PQV
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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}
​
​
kx\sqrt{kx}
​
​
xx
​
​
❌
​
PQV
​​​​
x\sqrt{x}
or 0,
ELE_L
​
​
ERE_R
​
​
xNx≤x\frac{x}{N}\sqrt{x}\leq \sqrt{x}
​
​
⭕
​
  • Ballots means Reflected votes
  • ​
    ELE_L
    : Refer to Equation 1​
  • ​
    ERE_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

EL=xNxE_{L}= \frac{x}{N}\sqrt{x}
Assume that total number of votes is
NN
and the number of votes a user has is
xx
. The expected value of votes when
xx
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}}
[Equation 2] is expected value of votes in a situation when
xx
votes are divided into
kk
to do Sybil attack.
pp
is the number of groups reflected in the vote out of
kk
.

Equation 3

EL>ERE_{L} \gt E_{R}
when
kNX≠0∨k>1\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
xx
,
NN
are not
00
and
kk
is always bigger than
22
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.