- Define weighted voting and distinguish it from equal voting
- Apply the Banzhaf power index and the Shapley-Shubik power Index to assess the relative power of individual voters in a weighted voting situation
Weighted Voting
In a corporate shareholders meeting, each shareholders’ vote counts proportional to the amount of shares they own. An individual with one share gets the equivalent of one vote, while someone with [latex]100[/latex] shares gets the equivalent of [latex]100[/latex] votes. This is called weighted voting, where each vote has some weight attached to it. Weighted voting is sometimes used to vote on candidates, but more commonly to decide “yes” or “no” on a proposal, sometimes called a motion. Weighted voting is applicable in corporate settings, as well as decision making in parliamentary governments and voting in the United Nations Security Council
In weighted voting, we are most often interested in the power each voter has in influencing the outcome.
We’ll begin with some basic vocabulary for weighted voting systems.
vocabulary for weighted voting
- Each individual or entity casting a vote is called a player in the election. They’re often notated as [latex]P_1,P_2,P_3,...,P_N[/latex] where [latex]N[/latex] is the total number of voters.
- Each player is given a weight, which usually represents how many votes they get.
- The quota is the minimum weight needed for the votes or weight needed for the proposal to be approved.
- A weighted voting system will often be represented in a shorthand form:
[latex][q: w_1, w_2, w_3, ... , w_N][/latex]
In this form, [latex]q[/latex] is the quota, [latex]w_1[/latex] is the weight for player 1, and so on.
In a small company, there are [latex]4[/latex] shareholders. Mr. Smith has a [latex]30\%[/latex] ownership stake in the company, Mr. Garcia has a [latex]25\%[/latex] stake, Mrs. Hughes has a [latex]25\%[/latex] stake, and Mrs. Lee has a [latex]20\%[/latex] stake. They are trying to decide whether to open a new location. The company by-laws state that more than [latex]50\%[/latex] of the ownership has to approve any decision like this. This could be represented by the weighted voting system:
[latex][51: 30, 25, 25, 20][/latex]
Here we have treated the percentage ownership as votes, so Mr. Smith gets the equivalent of [latex]30[/latex] votes, having a [latex]30\%[/latex] ownership stake. Since more than [latex]50\%[/latex] is required to approve the decision, the quota is [latex]51[/latex], the smallest whole number over [latex]50[/latex].
In order to have a meaningful weighted voting system, it is necessary to put some limits on the quota.
limits on the quota
The quota must be more than [latex]\frac{1}{2}[/latex] the total number of votes.
The quota can’t be larger than the total number of votes.
You may be wondering why these limitations are set. Let’s consider the voting system: [latex][q; 3, 2, 1][/latex]
Here there are [latex]6[/latex] total votes. If the quota was set at only [latex]3[/latex], then player 1 could vote yes, players 2 and 3 could vote no, and both would reach quota, which doesn’t lead to a decision being made. In order for only one decision to reach quota at a time, the quota must be at least half the total number of votes. If the quota was set to [latex]7[/latex], then no group of voters could ever reach quota, and no decision can be made, so it doesn’t make sense for the quota to be larger than the total number of voters.
In a committee there are four representatives from the management and three representatives from the workers’ union. For a proposal to pass, four of the members must support it, including at least one member of the union. Find a voting system that can represent this situation.