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What's the reasoning behind choosing the power of 2 for this?

It could be tuned to a power of 1.9 or 2.1 or 3 or 1.5.

Aside from being a round number, how do we know 2 is best? What would be the pro/con of a higher exponent? What about the pro/con of a lower exponent?

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As the paper's abstract states, "Quadratic cost uniquely makes the marginal cost proportional to votes purchased". The full paper shows their proof and the assumptions behind it, such as "an individuals’ value for votes is likely to be approximately linear in the number of votes she casts so long as her utility is driven by the impact she has on the vote total, as argued by Mueller (1973, 1977) and Laine (1977)".

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