Approval voting was first used in cases where the number of people to be elected was not fixed. For example, if a committee wants to co-opt a few more members to help organize a function, they might add all those who they think would make a positive contribution, not a fixed number. In that case a member might vote for every suitable candidate, casting as many votes as she wishes; all the votes are considered equal. The number of votes received by a candidate is called his approval rating.

Candidates with an approval rating of at least 50% (or 60%, or some other agreed figure) are elected.

Sometimes there is a strict requirement that a certain number be elected, or a minimum or maximum is imposed. Ties are more common in approval voting than they are in some other systems, and some sort of runoff procedure is often necessary.

The following example illustrates these ideas.

Sample Problem 1.1 Ten board members vote on eight candidates by approval. The candidates are A,B,C,D,E,F,G,H, and the board members are q,r,s,t,u, v,w,x,y,z. They vote as follows (× represents approval).

(i) Which candidate is chosen if just one is to be elected?

(ii) Which candidates are chosen if the top four are to be elected?

(iii) Which candidates are chosen if the top two are to be elected?

(iv) Which candidate or candidates are chosen if at most four are to be elected and 80% approval is required?

Solution. Here is a summary of the votes received:

A−7 B−8 C−2 D−8 E −6 F −9 G−5 H −5.

(i) F, who received the most votes.

(ii) A,B,D,F.

(iii) F is elected; there will need to be a runoff election between B and D.

(iv) B, D and F are elected.

The array in the example is called an approval table; we shall always denote approval with a cross in such tables.