Selected topics in finite mathematics/Weighted voting

Weighted voting is a means of voting or pass (or fail to pass) a motion when the voters may have differing power.

Objectives

Understand weighted voting and critical voters.

Details

A weight is the number of votes that each participant is given and in order for a motion to pass, the number of in votes in favor of a decision must reach a quota. With weighted voting systems there are two important types of voters, one who has veto power and a dummy voter. If one has veto power, their vote is required to pass or gail any motion, and a dummy voter has not influence in any part of the election.

Unreasonable: when it can't pass due to the fact that they pass and fail at the same time

Reasonable: when the motion either fails or passes

Examples

Three entrepreneurs found a company together. But they all invest different amounts. One invests $20,000, another invests $30,000, and the third invests $40,000. Accordingly they own 20, 30, and 40 shares of stock in the company. In case of dispute, a decision is made based on stock, using the weighted voting system [45:20,30,40]. While each founder owns a different amount of stock, it turns out that their voting system is nothing more than a simple two-out-of-three majority.

Homework

Describe the weighted voting system [9:5,4,3,2,2,1].

Solution [You can add the solution to the problem here!]

In the weighted voting system [9:5,4,3,2,2,1], identify any voters with veto power and any voters that are dummies.

Solution ['There are no voters with veto power or that are dummies.]--Lww11 (discuss • contribs) 03:37, 6 March 2013 (UTC)

Construct a weighted voting system in which everybody has veto power.

Solution [You can add the solution to the problem here!]

Construct a weighted voting system in which exactly half the voters have veto power.

Solution [You can add the solution to the problem here!]

Construct a weighted voting system in which exactly half the voters are dummies.

Solution [You can add the solution to the problem here!]

Construct three weighted voting systems which are equivalent to [67:24,24,24,24,3,1].

Solution [You can add the solution to the problem here!]

Consider the voting system [7:A=3,B=3,C=2,D=2,E=2]. If A opposes a motion, but B, C, D, and E support it, does the motion pass? Which voters are critical?

Solution [Motion does pass and B is a critical voter]

Consider the voting system [7:A=4,B=3,C=3]. Construct a table of all possible elections and for each election identify which voters are critical.

Solution [You can add the solution to the problem here!]

The United Nations security council has 15 voting members, and a motion requires the support of 9 members to pass. However, 5 of the voting members have veto power. Construct a weighted voting system that would model the United Nations security council.

Solution [You can add the solution to the problem here!]