
Multiplication Principle
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)

Dictators
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.

Dummy Voter
any voter whose votes are not needed to pass or black the motion

Veto Power
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.

Pivotal Voters
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.

ShapleyShubik Power Index(SSPI)
ShapleyShubik 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].

Critical voter
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.

Coalition
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

Sequential Coalitions
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.

Winning Coalition
is a subset (nonempty) 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.

Blocking Coalition
is a subset (nonempty) of voters who oppose a resolution and collectively have enough votes to block it. The threshold value of a blocking coalition is WQ +1 , where W= the total weight of the system.

Losing Coalition
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

Chair Veto
contains exactly one voter with veto power. All MWC contian the voter with veto power

Consensus
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

