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# Mathematical Proof

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`

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

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

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

**[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 $k$

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

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.

Last modified 11mo ago