# Quadratic Voting in Blockchain Governance

## Abstract

**:**

## 1. Introduction

## 2. The Framework

#### 2.1. A Benchmark Framework with Simultaneous Selection of ${s}_{ir}$

**Proposition**

**1.**

**Proof.**

#### 2.2. An “Alternative” Quadratic Voting (AQV)

**Proposition**

**2.**

**Proof.**

#### 2.3. Extending Quadratic Voting to “Any Power” Voting

**Corollary**

**1.**

**Proof.**

#### 2.4. An Asymmetric Model with Simultaneous Choice

- (i)
- $\alpha >1$ and ${s}_{1}-{s}_{2}\alpha >0,$ then expression (20) is never satisfied;
- (ii)
- $\alpha >1$ and ${s}_{1}-{s}_{2}\alpha <0$, then expression (20) can be satisfied;
- (iii)
- $\alpha <1$ and ${s}_{1}-{s}_{2}\alpha >0$, then expression (20) can be satisfied;
- (iv)
- $\alpha <1$ and ${s}_{1}-{s}_{2}\alpha <0$, then expression (20) is always satisfied.

#### 2.5. The General Model of QV

#### 2.5.1. A Linear Success Probability

## 3. Sybil Attacks

**Proposition**

**3.**

**Proof.**

## 4. Simultaneous versus Sequential Staking

- (1)
- at some round, when (at least one) user chooses the number of votes to allocate for that item, they are able to observe the number of votes chosen by the other users for that round.
- (2)
- at some round, users are able to observe the votes chosen by the other users in previous rounds and possibly change their plans made at the beginning of the voting session.

## 5. A More Detailed Specification of the Users’ Preferences

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Rounds | Total Value | ||||
---|---|---|---|---|---|

1 | 2 | 3 | |||

Users | 1 | ${v}_{11A}=10$ | ${v}_{12B}=15$ | ${v}_{13A}=2$ | ${V}_{1}=27$ |

2 | ${v}_{21B}=20$ | ${v}_{22A}=10$ | ${v}_{23B}=5$ | ${V}_{2}=35$ | |

3 | ${v}_{31A}=8$ | ${v}_{32A}=8$ | ${v}_{33A}=4$ | ${V}_{3}=20$ | |

${A}_{r}/{B}_{r}$ round value | ${A}_{1}=18,{B}_{1}=20$ | ${A}_{2}=18{B}_{2}=15$ | ${A}_{3}=6,{B}_{3}=5$ | ||

Total round value | $38$ | $33$ | $11$ | $82$ | |

Number of users | ${A}_{1}=2,{B}_{1}=1$ | ${A}_{2}=2,{B}_{2}=1$ | ${A}_{3}=2,{B}_{3}=1$ |

Rounds | % Value | ||||
---|---|---|---|---|---|

1 | 2 | 3 | |||

Value obtained by the users | 1 | $10$ | $0$ | $2$ | $12/27=0.44$ |

2 | $0$ | $10$ | $0$ | $10/35=0.28$ | |

3 | $8$ | $8$ | $4$ | $20/20=1$ |

Rounds | Total Value | ||||
---|---|---|---|---|---|

1 | 2 | 3 | |||

Users | 1 | ${v}_{11B}=10$ | ${v}_{12B}=13$ | ${v}_{13B}=4$ | ${V}_{1}=27$ |

2 | ${v}_{21B}=20$ | ${v}_{22B}=10$ | ${v}_{23B}=5$ | ${V}_{2}=35$ | |

3 | ${v}_{31A}=6$ | ${v}_{32A}=6$ | ${v}_{33A}=8$ | ${V}_{3}=20$ | |

${A}_{r}/{B}_{r}$ round value | ${A}_{1}=6{B}_{1}=30$ | ${A}_{2}=6{B}_{2}=23$ | ${A}_{3}=8,{B}_{3}=9$ | ||

Total round value | $36$ | $29$ | $17$ | $82$ | |

Number of users | ${A}_{1}=1,{B}_{1}=2$ | ${A}_{2}=1,{B}_{2}=2$ | ${A}_{3}=1,{B}_{3}=2$ |

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Dimitri, N.
Quadratic Voting in Blockchain Governance. *Information* **2022**, *13*, 305.
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Dimitri N.
Quadratic Voting in Blockchain Governance. *Information*. 2022; 13(6):305.
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**Chicago/Turabian Style**

Dimitri, Nicola.
2022. "Quadratic Voting in Blockchain Governance" *Information* 13, no. 6: 305.
https://doi.org/10.3390/info13060305