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
editUnderstand weighted voting and critical voters.
Details
editA 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
editThree 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.
Nonexamples
editFAQ
editHomework
editDescribe 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. |
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!] |