# Independence of Irrelevant Alternatives

## What is Independence of Irrelevant Alternatives?

Independence of Irrelevant Alternatives (IIA) is a condition that states that the relative likelihood of choosing from A from B won’t change if a third choice is placed into the mix.

To put that another way: Let’s say an election was held between two candidates A, B, and C. Candidate A wins the election. Candidate C is later disqualified and removed; Candidate A should still win the election. If the original winner (A) loses the modified election (with candidate C Removed), then the Independence of Irrelevant Alternatives has been violated.

First proposed by Arrow in 1951, Independence of Irrelevant Alternatives is widely attributed to Luce (1959) and often called Luxe’s Axiom of Choice.

## Violations of Independence of Irrelevant Alternatives

Benson et al. (2016) offer an anecdote of a dramatic violation of the axiom: A man asks his date if she would prefer rock music or classical music. She chooses rock music. When he mentions that he also has jazz (an irrelevant alternative), she changes her mind and says “In that case, I’d rather have classical.”

In an election, ideally the “best candidate” would win, but as we known—that isn’t always the case. Elections can be affected by many things, including the curveball candidate who doesn’t stand a chance of winning, yet appears on the ballet and affects results.

Let’s say the next presidential election has Mrs. Bidet (Democrat) and Mr. Crumb (Republican). These are the two viable candidates; It’s nearly 100% probability that one of these two will win the election. Let’s assume that Mrs. Bidet wins with a majority vote of 55%. The “best candidate” has won. Now let’s throw an “irrelevant alternative” into the mix: Mr. Rap, who has a practically zero chance of winning the election. However, his ideals are closely aligned with Mrs. Bidet, resulting in a 50/50 split between the two candidates. Even though Mr. Rap didn’t stand a chance of winning, his presence on the ballot took away Mrs. Bidet’s chance of winning, therefore violating the Independence of Irrelevant Alternatives.

## References

Arrow, K. Social Choice and Individual Values Paperback. Martino Fine Books, 2012.
Benson, A. et al. (2016). On the Relevance of Irrelevant Alternatives.
R. D. Luce. On the possible psychophysical laws.
Psychological Review, 66(2):81, 1959.
Ray, P. Independence of Irrelevant Alternatives. Econometrica. Vol. 41, No. 5 (Sep., 1973), pp. 987-991.