<p>To see that voting <span class="math-container">$C$</span> for a type <span class="math-container">$A$</span> 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 <span class="math-container">$A$</span> would be a natural candidate. There are three different scenarios if voting <span class="math-container">$C$</span>: <span class="math-container">$A$</span> wins, <span class="math-container">$B$</span> wins, <span class="math-container">$C$</span> wins. First, suppose <span class="math-container">$A$</span> wins, voting <span class="math-container">$A$</span> instead of <span class="math-container">$C$</span> does not change the outcome. Second, suppose <span class="math-container">$B$</span> wins, voting <span class="math-container">$A$</span> instead of <span class="math-container">$C$</span> will either not change the outcome or yield a win for <span class="math-container">$A$</span>, which the voter prefers. Third, suppose <span class="math-container">$C$</span> is the outcome, voting <span class="math-container">$A$</span> either does not change the outcome or yields that <span class="math-container">$B$</span> or <span class="math-container">$A$</span> wins, which the voter prefers to <span class="math-container">$C$</span>. Thus, <span class="math-container">$A$</span> weakly dominates <span class="math-container">$C$</span>.</p>
<p>Voting <span class="math-container">$B$</span> is not dominated. Suppose <span class="math-container">$B$</span> is the outcome, voting either <span class="math-container">$C$</span> or <span class="math-container">$A$</span> can result in candidate <span class="math-container">$C$</span> winning, which is worse. Thus, neither <span class="math-container">$A$</span> nor <span class="math-container">$C$</span> dominate <span class="math-container">$B$</span>.</p>

CC BY-SA 4.0
2022-05-06T10:21:27.953Z
2022-05-06T10:30:44.337Z
2022-05-06T10:30:44.337Z
GameTheorist1982
<p>I don't fully understand why voting for your worst alternative is a weakly dominated action.</p>
<p>The question comes from a question I'm working on:
&quot;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.</p>
<p>..</p>
<p>b) Show that for voter of type A, voting for B is not weakly dominated</p>
<p>c) Show, for a voter of type A, voting for C is weakly dominated</p>
<p>d) Can you construct an equilibrium in which i) Candidate C wins the election and ii) no voter is using a weakly dominated action?&quot;</p>

CC BY-SA 4.0
2022-04-30T17:12:26.143Z
Do you understand what weakly dominated action means?
Herr K.
<game-theory><nash-equilibrium><political-economy><voting>
(Game Theory) Why is voting for your worst alternative a weakly dominated action?
christinaf
# (Game Theory) Why is voting for your worst alternative a weakly dominated action?

(Asked by: username_0 on 2022-04-30)

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.

..

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?"

**Tags:** game-theory, nash-equilibrium, political-economy, voting

> **Comments:**
> **username_1**: Do you understand what weakly dominated action means?  

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## 1 answers

### Answer by username_2 on 2022-05-06. Score: 1

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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