below is taken from Blum and Zuber’s (Blum, Christian and Zuber, Christina Isabel, 2016) formalization of Liquid Democracy.

1. directly vote on all policy issues (direct democratic component);

2. delegate their votes to a representative to vote on their behalf on

   (1) a singular policy issue, or (2) all policy issues in one or more policy areas, or (3) all policy issues in all policy areas

   (flexible delegation component);

3. delegate those votes they have received via delegation to another representative (meta-delegation component);
4. terminate the delegation of their votes at any time (instant recall component).

The core distiction is this is referring to several policies.

but not fractional voting.

but is liquid democracy “representative”, “democratic”, “legitimite”? (Landemore, H{\'e}l{\`e}ne, 2020)

We are familiar with electoral representation, what other forms of democratic representative is out there.

• lottocratic
• self-appointed (only if its open to everyone)

is electoral representation open? secure⁴?

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:

-	Minori	Yasushi	Ray	* Rice	* Bread
Minori	0.0	0.2	0.4	0.0	0.0
Yasushi	0.1	0.0	0.4	0.0	0.0
Ray	0.1	0.3	0.0	0.0	0.0
* Rice	0.1	0.5	0.0	1.0	0.0
* Bread	0.7	0.0	0.2	0.0	1.0

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

1. results:

Plan	Votes
Rice	1.1725
Bread	1.8275

so we have Bread this morning.

1. influences:

Representative	influence
Minori	11.0
Yasushi	4.625
Ray	3.5

Minori has the most influence for deciding breakfast.

Other than delegating to humans, we can think of delegeting to categories or parties, which usually a combination of policies or delegates. For instance when deciding dinner, there might be options like Sushi, Ramen, Tacos, but I might be indifferent between the two Japanese dishes. in other words:

\[ \textrm{sushi} \sim \textrm{ramen} \succ \textrm{tacos} \]

We can express political parties if it splits the votes to the party members. Or another way to use this is deciding songs, we usually have the option to choos between songs, albums, artists, genres or eras. But this can lead to a whole new can of worms, when we can vote to vote results, since a vote result is just a intermediary that holds some weights….

based on (Suzuki, Ken, 2013) pp 141-148 the book shows a situation where there is only policies and voters(or delegates), here we elaborate to incorporate intermediaries.

Given a set of voters \(X = \{x_1, \cdots, x_{|X|}\}\) and intermediaries, \(Y = \{y_1 ,\cdots , y_{|Y|}\}\), and policies \(Z = \{z_1, \cdots, z_{|Z|}\}\). The total set will be \(S = \{s_1, \cdots, s_{|X \cup Y \cup Z|}\}\). Each element \(s_i\) will have a value pointed to elements in \(\{s_j \in S\}\) this could be frankly refered as votes from other nodes (\(v_{ij}\)). this will form a voting matrix \(V\):

\begin{bmatrix} \ v_{XX} & v_{YZ} & v_{ZX} \\ \ v_{XY} & v_{YY} & v_{ZY} \\ \ v_{XZ} & v_{YZ} & v_{ZZ} \\ \end{bmatrix}

The actual matrix can also elaborated as this:

\begin{bmatrix} \ v_{x_1x_1} & \cdots & v_{x_{|X|}x_1} & v_{y_1x_1} & \cdots & v_{y_{|Y|}x_1} & v_{z_1x_1} & \cdots & v_{z_{|Z|}x_1} \\ \ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ \ v_{x_1x_{|X|}} & \cdots & v_{x_{|X|}x_{|X|}} & v_{y_1x_{|X|}} & \cdots & v_{y_{|Y|}x_{|X|}} & v_{z_1x_{|X|}} & \cdots & v_{z_{|Z|}x_{|X|}} \\ \ v_{x_1y_1} & \cdots & v_{x_{|X|}y_1} & v_{y_1y_1} & \cdots & v_{y_{|Y|}y_1} & v_{z_1y_1} & \cdots & v_{z_{|Z|}y_1} \\ \ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ \ v_{x_1y_{|Y|}} & \cdots & v_{x_{|X|}y_{|Y|}} & v_{y_1y_{|Y|}} & \cdots & v_{y_{|Y|}y_{|Y|}} & v_{z_1y_{|Y|}} & \cdots & v_{z_{|Z|}y_{|Y|}} \\ \ v_{x_1z_1} & \cdots & v_{x_{|X|}z_1} & v_{y_1z_1} & \cdots & v_{y_{|Y|}z_1} & v_{z_1z_1} & \cdots & v_{z_{|Z|}z_1} \\ \ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ \ v_{x_1z_{|Z|}} & \cdots & v_{x_{|X|}z_{|Z|}} & v_{y_1z_{|Z|}} & \cdots & v_{y_{|Y|}z_{|Z|}} & v_{z_1z_{|Z|}} & \cdots & v_{z_{|Z|}z_{|Z|}} \\ \end{bmatrix}

a visual representation will be something like this:

To form this voting matrix, there will be special restrictions. Namely the rules below:

\[v_{x_ix_i} = 0 \; \textrm{where} \; i = \{1, \cdots, |X|\} \]

\[v_{y_jy_j} = 0 \; \textrm{where} \; j = \{1, \cdots, |Y|\} \]

Avoiding self voting for voters and intermediaries, and

\[v_{z_iX \cup Y}=0 \; v_{ZZ} = I\]

to ultimately collect all votes to policies.

Another visualization on the rules will be:

\[ \ A_{t+1} = V \cdot A_t \]

where the \(V\) is the voting matrix and \(A_0\) is \(I\). when \(A'\) is the result with enough recusive operations…

While this iteration, the votes will be gathered into the left bottom part of the matrix.

When I say ’converge’ this means that all votes land to this area, not forming a endless loop between voters (or intermediaries) In the case of not converging, meaning that some votes just circulate between users or intermediaries, we might want to check if the whole network is a DAG (Gupta, Anshul and Suzumura, Toyotaro, 2021).

The result will be summing up each row of that rectangle area.

\[ a_{z_j}=\sum_{i=1}^X A'_{x_iz_j} \; \textrm{for all} \; j \in Z \]