1 Introduction
The aggregation of individual preferences by some form of voting is common in politics, business, and everyday life. Members of a board, council, or committee are rarely aware how much their collective choices depend on the adopted aggregation rule. The importance of the method may be identified a posteriori by comparing voting outcomes for a given preference configuration. For instance, suppose a hiring committee involved three groups with , , and members each and the following preferences over five candidates : , , and . If the groups voted sincerely (for informational, institutional, or other reasons) then candidate would have received the position under plurality rule with 6 vs. 5 vs. 3 votes. However, requiring a runoff between the frontrunners, given that none has majority support, would have made the winner with 8 to 6 votes. Candidate would have won every pairwise comparison and been the Condorcet winner; Borda rule would have singled out ; candidate could have been the winner if approval voting had been used.
With enough posterior information, each voter group can identify a ‘mostpreferred voting method’ for the decision at hand: group 1 should have tried to impose plurality rule in order to have its way; group 2 should have argued for plurality runoff; and group 3 for pairwise comparisons. It is less obvious, though, how adoption of one aggregation method rather than another will affect a group’s success and influence a priori, i.e., not yet knowing what will be the applicable preferences, or averaged across many decisions. More generally, can we verify if small groups are enjoying a greater say when committees fill a top position, elect an official, etc. by pairwise votes or by the plurality runoff method? Which rules from a given list of suggestions tend to maximize (or minimize) voting power of a particularly sized group in a committee? There is a huge literature on voting power but questions of this kind have to our knowledge not been addressed by it yet.^{1}^{1}1The most closely related analysis seems to be the investigation of effectivity functions; cf. Peleg:1984. See Felsenthal/Machover:1998, Laruelle/Valenciano:2008, or Napel:2018 for overviews on the measurement of voting power. We seek to change this by generalizing tools that were developed for analysis of simple voting games with binary options (‘yes’ or ‘no’) to collective choice from alternatives.
The most prominent tools for analyzing the former are the PenroseBanzhaf index (Penrose:1946; Banzhaf:1965) and the ShapleyShubik index [Shapley and ShubikShapley and Shubik1954]. They evaluate the sensitivity of the outcome to changes in a given voter’s preferences – operationalized as the likelihood of this voter being pivotal or critical: flipping its vote would swing the collective decision – under specific probabilistic assumptions. This paper applies the same logic to weighted committee games. These, more simply referred to as weighted committees, are tuples that specify a set of players, a set of alternatives, and the combination of an anonymous voting method
(e.g., Borda rule, plurality rule, and so on) with a vector
of integers that represents group sizes, voting shares, etc.As in analysis of (binary) simple voting games, high a priori voting power of player in a committee goes with high sensitivity of the outcome to
’s preferences. It can be quantified as the probability for a change in
’s preferences causing a change of the collective choice. Although other assumptions could be made, we focus on the simplest case in which all profiles of strict preference orderings are equally likely a priori. This corresponds to the PenroseBanzhaf index for . The respective power indications help to assess who gains and loses from institutional reforms or whether the distribution of influence in a committee satisfies some fairness criterion; they can also inform stakeholders, lobbyists, or other committee outsiders with an interest in who has how much say in the committee.2 Related Work
The distribution of power in binary weighted voting games has received wide attention ever since [Ch. 10]vonNeumann/Morgenstern:1953 formalized them as a subclass of socalled simple (voting) games. See, e.g., Mann/Shapley:1962, Riker/Shapley:1968, Owen:1975:presidential, or Brams:1978 for seminal investigations. The binary framework can be restrictive, however. Even for collective ‘yes’or‘no’ decisions, individual voters usually have more than two options. For instance they can abstain or not even attend a vote, and this may affect the outcome differently than casting a vote either way. Corresponding situations have been formalized as ternary voting games (Felsenthal/Machover:1997; Tchantcho/DiffoLambo/Pongou/Engoulou:2008; Parker:2012) and quaternary voting games [Laruelle and ValencianoLaruelle and Valenciano2012]. Players can also be allowed to express graded intensities of support: in games, studied by Hsiao/Raghavan:1993 and Freixas and Zwicker (Freixas/Zwicker:2003, Freixas/Zwicker:2009), each player selects one of ordered levels of approval. The resulting partitions of players are mapped to one of ordered output levels; suitable power indices have been defined by Freixas (Freixas:2005:Banzhaf, Freixas:2005:Shapley).
Linear orderings of actions and feasible outcomes, as required by games, are naturally given in many applications but fail to exist in others. Think of options that have multidimensional attributes – for instance, candidates for office or an open position, policy programs, locations of a facility, etc. Pertinent extensions of simple games, along with corresponding power measures, have been introduced as multicandidate voting games by Bolger:1986 and taken up as simple games by Amer/Carreras/Magana:1998b and as weighted plurality games by Chua/Ueng/Huang:2002. They require each player to vote for a single candidate. This results in partitions of player set that, in contrast to games, are mapped to a winning candidate without ordering restrictions.
We will draw on the yet more general framework of weighted committee games (Kurz/Mayer/Napel:2018). Winners in these games can depend on the entire preference rankings of voters rather than just the respective top. We conceive of player ’s influence or voting power as the sensitivity of joint decisions to ’s actions or likings. The resulting ability to affect collective outcomes is closely linked to the opportunity to manipulate social choices in the sense of Gibbard:1973 and Satterthwaite:1975. Our investigation therefore relates to computational studies by Nitzan:1985, Kelly:1993, Aleskerov/Kurbanov:1999, or Smith:1999 that have quantified the aggregate manipulability of a given decision rule. The conceptual difference between corresponding manipulability indices and the power index defined below is that the latter is evaluating consequences of arbitrary preference perturbations, while the indicated studies only look at strategic preference misrepresentation that is beneficial from the perspective of a player’s original preferences.^{2}^{2}2Nitzan:1985 also checked if outcomes could be affected by arbitrary variations of preferences before assessing manipulation. He tracked this at the aggregate level, while we break it down to individuals in order to link outcome sensitivity to voting weights. Voting power as we quantify it could be used for a player’s strategic advantage but it need not. A ‘preference change’ might also be purely idiosyncratic, result from logrolling or external lobbying (where costs of persuasion can relate more to preference intensity than a player’s original ranking of options), or could be a demonstration of power for its own sake.
Other conceptualizations of the influence derived from a given collective choice rule track the sets of outcomes that can be induced by partial coalitions. For instance, Moulin:1981 uses veto functions in order to describe outcomes that given coalitions of players could jointly prevent; Peleg’s Peleg:1984 effectivity functions describe the power structure in a committee by a list of all sets of alternatives that specific coalitions of voters can force the outcome to lie in. We, by contrast, follow the literature pioneered by Penrose:1946, Shapley/Shubik:1954, and Banzhaf:1965, and try to assess individual influence on outcomes concisely by a number between zero and one.
3 Preliminaries
3.1 Anonymous Voting Rules
We consider finite sets of voters or players such that each voter has strict preferences over a set of alternatives. We write in abbreviation of when the player’s identity is clear. The set of all strict preference orderings on is denoted by . A (resolute) voting rule maps each preference profile to a single winning alternative . Rule is anonymous if for any and any permutation with we have .
Rule  Winning alternative at preference profile 
Borda  
Copeland  
Plurality  
Plurality runoff  
Schulze  – see Schulze:2011 
We will restrict attention to truthful voting^{3}^{3}3In principle, power analysis could also be carried out for strategic voters. This would require specifying the mapping from profiles of players’ preferences to the element of
(or a probability distribution over
) which is induced by the selected voting equilibrium. Determination of the latter usually is a hard task in itself and here left aside. under one of the five anonymous rules summarized in Table 1, assuming lexicographic tie breaking. See Laslier:2012 on the pros and cons of a big variety of voting procedures. Our selection comprises two positional rules (Borda, plurality), two Condorcet methods (Copeland, Schulze), and a twostage procedure that is used for filling political offices in many European jurisdictions (plurality runoff).Under plurality rule , each voter simply names his or her topranked alternative and the alternative that is ranked first by the most voters is chosen. This is also the winner under plurality with runoff rule if the obtained plurality constitutes a majority (i.e., more than 50% of votes); otherwise a binary runoff between the alternatives and that obtained the highest and secondhighest plurality scores in the first stage is conducted.
Borda rule has each player assign , , …, points to the alternative that he or she ranks first, second, etc. These points equal the number of alternatives that ranks below . The alternative with the highest total number of points, known as its Borda score, is selected.
Copeland rule considers pairwise majority votes between the alternatives. They define the majority relation and the alternative that beats the most others according to is selected. Copeland rule is a Condorcet method: if some alternative beats all others, then . The same is true for Schulze rule . But while just counts the number of direct pairwise victories, also considers indirect victories and invokes majority margins in order to evaluate their strengths. then picks the alternative that has the strongest chain of direct or indirect majority support (see Schulze:2011 for details). The attention paid to margins makes more sensitive to voting weights than in case is cyclical. Despite its nontrivial computation,
has been applied, e.g., by the Wikimedia Foundation and Linux opensource communities.
3.2 Weighted Committees
Anonymous rules treat any components and of a preference profile like indistinguishable ballots. Still, when a committee conducts pairwise comparisons, plurality voting, etc., two individual preferences often feed into the collective decision rather asymmetrically because, e.g., stockholders have as many votes as they own shares or the relevant players in a political committee cast bloc votes in proportion to party seats. The resulting (nonanonymous) mapping from preferences to outcomes is a combination of anonymous voting rule with weights for players that is defined for all by
(1) 
The combination of a set of voters, a set of alternatives and a particular weighted voting rule defines a weighted committee (game). When the underlying anonymous rule is plurality rule , then is called a (weighted) plurality committee. Similarly, , , and are referred to as a plurality runoff committee, Borda committee, Copeland committee, and Schulze committee. See Kurz/Mayer/Napel:2018 on structural differences between them.
Weighted committee games and are equivalent if the respective mappings from preference profiles to outcomes coincide; that is, when for all . Equivalent games evidently come with equivalent expectations for individual players to influence the collective decision (voting power) and to obtain outcomes that match their own preferences (success). We will focus on voting power and nonequivalent committees that involve either the same rule but different weights and , or the same weights but different rules and . Example questions that we would like to address are: to what extent does a change of voting weights, implied for example by the recent reallocation of voting rights in the International Monetary Fund or partyswitching of a member of parliament, shift the respective balance of power? How does players’ attractiveness to lobbyists change when a committee replaces one voting method by another?
4 Measuring Influence in Weighted Committees
Some obvious shortcomings notwithstanding (see, e.g., Garrett/Tsebelis:1999, Garrett/Tsebelis:2001), voting power indices such as the PenroseBanzhaf and the ShapleyShubik index have received wide attention in theoretical and applied analysis of binary decisions. See, e.g., the contributions in Holler/Nurmi:2013. Most indices can be seen as operationalizing power of player as expected sensitivity of collective decisions to player ’s behavior [Napel and WidgrénNapel and Widgrén2004]. Sensitivity in the binary case means that, taking the behavior of other players as given, the collective outcome would have been different had player voted ‘no’ instead of ‘yes’, or ‘yes’ instead of ‘no’. Distinct indices reflect distinct probabilistic assumptions about the voting configurations that are evaluated. For instance, the PenroseBanzhaf index is predicated on the assumption that the preferences of all
players are independent random variables and the
different ‘yes’‘no’ configurations are equally likely.4.1 Power as (Normalized) Expected Sensitivity of the Outcome
It is not hard to generalize the idea of measuring influence as outcome sensitivity to weighted committees
. Continuing in the PenroseBanzhaf tradition, we will assume that individual preferences are drawn independently from the uniform distribution over
, i.e., all profiles are equally likely.^{4}^{4}4This is also known as the impartial culture assumption. It has limited empirical support (see, e.g., [Ch. 1]Regenwetter/Grofman/Marley/Tsetlin:2012) but is commonly adopted as a starting point for assessing links between voting weights and power a priori. We leave the pursuit of other options for future research (e.g., singlepeakedness along a common dimension). In order to assess the voting power of player with weight , we perturb ’s realized preferences to a random and check if this individual preference change would change the collective outcome.^{5}^{5}5One might also restrict attention to local perturbations, that is, only allow changes of to adjacent orderings. So when and , one would impose the constraint . This would not change the qualitative observations in the IMF section below. Specifically, using notation with , we are interested in the behavior of indicator function(2) 
We stay agnostic about the precise source of perturbations: the switch from to might reflect a spontaneous change of mind or intentional preference misrepresentation, e.g., because someone has bought ’s vote. Variations might also be the result of a mistake or of receiving lastminute private information about some of the candidates. Our important premise is only that: a committee member’s input to the collective decision process matters more, the more influential player is in the committee and vice versa.
We can then quantify player ’s a priori influence or power – and compare it to that of other players or for variations of – by taking expectations over all conceivable and all possible perturbations of at any given :
(3) 
A value of , for example, means that 25% of player ’s preference variations would change the outcome.
The expected value in (3) equals the probability that a change of player ’s preferences from to random at a randomly drawn profile affects the outcome. It is zero if and only if player is a null player, i.e., its preferences never make a difference to the committee decision. However, falls short of one for a dictator player, i.e., when if and only if player ranks top: since only changes of the dictator’s top preference matter, only out of perturbations of affect the outcome. We suggest to normalize power indications so that they range from zero to one. Specifically, we focus on the voting power index with
(4) 
denoting player ’s a priori influence or voting power in weighted committee .
The normalization destroys ’s interpretation as a probability but facilitates comparison across committees. Regardless of how many alternatives and players are involved, indicates how close player is to being a dictator in . , for instance, implies that ’s influence lies halfway between that of a null player and a dictator. So, on average, outcomes are half as sensitive to ’s preferences than they would be if commanded all votes.
4.2 Relationship to the PenroseBanzhaf Index
For alternatives, the denominators in (3) and (4) equal and . Moreover, for any of the rules introduced above, weighted committees coincide with the subclass of simple voting games where is given by with for all . If we consider the simple game induced by , its PenroseBanzhaf index turns out to coincide with . Namely, ’s usual definition then specializes to
(5)  
In this case, for with lexicographic tiebreaking,
(6) 
The last line substitutes where . Hence
(7)  
as every arises for two profiles , one involving and the other .
5 Illustration
5.1 A Toy Example
As a first illustration let us evaluate the distribution of voting power when our stylized hiring committee (see Introduction) with three groups of 6, 5, and 3 members adopts Borda rule , that is weighted committee . With candidates, the applicant ranked first by any two groups wins. So all three players are symmetric and have identical power according to the PenroseBanzhaf or any other established voting power index.
The symmetry is broken when three or more candidates are involved. Given , evaluates all strict preference profiles and checks whether a change of player ’s respective preference makes a difference to the Borda winner. Table 2 illustrates this for profile . The Borda winner at has a score of points vs. 10 for vs. 12 for (first block of table). When preferences of group 1 are varied (second block), changes to result in a new Borda winner (indicated by an asterisk) while does not. Similarly, three out of five perturbations of player 2’s preferences would change the outcome (third block); however, no variation of affects the committee choice (last block). The considered profile therefore contributes to .
*  14  6          16  6  
*  8  12  10  15  *  16  9  
16  6  5  12  13  6  
10  12          0  17  10  9  
16  8  *  5  15  *  13  12  
10  14  *  0  20  *         
Aggregating the corresponding figures for all yields
(8) 
So for the weights at hand, group 1 has almost 70% of the opportunities to swing the collective choice that it would have when deciding alone. This figure is roughly halved for group 3 even though both are symmetric when choosing between two candidates. The comparison shows that traditional voting power indices for simple voting games, such as , can yield highly misleading conclusions when decisions of the investigated institution involve more than alternatives. (This is one of the shortcomings alluded to earlier.) One can also see from the numbers in (8) that need not add to one: the collective outcome at a given may be sensitive to the preferences of several players at the same time, or to those of none.^{6}^{6}6Therefore, normalization is sometimes applied in binary voting analysis.
(0.6667, 0.4444, 0.4444)  (0.7500, 0.3750, 0.3750)  (0.8000, 0.3200, 0.3200)  
(0.5556, 0.5556, 0.5000)  (0.5833, 0.5833, 0.5000)  (0.6000, 0.6000, 0.5000)  
(0.6806, 0.5972, 0.3611)  (0.7372, 0.6246, 0.3644)  (0.7631, 0.6462, 0.3839)  
(0.5509, 0.5509, 0.5509)  (0.5851, 0.5851, 0.5851)  (0.6098, 0.6098, 0.6098)  
(0.5972, 0.5278, 0.5278)  (0.6584, 0.5426, 0.5426)  (0.7011, 0.5515, 0.5515) 
Of course, manual computations as in Table 2 are tedious. It is not difficult, though, to evaluate with a standard desktop computer for up to five alternatives; and to compare the respective distribution of voting power to that arising from other voting rules. Findings are summarized for in Table 3. As the comparison between and 3 showed already, voting powers vary in the number of alternatives. Under plurality rule, for instance, player 1 is closer to having dictatorial influence, the more alternatives split the vote of players 2 and 3. tends to as (given sincere voting and independent preferences).^{7}^{7}7Comparative statics are more involved for other rules: bigger tends to raise the share of profiles at which some perturbation of affects the outcome but lowers the fraction of perturbations that do so for both player and the hypothetical dictator used as normalization. Superposition of these effects here increases power for all players and , but this is not true in general. That power of all three players coincides under confirms that Copeland rule extends the symmetry relation between players for alternatives to any number (see [Prop. 3]Kurz/Mayer/Napel:2018).
5.2 Election of the IMF’s Managing Director
Evaluation of is, of course, more interesting for realworld institutions than toy examples. We pick the International Monetary Fund (IMF) as a case in point. Binary power indices have been applied to it many times. See, e.g., Leech Leech:2002,Leech:2003, Aleskerov/Kalyagin/Pogorelskiy:2008, or Leech/Leech:2013. We extend the analysis to election of the IMF’s Managing Director from three candidates by the Executive Board.
The Executive Board consists of 24 members whose voting weights reflect financial contributions to the IMF, socalled quotas. The six largest contributors – USA, China, Japan, Germany, France and the UK – and Saudi Arabia currently provide one Executive Director each. The remaining 182 countries are grouped into seventeen constituencies. Each supplies one Executive Director who represents the group members and wields their combined voting rights.
Various changes to the distribution of quotas have taken place since the IMF’s inception in 1944 at Bretton Woods. The most recent reform was agreed in 2010 and started to be implemented in 2016. A significant share of votes has shifted from the USA and Europe to emerging and developing countries. In particular, China’s vote share has gone up to 6.1% (compared to 3.8% before). India’s share increased to 2.6% (2.3%), Russia’s to 2.6% (2.4%), Brazil’s to 2.2% (1.7%) and Mexico’s to 1.8% (1.5%). On the occasion, the IMF also modified the election process for its key representative, the Managing Director (currently: Christine Lagarde).
Prior to the reform, the process was criticized as intransparent and undemocratic: the Managing Director used to be a European chosen in backroom negotiations with the US. The new process is advertised as “open, merit based, and transparent” (IMF Press Release 16/19): all Executive Directors and IMF Governors may nominate candidates. If the number of nominees is too big, a shortlist of three candidates is drawn up based on indications of support. From this shortlist the new Managing Director is elected “by a majority of the votes cast” in the Executive Board.^{8}^{8}8IMF Press Release 16/19, Part 4, holds that “Although the Executive Board may select a Managing Director by a majority of the votes cast, the objective of the Executive Board is to select the Managing Director by consensus …”. The same is said in Part 3 about adoption of the “shortlist”. Our analysis presumes that a consensus may not always be found – or it actually arises in the shadow of the anticipated outcome of voting.
Vote share (%)  
USA  16.72  16.47  0.7126  0.7030  0.6740  0.6653  0.6880  0.6790 
Japan  6.22  6.13  0.1986  0.1989  0.2239  0.2233  0.2164  0.2159 
China  3.80  6.07  0.1216  0.1967  0.1404  0.2209  0.1340  0.2135 
Netherlands  6.56  5.41  0.2092  0.1755  0.2350  0.1983  0.2277  0.1910 
Germany  5.80  5.31  0.1851  0.1720  0.2097  0.1950  0.2024  0.1876 
Spain  4.90  5.29  0.1567  0.1718  0.1789  0.1945  0.1717  0.1871 
Indonesia  3.93  4.33  0.1254  0.1403  0.1448  0.1607  0.1382  0.1538 
Italy  4.22  4.12  0.1349  0.1337  0.1551  0.1533  0.1482  0.1465 
France  4.28  4.02  0.1370  0.1306  0.1574  0.1499  0.1507  0.1432 
United Kingdom  4.28  4.02  0.1369  0.1304  0.1574  0.1498  0.1506  0.1431 
Korea  3.48  3.78  0.1114  0.1226  0.1291  0.1410  0.1230  0.1345 
Canada  3.59  3.37  0.1150  0.1093  0.1332  0.1265  0.1268  0.1203 
Sweden  3.39  3.28  0.1085  0.1063  0.1259  0.1231  0.1198  0.1171 
Turkey  2.91  3.22  0.0932  0.1044  0.1088  0.1209  0.1032  0.1149 
South Africa  3.41  3.09  0.1091  0.1001  0.1267  0.1162  0.1205  0.1104 
Brazil  2.61  3.06  0.0835  0.0993  0.0979  0.1154  0.0927  0.1096 
India  2.80  3.04  0.0898  0.0988  0.1048  0.1147  0.0993  0.1089 
Switzerland  2.94  2.88  0.0941  0.0935  0.1097  0.1087  0.1041  0.1030 
Russian Federation  2.55  2.83  0.0817  0.0920  0.0957  0.1070  0.0905  0.1015 
Iran  2.73  2.54  0.0874  0.0823  0.1024  0.0962  0.0970  0.0910 
Utd. Arab Emirates  2.57  2.52  0.0822  0.0817  0.0963  0.0955  0.0911  0.0904 
Saudi Arabia  2.80  2.01  0.0896  0.0652  0.1046  0.0767  0.0992  0.0723 
Dem. Rep. Congo  1.46  1.62  0.0465  0.0526  0.0555  0.0621  0.0521  0.0584 
Argentina  1.84  1.59  0.0587  0.0515  0.0695  0.0610  0.0654  0.0573 
(groups as of Dec. 2018 indicated by largest member, Nauru included in )
The IMF has neither publicly nor upon our email request specified what “majority of the votes cast” exactly means to it for three candidates. We take the resulting room for interpretation as an opportunity to simultaneously investigate the voting power effects of the weight reform and of a procedural choice between using plurality rule, plurality with a runoff if none of three shortlisted candidates secures 50% of the votes, or Copeland rule. In the spirit of earlier a priori analysis of the IMF, we maintain the independence assumption that underlies and
. This provides an a priori assessment of how level is the playing field created by weights as such rather than an estimate of who wields how much influence on the next decision given prevailing political ties, economic interdependencies, etc.
Influence figures in Table 4 are based on Monte Carlo simulations with sufficiently many iterations so that differences within rows are significant with confidence.^{9}^{9}9 The only exception is that the difference between and is not signficant. The large number of preference profiles renders exact calculation of impractical. We find that 2016’s increase of vote shares for emerging market economies has indeed raised their voting power, no matter which voting rule we consider. This is most pronounced for China, with an increase of more than 50%. Influence of the groups led by Brazil and Russia (incl. Syria) increased by about 18% and 12%, respectively; that of the Turkish and Indonesian group by about 11% each; the Indian and Spanish (incl. Mexico and others) groups gained about 10% and 9%, respectively. Intended or not, the South African group lost about 8% of its a priori voting power; Saudi Arabia is the greatest loser with roughly 27%. Germany, France and UK each lost between 5% and 7% while voting power of the USA stayed largely constant.
The computations exhibit a simple pattern regarding the adopted interpretation of ‘majority’: voting power of the USA is higher for plurality rule than for Copeland than for plurality runoff; the opposite applies to all other (groups of) countries. This echoes findings for our toy example: the largest player’s influence was highest for ; small and medium players were more influential under and .
6 Towards More General Rule Comparisons
It seems worthwhile to check whether observations like the ones above are robust: does the largest group benefit from plurality votes or the smallest from pairwise comparisons in general? And can recommendations for maximizing a player’s influence be given also if information about the exact distribution of voting weights is vague or fluctuates? We take a first step beyond specific examples and look for possible size biases of the rules. Attention is restricted to small numbers of players and alternatives, namely . The respective intuitions may still apply more generally to shareholder meetings, weighted voting in political bodies, etc.
We use the standard projection of the 3dimensional simplex of relative weights to the plane in order to represent all possible weight distributions among three players: vertices give 100% of voting weight to the indicated player, the midpoint corresponds to , etc. Each figure presents the result of comparing influence of player 1 under some rule vs. rule . Areas colored green (red) indicate weight distributions for which . Borda rule was found to be particularly sensitive to weight differences by Kurz/Mayer/Napel:2018; when it is involved in a comparison, we use darker tones of green or red to indicate greater influence differences.
6.1 Borda vs. Plurality
The major cases that arise when we compare player 1’s influence in Borda vs. plurality committees are numbered in Figure 1. We focus on and write , , and . The following recommendations could be given to an influencemaximizing player 1 if the procedural choice between and is at this player’s discretion:

If you wield the majority of votes (region 1) impose plurality rule.
Namely, makes you both a plurality and Borda dictator (region 1a); implies dictatorship only under plurality rule (region 1b). 
Also impose plurality rule (region 5)

if your weight is smallest and both other players have a third to half of the votes each (), or

if you have less than a third of votes and the largest player falls short of the majority by no more than a quarter of the remaining player’s votes ().


Otherwise, as a good ‘rule of thumb’, impose Borda rule.
Namely, when some other player holds the majority (region 2, ) the observations for region 1 essentially get reversed. In case that nobody holds the majority, Borda comes with greater influence if you are secondlargest with at least a third of votes (region 4, ). This extends weakly to when you are largest (region 3). The only exception to the rule are two small subregions where all weights are similar but .
6.2 Further Comparisons
Analogous pairwise influence comparisons are depicted in the Appendix for all possible weight distributions and . Again, Borda’s high responsiveness to weight differences makes for more detailed case distinctions (see Figures A1–A4). By contrast, when plurality rule is compared to either Copeland, plurality runoff, or Schulze rule (Figure A5), the recommendation always is simple and intuitive: plurality rule maximizes influence if you have the most votes. If anyone else has more votes, your influence is greater (at least weakly) under the respective other rule.
Recommendations to an influencemaximizing player are similar for Copeland vs. Schulze rule (Figure A6): if you wield a plurality of votes, Schulze comes with greater influence; in case someone else has more votes, it is the opposite. For Copeland vs. plurality runoff (Figure A7), the former gives greater influence to you if you have at least the secondmost votes. If player 1 is to choose between plurality runoff and Schulze rule (Figure A8), Schulze rule gives greater influence if is either largest or smallest; otherwise it is better to adopt plurality runoff.
6.3 Influencemaximizing Voting Rules
Complementing pairwise comparisons as in Figures 1 and A1–A8, one can also check directly which of the considered voting rules maximizes a specific player’s a priori voting power at any given weight distribution. This is done in Figure 2: configurations of same color indicate the same set of influencemaximizing voting rules for player 1. (When the respective weight regions are lines or single points, we have manually enlarged them.) Tongueincheek, Figure 2 provides a ‘map’ for influencemaximizing chairpersons – or for whoever has a say on the adopted voting rule and cares about a priori influence on decisions between three candidates. It also gives players 2 or 3 reason for criticizing adoption of a particular rule as biased.
7 Concluding Remarks
We have investigated how the distribution of voting power in a committee depends not only on voting weights but which of the many possible aggregation procedures for alternatives is adopted – from simple plurality voting to the elaborate computation of Schulze winners. Traditional measures of voting power, such as the PenroseBanzhaf or ShapleyShubik indices, fail to capture this. They should be accompanied by power indices for three or more alternatives.
One such index has been proposed and illustrated here. It evaluates how sensitive the collective choice under the given aggregation rule and weights is to preference changes by an individual player. The index is proportional to the probability for a random individual preference perturbation to affect the outcome, assuming preferences are independently uniformly distributed a priori.^{10}^{10}10It as an open challenge to find properties or ‘axioms’, like those used by Dubey/Shapley:1979 or Laruelle/Valenciano:2001 for the PenroseBanzhaf index, that would characterize the proposed index without making specific probability assumptions. Our attempts have failed so far. A dictator player’s voting power is normalized to one; a null player’s power is zero. The extent to which the distribution of weights matters differs across rules. So do comparative statics regarding the number of alternatives. How the adopted rule affects the distribution of voting power has been illustrated for (nonconsensual) election of the IMF’s Managing Director by its Executive Board. Similar analysis might be conducted for multicandidate election primaries, party conventions, corporate boards, etc.
Several case studies have shown with the benefit of hindsight how choice of a particular voting method may have affected big political decisions (cf., e.g., Leininger:1993; Tabarrok/Spector:1999; or Maskin/Sen:2016). We deem it worthwhile to evaluate collective choice methods also from an a priori perspective and ‘on average’. For the simplest case with three options, we have considered the power implications of all conceivable weight distributions among three players and identified differences in how several prominent rules translate weights into voting power. Except for Borda rule, precise knowledge about the distribution of voting weights is typically not needed for gaging the sensitivity of outcomes to preferences of a large, middle, or small player respectively. It is of course desirable to obtain results also for bigger numbers of players and alternatives in the future.
Our analysis hopefully encourages the extension of voting power analysis to richer choice settings. There is a great variety of singlewinner rules that could be added to the influence map in Figure 2 (see, e.g., Aleskerov/Kurbanov:1999; [Ch. 7]Nurmi:2006; or Laslier:2012). And nothing in principle would preclude similar analysis for multiwinner elections (see, e.g., Elkind/Faliszewski/Skowron/Slinko:2017) or strategic voting equilibria, at least for restricted preference domains. It also seems worthwhile to investigate weight apportionment for twotiered voting systems, such as US presidential elections, with more than two promising candidates. It is unknown, for instance, how the Penrose square root rule for the independent binary case (see, e.g, [Ch. 3.4]Felsenthal/Machover:1998) or Shapley linear rule for affiliated spatial preferences (cf. Kurz/Maaser/Napel:2018) extend to twotiered plurality decisions or plurality voting with a runoff.
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