In the below table, Colbert, Fallon, and Kimmel are ranking three public goods (Courthouse, School, and Park) according to their preferences.
| Ranking |
Public good
| Colbert
| Fallon
| Kimmel
|
Courthouse
| 2nd Choice
| 1st Choice
| 3rd Choice
|
School
| 3rd Choice
| 2nd Choice
| 1st Choice
|
Park
| 1st Choice
| 3rd Choice
| 2nd Choice
|
The first look at the table does not show any dominant preference for any of the public goods. Thus, you will use paired-choice majority voting in which a first vote is held between any two of the three goods, and then the winner is compared with the left public good to see which one wins.
You will start by taking the pair of Courthouse and School and observing each individual's ranking. The public good, which will have a better ranking, will be preferred. For example, Colbert gives Courthouse a second choice and School a third choice; comparing these two, Colbert prefersCourthouse over School. Similarly, you can find this for others, as done below.
S.No
| Election Pairs
| Supporter
| Outcome
|
| 1. | Courthouse-School
| Colbert and Fallon prefer Courthouse and Kimmel prefers School
| Courthouse
|
| 2. | School-Park
| Fallon and Kimmel prefer School and Colbert prefers Park
| School
|
| 3. | Courthouse-Park
| Fallon prefers Courthouse but Colbert and Kimmel prefer Park
| Park
|
You can see that there is no overall majority for one particular public good. There is only a pairwise majority. For example, the Courthouse is preferred over School, and School is preferred over Park. This should mean that Courthouse should be preferred over Park as well, but this is not the case above. This is the paradox of voting, where ranking preferences can lead to irrational outcomes.
