# Liquid Democracy

## The third ‘Democracy’

### Current known democratic processes

#### Direct Democracy

poeple vote for policies directly. This type of democracy only works when each of the members are informed and can estimate the impact of each policy.

similar but different Approval Voting

(最高裁判所裁判官国民審査)

#### Representative Democracy

you choose a representative (usually someone) to decide for you. Cuts costs for informed concent, but trades off control.

### different democracies and their flaws

アリストテレスが偉大だったのは、この問いかけについて、「この政治体制こそが最高である！」と自分の信念を主張するのではなく、「そもそも、どういう政治体制がありえるのか、そして、それぞれどんな特徴を持っているのか、まずは分析してみよう」

#### TODO redraw diagram into english

Liquid Democracy tries to include all three, the intention is to have a smooth transition between them.

### Liquid Democracy

Democracy that you can vote for both policies and people.

#### Example1: Deciding Breakfast

As a family we need to decide what to have for breakfast. The options are Rice or Bread.

Minori put 0.1 points to Yasushi and Ray to deicide, and voted directy 0.1 and 0.7 points respectively to Rice and Bread. Yasushi voted 0.2 points to Minori and 0.3 points to Ray. He gave half of his vote to Rice. Ray gave 0.4 points to her parents and gave the remainder for eating Bread.

some points to consider

• Minori contradics. She is giving 0.1 points to rice while giving it 0.7 to bread.
• Yasushi is the only one directly voting for Rice
• Ray has limited knowledge and may not know how Rice tastes.

• Voting matrix

The Voting Matrix ($$V$$) will be:

Minori0.00.20.40.00.0
Yasushi0.10.00.40.00.0
Ray0.10.30.00.00.0
* Rice0.10.50.01.00.0

\begin{bmatrix} 0.0 & 0.2 & 0.4 & 0.0 & 0.0 \\
0.1 & 0.0 & 0.4 & 0.0 & 0.0 \\
0.1 & 0.3 & 0.0 & 0.0 & 0.0 \\
0.1 & 0.5 & 0.0 & 1.0 & 0.0 \\
0.7 & 0.0 & 0.2 & 0.0 & 1.0 \\
\end{bmatrix}

• results:

Rice1.1725

so we have bread this morning.

• influences:

Representativeinfluence
Minori11.0
Yasushi4.625
Ray3.5

Minori has the most influence for deciding breakfast.

#### Math

based on [1]. ref pp 141-148

\begin{bmatrix} \ v_{11} & v_{12} & v_{13} & \cdots & v_{1n} & 0 & 0 & 0 & \cdots & 0 \\
\ v_{21} & v_{22} & v_{23} & \cdots & v_{2n} & 0 & 0 & 0 & \cdots & 0 \\
\ v_{31} & v_{32} & v_{33} & \cdots & v_{3n} & 0 & 0 & 0 & \cdots & 0 \\
\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
\ v_{n1} & v_{n2} & v_{n3} & \cdots & v_{nn} & 0 & 0 & 0 & \cdots & 0 \\
\ v_{(n+1)1} & v_{(n+1)2} & v_{(n+1)3} & \cdots & v_{(n+1)n} & 1 & 0 & 0 & \cdots & 0 \\
\ v_{(n+2)1} & v_{(n+2)2} & v_{(n+2)3} & \cdots & v_{(n+2)n} & 0 & 1 & 0 & \cdots & 0 \\
\ v_{(n+3)1} & v_{(n+3)2} & v_{(n+3)3} & \cdots & v_{(n+3)n} & 0 & 0 & 1 & \cdots & \vdots \\
\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & 0 \\
\ v_{(n+m)1} & v_{(n+m)2} & v_{(n+m)3} & \cdots & v_{(n+m)n} & 0 & 0 & \cdots & 0 & 1 \\
\end{bmatrix}

$\ A_{t+1} = VA_t$

where the $$V$$ is the voting matrix and $$A_0$$ is $$I$$. when $$A’$$ is the result with enough recusive operations that convergeses, the vote result is.

$\ a_i=\sum_{j=1}^{M} A’_{ij} \; \textrm{for all} \; i$

the infulence of each member can be calculated by:

$\ \frac{\sum_{j=1}^{H} \sum_{t=0}^{\infty} A_{ij}(t)}{\sum_{t=0}^{\infty} A_{ii}(t)} \; \textrm{for all} \; 1 \leq i \leq H$

code can be found at github

## Reference

1. なめらかな社会とその敵 - Ken Suzuki

2. Super minds - Thomas W. Malone