Arrow’s Impossibility Theorem and Voters’ Transitivity (Post - 4)

“Democracy is the worst form of government, except all of others that have been tried.”

Winston Churchill!

Preferences plays an important role in everyone’s daily life. Consumers buy what they prefer. If one prefers chocolate over vanilla, he or she buys chocolate if both items priced equally. If the same person prefers vanilla over strawberry, he or she buys vanilla over strawberry if the prices of vanilla and strawberry priced equally. Furthermore, this individuals chose chocolate over strawberry if these two priced equally. This sequences of preferences called a transitive preferences. This simply illustrates that, if one prefers A to B and B to C, a rational person expected to choose A over C.

This is not the case in democracy. If there are three candidates to vote from, it is possible that the candidate who received the most vote might not be the winner, and it demonstrated by “Arrow’s Impossibility Theorem.” In social choice theory, Arrow's Impossibility Theorem states that a clear order of preferences can’t be determined while adhering to mandatory principles of fair voting procedure.

Here is an illustration of typical election problems. Consider that voters asked to rank their preferences of candidates A, B, and C.

• 45 votes A > B> C (45 people prefer A over B and B over C.
• 40 votes B > C >A (40 people prefer B over C and C over A).
• 30 votes C > A > B (30 people prefer C over A and prefer A over B).

Candidate A has the most votes, and he or she would be the winner. However, if B is not running, C would be the winner, as more people prefer C over A. A would have 45 votes, and C would have 70 votes. Since only one has to represent one particular constituent, election rules are set for an outcome of one winner. When three candidates run for the same seat, the top two often go on for run off. The third candidates determined as a loser automatically. There is a possibility that the loser could be the winner according to the Arrow’s Impossibility Theorem. The above example illustrates this finding, when we follow one rule, candidate A wins and using different rule, candidate C would be the winner.

This outcome violates the transitivity rule which noted above. When one presented with chocolate, vanilla, and strawberry, a rational individual would choose always chocolates if he or she prefers chocolate over vanilla and strawberry doesn’t matter how one presented it. Voter’s preferences doesn’t follow this pattern and voter’s preferences are not transitive. Transitivity is the assumption that if one prefers candidate A to candidate B and candidate B over candidate C, it follows that this voters must prefer candidate A over candidate C. As noted in the example above, this transitivity assumption cannot guaranteed under democratic voting system. In simple term, consumers can get what they prefer at the grocery store; on the other hand, voters can’t get the candidate they prefer given that there are three or more candidates running. That is the outcome Kenneth Arrow proved long ago.

Man, Priscilla T Y ¹ ; Takayama, Shino

A Unifying Impossibility Theorem

Social Choice, Clubs, Committees, Associations (D71);
Impossibility Theorem;
Social Choice