# Why does the voter paradox entail 0 turnout?

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?