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Mathematical Proof

Comparison Between QV and PQV
Voting Method
Ballots
Ballots in Sybil Attack
Max Ballots
Sybil Resistancy
QV
​x\sqrt{x}x​
​
​kx\sqrt{kx}kx​
​
​xx x
​
​❌
​
PQV
​​​​x\sqrt{x} x​
 or 0, ELE_LEL​
​
​ERE_RER​
​
​xNx≤x\frac{x}{N}\sqrt{x}\leq \sqrt{x}Nx​x​≤x​
​
​⭕
​
Ballots means Reflected votes
​ELE_LEL​
: Refer to Equation 1​
​ERE_RER​
: 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
EL=xNxE_{L}= \frac{x}{N}\sqrt{x}EL​=Nx​x​
Assume that total number of votes is NNN
 and the number of votes a user has is xxx
. The expected value of votes when xxx
 votes are honestly voted at once from one account can be expressed as [Equation 1].
Equation 2
ER=∑p=0k(kp)(x/kN)p(1−x/kN)k−ppxkE_{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}}ER​=p=0∑k​(pk​)(Nx/k​)p(1−Nx/k​)k−ppkx​​
[Equation 2] is expected value of votes in a situation when xxx
 votes are divided into kkk
 to do Sybil attack. ppp
 is the number of groups reflected in the vote out of kkk
.
Equation 3 
EL>ERE_{L} \gt E_{R}EL​>ER​
when kNX≠0∨k>1\frac{kN}{X}\neq 0 \vee k\gt 1XkN​=0∨k>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 xxx
, NNN
 are not 000
 and kkk
 is always bigger than 222
 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.

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

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Last modified 11mo ago