<p>The probability of changing the outcome of the election with your vote (p) does not decrease with the population size but instead with the total number of votes cast. As the population increases, the model you've described predicts that voter turnout (ie. votes/population) will approach zero not because the number of votes cast is decreasing but because it's levelling off while population continues to increase.</p>
<p>In any case, Down's paradox is not really a paradox but just the fact that voter turnouts are generally far higher than what this model predicts (<a href="https://www.jstor.org/stable/3216877" rel="nofollow noreferrer">https://www.jstor.org/stable/3216877</a>). There are several proposed explanations for this discrepancy but what they all have in common is that prospective voters are not making their decision as traditionally &quot;rational&quot; actors.</p>

CC BY-SA 4.0
2024-03-21T19:06:13.723Z
2024-03-21T19:06:13.723Z
H Rogers
<p>The decision to cast a vote (under a simplistic FPTP two-candidate model), as formalised by Downs can be written as follows:</p>
<p><span class="math-container">$pB \geq C$</span></p>
<p>where <span class="math-container">$p$</span> is the probability of you changing the outcome, <span class="math-container">$B$</span> the benefit received from the outcome you were aiming for, and <span class="math-container">$C$</span> the cost of casting.</p>
<p>The paradox states that, as the population of a constituency increases, the probability of your vote having an impact must decrease, such that we ought to expect no votes in a sufficiently large constituency.</p>
<p>I am a little confused by this reasoning, as for each voter who decides not to vote due to this condition, we would expect the probability to rise again, such that we should reach an equilibrium with some turnout greater than 0.</p>
<p>Whether this is a function of the cost threshold, or implies a turnout of 100%, or some other value, I do not feel confident enough to tell, but the 0 turnout seems very suspect to me. Might anyone be able to help me with the reasoning?</p>
<p>My sincere thanks in advance.</p>

CC BY-SA 4.0
2023-12-06T07:57:26.607Z
Hi! Can you please provide a proper reference to the claim "*The paradox states that, as the population of a constituency increases, the probability of your vote having an impact must decrease, such that we ought to expect no votes in a sufficiently large constituency.*"?
Giskard
<game-theory><political-economy>
Why does the voter paradox entail 0 turnout?
fruitless fruit juice
# Why does the voter paradox entail 0 turnout?

(Asked by: username_0 on 2023-12-05)

The decision to cast a vote (under a simplistic FPTP two-candidate model), as formalised by Downs can be written as follows:

$pB \geq C$

where $p$ is the probability of you changing the outcome, $B$ the benefit received from the outcome you were aiming for, and $C$ the cost of casting.

The paradox states that, as the population of a constituency increases, the probability of your vote having an impact must decrease, such that we ought to expect no votes in a sufficiently large constituency.

I am a little confused by this reasoning, as for each voter who decides not to vote due to this condition, we would expect the probability to rise again, such that we should reach an equilibrium with some turnout greater than 0.

Whether this is a function of the cost threshold, or implies a turnout of 100%, or some other value, I do not feel confident enough to tell, but the 0 turnout seems very suspect to me. Might anyone be able to help me with the reasoning?

My sincere thanks in advance.

**Tags:** game-theory, political-economy

> **Comments:**
> **username_1**: Hi! Can you please provide a proper reference to the claim "*The paradox states that, as the population of a constituency increases, the probability of your vote having an impact must decrease, such that we ought to expect no votes in a sufficiently large constituency.*"?  

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## 1 answers

### Answer by username_2 on 2024-03-21. Score: 2

The probability of changing the outcome of the election with your vote (p) does not decrease with the population size but instead with the total number of votes cast. As the population increases, the model you've described predicts that voter turnout (ie. votes/population) will approach zero not because the number of votes cast is decreasing but because it's levelling off while population continues to increase.

In any case, Down's paradox is not really a paradox but just the fact that voter turnouts are generally far higher than what this model predicts (<https://www.jstor.org/stable/3216877>). There are several proposed explanations for this discrepancy but what they all have in common is that prospective voters are not making their decision as traditionally "rational" actors.


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