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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?"

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  • $\begingroup$ Do you understand what weakly dominated action means? $\endgroup$
    – Herr K.
    Commented Apr 30, 2022 at 17:12

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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 $B$.

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