In this note, we demonstrate using two simple examples that generalization of the Banzhaf measure of voter influence to non-binary voting games that requires as starting position a voter’s membership in a winning coalition is likely to incompletely reflect the influence a voter has on the outcome of a game. Generalization of the Banzhaf measure that takes into consideration all possible pivot moves of a voter including those moves originating from a losing coalition will, on the other hand, result in a measure that is proportional to the Penrose measure only in the ternary case.Penrose measure, Banzhaf index, ternary games, multicandidate weighted voting games

C H Ueng

Vincent C H Chua

English

Research Papers in Economics

 
 
ANY OPINIONS EXPRESSED ARE THOSE OF THE AUTHOR(S) AND NOT NECESSARILY THOSE OF 
THE SCHOOL OF ECONOMICS & SOCIAL SCIENCES, SMU 
 
S S SM MU U U    E E EC C CO O ON N NO O OM M MI I IC C CS S S    & & &    S S ST TA M TA AT T TI I IS S ST T TI I IC C CS S S    
W W WO O OR R RK K KI I IN N NG G G    P P PA A AP P PE E ER R R    S S SE E ER R RI I IE E ES S S 
 
 
 
 
 
 
 
 
On Measuring Influence in                                              
Non-Binary Voting Games 
 
 
 
 
 
 
 
 
 
 
Vincent C H Chua and C H Ueng  
  December 2004 
 
 
 
 
 
 
   Paper No. 25-2004   
 
 
 
 
 
ON MEASURING INFLUENCE IN NON-BINARY VOTING GAMES
*
 
 
 
Vincent C H CHUA and C H UENG  
School of Economics and Social Sciences  
Singapore Management University 
 
 
 
 
Abstract 
 
In this note, we demonstrate using two simple examples that generalization of the 
Banzhaf measure of voter influence to non-binary voting games that requires as 
starting position a voter’s membership in a winning coalition is likely to incompletely 
reflect the influence a voter has on the outcome of a game. Generalization of the 
Banzhaf measure that takes into consideration all possible pivot moves of a voter 
including those moves originating from a losing coalition will, on the other hand, 
result in a measure that is proportional to the Penrose measure only in the ternary case.  
 
 
JEL Classification: C6, D7 
 
Keywords:  Penrose measure, Banzhaf index, ternary games, multicandidate weighted 
voting games 
 
 
 
 
Address:   Vincent CH CHUA 
School of Economics and Social Sciences 
  Singapore  Management  University 
  469  Bukit  Timah  Road 
Singapore 259756   
 
Fax No:  (65)-6822-0833 
e-mail:   vincentchua@smu.edu.sg 
 
                                                 
* The authors gratefully acknowledge the help and very insightful comments of Professor Dan 
Felsenthal and Professor Moshé Machover on an earlier draft of this note. All remaining errors are ours. 1. Introduction 
In a seminal paper in 1946, LS Penrose posited a measure of voter influence in binary 
majority voting games in which a given voter’s influence is proportional to the 
number of divisions of other voters in which the given voter can make a difference. 
This is directly related to the number of instances that the given voter is a member of 
the “winning side” given that all other voters vote in a random manner. It is well 
known that Penrose’s measure is directly equivalent to the absolute Banzhaf index, 
one based on the ability of a voter in a “winning” coalition to affect the passage or 
defeat of an issue [see Banzhaf (1965)]. Thus, it is a matter of indifference when 
measuring influence whether one focuses attention on the number of divisions of 
other voters in which the given voter can make a difference or on the number of pivot 
(or outcome-changing) moves the given voter has given his membership in a winning 
coalition.  
  In his generalization of the Banzhaf measure to non-binary voting games, 
Bolger (1983, 1986, 1990, 1993, SCW2002) proceeded along the latter line by 
requiring as starting position membership of the given voter in a winning coalition 
and modifying the definition of a pivot move to take into consideration the destination 
of the move since there is now more than one destination for a voter who defects from 
the winning coalition. In enumerating the number of pivot moves of a voter, the 
requirement that the given voter must be initially a member of the winning coalition, 
however, results in the exclusion of outcome-changing moves relating to those moves 
of voter when he is originally a member of a losing coalition. In the binary case, 
defection of a member from the losing coalition cannot be pivotal. In the r-alternative 
case, this continues to be true only if the winning coalition wins unconditionally, that 
is, when the win is independent of how other voters outside of the winning subset is   2
distributed across the other (r-1) alternatives. Otherwise, such pivotal moves become 
possible. We demonstrate this point by appealing to two simple examples: the UN 
Security Council game, a game discussed in Bolger’s work including Bolger (2002) 
and the set of non-dictatorial 3-voter 3 –candidate weighted plurality games. 
  
2. The UN Security Council Game  
The procedure for computing the Banzhaf values for voters in ternary games and the 
Banzhaf values for a permanent (and rotating) member in the UN Security Council 
game are already a part of the established literature [see Felsenthal and Machover 
(1997, 1998)]. It is thus not the purpose of this example to belabor these points further. 
Rather, the intention here is to highlight the problematic nature of the Bolger measure 
that enumerates pivot moves only when such moves emanates from a winning 
coalition.  
 On  non-procedural  issues,  the practical implementation of Article 27c of the 
UN Charter suggests that securing 9 affirmative votes out of the possible 15 is 
sufficient to secure passage of a resolution provided no permanent member vetoes the 
resolution. Under such a voting structure, a rotating member is pivotal for a given 
partition of voters if and only if it is a member of a strictly minimal winning coalition. 
This occurs when exactly nine members of the Council, inclusive of the rotating 
member in question, vote in the affirmative and no permanent member vote against 
the proposal. Accordingly, the total number of pivot moves of such a member is 
106,308 as reported in Bolger (2002, p.711).   
  If a permanent member is a member of a strictly minimal winning coalition, 
then any move by such a member will be pivotal. Otherwise, the member is pivotal 
only if the move involves a veto. Again as enumerated in Bolger (2002, p.711), such   3
moves total 178,212.  Any measure that relies solely on these values to infer the 
relative influence a permanent member has in this game, however, is likely to be 
inappropriate since it does not fully account for all the instances in which a permanent 
member may influence the outcome of a vote in the Security Council.  
  In addition to the 178,212 pivot moves enumerated above, there are altogether 
38,460 other instances in which a permanent member may change the outcome of a 
vote.  All such instances invariably involve the situation whereby the permanent 
member in question had abstained but nonetheless the motion will be carried because 
nine or more affirmative votes have been obtained and there is no veto vote. In such 
instances, a move by the permanent member in question from an abstention to a nay 
vote would veto the resolution. We list these scenarios below using the following 
notation. The letter P denotes a permanent member, the letter R denotes a rotating 
member and the permanent member in focus is denoted by the letter P*. The letters Y, 
N and A refer to Yes, No and Abstain respectively. 
(1)  Y(4Ps, 5Rs), A(P*), 5Rs (either A or N)   : 
10C5x2
5   =  8,064 instances 
(2)  Y(4Ps, 6Rs), A(P*), 4Rs (either A or N)   : 
10C6x2
4   =  3,360 instances 
(3)  Y(4Ps, 7Rs), A(P*), 3Rs (either A or N)   : 
10C7x2
3   =   960 instances 
(4)  Y(4Ps, 8Rs), A(P*), 2Rs (either A or N)   : 
10C8x2
2   =   180 instances 
(5)  Y(4Ps, 9Rs), A(P*), 1R (either A or N)    : 
10C9x2
1   =     20 instances 
(6)  Y(4Ps, 10Rs), A(P*)         : 
10C10x2
0   =       1 instance 
(7)  Y(3Ps, 6Rs), A(P*, P), 4Rs (either A or N)   : 
4C3x
10C6x2
4  = 13,440 instances 
(8)  Y(3Ps, 7Rs), A(P*, P), 3Rs (either A or N)   : 
4C3x
10C7x2
3  =  3,840 instances 
(9)  Y(3Ps, 8Rs), A(P*, P), 2Rs (either A or N)   : 
4C3x
10C8x2
2  =    720 instances  
(10) Y(3Ps, 9Rs), A(P*, P), 1R (either A or N)   : 
4C3x
10C9x2
1  =      80 instances 
(11) Y(3Ps, 10Rs), A(P*, P)         : 
4C3x
 10C10x2
0 =       4 instances 
(12) Y(2Ps, 7Rs), A(P*, 2Ps), 3Rs (either A or N)   : 
4C2x
 10C7x2
3  =  5,760 instances 
(13) Y(2Ps, 8Rs), A(P*, 2Ps), 2Rs (either A or N)   : 
4C2x
 10C8x2
2  =  1,080 instances 
(14) Y(2Ps, 9Rs), A(P*, 2Ps), 1R (either A or N)   : 
4C2x
 10C9x2
1  =   120 instances 
(15) Y(2Ps, 10Rs), A(P*, 2Ps)       : 
4C2x
 10C10x2
0 =       6 instances 
(16)  Y(P, 8Rs), A(P*, 3Ps), 2Rs (either A or N)   : 
4C1x
10C8x2
2  =   720 instances  
(17) Y(P, 9Rs), A(P*, 3Ps), 1R (either A or N)   : 
4C1x
10C9x2
1  =     80 instances 
(18) Y(P, 10Rs), A(P*, 3Ps)         : 
4C1x
 10C10x2
0 =      4 instances 
(19) Y(9Rs), A(P*, 4Ps), 1R (either A or N)    : 
10C9x2
1   =     20 instances 
(20) Y(10Rs), A(P*, 4Ps)         : 
10C10x2
0   =
           1 instance 
____________________ 
38,460 instances 
____________________ 
    4
Thus, focusing solely on pivot moves when the member is originally a member of the 
winning coalition will not result in a measure that fully reflects the ability a 
permanent member has in influencing the outcome of a vote in the Security Council. 
Under Bolger’s definition, the influence a permanent member has relative to that of a 
rotating member is 
308 , 106
212 , 178 (or 1.676 times) whereas under the alternative definition, 
this ratio is 
308 , 106
672 , 216 (or 2.038 times). This latter number coincides with that obtained 
in Felsenthal and Machover (1997, p. 348). In this 1997 paper, the reported Banzhaf 
value for a permanent member is 0.1009 and that for a rotating member is 0.0495 and 
the ratio of these numbers yields the value 2.038. That the value obtained, based on a 
complete enumeration of all pivot moves, is consistent with the Penrose measure may 
be directly inferred from Felsenthal and Machover (1997) and thus requires no further 
elaboration.  
  At this point, it is important to highlight an alternative extension of the 
Banzhaf measure to the UN Security Council game, one that is probably more 
appropriate given the particular structure of this ternary game. This is the hybrid 
ternary-binary model proposed in Felsenthal and Machover (1997, 1998). As noted in 
these works, “since abstention by an ordinary member has exactly the same effect as a 
‘no’ vote, these members have in effect just two voting options -- `no' and ‘yes’ -- 
whereas only for the permanent members is abstention a distinct tertium quid.” 
Explicit recognition of this fact suggests that the total number of relevant and 
“distinct” divisions of the voter set for this game is 3
52
10 or 248,832 divisions, not 3
15 
or 14,348,907 divisions. Thus, a member’s ability to influence outcomes in these 
248,832 divisions becomes the relevant indication of his influence in the UN Security 
Council game. This line of reasoning will lead to a measure that assigns   5
approximately 0.1038 to a permanent member and 0.0482 to a rotating member, thus 
giving rise to the outcome that a permanent member is 
0482 . 0
1038 . 0  or 2.154 times more 
influential than a rotating member.  
 
3. The set of non-dictatorial 3-voter 3-candidate weighted plurality games 
In this second example, we demonstrate once again that Bolger’s measure fails to 
fully account for all the instances in which a voter is pivotal. In addition, we would 
like to also highlight that in the proper (n, r) multi-candidate setting, such as in this 
example, both the Bolger measure and the measure derived from a complete 
enumeration of all pivot moves do not coincide with the Penrose measure, one that 
enumerates, for a given voter, the number of r-partitions of the other (n-1) voters in 
which the given voter may make a difference.  
   Let N={1, 2, 3} be a set of voters and let the set of coalitions be denoted by N 
={{123},{12},{13},{23},{1},{2},{3}, ∅}. Let the weight vector associated with the 
voter set be (w1, w2, w3) where, without loss of generality, we let w1>w2>w3 and 
w2+w3>w1. Whether a coalition is winning or otherwise will depend on how the 3 
voters are distributed across the 3 alternatives. The following list completely 
enumerates the 3-partitions and the corresponding embedded winning coalitions 
(bold-faced) for this set of games up to a permutation of alternatives: {({1,2,3},∅,∅), 
({1,2},{3},∅), ({1,3},{2},∅), ({2,3},{1},∅), ({1},{2},{3})}. 
  Using Bolger’s definition of an ordered pivot move, we obtain the number of 
pivot moves for each voter corresponding to each of these partitions as follows. Voter 
1 has no pivot move in partitions ({1,2,3},∅,∅) and ({2,3},{1},∅), two pivot moves 
each in partitions ({1,2},{3},∅), ({1,3},{2},∅) and ({1},{2},{3}) making the total   6
count 6.  Voter 2 has one pivot move in partition ({1,2},{3},∅) and two pivot moves 
in partition ({2,3},{1},∅) while Voter 3 has one pivot move in partition ({1,3},{2},∅) 
and two pivot moves in partition ({2,3},{1},∅). There are no other pivot moves for 
Voter 2 and Voter 3. Thus, under Bolger’s definition, the vector of pivot moves for 
the voters are (6, 3, 3) giving rise to a measure of relative influence for the voters 
denoted by the vector (0.5, 0.25, 0.25).  
  Recall the partition ({1},{2},{3}) and note that, for this partition, although 
Voter 2 and Voter 3 are not members of the winning coalition {1}, both voters can 
influence the outcome of the game by forming the coalition {2,3} either by Voter 2 
moving to the third alternative or by Voter 3 moving to the second. In each of these 
instances, the resulting outcome will differ from alternative 1. If these extra pivotal 
moves of Voter 2 and Voter 3 are taken into consideration when computing the 
measure of influence, the resulting vector of pivot moves for the voters would have 
been (6, 4, 4) giving rise to a normalized influence vector equal to (0.428, 0.286, 
0.286) as opposed to Bolger’s (0.5, 0.25, 0.25). Thus, for this set of games, by 
ignoring the pivot moves of the smaller voters originating from a losing coalition, 
Bolger has inadvertently overstated the influence of the largest voter. Under the 
Bolger measure, Voter 1 has twice as much influence as Voter 2 or Voter 3. Under 
complete enumeration, the influence of Voter 1 is only 1.5 times that of Voter 2 or 
Voter 3.  
  On the Penrose measure, it may be deduced from the following set of 3-
partitions (up to a permutation) {({2,3},∅,∅), ({2},{3},∅)} that Voter 1 is influential 
in the partition ({2},{3},∅). Again, up to a permutation of alternatives, the 3-
partitions in which Voter 2 and Voter 3 are, respectively, influential are ({1},{3},∅) 
and ({1},{2},∅). Thus, the number of 3-partititons inclusive of permutations in which   7
the different voters are influential are respectively (6, 6, 6) yielding a relative 
influence vector equal to (0.33, 0.33, 0.33). This is clearly different from the 
preceding measures. In this instance, Voter 2 and Voter 3 are just as influential. This 
outcome arises because in this truly multicandidate situation, when a given voter is 
influential for a particular 3-partition of the other voters, the influence may be to a 
different degree. For instance, consider Voter 1 and the 3-partition ({2},{3},∅). 
Without Voter 1, the outcome is the first alternative. Here, Voter 1 is influential and 
may be able to effect the second alternative or the third alternative. This is because he 
is the largest voter and the other voters are divided. On the other hand, consider Voter 
2 and the 3-partition ({1},{3},∅). Without Voter 2, the outcome is the first alternative. 
Here, Voter 2 is influential but is able to change the outcome only to the second 
alternative, not the third. For the above set of (3, 3) games, therefore, because the 
Penrose measure does not make the distinction between the two different degree of 
influence, the resulting measure is biased against the largest voter, making him only 
just as influential as each of the other two smaller voters.  
 
4. Some concluding remarks 
In both examples discussed above, failure to consider pivot moves by voters 
originally in a losing coalition will cause the resulting measure of influence to be 
inappropriate. Loosely speaking, the underlying reason for the inappropriateness has 
to do with the fact that, in some instances, the winning coalition “wins” only because 
of the particular way in which the other voters are distributed over the remaining 
alternatives -- a conditional win.  In the UN Security Council game, this difficulty 
would not arise if all permanent members and at least four other rotating members 
vote in the affirmative because, in this instance, the win is unconditional and how the   8
other voters are distributed over the remaining alternatives would be largely irrelevant 
to the outcome.  Likewise, in the set of (3, 3) weighted plurality games, if the winning 
coalition has a simple majority of the votes, this difficulty will also not arise as the 
win would be unconditional. The difficulty with Bolger’s generalization highlighted 
in this note is not confined to the games exposited above but applies generally across 
(n, r) games.   9
REFERENCES  
Banzhaf JF (1965) Weighted voting doesn't work: A mathematical analysis. Rutgers 
Law Review 19: 317-343 
Bolger EM (1983) The Banzhaf index for multicandidate presidential elections. SIAM 
Journal of Algebraic Discrete Methods 4: 422-458 
Bolger EM (1986) Power indices for multicandidate voting games. International 
Journal of Game Theory 15: 175-186 
Bolger EM (1990) A characterization of an extension of the Banzhaf value to 
multicandidate voting games. SIAM Journal of Discrete Mathematics 3: 466-477 
Bolger EM (1993) A value of games with n players and r alternatives. International 
Journal of Game Theory 22: 319-334 
Bolger EM (2002) Characterization of two power indices for voting games with r 
alternatives. Social Choice and Welfare 19: 709-721 
Felsenthal DS, Machover M (1997) Ternary voting games. International Journal of 
Game Theory 26: 335-351 
Felsenthal DS, Machover M (1998) The measurement of voting power. Cheltenham: 
Elgar 
Penrose LS (1946) The elementary statistics of majority voting. Journal of the Royal 
Statistical Society 109:53-57 
 

Weighted voting doesn't work: A mathematical analysis. Rutgers Law Review 19: 317-343 Bolger EM

On Measuring Influence in Non-Binary Voting Games

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On Measuring Influence in Non-Binary Voting Games

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