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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.

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  • $\begingroup$ 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."? $\endgroup$
    – Giskard
    Commented Dec 6, 2023 at 7:57

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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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