Transcript Conjecture: Equal weight implies equal power but not the

Theorem:
Equal weight implies equal power
but not the converse.
Equal weight implies equal power
• Let’s consider the Shapley-Shubik power index first.
• Suppose we have a weighted voting system for n voters given by
[ q : w1, w2, w3, …, wn ] .
• Suppose voter i, with i {1,2,3,...,n} has weight wi and is pivotal
k times. Note that k is any number from 0 to n!.
• By definition, k is the number of permutations of all voters in which
the sum of the weights of voters preceding voter i is less than q and
the sum of the weights of the voters preceding voter i plus the
weight of voter i is greater than or equal to q.
• By definition, the Shapley-Shubik power index for voter i is
k
n!
.
Equal weight implies equal power
n voters
___ ___ ___
sum < q
sum > q
wi
___
___
___
___
If voter i has power k/n! then there are k
different permutations of this type.
Now consider any other voter j with the
same weight as voter i. That is, assume
wJ = w i.
Because wJ = wi, the number of
permutations in which voter j is pivotal is
also equal to k.
Equal weight implies equal power
• In other words, given any voter j, with
we may conclude the following:
j {1,2,3,...,n}
and wJ = wi
• k is also the number of permutations of all voters in which the sum of
the weights of voters preceding voter j is less than q and the sum of
the weights of the voters preceding voter j plus the weight of voter j
is greater than or equal to q.
• Thus, we may conclude the Shapley-Shubik power index for voter j
is also given by k .
n!
• Finally, we may conclude that any voters with equal weight will have
equal power as measured by the Shapley-Shubik index.
Equal weight implies equal power
• Now, let’s consider the Banzhaf power index. As before, suppose we
have a weighted voting system for n voters given by
[ q : w1, w2, w3, …, wn ] .
• Suppose voter i, with i {1,2,3,...,n} has weight wi with wi > 1 and
is critical to k winning coalitions. Note that now k is any integer from
0 to 2n-1.
• Remember that to compute the Banzhaf power for voter i we
consider only combinations of voters and not permutations.
• Because voter i is critical to k winning coalitions, there are k distinct
coalitions of voters with weight wc that satisfy the following inequality:
q < wc < q + wi – 1.
Equal weight implies equal power
• Now, consider any other voter j, with
j {1,2,3,...,n}.
• Suppose that voter i and j have equal weight, that is, assume that
w J = wi.
• Now consider the number of distinct coalitions of voters with weight
wc that satisfy the following inequality: q < wc < q + wJ – 1.
• Because wJ = wi then also q + wJ – 1 = q + wi – 1.
• Therefore, the number of distinct coalitions satisfying the inequality
q < wc < q + wJ – 1 is also equal to k.
• This is because, if wJ = wi , then the inequalities q < wc < q + wi – 1
and q < wc < q + wJ – 1 are equivalent.
Equal weight implies equal power
• We may conclude that any other voter j , with weight wJ = wi, is also
critical to k distinct coalitions.
• And finally, by definition, both voters i and j have Banzhaf power 2k.
• Thus any voters with equal weight will have equal power as
measured by the Banzhaf analysis of power.
Equal weight implies equal power
• Clearly, any two voters with equal weight will have equal nominal
power.
• Suppose we have a weighted voting system for n voters given by
[ q : w1, w2, w3, …, wn ] . Suppose the total weight of the system
is w = w1 + w2 + w3 + … + wn.
• Suppose two voters have equal weight. That is, assume wJ = wi.
• Therefore the nominal power of these voters is equal, that is,
because wJ = wi then it is also true that
wi w j

.
w w
• Conclusion: Voters with equal weight have equal nominal power.
The Converse is Not True
• For the Shapley-Shubik, Banzhaf and nominal measures of power
for a weighted voting system, we have shown that the following
statement is true:
• If voters have equal weight, then they have equal power.
• The converse of this statement is: If voters have equal power then
they have equal weight.
• The converse is not true.
• To prove a statement is not true, we need only provide a counterexample to the statement.
Equal Power Does Not Imply Equal Weight
• Consider the statement: If voters have equal power, then they have
equal weight.
• This can be shown to be false by the following counter-example:
• Consider the weighted voting system [ 100 : 60, 39, 1 ].
• The Banzhaf index for this system is ( 2, 2, 2 ) and the ShapleyShubik index is ( 1/3, 1/3, 1/3 ).
• Thus, we have an example where power is equal among voters but
none have equal weight.
• In summary, if voters have equal weight then they have equal power
but if they have equal power, they might not have equal weight.