Mathematical Definitions
For more details on the mathematics of elections, see Decision theory and Social choice theory.Possible mathematical definitions for the spoiler effect include failure of the independence of irrelevant alternatives (IIA) axiom, and vote splitting.
Arrow's impossibility theorem states that rank-voting systems are unable to satisfy the independence of irrelevant alternatives criterion without exhibiting other undesirable properties as a consequence. However, different voting systems are affected to a greater or lesser extent by IIA failure. For example, instant runoff voting is considered to have less frequent IIA failure than First Past the Post (also known as Plurality Rule). The independence of Smith-dominated alternatives (ISDA) criterion is much weaker than IIA; unlike IIA, some ranked-ballot voting methods can pass ISDA.
A possible definition of spoiling based on vote splitting is as follows: Let W denote the candidate who wins the election, and let X and S denote two other candidates. If X would have won had S not been one of the nominees, and if (most of) the voters who prefer S over W also prefer X over W (either S>X>W or X>S>W), then S is a spoiler. Here is an example to illustrate: Suppose the voters' orders of preference are as follows:
| 33%: S>X>W | 15%: X>S>W | 17%: X>W>S | 35%: W>X>S |
The voters who prefer S over W also prefer X over W. W is the winner under Plurality Rule, Top Two Runoff, and Instant Runoff. If S is deleted from the votes (so that the 33% who ranked S on top now rank X on top) then X would be the winner (by 65% landslide majority). Thus S is a spoiler with these three voting methods.
Read more about this topic: Spoiler Effect
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