings becomes crucial. A central possibility is that the voting order is determined by application of more or less general rules. Examples of rules re/~ulating the choice of voting orders are: "vote upon status quo alternatives last", and "begin the voting process with the two most extreme alternatives, and introduce successively less extreme alternatives". To highlight some of the problems voting rules create in power analysis, we again turn to the example. Suppose now that a is a status quo proposal. Alternative b is a governmental bill, and c is an amended bill. The rules of the American Congress, where an amendment procedure is used, would invariably lead to BC as the winning coalition. There is no admissible voting order but eba. In the Swedish Riksda$, it is not required that status quo alternatives are voted upon last (Bjurulf and Niemi, 1978). Consequently, AB becomes a possibleor even likely - winning coalition. Hence, differences in winning coalitions may be due to differences in voting order rules (and not to differences in voting strength or preferences~. Voting rules clearly influence the likelihood of the occurrence of the possible winning coalitions in parliamentary voting. Consequently,
288
Real Voting Power Index
the real voting power of the actors is also affected. It seems strange and counter-intuitive to assign real voting power to an actor whose membership in a winning coalition simply rests on the mechanical functioning of a specific rule. To be powerful it is not sufficient just to be member of winning coalitions. Thus, the differences between formal and real voting power in the Swedish Riksdag may not solely or sufficently reflect "how political parties in their coalition behavior restrict the numbers and types of coalition possibilities in accordance with ideological preferences" (SLB, p.73). What happens if voting order formation is not restricted by rules, and parties a~'e free to select whatever voting order they want? This requires that the parties have preferences over voting orders or winning coalitions in addition to or in combination with preferences over alternatives. (Otherwise, the situation would be indeterminate.) Again, we can turn to the above example. For simplicity, strategic voting is excluded, and it is assumed that the actors are indifferent between (i) winning coalition BC as a result of voting order acb, and (ii) winning coalition BC as a result of voting order bca. (Remember that alternative b is adopted regardless of the voting order chosen or the winning coalition finally realized.) The possible strict preferences in this (over)simplified setting is shown in Figure 1. From Figure 1 we see that in two cases (1 and 8) the actors unanimously prefer one winning coalition to the other. In two other cases, as long as majority rule guides the choice of voting orders, the voting order is determined by the two actors who will end up as members of the winning coalition. Case 2 (coalition AB) and Case 4 (coaltion BC) are of this type. So far, then, winning coalitions also seem to be "ruling" coalitions or powerful actors. The real voting power index will probably function quite well in these cases. The remaining cases, however, are more problematical. Consider first cases 5 and 7. The voting order is determined by B, who is a member of the winning coalition, a~d by C, who is outside the winning coalition - a loser at the voting stage. In the two final cases, 3 and 6, actor B opposes the choice of voting order and yet is part of the resulting winning coalition. Thus actor B's first preference is adopted, but the actor's powerhflness can indeed be questioned. Here we are dealing with power relations too complex and ambiguous to be grasped in the real voting power model, and I believe, too common in real-world legislatures to be overlooked by the analyst (cf., Rasch, 1984). It must at least be shown in every application of the real voting power index that these difficulties represent no problem (and this is not done by SLB in their study).
Real Voting Power Index
289
It might be argued that it is not surprising that strange results occur if we allow party A to prefer the winning of coalition BC to the winning of AB.I Existing power indices are based on the assumption that actors strive to be members of winning coalitions. However, because we are dealing with real voting power measurement, it is crucial that the assumptions of the model utilized are realistic. The assumption that actors always waist membership in the winning coalition oversimplifies the matter, and may, accordingly, give rise to misleading index measures. The conclusion in the wake of the discussion in this section can now be formulated, assuming, that the actual voting body at least from time to time votes upon issues where more than two alternatives exist: Proposition $: The measurement of real voting power (i) is possibly distorted if the voting order is formed by application of general voting order rules, or (ii) may not grasp the real power relations, at work if the formation of voting orders is "free" (all possible voting orders are equally admissible).
4. Conclusion Although this article has been mainly critical or negative to the real voting power index, it is thereby not implied that the models of formal or a pr/or/voting power need to be equally dubious. It all depends on what problems the analyst wants to illuminate, and on how the results of calculations are interpreted. Thus, formal power indices may be suitable analytical instruments. It does, however, at least require that the actors exclusively strive for membership in winning coalitions. Moreover, it is probably necessary that there exist strong bargaining mechanJ~sms in the voting bodies under study. Such mechanisms may work effectively to reduce the alternative set; it becomes likely that the final vote on any issue will contain no more than two alternatives.
290
Real Voting Power Index
Figure I:
A
Alternative preferences of actors A, B, and C over winning coalitions AB and BC.
B
C
Winning coalition
Case
AB p BC
AB
Case I
BC p AB
AB
Case 2
AB p BC
AB
Case 3
BC p AB
BC
Case
AB p BC
AB
Case 5
BC p AB
BC
Case 6
AB p BC
BC
Case 7
BC p AB
BC
Case 8
AB p BC AB p BC a BC p AB
AB p BC BC p AB BC p AB
a: "AB p BC" means that winning coalition AB is (strictly) preferred to winning coalition BC.
No~es: * University of Oelo, Institute of Political Science, P.O.Box 1097, Blindem, 0317 Oslo 3, Norway. I am grategul to Richard Niemi, Manfred Holler, 35rgen Hermansson and an anonymous referee for most helpful suggestions and comments. 1 The reason for such a "perverse" preference may be that the decision is assumed to be unpopular. A party can free itself from any responsibility for the outcome by acting on the perverse preference when the voting order is formed.
Real Voting Power Index
291
References Bjurulf, B. and Niemi, R. (1978). "Strategic voting in Scandinavian parliaments", Scandinavian Political Studies, 1, pp. 5-22. McKelvey, R. and Niemi, R. (1978). "A multistage game representation of sophisticated voting for binary procedures", Journal of Economic Theory, 18, pp. 1-22. Rasch, B.E. (1984). "Et spillperspektiv p~ voteringsstyrke (A Game---Theoretical Perspective on Voting Strength)", Tidsskri~ for samfunnsforskning, 25, pp. 563--582. Stenlund, H., Lane, J.E. and Bjurulf, B. (1985). "Formal and real voating power", European Journal of Political Economy, 1, pp. 59-75.
Abstract: Both formal and real voting power indices completely disregard the significance of voting methods to processes of coalition formation. This may result in misleading figures of voting strength if estimated probabilities of certain coalitions are utilized in the indices. It is shown that application of the real voting power index of Stenlund, Lane and Bjurulf (a measure closely related to the Banzhaf index) to parliaments using the amendment voting procedure, is problematical. If a parliament votes upon three or more alternatives, the winning coalition on the issue depends more or less on the order of voting. The measurement of real voting power is distorted both if the formation of the voting order is based on application of general voting order rules, or if it is totally unregulated.