> For the complete documentation index, see [llms.txt](https://d3lab-dao.gitbook.io/pqv/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://d3lab-dao.gitbook.io/pqv/solution-pqv/mathematical-proof.md).

# Mathematical Proof

## Comparison Between QV and PQV

<table><thead><tr><th width="150" align="center">Voting Method</th><th width="150" align="center">Ballots</th><th width="150" align="center">Ballots in Sybil Attack</th><th width="150" align="center">Max Ballots</th><th width="150">Sybil Resistancy</th></tr></thead><tbody><tr><td align="center"><strong>QV</strong></td><td align="center"><span class="math">\sqrt{x}</span></td><td align="center"><span class="math">\sqrt{kx}</span></td><td align="center"><span class="math">x </span></td><td><span data-gb-custom-inline data-tag="emoji" data-code="274c">❌</span></td></tr><tr><td align="center"><strong>PQV</strong></td><td align="center">​​​​<span class="math">\sqrt{x} </span> or 0, <span class="math">E_L</span></td><td align="center"><span class="math">E_R</span></td><td align="center"><span class="math">\frac{x}{N}\sqrt{x}\leq \sqrt{x}</span></td><td><span data-gb-custom-inline data-tag="emoji" data-code="2b55">⭕</span></td></tr></tbody></table>

* `Ballots` means `Reflected votes`
* $$E\_L$$: Refer to [Equation 1](#equation-1)
* $$E\_R$$: Refer to [Equation 2](#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&#x20;

$$
E\_{L} \gt E\_{R}
$$

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.