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# 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
$x$
$k$
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.