Question

The voting paradox (10 points) Consider an election with three candidates, George, Lee, and Ray. There are 12 voters with the following preferences:

Number of Voters	Best	Second	Last
5	G	L	R
4	R	G	L
3	L	R	G

In the plurality voting, where 12 voters vote for the three candidates and each voter is allowed to vote for only one candidate, who will become the winner? How many voters actually prefer G over R and how many voters prefer R over G? What are the advantages and disadvantages of the plurality voting?

2). In the pairwise voting following the majority rule, is there any candidate who is a dominate winner (one wins in every pairwise comparison) or a sure loser (one loses in every pairwise comparison)? Explain the voting paradox in this example if there is any.

Answer 1

1. At first it seems that that George is the clear winner. But a detailed pairwise comparison shows 7 voters prefer Ray over George. Now the final result depends on Ray and Lee. It is seen that 8 voters prefer Lee over Ray. But again 9 prefers george over Lee. So, finally it can be said that though Lee is prefered over Ray but George is prefered over Lee. Therefore, George is the winner.

5 voters prefer G over R

7 voters prefer R over G

Advantages:

1. The idea behind Plurality voting is easy to understand

2. Decision can be made readily.

3. Less costly.

Disadvantages:

1. In case of more than 2 candidates, the result may be in favour of an individual who is not the first preference of majority voters.

2) In the above case, the result is a 3 way tie

G vs L = G is the winner ( 9 votes for G and 3 votes for L )

L vs R = L is the winner ( 8 votes for L and 4 votes for R )

R vs G = R is the winner ( 8 votes for R and 4 votes for G )

The voting paradox in this example is that it does not produce a clear winner, even though each individual voter has a clear order of preference over the alternative options. The preference is cyclical by nature.