# (Game Theory) Why is voting for your worst alternative a weakly dominated action?

I don't fully understand why voting for your worst alternative is a weakly dominated action.

The question comes from a question I'm working on: "Assume there are three candidates, A,B and C,running in an election. The voters have differing preferences between the three candidates. Assume that there are 10 voters of type A who prefer A to B, and B to C. Assume there are 10 voters of type B who prefer B to A, and A to C. Finally, assume there are 3 voters of type C, who prefer C to B, and B to A.

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b) Show that for voter of type A, voting for B is not weakly dominated

c) Show, for a voter of type A, voting for C is weakly dominated

d) Can you construct an equilibrium in which i) Candidate C wins the election and ii) no voter is using a weakly dominated action?"

• Do you understand what weakly dominated action means? Commented Apr 30, 2022 at 17:12

To see that voting $$C$$ for a type $$A$$ voter is weakly dominated you need to find a strategy that results in a weakly better outcome irrespective of the behavior of the other voters. Voting $$A$$ would be a natural candidate. There are three different scenarios if voting $$C$$: $$A$$ wins, $$B$$ wins, $$C$$ wins. First, suppose $$A$$ wins, voting $$A$$ instead of $$C$$ does not change the outcome. Second, suppose $$B$$ wins, voting $$A$$ instead of $$C$$ will either not change the outcome or yield a win for $$A$$, which the voter prefers. Third, suppose $$C$$ is the outcome, voting $$A$$ either does not change the outcome or yields that $$B$$ or $$A$$ wins, which the voter prefers to $$C$$. Thus, $$A$$ weakly dominates $$C$$.
Voting $$B$$ is not dominated. Suppose $$B$$ is the outcome, voting either $$C$$ or $$A$$ can result in candidate $$C$$ winning, which is worse. Thus, neither $$A$$ nor $$C$$ dominate $$B$$.