if there are n1 ways to perform task 1, n2 ways to perform task 2, n3 ways to perform task3…. And nk ways to perform task(k), then there are n1 x n2 x n3 x……..nk different ways to perform all K tasks in succession.
Weighted Voting System
voter’s weight consist of two alternatives and a group of voters that are allowed more than one vote. The two alternatives are propositions, which require accept/ or reject.
Restriction on the Quota
The value of q is restricted so that no “ridiculous” or impossible resolution passes or so that it is always possible to either pass or block a resolution.·
The value of q must be greater than half of the total number of votes(sum of the votes of all participants) and no more than the number of votes (added together)
a WVS a voter with weights equal or greater than the quota q is said to be a dictator. This voter has all the power and can pass or block any resolution.
any voter whose votes are not needed to pass or black the motion
a voter is said to have veto power if any only if and only if her/his votes are absolutely necessary for a motion to pass.
Paradox of New members
occurs when the addition of a new member to WVS increases, (instead of decreases) the voting power of some of the original members.
a voter is said to be a pivotal in a sequential collation (permutation) if and only if the cumulative weight of that voter plus the weights of all her predecessors is equal or greater than quota q.
Shapley-Shubik Power Index(SSPI)
Shapley-Shubik Power Index(SSPI): is the ration of the number of times that a voter is pivotal in a permutation over the number of permutations of all voters in the system.
Properties of SSPI
Dummy voter have SSPI =0 and dictators have SSPI= 1 · The sum of SSPI values for each WVS equals 1. · Voters with equal weights have equal SSPI
SSPI values are positive real number in the closed interval [0,1].
a voter is said to be critical in a coalition if and only if the voter’s weight is absolutely necessary for the coalition to win. In a blocking coalition, a voter is said to be critical in a coalition if and only if the voter’s weight is absolutely necessary for the coalition to block. Without the votes of a critical voter, the coalition cannot get its way.
Banzhaf Power Index (BPI):
Banzhaf Power Index (BPI): is the number of times that a voter is critical in winning and blocking coalitions(subsets of voters).
Propeties of BPI values;
· Voters with equal weights have equal BPI values.
· The sum of BPI values is not equal to any particular number , but to any whole number more than zero. · The number of times a voter is critical in winning coalitions is the same as the number of times that the voter is critical in blocking coaltions.
a group of voters who are either (all in favor) or all against a resolution. In addition, they have enough votes to win or enough votes to block the resolution. Therefore a coalition is a subset of all voters in a WVS. Note subset is found by 2 to the N power= 2
is simply any permutation of all voters in WVS. The total number of permutations of all N elements of a set is given by N!. If a WVS consists of N voters then the total number of permutations of all voters in the system is N!. Ex: 3 voters = 3! = 6 permutations. And then write down all the subsets of it.
is a subset (non-empty) of voters in which every voter favors a resolution and the sum of the weights is enough to make the resolution pass. A resolution pass whenever the total number of votes of all voters in favor is equal or greater than the q (quota). The value of q is the threshold value of a winning collation.
is a subset (non-empty) of voters who oppose a resolution and collectively have enough votes to block it. The threshold value of a blocking coalition is W-Q +1 , where W= the total weight of the system.
a coalition that cannot get its way (not enough votes to win/ or block).
Minimal Winning Coalition
a coalition were every member is a critical voter
Minimal blocking Coalition
are coalition in which every member is a critical voter in blocking
contains exactly one voter with veto power. All MWC contian the voter with veto power
every voter has veto power ex: [9:5,3,2]
Equivalent weighted voting system
systems for which the set of winning coalitions are the same , execpt for the symbols. Ex: [5:4,3,1] to [20:15,12,6] is writen X=A, Y=B, Z=C