🚧Mathematical Proof

Comparison Between QV and PQV

Voting Method	Ballots	Ballots in Sybil Attack	Max Ballots	Sybil Resistancy
QV	$\sqrt{x}$	$\sqrt{kx}$	$x$	❌
PQV	​​​​$\sqrt{x}$ or 0, $E_L$	$E_R$	$\frac{x}{N}\sqrt{x}\leq \sqrt{x}$	⭕

• Ballots means Reflected votes

• $E_L$: Refer to Equation 1

• $E_R$: Refer to Equation 2

Sybil Resistancy

PQV can prove its quality as a new voting system by mathematically proving its Sybil resistancy without (less) harming the result of existing QV.

Equation 1

$$E_{L}= \frac{x}{N}\sqrt{x}$$

Assume that total number of votes is $N$ and the number of votes a user has is $x$. The expected value of votes when $x$ votes are honestly voted at once from one account can be expressed as [Equation 1].

Equation 2

$$E_{R}= \sum_{p=0}^{k}\binom{k}{p}\left( \frac{x/k}{N} \right)^p\left(1- \frac{x/k}{N}\right)^{k-p}p\sqrt{\frac{x}{k}}$$

[Equation 2] is expected value of votes in a situation when $x$ votes are divided into $k$ to do Sybil attack. $p$ is the number of groups reflected in the vote out of $k$.

Equation 3

$$E_{L} \gt E_{R}$$

when $\frac{kN}{X}\neq 0 \vee k\gt 1$

In order for a Sybil attack to always less beneficial than an honest vote, the condition of [Equation 3] needs to be satisfied. This condition is always satisfied because $x$, $N$ are not $0$ and $k$ is always bigger than $2$ to do Sybil attack.

PQV makes it always a loss to do sybil attack by applying probabilistic element on quadratic voting. Splitting voting power makes the expected value of voting power lower that executing 1 voting power. Therefore, rational users who want to maximize their voting influence will honestly exercise their votes at once by using one account. This means PQV can prevent Sybil attacks.