# Generalizing Liquid Democracy to multi-agent delegation: A Voting Power
Measure and Equilibrium Analysis

Francisco M. Bersetche  Departamento de Matemática, Universidad Técnica
Federico Santa María, Av. España 1680, Valparaíso, Chile<EMAIL_ADDRESS>
###### Abstract.

Liquid democracy has gained popularity in recent years due to its ability to
balance representation and delegation of power. In this work, we propose a
generalization of the classic model that allows for fractional delegation of
voting weight. Our approach enables agents to divide and delegate their votes
to multiple agents, while retaining a portion of the voting power for
themselves. We discuss the desirable properties of a reasonable generalization
of the classic model and introduce a set of simpler voting measures that
include a penalty factor on the length of delegation chains. We demonstrate
that the proposed voting measure is a well-defined limit of these simpler
measures when the penalty approaches zero, and inherits key features of the
classic model. In the second part of the article, we investigate the existence
of equilibrium states in a delegation game that employs the suggested
measures. We show that this game has pure strategy Nash equilibria as long as
a penalty on the length of delegation chains is enforced.

The author was partially supported by ANID through FONDECYT Project 3220254.
Part of this work was performed in Centro Atómico Bariloche under a
postdoctoral fellowship funded by CONICET

##  1\. Introduction

Liquid democracy (LD) is a collective decision making method that lies between
representative democracy and direct democracy. In LD, an agent can choose to
either vote directly or to select another agent as their proxy. Unlike classic
proxy voting, the delegation is transitive; this means that the proxy may
delegate their voting weight further, creating delegation paths along which
the voting weight flows and accumulates.

To illustrate the mechanics of LD, let $\\{a,b,c,d,e\\}$ be a set of five
agents. If $a$ delegates her voting weight to $b$, $b$ delegates to $c$, and
$d$ and $e$ do not delegate their vote to anyone, then the distribution of
voting weight in the set is: agent $d$ has a voting weight of 4 (the votes of
agents $c$, $b$, and $a$, plus her own vote), agent $e$ has a voting weight of
1, and the rest have 0 voting weight. If instead agent $b$ delegates to agent
$e$, then the distribution is 2 for agent $d$, 3 for agent $e$, and 0 for the
rest. If further agent $e$ delegates to agent $a$, then a delegation cycle is
produced since each one in turn delegates back to another one in the cycle. In
most articles and practical implementations of LD such cycles are usually
ignored with their votes considered as null.

Over the past decade, several practical implementations of LD have been
conducted [23]. Notable examples include Zupa [22], which was employed by the
Student Union of the Faculty of Information Studies in Novo mesto, Slovenia;
LiquidFeedback [2], which was used by the German Pirate Party; and Sovereign
[25][Section 2], a blockchain-based platform developed and utilized by the
Democracy Earth Foundation in Argentina. Of these implementations, the use of
LiquidFeedback by the German Pirate Party is the best documented in literature
and will be considered as a benchmark for LD practical applications throughout
this work 111It is worth noting the findings obtained in [24], wherein the
authors experimentally implement liquid democracy systems focused on binary
problems. This study found that higher-expertise voters are less likely to
delegate their vote in liquid democracy than lower-expertise voters, and the
average expertise of voters who delegate is lower than the expertise of those
receiving delegations, suggesting that delegation behaviors meet the
conditions for positive theoretical guarantees..

Proxy voting models have been studied since Carroll’s work in 1884 [13]. In
the last decade, these models have been investigated by political science [18]
and artificial intelligence communities [9]. Subsequent research has focused
on potential improvements and modifications, as well as possible weaknesses of
these models [5, 11, 12, 19, 20], among others. For a comprehensive review of
this topic see [23] and the references therein.

### Contribution

Most implementations of liquid democracy and much of the literature consider a
paradigm of simple delegation, wherein each agent can either delegate all
their voting weight to a single agent or cast their vote directly 222
Sovereign [25] provides each user with a finite number of votes that can be
delegated independently, constituting a multi-agent delegation. However, each
received vote must then be dispensed independently by each proxy.. This paper
generalizes this model by allowing agents to divide their voting weight
arbitrarily and delegate it to multiple agents, while also retaining part of
it for themselves. This is achieved through a new voting weight measure which
preserves the main properties of classic delegation. This measure is
constructed from a set of simpler voting measures that incorporate a penalty
factor on the length of delegation chains. We then define the proposed voting
measure as the limiting case when this penalty goes to zero. As in most
platforms and available works, we consider the voting weight trapped in
delegation cycles as null votes (see [1]).

The proposed measure has the following properties:

  * (P1)

It generalizes the concept of simple delegation, whereby classic delegation is
recovered when agents choose to delegate their voting weight without dividing
it.

  * (P2)

Delegation makes sense. If agent $a$ delegates her voting weight to agent $b$,
the distribution of voting weight is equivalent to what would be obtained if
agent $a$ replicated the delegation profile of agent $b$. This implies that
delegating to an agent is equivalent to replicating their actions. Although we
are being advisedly vague here, this property is formalized and proven in
Theorem 3.7.

  * (P3)

If an agent chooses to retain a fraction $q\in(0,1]$ of the vote for herself,
then her voting weight should be no less than $q$. Conversely, if an agent
opts to delegate all the power, her voting weight should be rendered null.
This property is referred to as consistency.

Note that property (P2) implies that, for an agent, delegating is equivalent
to allowing their proxy to make decisions on their behalf. This intuitively
corresponds with the expected outcome of delegation. To the best of our
knowledge, no existing multi-agent delegation models which allow agents to
retain a portion of voting weight have been able to achieve this key property.
We refer to (P2) as the delegation property. We emphasize that the properties
(P1)-(P3) are desirable, but do not necessarily provide a comprehensive
characterization of a reasonable generalization of the classic LD model. We
prove that our proposed measure has desirable features in addition to
(P1)-(P3).

On the other hand, the temporal variable is a crucial factor to consider in
our analysis. As reported in [3], the experience with LiquidFeedback reveals
that the behavior of agents changes over time due to their observations of
other agents’ actions, which is essential for the development of decision-
making processes. Therefore, we assume that an agent’s delegation profile can
be represented as a function that evolves over time and depends on real-time
feedback received by the agent. This leads us to ask whether this process can
reach an equilibrium state, then in the second part of this work, we introduce
a game-theoretic model where agents serve as players and their delegation
profiles represent their strategies. Each agent $i$ is endowed with a utility
function that measures the amount of their own vote attributed to a specific
player $j$, weighted by a satisfaction factor which accounts for $i$’s
contentment with the proportion of their voting power ceded to $j$. We
demonstrate the existence of Nash equilibria in this game under the condition
of non-zero penalties imposed on extended delegation chains. Furthermore, we
establish the main result of this work, where we establish the possibility of
defining a voting power measure such that the fundamental properties of
classic delegation are upheld within an acceptable margin of error, while
simultaneously attaining pure Nash equilibria for the examined game. The
voting power measures employed in this result are identical to those used in
constructing the measure introduced in the first part of this work, with a
suitably small penalty factor.

### Related work

The model proposed in [15] is similar in spirit to the one presented in this
work, as it is a generalization of classic LD and allows for the division and
delegation of voting weight to multiple agents. However, this model does not
allow agents to retain a fraction of voting weight for themselves. It should
be noted that this model coincides with the one proposed here when agents are
not allowed to preserve a fraction different from 1 or 0 of voting weight for
themselves (see Remark 3). Spectral type centrality measures have been
proposed for measuring voting weight; [26] explores the use of PageRank as a
voting measure, while [7] propose another voting measure based on a PageRank-
type centrality which seeks to penalize long delegation paths. Although simple
delegation is used here for theoretical analysis purposes, these techniques
can easily be generalized to multi-agent delegation. On the other hand,
following a different approach, in [14], the authors propose and analyze a
model based on multi-agent ranked delegation, in which agents provide a
ranking over multiple delegations to be used as backups when solving possible
delegation cycles. Similarly, [10] and [11] study the delegation of vote
through a ranking of preferences, while [17] proposes a technique based on
fluid dynamics that attempts to minimize concentration of votes by choosing an
appropriate delegation graph according to the agents’ preferences.

In Section 4, we define a game in which the delegation profile of each agent
is considered as a strategy, and an utility function based on the preferences
of each agent over the others. This type of game has been studied in the
literature with regard to simple delegation, e.g., [16] which analyzes this
kind of games with three different utility functions. It can be seen that the
problem addressed in Section 4 is a generalization of the MINDIS problem
defined in [16, Section 3.2]. Other works dealing with similar topics include
[4, 21, 27].

### Organization of the paper

The rest of the paper is organized as follows. Section 2 provides an intuitive
description of the proposed voting power measure, beginning with its
probabilistic interpretation. These ideas are formalized in Section 3, where
we define the measure and prove its main properties, as well as provide an
efficient algorithm for computation. In Section 4, game theory is employed to
investigate the existence of equilibrium states in the context of agents’
ability to modify their delegation profile according to system outcomes.
Ultimately, our main result is demonstrated, which establishes the feasibility
of constructing a voting power measure through imposition of penalties on long
delegation chains, that both preserves classic delegation properties and
allows for pure strategy Nash equilibria.

##  2\. Preliminaries and main ideas

###  2.1. Preliminaries

Let $N=\\{1,...,n\\}$ be the set of agents. Each agent $i$ expresses to whom
they want to delegate their voting weight, and the fraction they want to
delegate, through the delegation profile $x_{i}\in X$, with
$X:=\\{x\in\mathbb{R}^{n}_{\geq 0}:\sum_{k}x_{k}=1\\}$. Here $x_{ij}$
represents the fraction of voting weight that the agent $i$ wants to delegate
to agent $j$. We define the matrix $P:=(x_{1}|...|x_{n})\in\mathbb{R}^{n\times
n}$ with the delegation profiles as columns, and we call it delegation matrix.
Clearly $P$ is a (left)stochastic matrix (since $\sum_{i}P_{ij}=1$), with the
delegation profile of agent $j$ stored in its $j$-th column. We denote by
$\mathcal{P}^{n}\subset\mathbb{R}^{n\times n}$ the set of all possible
delegation matrices. Note that while the delegation profiles from set X and
delegation matrices from $\mathcal{P}^{n}$, are strongly related, it is
convenient to treat these two concepts separately, especially in Section 4,
where delegation profiles represent agents’ strategies.

Given $i$ and $j\in N$, we say that there is a delegation path between $i$ and
$j$ if there is a strictly positive sequence
$(x_{ik_{1}},x_{k_{1}k_{2}},...,x_{k_{q-1}k_{q}},x_{k_{q}j})$ with
$k_{1},...,k_{q}\in N$. Given a vector $v\in\mathbb{R}^{n}$, we define the
matrix $diag(v)\in\mathbb{R}^{n\times n}$ as $diag_{ii}(v):=v_{i}$ for all
$1\leq i\leq n$, and $diag_{ij}(v):=0$ for all $i\not=j$. As is common in
packages designed for numerical methods, if $A\in\mathbb{R}^{n\times n}$, we
define $diag(A)\in\mathbb{R}^{n}$ as $diag_{i}(A):=A_{ii}$ for all $1\leq
i\leq n$. Unless specified otherwise, throughout this paper vectors will
always be regarded as column vectors.

###  2.2. The problem of multiple delegation

Let $N=\\{1,...,n\\}$ denote the set of agents in the classic model of liquid
democracy (LD), where agents can delegate their voting power to a single
agent, including themselves, and we refer to agents who preserve all their
voting power as candidates. The delegation matrices obtained under this
paradigm correspond to adjacency matrices associated with delegation graphs
defined by agent preferences. Defining
$\mathcal{A}^{n}:=\\{A\in\mathcal{P}^{n}:A_{ij}\in\\{0,1\\}\text{ for all
}1\leq i,j\leq n\\}$, the classic measure of voting power or voting weight is
a function $V^{c}:\mathcal{A}^{n}\to\mathbb{R}^{n}_{\geq 0}$, where
$V^{c}_{i}(A)$ represents the voting power of agent $i$ under the delegation
configuration given by the matrix $A\in\mathcal{A}^{n}$. In the classic model
of LD, the number of votes received by a candidate is determined as the sum of
votes received by all agents who delegate their power to the candidate, plus
the votes of all agents who delegate to these agents, and so on. Although this
procedure can be easily described recursively, we provide an equivalent
vectorial definition, which is more manageable in the context of this work. To
this end, for a given $A\in\mathcal{A}^{n}$, we can provide a vectorial
definition of $V^{c}:\mathcal{A}^{n}\to\mathbb{R}^{n}_{\geq 0}$ as follows:

(2.1) $V^{c}(A):=\lim_{k\to\infty}(A^{k}\mathbf{1})\odot diag(A),$

where $\mathbf{1}=(1,...,1)\in\mathbb{R}^{n}$ and $\odot$ denotes the element
wise product. Given the known properties of adjacency matrices, it can be
easily observed that successive products $A^{k}\mathbf{1}$ flow and accumulate
votes from agents to candidates through delegation chains. Note that
$diag_{i}(A)\in\\{0,1\\}$ for all $i$, and $diag_{i}(A)=1$ if and only if $i$
is a candidate. Thus, the element-wise product with $diag(A)$ does not affect
the entries of $A^{k}\mathbf{1}$ corresponding to candidates and nullifies the
remaining entries. This ensures that the limit in (2.1) is well-defined as the
votes trapped in delegation cycles are multiplied by zero. We now consider the
problem of multi-agent delegation, that is, we aim to extend the function
$V^{c}$ to the entire set $\mathcal{P}^{n}$ in such a way that this function
constitutes a reasonable generalization of the classic LD model. The first
natural idea is to use the definition (2.1) by considering
$A\in\mathcal{P}^{n}$, since the limit in this expression remains well-defined
333It can be easily seen that this generalization is closely related to the
Eigenvector Centrality Measure [8], and related in general to spectral
centrality measures, as the PageRank used in [6, 7, 26].. However, it is
possible to see that this generalization does not satisfy the properties
stated in the previous section. In particular, we can consider the following
example: Let the system be composed of $N=\\{1,2\\}$, where agent $1$
delegates $1/2$ of their voting power to agent $2$ and retains $1/2$ for
themselves, while agent $2$ retains all their voting power. In this example,
it is easy to see that $V^{c}(A)=(0,2)$, while the desired result would be
$(1/2,2/3)$. Therefore, this generalization does not satisfy the consistency
property (P3), and it is also possible to see that it does not satisfy (P2)
(see Appendix A.1). The issue with this generalization stems from the
asymmetry between an agent who chooses to preserve all her voting weight for
herself and one who chooses to only preserve part of her voting power. From
the perspective of the Markov chain given by the delegation matrix, the former
corresponds to an absorbing state, while the latter is associated with a
transient state. Furthermore, it should be mentioned that if we prohibit
agents from preserving a fraction of votes for themselves that is different
from 1 or 0, the aforementioned generalization preserves the properties stated
in the previous section.

The main goal of the first part of this paper is to extend the function
$V^{c}$ to the entire set $\mathcal{P}^{n}$, so that it constitutes a
reasonable generalization of the classic LD model. To achieve this goal, we
will propose a generalization that preserves properties (P2) and (P3), as well
as other relevant characteristics.

In the following section, we informally introduce the proposed voting weight
measure through its probabilistic interpretation. We will see that this
generalization can be defined by approximation using a family of simpler
measures that involve a penalty on the length of delegation chains. This
construction offers two main advantages: firstly, the probabilistic model
provides an intuitive interpretation that allows us to understand issues such
as the treatment of delegation cycles and the conservation of the sum of
votes. Secondly, defining this measure as the limit of simpler functions
allows us to demonstrate properties for these functions first and then extend
them by approximation to the proposed measure. This plays a central role in
the second part of this work, where we analyze the existence of Nash
equilibria in a game where the agents’ strategies are given by their
delegation profile. Here, the continuity of the approximating functions allows
us to use classic results from game theory to prove the existence of
equilibrium states.

######  Remark 1.

The available multi-agent delegation models in the literature exhibit
limitations related to agents’ retention of a fraction of the voting power for
themselves. Either the models are not defined for these instances, or upon
encountering them, desirable properties, such as those discussed in the
previous section, cease to hold. Therefore, it is important to consider in
which situations it is appropriate to grant agents the ability to retain an
arbitrary proportion of the voting power. For example, let us consider a
system of agents making a difficult binary decision with serious consequences
for all members of the system. Suppose agent $i$ have a degree of knowledge
relevant to making this decision, which can be encoded in a certain
probability $p_{i}>0.5$ of making the correct choice. However, the agent may
not want to bear the responsibility of making the wrong choice. Given that
agent $i$ can positively contribute to the decision-making process since
$p_{i}>0.5$, it is reasonable to allow the agent to retain a fraction of the
voting power to impact the decision-making process positively. Meanwhile, the
agent can delegate the remaining voting power to other agents whom she trusts,
reducing her degree of responsibility in the decision-making process. The
fraction of voting power that the agent seeks to preserve for herself will be
the one that optimizes the trade-off between maximizing the benefit of
contributing positively to the decision-making and minimizing the cost
associated with responsibility.

######  Remark 2.

It can be observed that the voting measures relying on PageRank fail to meet
the criteria stated in the previous section. In particular, the consistency
property is not upheld by this measure, as per in [7, Proposition 2].

###  2.3. The voting weight measure

Our goal is to construct a function $V:\mathcal{P}^{n}\to\mathbb{R}^{n}$ that
represents the voting weight of each agent $i$, which is a reasonable
generalization of $V^{c}$. In Section 3, we provide a formal definition of the
proposed function $V$, but in this section, we first introduce its
probabilistic interpretation. Specifically, we construct $V$ using a particle
system that depends on the agents and their delegation preferences, governed
by the following rules:

Given a delegation matrix $P\in\mathcal{D}$, within a time interval of length
$\delta t>0$:

  * •

Each agent $j$ receives 1 particle with probability $\delta t$.

  * •

A particle in position $j$ has a probability $\delta tP_{ij}$ to jump to
position $i$, with $j\not=i$.

  * •

A particle in position $j$ leaves the system with probability $\delta
tP_{jj}$. In such a case we shall say that the particle was consumed by agent
$j$.

We consider the limit as $\delta t\to 0$, and define $V_{i}(P)$ as the mean
rate of particles consumed by agent $i$ when the system reaches equilibrium.

To compute $V_{i}(P)$, we introduce $u_{i}$ as the average number of particles
occupying position $i$ in the equilibrium state. We can then estimate
$V_{i}(P)$ as the expected number of particles consumed by agent $i$ per unit
time, given by $P_{ii}u_{i}$. To determine $u_{i}$, we set up a set of balance
equations for the particle system described in the previous section. Using the
rules governing the system, the evolution of the number of particles in
position $i$ is given by:

$\displaystyle u_{i}(t+\delta t)$ $\displaystyle=u_{i}(t)+\delta
t\Big{(}\sum_{j\not=i}u_{j}(t)P_{ij}+1-u_{i}(t)\Big{)},$
$\displaystyle\frac{u_{i}(t+\delta t)-u_{i}(t)}{\delta t}$
$\displaystyle=\sum_{j\not=i}u_{j}(t)P_{ij}+1-u_{i}(t).$

Since we look for $u_{i}$ at the equilibrium state, we have

(2.2) $0=\sum_{j\not=i}u_{j}P_{ij}+1-u_{i}.$

The above expression can be written in matrix form:

$0=\tilde{P}u-u+\mathbf{1},$ (2.3) $u=(I-\tilde{P})^{-1}\mathbf{1},$

with $\tilde{P}=P-diag((P_{11},...,P_{nn}))$.

At this point, we can verify that $V(P)$ satisfies some of the properties
enlisted in the preceding section. In particular, part of the property (P3)
can be readily established, as when an agent $i$ delegates all their voting
weight, we get $P_{ii}=0$, and thus $V_{i}(P)=P_{ii}u_{i}=0$. Furthermore, we
can observe that to determine which agents consume a part of the voting weight
of a certain agent $i$, we may use the same particle system defined above,
with the constraint that the influx of particles is only through the agent
$i$. That is, we calculate $V(P)$ from $u=(I-\tilde{P})^{-1}\bm{\delta}_{i}$,
where $\bm{\delta}_{i}\in\mathbb{R}^{n}$, $\bm{\delta}_{ii}=1$ and
$\bm{\delta}_{ij}=0$ if $i\not=j$. Consequently, $V_{j}(P)$ will quantify the
amount of voting weight consumed by agent $j$, coming from agent $i$.

###  2.4. Solving delegation cycles

It is clear that equation (2.3) is not always solvable. This is due to the
possible existence of delegation cycles. We formally define a delegation cycle
as follows:

######  Definition 2.1 (Delegation cycle).

Given a set of agents $N=\\{1,...,n\\}$ and their delegation profiles
$x_{1},...,x_{n}$, we say that $C\subseteq N$ is a delegation cycle if
$x_{ii}=0$ for all $i\in C$, and given $i\in C$, $x_{ij}=0$ for all $j\in
N\setminus C$.

Based on this definition, it follows that for a cycle $C$, any particle that
reaches an agent $i\in C$ cannot exit the system. To address this issue, we
propose the introduction of an artificial agent $n+1$ into the system, where
each particle has a probability $\delta t\varepsilon$ of transitioning to
agent $n+1$ from any position within a time interval $\delta t$, where
$0<\varepsilon<1$ and $P_{n+1n+1}=1$. Specifically, we add agent $n+1$ with
delegation profile $x_{n+1}=(0,...0,1)\in\mathbb{R}^{n+1}$, and redefine the
profiles of the remaining agents as
$x_{i}:=((1-\varepsilon)x_{i},\varepsilon)$, where $i\in N$. It is worth
noting that every particle that reaches agent $n+1$ is absorbed. We denote the
voting weight measure obtained through this method as $V^{\varepsilon}$.

Following the modification introduced, delegation cycles no longer pose a
challenge, as any particle trapped in a cycle is consumed by agent $n+1$.
However, this modification leads to the penalization of extended delegation
paths. To illustrate, consider a classic delegation chain of length $k$
between agent $i$ and agent $j$. For a particle originating from agent $i$,
the probability of exiting the system through agent $n+1$ before reaching
agent $j$ is given by $p^{k}_{\varepsilon}=1-(1-\varepsilon)^{k}>0$.
Consequently, the voting weight of agent $i$ dissipates through agent $n+1$,
which results in the penalization of long delegation paths 444This damping
factor is similar to the one employed in PageRank computation, and is examined
in [6, 7]. Moreover, $V_{n+1}^{\varepsilon}$ denotes the aggregate of all the
voting weight dissipated through cycles and extended delegation paths. Note
that $p^{k}_{\varepsilon}\to 0$ as $\varepsilon\to 0$, yet the voting weight
dissipated by cycles remains invariant. To illustrate this, classic delegation
cycles can be considered infinite-length delegation chains, and
$p^{\infty}_{\varepsilon}:=\lim_{k\to\infty}p^{k}_{\varepsilon}=1$, which
establishes the independence of $\varepsilon$. In order to eliminate all
penalties in long but finite chains, we may define

(2.4) $V(P):=\lim_{\varepsilon\to 0}V^{\varepsilon}(P),$

where $V_{n+1}(P)$ accumulates the total amount of voting weight lost in
delegation cycles.

As previously noted, this section aims to provide an intuitive understanding
of $V(P)$ without delving into formal details. Notably, issues such as the
existence of the limit in (2.4) or the preservation of properties (P1)-(P3) as
$\varepsilon\to 0$ remain outstanding. In the subsequent section, we address
these concerns and provide a rigorous exposition of the concepts introduced
above.

##  3\. Formal framework

Let $N=\\{1,...,n\\}$ be a set of agents with delegation profiles
$x_{1},...,x_{n}$, and $P$ their delegation matrix. We define

$P^{\varepsilon}:=\begin{pmatrix}(1-\varepsilon)P&\mathbf{0}\\\
\bm{\varepsilon}&1\\\ \end{pmatrix},$ (3.1)
$u^{\varepsilon}:=(I-\tilde{P}^{\varepsilon})^{-1}\bm{1}_{0},$

and

(3.2)
$V^{\varepsilon}(P):=(P^{\varepsilon}_{11}u^{\varepsilon}_{1},...,P^{\varepsilon}_{nn}u^{\varepsilon}_{n},u^{\varepsilon}_{n+1}),$

where $\bm{\varepsilon}=(\varepsilon,...,\varepsilon)\in\mathbb{R}^{1\times
n}$,
$\tilde{P}^{\varepsilon}=P^{\varepsilon}-diag\big{(}(P_{11}^{\varepsilon},...,P_{nn}^{\varepsilon},P_{n+1n+1}^{\varepsilon})\big{)}$
and $\bm{1}_{0}=(1,...,1,0)\in\mathbb{R}^{n+1}$. Note that, since
$\varepsilon>0$ we have $\|\tilde{P}^{\varepsilon}\|<1$, as a consequence
$(I-\tilde{P}^{\varepsilon})^{-1}$ exists and $V^{\varepsilon}(P)$ is well
defined.

We define the voting power (or voting weight) $V(P)$ associated to a
delegation matrix $P$ as

(3.3) $V(P):=\lim_{\varepsilon\to 0}V^{\varepsilon}(P).$

###  3.1. Well-definedness

To establish the existence of the limit in (3.3), it is necessary to introduce
the following auxiliary result.

######  Lemma 3.1.

Let $\\{a_{k}\\}_{k\in\mathbb{N}_{0}}$ and
$\\{b_{k}\\}_{k\in\mathbb{N}_{0}}\in\ell^{\infty}$ be two uniformly bounded
sequences, where $a_{k}$ is periodic with period $q\in\mathbb{N}_{0}$, and
$\sum_{k}|b_{k}|<\infty$. Then:

  * $(i)$

$\lim_{\varepsilon\to 0}\sum_{k\geq 0}\varepsilon(1-\varepsilon)^{k}b_{k}=0,$

  * $(ii)$

$\lim_{\varepsilon\to 0}\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}a_{k}=\ell$, for some $\ell$ depending on
$\\{a_{k}\\}_{k\in\mathbb{N}_{0}}$.

###### Proof.

Since $\sum_{k\geq 0}|b_{k}|=c$ for some constant $c$, we proceed

$\big{|}\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}b_{k}\big{|}\leq\varepsilon\sum_{k\geq
0}|b_{k}|=\varepsilon c\to 0$

when $\varepsilon\to 0$. And then $(i)$ is proved.

On the other hand, since $\\{a_{k}\\}$ is periodic with period $q$, there are
constants $c_{j}$, with $j=0,..,q-1$ such that $a_{qk+j}=c_{j}$ for all $k\geq
0$. Then, for every fixed $j$ we have

$\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{qk+j}a_{qk+j}=c_{j}(1-\varepsilon)^{j}\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{qk}=\frac{c_{j}(1-\varepsilon)^{j}\varepsilon}{1-(1-\varepsilon)^{q}}.$

Observe that the denominator in the right-hand side corresponds to a
polynomial $S(\varepsilon)$ of degree $q$. Additionally, one can readily
confirm that $S(0)=0$ and $S^{\prime}(0)\neq 0$. Therefore, it follows that
$S(\varepsilon)=\varepsilon R(\varepsilon)$, where $R$ denotes a polynomial of
degree $q-1$, and satisfies $R(0)\neq 0$. Then

$\frac{c_{j}(1-\varepsilon)^{j}\varepsilon}{1-(1-\varepsilon)^{q}}=\frac{c_{j}(1-\varepsilon)^{j}}{R(\varepsilon)}\to\frac{c_{j}}{R(0)}$

when $\varepsilon\to 0$. Therefore, we can infer that $\lim_{\varepsilon\to
0}\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}a_{k}=\sum_{j=0}^{q-1}\frac{c_{j}}{R(0)}$,
establishing the proof of $(ii)$.

∎

Using the previous lemma, we can prove the main result of this section.

######  Theorem 3.1.

Let $N=\\{1,...,n\\}$ be a set of agents with delegation profiles
$x_{1},...,x_{n}$, and $P$ their delegation matrix. Then the limit in (3.3) is
well defined.

###### Proof.

We have

$I-\tilde{P}^{\varepsilon}=\begin{pmatrix}I-(1-\varepsilon)\tilde{P}&\mathbf{0}\\\
-\bm{\varepsilon}&1\\\ \end{pmatrix},$

with $\tilde{P}=P-diag\big{(}(P_{11},...,P_{nn})\big{)}.$ For the sake of
simplicity we define $A_{\varepsilon}:=I-(1-\varepsilon)\tilde{P}$. Since
$\varepsilon>0$ and $P$ is a stochastic matrix, we have
$\|(1-\varepsilon)\tilde{P}\|<1$. Thus $A_{\varepsilon}^{-1}$ exists, and can
be computed using the identity

(3.4) $A^{-1}_{\varepsilon}=\sum_{k\geq 0}(1-\varepsilon)^{k}\tilde{P}^{k}.$

Utilizing the blockwise inversion formula yields:

$(I-\tilde{P}^{\varepsilon})^{-1}=\begin{pmatrix}A_{\varepsilon}&\mathbf{0}\\\
-\bm{\varepsilon}&1\\\
\end{pmatrix}^{-1}=\begin{pmatrix}A^{-1}_{\varepsilon}&\mathbf{0}\\\
\bm{\varepsilon}A^{-1}_{\varepsilon}&1\\\ \end{pmatrix}.$

from we can derive

$u^{\varepsilon}=\begin{pmatrix}A^{-1}_{\varepsilon}&\mathbf{0}\\\
\bm{\varepsilon}A^{-1}_{\varepsilon}&1\\\
\end{pmatrix}\bm{1}_{0}=\begin{pmatrix}A^{-1}_{\varepsilon}\bm{1}\\\
\bm{\varepsilon}A^{-1}_{\varepsilon}\bm{1}\end{pmatrix},$

where $\bm{1}=(1,...,1)\in\mathbb{R}^{n}$.

Now we focus on the computation of $A^{-1}_{\varepsilon}\bm{1}$. Using (3.4)
we obtain $A^{-1}_{\varepsilon}\bm{1}=\sum_{k\geq
0}(1-\varepsilon)^{k}\tilde{P}^{k}\bm{1}$. In order to analyze the powers of
$\tilde{P}$ we define:

(3.5) $\displaystyle N_{1}$ $\displaystyle:=\\{i\in N\text{ such that
}P_{ii}\not=0\\},$ (3.6) $\displaystyle N_{2}$ $\displaystyle:=\\{i\in
N\setminus N_{1}\text{ such that exists a delegation path from }i\text{ to
}j\in N_{1}\\},$ (3.7) $\displaystyle N_{3}$
$\displaystyle:=N\setminus(N_{1}\cup N_{2}).$

Given that $N=N_{1}\overset{d}{\cup}N_{2}\overset{d}{\cup}N_{3}$, we adopt,
without loss of generality, an indexing of the agents such that those
belonging to $N_{1}$ appear first, those belonging to $N_{2}$ appear second,
and those belonging to $N_{3}$ appear last. From this, we observe that
$\tilde{P}$ can be written as

(3.8) $\tilde{P}=\begin{pmatrix}P_{1}&\bm{0}\\\ P_{2}&P_{3}\\\ \end{pmatrix},$

with $P_{1}$ being a sub-stochastic matrix, and $P_{3}$ a stochastic matrix.
It is well known that the powers of $\tilde{P}$ can be computed as

(3.9) $\tilde{P}^{k}=\begin{pmatrix}P_{1}^{k}&\bm{0}\\\ B_{k}&P_{3}^{k}\\\
\end{pmatrix},$

with $k\in\mathbb{N}$, and
$B_{k}=\sum^{k-1}_{j=0}P_{3}^{j}P_{2}P_{1}^{k-j-1}$. Then we have

$\tilde{P}^{k}\bm{1}=\begin{pmatrix}P_{1}^{k}&\bm{0}\\\ B_{k}&P_{3}^{k}\\\
\end{pmatrix}\bm{1}=\begin{pmatrix}P_{1}^{k}\bm{1}\\\
B_{k}\bm{1}+P^{k}_{3}\bm{1}\\\ \end{pmatrix},$

as a consequence

(3.10) $A^{-1}_{\varepsilon}\bm{1}=\sum_{k\geq
0}(1-\varepsilon)^{k}\tilde{P}^{k}\bm{1}=\begin{pmatrix}\sum_{k\geq
0}(1-\varepsilon)^{k}P_{1}^{k}\bm{1}\\\ \sum_{k\geq
0}(1-\varepsilon)^{k}(B_{k}\bm{1}+P^{k}_{3}\bm{1})\\\ \end{pmatrix}.$

Since $P_{1}$ is a sub-stochastic matrix, its spectral radium $\rho(P_{1})<1$.
Then, each entry of $P_{1}^{k}\bm{1}$ decays exponentially, and
$\lim_{\varepsilon\to 0}\sum_{k\geq 0}(1-\varepsilon)^{k}P_{1}^{k}\bm{1}$
exists. Note that we do not need to compute $\sum_{k\geq
0}(1-\varepsilon)^{k}(B_{k}\bm{1}+P^{k}_{3}\bm{1})$ since these entries
correspond to agents $i\in N_{3}$ with $P_{ii}=0$, and then
$V^{\varepsilon}_{i}(P)=0$ for all $\varepsilon>0$.

Now our aim is to compute the limit of $\lim_{\varepsilon\to
0}\bm{\varepsilon}A^{-1}{\varepsilon}\bm{1}$. By exploiting the fact that
$\bm{\varepsilon}A^{-1}{\varepsilon}\bm{1}=\bm{1}\varepsilon
A^{-1}{\varepsilon}\bm{1}$, our focus is solely on studying the behavior of
$\varepsilon A^{-1}{\varepsilon}\bm{1}$. Specifically, we obtain

$\varepsilon A^{-1}_{\varepsilon}\bm{1}=\begin{pmatrix}\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}P_{1}^{k}\bm{1}\\\ \sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}(B_{k}\bm{1}+P^{k}_{3}\bm{1})\\\
\end{pmatrix}.$

As previously established, the limit of $\lim_{\varepsilon\to 0}\sum_{k\geq
0}(1-\varepsilon)^{k}P_{1}^{k}\bm{1}$ exists. In this regard, we proceed to
examine $\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}(B_{k}\bm{1}+P^{k}_{3}\bm{1})$. Notably, the
definition of $B_{k}$, coupled with the sub-stochasticity of $P_{1}$, enables
us to confirm that each element of $B_{k}\bm{1}$ is an exponentially
convergent sequence. We define $\lim_{k\to\infty}[B_{k}\bm{1}]_{i}=:\ell_{i}$,
where $i\in\\{1,\ldots,N+1\\}$, and further introduce the vector
$L\in\mathbb{R}^{N}$ such that $L_{i}=\ell_{i}$. Consequently, each element of
the sequence $B_{k}\bm{1}-L$ converges exponentially to zero. This leads to
the expression:

$\sum_{k\geq 0}\varepsilon(1-\varepsilon)^{k}B_{k}\bm{1}=\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}(B_{k}\bm{1}-L)+\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}L,$ $=\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}(B_{k}\bm{1}-L)+L,$

where in the last step we computed the geometric summation. Since every entry
of $B_{k}\bm{1}$ is a positive summable sequence, from Lemma 3.1 we conclude
that $\lim_{\varepsilon\to 0}\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}(B_{k}\bm{1}-L)=0$, and then

$\lim_{\varepsilon\to 0}\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}B_{k}\bm{1}=L.$

We proceed to compute the expression $\lim_{\varepsilon\to 0}\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}P^{k}_{3}\bm{1}$. It is pertinent to note that
the only information available regarding $P_{3}$ is that it is a stochastic
matrix. It is a well-established fact that the powers of $P_{3}$ can be
represented as:

$P^{k}_{3}=C_{1}+C^{k}_{2}+C^{k}_{3},$

where $C_{i}=XJ_{i}X^{-1}$, being $J$ the Jordan normal form $P_{3}=XJX^{-1}$,
with $J=J_{1}+J_{2}+J_{3}$. Here, $J_{1}$ consists solely of the Jordan block
corresponding to the eigenvalue $\lambda=1$, $J_{2}$ encompasses the Jordan
blocks of the eigenvalues $\lambda$ satisfying $|\lambda|=1$ and
$\lambda\not=1$, and $J_{3}$ comprises the remaining blocks. Since $P_{3}$ is
a stochastic matrix, the geometric multiplicity of every eigenvalue $\lambda$
with $|\lambda|=1$ is $1$. Consequently, the sequence $C_{2}^{k}\bm{1}$ is
periodic. Further, as per the definition of $J_{3}$, it follows that the
sequence $C_{3}^{k}\bm{1}$ converges exponentially to $\bm{0}$.

From this we have

$\sum_{k\geq 0}\varepsilon(1-\varepsilon)^{k}P^{k}_{3}\bm{1}=\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}C_{1}\bm{1}+\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}C^{k}_{2}\bm{1}+\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}C^{k}_{3}\bm{1},$ $=C_{1}\bm{1}+\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}C^{k}_{2}\bm{1}+\sum_{k\geq
0}\varepsilon(1-\varepsilon)^{k}C^{k}_{3}\bm{1}.$

Based on the findings in Lemma 3.1, it is concluded that the limit as
$\varepsilon$ tends to zero exists for both the second and third terms. Thus,
the theorem is proven. ∎

###  3.2. Generalization

Up to this point, we have made the assumption that each agent holds an equal
role in the decision-making process. In the context of the particle system
introduced in Section 2.3, this assumption is reflected in the distribution of
one particle to each agent per unit time. This is formalized in the definition
of $V$, where the vector $\bm{1}_{0}$ is used to construct $u^{\varepsilon}$
in (3.1). To modify the inherent voting weight of each agent, we need only
adjust the definition of $u^{\varepsilon}$ by replacing the vector
$\bm{1}_{0}$ with an alternative vector $f\in\mathbb{R}^{n+1}$, where $f_{i}$
represents the inherent voting weight of agent $i$. We can then generalize the
definitions of $u^{\varepsilon}$, $V^{\varepsilon}(P)$, and
$V^{\varepsilon}(P)$ as follows: Let $N=\\{1,...,n\\}$ denote a set of agents
with delegation profiles $x_{1},...,x_{n}$, $P$ their delegation matrix, and
$f\in\mathbb{R}^{n+1}$. We define

(3.11) $u^{\varepsilon}:=(I-\tilde{P}^{\varepsilon})^{-1}f,$ (3.12)
$V^{\varepsilon}(P,f):=(P^{\varepsilon}_{11}u^{\varepsilon}_{1},...,P^{\varepsilon}_{nn}u^{\varepsilon}_{n},u^{\varepsilon}_{n+1}),$

and

(3.13) $V(P,f):=\lim_{\varepsilon\to 0}V^{\varepsilon}(P,f).$

It is noteworthy that the proof of Theorem (3.1) can be straightforwardly
adapted to encompass the extended definitions. Therefore, we obtain:

######  Theorem 3.2.

Let $N=\\{1,...,n\\}$ be a set of agents with delegation profiles
$x_{1},...,x_{n}$, $P$ their delegation matrix, and $f\in\mathbb{R}^{n+1}$.
Then the limit in definition (3.13) is well defined.

While it is possible to consider an arbitrary vector $f\in\mathbb{R}^{n+1}$,
not all choices of $f$ have a clear interpretation within the proposed model.
For instance, some components of $f$ could be negative, or we might have
$f_{n+1}\neq 0$, which would assign voting weight to the auxiliary agent
$n+1$. Nevertheless, we opt for generality in our theory by developing it for
any $f\in\mathbb{R}^{n+1}$, leaving the interpretation of certain values of
$f$ for future analysis.

###  3.3. Consistency

The following lemma asserts that a positive value of the term $f$ implies a
positive voting weight for each agent.

######  Lemma 3.2.

Let $N=\\{1,...,n\\}$ be a set of agents with delegation profiles
$x_{1},...,x_{n}\in X$, $P$ their delegation matrix, and
$f\in\mathbb{R}^{n+1}$ with $f_{i}\geq 0$ for all $i\in N$. Then
$V_{i}^{\varepsilon}(P,f)\geq 0$ for all $i\in N$, for all
$\varepsilon\in[0,1)$.

###### Proof.

Given $\varepsilon\in(0,1)$, consider $u^{\varepsilon}$ as in (3.11). Define
$u^{m}=\min_{1\leq i\leq n}u^{\varepsilon}_{i}$, and observe that from (3.11)
we can estimate

$\displaystyle u^{m}$
$\displaystyle=f_{k}+\tilde{P}^{\varepsilon}_{k1}u_{1}+...+\tilde{P}^{\varepsilon}_{kk-1}u_{k-1}+\tilde{P}^{\varepsilon}_{kk+1}u_{k+1}+...+\tilde{P}^{\varepsilon}_{kn}u_{n}$
$\displaystyle\geq
f_{k}+\tilde{P}^{\varepsilon}_{k1}u^{m}+...+\tilde{P}^{\varepsilon}_{kk-1}u^{m}+\tilde{P}^{\varepsilon}_{kk+1}u^{m}+...+\tilde{P}^{\varepsilon}_{kn}u^{m},$
$\displaystyle u^{m}$
$\displaystyle\geq\frac{f_{k}}{1-\tilde{P}^{\varepsilon}_{k1}-...-\tilde{P}^{\varepsilon}_{kk-1}-\tilde{P}^{\varepsilon}_{kk+1}-...-\tilde{P}^{\varepsilon}_{kn}}\geq
0,$

for some $k\in N$ such that $u_{k}=u^{m}$. Henceforth, it is established that
$u_{i}^{\varepsilon}\geq 0$ for all $i\in N$. As a consequence, it can be
deduced that $V_{i}^{\varepsilon}(P,f)\geq 0$ holds true for all $i\in N$ and
for all $\varepsilon\in(0,1)$. Furthermore, the scenario $\varepsilon=0$ can
be trivially derived by taking the limit as $\varepsilon$ approaches 0. ∎

The application of the aforementioned result allows us to establish the
generalized version of the consistency property (P3). The subsequent theorem
states that an agent’s voting weight is zero if they choose to delegate their
entire voting power, whereas if they retain a fraction of their voting power
for themselves, then their voting weight is no less than their inherent voting
weight scaled by this fraction. The theorem is stated as follows.

######  Theorem 3.3 (Consistency).

Let $N=\\{1,...,n\\}$ be a set of agents with delegation profiles
$x_{1},...,x_{n}\in X$, $P$ their delegation matrix, and
$f\in\mathbb{R}^{n+1}$ with $f_{i}\geq 0$ for all $i\in N$. Then we have that
$P_{ii}=0$ implies $V_{i}^{\varepsilon}(P,f)=0$, and
$V_{i}^{\varepsilon}(P,f)\geq(1-\varepsilon)P_{ii}f_{i}$ for all $i\in N$ and
$\varepsilon\in[0,\varepsilon)$.

###### Proof.

Given that $V_{i}^{\varepsilon}(P,f)=P_{ii}u_{i}^{\varepsilon}$, it can be
readily verified that $P_{ii}=0$ implies $V_{i}^{\varepsilon}(P,f)=0$ for all
$1\leq i\leq n$ and all $\varepsilon\in(0,\varepsilon)$. Moreover, from Lemma
3.2, which establishes that $u^{\varepsilon}_{i}\geq 0$ for all $i\in N$,
since $f_{i}\geq 0$ for all $i\in N$, we can estimate using (3.11) that:

$u^{\varepsilon}_{i}=f_{i}+\tilde{P}^{\varepsilon}_{i1}u_{1}+...+\tilde{P}^{\varepsilon}_{in}u_{n}\geq
f_{i},$

As a result, we obtain
$V_{i}^{\varepsilon}(P,f)=P^{\varepsilon}_{ii}u^{\varepsilon}_{i}\geq(1-\varepsilon)P_{ii}f_{i}$
for all $i\in N$, from which we can deduce the desired outcome for all
$\varepsilon\in(0,\varepsilon)$. The scenario $\varepsilon=0$ can be easily
handled by means of the aforementioned derivation, as we take the limit
$\varepsilon\to 0$. ∎

###  3.4. Conservation of voting power

The preservation of voting power is a desired property in the generalization
of the classic LD model, which requires that the sum of the voting power of
each agent and the lost voting power in delegation cycles equals the number of
agents. Although this property is usually achieved through the normalization
of the voting weight vector, it is demonstrated in our case that this step is
unnecessary and, therefore, preserves the interpretation of the proposed
voting measure discussed in Section 2. The proof of this property is presented
in the following result.

######  Theorem 3.4.

Let $N=\\{1,...,n\\}$ be a set of agents with delegation profiles
$x_{1},...,x_{n}\in X$, $P$ their delegation matrix, and
$f\in\mathbb{R}^{n+1}$. Then
$\sum^{n+1}_{i=1}V^{\varepsilon}_{i}(P,f)=\sum^{n+1}_{i=1}f_{i}$ for all
$\varepsilon\in[0,1)$.

###### Proof.

We consider $V^{\varepsilon}(P,f)\in\mathbb{R}^{n+1}$, defined as in (3.12),
for some $\varepsilon>0$. Firstly, our aim is to demonstrate that the equality
$\sum^{n+1}_{i=1}V^{\varepsilon}_{i}(P,f)=\sum^{n+1}_{i=1}f_{i}$ holds for all
$\varepsilon>0$. To this end, using the definitions provided in equations
(3.12) and (3.11), we can write

(3.14) $(I-\tilde{P}^{\varepsilon})u^{\varepsilon}=f.$

Let $D\in\mathbb{R}^{n+1\times n+1}$ be the diagonal matrix
$D:=diag\big{(}(P_{11}^{\varepsilon},...,P_{nn}^{\varepsilon},P_{n+1n+1}^{\varepsilon})\big{)}$.
Since $\tilde{P}^{\varepsilon}=P^{\varepsilon}-D$, from (3.14) we obtain

$(I-\tilde{P}^{\varepsilon})u^{\varepsilon}=u^{\varepsilon}-P^{\varepsilon}u^{\varepsilon}+Du^{\varepsilon}=f.$

By summing all the equations of the preceding linear system, we obtain

$\sum^{n+1}_{i=1}u_{i}^{\varepsilon}-\sum^{n+1}_{i=1}\sum^{n+1}_{j=1}P_{ij}^{\varepsilon}u_{j}^{\varepsilon}+\sum^{n+1}_{i=1}V^{\varepsilon}_{i}(P,f)=\sum^{n+1}_{i=1}f_{i}.$

Since $P^{\varepsilon}$ is a stochastic matrix,
$\sum^{n+1}_{i=1}P_{ij}^{\varepsilon}=1$ for all $j$, and then

$\sum^{n+1}_{i=1}\sum^{n+1}_{j=1}P_{ij}^{\varepsilon}u_{j}^{\varepsilon}=\sum^{n+1}_{j=1}u_{j}^{\varepsilon}\sum^{n+1}_{i=1}P_{ij}^{\varepsilon}=\sum^{n+1}_{j=1}u_{j}^{\varepsilon}.$

By combining this with the previous equality, we obtain:

$\sum^{n+1}_{i=1}V^{\varepsilon}_{i}(P,f)=\sum^{n+1}_{i=1}f_{i}.$

Given that
$W:=\\{w\in\mathbb{R}^{n+1}:\sum^{n+1}_{i=1}w_{i}=\sum^{n+1}_{i=1}f_{i}\\}$ a
closed set and $V^{\varepsilon}(P,f)\in W$ for all $\varepsilon>0$, it follows
that $V(P,f)=\lim_{\varepsilon\to 0}V^{\varepsilon}(P,f)$, and therefore,
$V(P,f)\in W$. This concludes the proof.

∎

###  3.5. Numerical computation

The computation of $V^{\varepsilon}$ requires the inverse of a badly
conditioned matrix when $\varepsilon$ is small. However, from the proof of
Theorem 3.1, it is possible to obtain a procedure for computing $V$ that
avoids the need for computing $V^{\varepsilon}$. Specifically, (3.8) and
(3.10) imply that only the inverse of $(I-P_{1})$ needs to be computed, where
$P_{1}$ is a sub-stochastic matrix. This is due to the fact that the voting
weight of agents in $N_{3}$ (and $N_{2}$) is zero, and therefore, it suffices
to obtain the voting weight of agents in $N_{1}$. This observation is
summarized in Algorithm 1.

Algorithm 1 Computation of $V(P,f)$

Input: A set of agents $N=\\{1,...,n\\}$, their delegation profiles
$x_{1},...,x_{n}$, and a vector $f\in\mathbb{R}^{n+1}$;

$\bm{1}$: Define the delegation matrix $P$ using the profiles
$x_{1},...,x_{n}$, and $N_{1}$ and $N_{2}$ as in (3.5) and (3.6) respectively.
Define $I_{i}$ as the $i$-th element of $N_{1}\cup N_{2}$, with $1\leq
i\leq\\#(N_{1}\cup N_{2})$;

$\bm{2}$: Define $\tilde{P}:=P-diag((P_{11},...,P_{nn}))$, and $\tilde{P}_{r}$
as the restriction of $\tilde{P}$ to $N_{1}\cup N_{2}$. That is, remove from
$\tilde{P}$ the $i$-th row and the $i$-th column, for all $i\not\in N_{1}\cup
N_{2}$. Analogously, define $f_{r}$ removing the $i$-th entry of $f$ for all
$i\not\in N_{1}\cup N_{2}$.

$\bm{3}$: If $\tilde{P}_{r}\not=\emptyset$, compute
$u_{r}=(I-\tilde{P}_{r})^{-1}f_{r}$;

$\bm{4}$: Compute $V_{I_{i}}(P,f)=P_{I_{i}I_{i}}[u_{r}]_{i}$ for all $1\leq
i\leq\\#(N_{1}\cup N_{2})$, and $V_{i}(P,f)=0$ for all $i\not\in N_{1}\cup
N_{2}\cup\\{n+1\\}$. Finally, using Theorem 3.4, compute
$V_{n+1}(P,f)=\sum^{n+1}_{i=1}f_{i}-\sum^{n}_{i=1}V_{i}(P,f)$;

###  3.6. An alternative definition of $V$

As previously stated, the definition of $V$, as given by (3.3) and (3.13),
involves an approximation using simpler measures $V^{\varepsilon}$, which can
be obtained from the particle system discussed in Section 2.1. This approach
is advantageous in providing an intuitive interpretation of the measure $V$
and its characteristics. Furthermore, the use of simpler measures is helpful
in establishing properties of $V$, as they can first be proven for
$V^{\varepsilon}$ and then extended to $V$ by taking the limit as
$\varepsilon\to 0$. This methodology plays a crucial role in Section 4, where
the main result of this work is presented. Nonetheless, $V$ can be defined
directly without relying on the functions $V^{\varepsilon}$. Algorithm 1
provides a way to compute $V$ directly. It is worth noting that this procedure
removes agents belonging to delegation cycles from the system to ensure that
the matrix $I-P_{1}$ is invertible. However, considering that entries of $V$
that correspond to agents belonging to cycles are be multiplied by zero in the
last step, $V$ can be defined in terms of powers of the delegation matrix.
Specifically, let $P\in\mathcal{P}^{n}$ and $f\in\mathbb{R}^{n}$, then
$V:\mathcal{P}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ can be defined as:

(3.15)
$V(P,f):=\lim_{k\to\infty}\big{(}\sum^{k}_{\ell=0}\tilde{P}^{k}f\big{)}\odot
diag(P),$

where $\tilde{P}=P-diag((P_{11},...,P_{nn}))$. It should be noted that the
value associated with node $n+1$ may be computed utilizing Theorem 3.4, as
stipulated in step 4 of Algorithm 1.

###  3.7. The generalization property

Expanding upon the previous section’s alternative definition of $V$, we
establish that this voting measure is a generalization of the classic LD
model, as it satisfies the property (P1) outlined in Section 1. Moreover, the
following result demonstrates that the proposed measure $V$ and the standard
generalization of the classic model defined in (2.1) are equivalent, provided
that agents are unable to retain a portion $q\in(0,1)$ of their vote.

######  Theorem 3.5.

Let $\mathcal{B}^{n}:=\\{B\in\mathcal{P}^{n}:B_{ii}\in\\{0,1\\}\text{ for all
}1\leq i\leq n\\}$, and let $V$ and $V^{c}$ the voting measures defined in
(3.3), and (2.1) respectively. Then $V_{i}(P)=V_{i}^{c}(P)$ for all
$P\in\mathcal{B}^{n}$, and $i\in N$.

###### Proof.

Let $P\in\mathcal{B}^{n}$ and $N=\\{1,\dots,n\\}$ be the set of agents
associated with this delegation matrix. From the definitions of $V$ and
$V^{c}$, we observe that $V_{i}(P)=V^{c}_{i}(P)=0$ for every agent $i$ that
belongs to a delegation cycle (according to Definition 2.1). Therefore,
without loss of generality, we can assume that the delegation matrix $P$ does
not contain cycles. This allows us to omit the elementwise product in the
definition of $V^{c}$ (2.1). Starting from the alternative definition of $V$
given in (3.15), it suffices to prove that

(3.16)
$\lim_{k\to\infty}P^{k}\bm{1}=\lim_{k\to\infty}\big{(}\sum^{k}_{\ell=0}\tilde{P}^{\ell}\bm{1}\big{)}\odot
diag(P),\quad\forall P\in\mathcal{B}^{n}.$

To this end, without loss of generality, we can assume that the candidate
agents are listed last. In this way, the matrix $P$ can be written as

$P=\begin{pmatrix}P_{1}&\bm{0}\\\ P_{2}&I\\\ \end{pmatrix},$

with $P_{1}$ being a sub-stochastic matrix. As in (3.9), the powers of $P$ can
be computed as

$P^{k}=\begin{pmatrix}P_{1}^{k}&\bm{0}\\\ B_{k}&I\\\ \end{pmatrix},$

with $k\in\mathbb{N}$, and
$B_{k}=\sum^{k-1}_{\ell=0}I^{\ell}P_{2}P_{1}^{k-\ell-1}=\sum^{k-1}_{\ell=0}P_{2}P_{1}^{\ell}$.
Applying the same idea, we can see that

$\tilde{P}=\begin{pmatrix}P_{1}&\bm{0}\\\ P_{2}&\bm{0}\\\ \end{pmatrix},$

and the powers of this matrix can be computed as

$\tilde{P}^{\ell}=\begin{pmatrix}P_{1}^{\ell}&\bm{0}\\\
P_{2}P_{1}^{\ell-1}&\bm{0}\\\ \end{pmatrix},$

with $\ell\in\mathbb{N}$.

Now, we have

(3.17) $P^{k}\bm{1}=\begin{pmatrix}P_{1}^{k}&\bm{0}\\\
\sum^{k-1}_{\ell=0}P_{2}P_{1}^{\ell}&I\\\
\end{pmatrix}\bm{1}=\begin{pmatrix}P_{1}^{k}\bm{1}\\\
(\sum^{k-1}_{\ell=0}P_{2}P_{1}^{\ell})\bm{1}+\bm{1}\\\ \end{pmatrix}.$

Also we have

(3.18)
$\sum^{k}_{\ell=0}\tilde{P}^{\ell}\bm{1}=\bm{1}+\sum^{k}_{\ell=1}\tilde{P}^{\ell}\bm{1}=\bm{1}+\begin{pmatrix}\sum^{k}_{\ell=1}P_{1}^{\ell}&\bm{0}\\\
\sum^{k}_{\ell=1}P_{2}P_{1}^{\ell-1}&\bm{0}\\\
\end{pmatrix}\bm{1}=\begin{pmatrix}(\sum^{k}_{\ell=0}P_{1}^{\ell})\bm{1}\\\
(\sum^{k-1}_{\ell=0}P_{2}P_{1}^{\ell})\bm{1}+\bm{1}\\\ \end{pmatrix}.$

Note that $\lim_{k\to\infty}P_{1}^{k}\bm{1}=0$ since $P_{1}$ is a sub-
stochastic matrix. This, together with (3.17), yields

(3.19)
$\lim_{k\to\infty}P^{k}\bm{1}=\begin{pmatrix}\lim_{k\to\infty}P_{1}^{k}\bm{1}\\\
\lim_{k\to\infty}(\sum^{k-1}_{\ell=0}P_{2}P_{1}^{\ell})\bm{1}+\bm{1}\\\
\end{pmatrix}=\begin{pmatrix}\bm{0}\\\
\lim_{k\to\infty}(\sum^{k-1}_{\ell=0}P_{2}P_{1}^{\ell})\bm{1}+\bm{1}\\\
\end{pmatrix}.$

We now observe that by listing the candidates last, we have
$diag(P)=\begin{pmatrix}\bm{0}\\\ \bm{1}\\\ \end{pmatrix}$, combining this
with (3.18) gives us

(3.20) $\lim_{k\to\infty}(\sum^{k}_{\ell=0}\tilde{P}^{\ell}\bm{1})\odot
diag(P)=\begin{pmatrix}\bm{0}\\\
\lim_{k\to\infty}(\sum^{k-1}_{\ell=0}P_{2}P_{1}^{\ell})\bm{1}+\bm{1}\\\
\end{pmatrix}.$

Finally, from (3.19) and (3.20), we can deduce (3.16). ∎

Analogous results hold for $V^{\varepsilon}$, which establish that this voting
power measure can be interpreted as a generalization of the classic LD model
with a specific margin of error, subject to the penalty parameter
$\varepsilon$.

######  Theorem 3.6.

Let $\mathcal{A}^{n}:=\\{A\in\mathcal{P}^{n}:A_{ij}\in\\{0,1\\}\text{ for all
}1\leq i,j\leq n\\}$, and let $V^{\varepsilon}$ and $V^{c}$ the voting
measures defined in (3.2), and (2.1) respectively. Given $\delta>0$ there
exist $\varepsilon_{\delta}>0$ such that if $\varepsilon<\varepsilon_{\delta}$
then $\max_{1\leq i\leq n}|V^{\varepsilon}_{i}(P)-V_{i}^{c}(P)|<\delta$ for
all $P\in\mathcal{A}^{n}$.

###### Proof.

We posit the assumption that not all the agents within the system are part of
a cycle, otherwise, there would be nothing to demonstrate. By applying Theorem
3.5, we obtain
$|V^{\varepsilon}_{i}(P)-V_{i}^{c}(P)|=|V^{\varepsilon}_{i}(P)-V_{i}(P)|$ for
all $P\in\mathcal{A}^{n}$. Therefore, it suffices to establish that the
convergence addressed in Theorem 3.1 uniformly holds in $\mathcal{A}^{n}$. The
proof of this result reveals that the convergence of $V^{\varepsilon}_{i}$,
where $1\leq i\leq n$, relies on the powers of the matrix
$P_{1}\in\mathbb{R}^{r\times r}$, $1\leq r\leq n$, defined in (3.8). It is
noteworthy that this matrix can be represented as:

$P_{1}=\begin{pmatrix}\bm{0}&B\\\ \end{pmatrix},$

with $B_{ij}\in\\{0,1\\}$, and $\bm{0}\in\mathbb{R}^{s\times r}$ with $s\geq
1$, since we are considering $P\in\mathcal{A}^{n}$. Furthermore, in view of
the fact that $P_{1}$ accounts for interactions among agents who not belong to
any delegation cycle, it can be deduced based on the construction of $P_{1}$
that this matrix is nilpotent. Given that $P_{1}\in\mathbb{R}^{r\times r}$
with $r\leq n$, it follows that $\sum_{k\geq
0}(1-\varepsilon)^{k}P^{k}_{1}\bm{1}=\sum_{0\leq k\leq
n+1}(1-\varepsilon)^{k}P^{k}_{1}\bm{1}$. Thus, the uniform convergence of
$V^{\varepsilon}_{i}(P)$ in $\mathcal{A}^{n}$ can be readily inferred.

∎

######  Remark 3.

It is worth noting that the model proposed in [15] is equivalent to equation
(3.15) only under the condition that agents are not allowed to retain any
fraction of their voting power other than 1 or 0. This condition yields the
voting weight measure proposed in [15] as
$\lim_{k\to\infty}\sum^{k}_{\ell=0}\tilde{P}^{k}\bm{1}$, where $P$ denotes the
delegation matrix. This work only considers the weight of votes corresponding
to candidates, while the non-zero values obtained from other agents are
disregarded. This is equivalent to setting the corresponding entries of non-
candidate agents to zero. Hence, the computation of the proposed weight of
votes in [15] can be expressed exactly as in equation (3.15) by element-wise
multiplication of the above expression by $diag(P)$. By utilizing Theorem 3.5,
it is observed that the proposed measure $V$, the standard generalization of
the classic LD model defined in equation (2.1), and the model proposed in
[15], are equivalent whenever delegation matrices belong to the space
$\mathcal{B}^{n}$ defined therein.

###  3.8. The delegation property

This section presents the formalization and proof of the delegation property
(P2), which establishes the legitimacy of delegation. Specifically, the
property posits that if agent $i$ delegates her voting weight to agent $j$,
who subsequently delegates all of her voting weight to other agents, the
resulting weight distribution would be equivalent to that obtained if agent
$i$ had utilized delegation profile $x_{j}$ instead of $x_{i}$. Furthermore,
when agent $i$ assigns $\frac{1}{4}$ and $\frac{3}{4}$ of her voting weight to
agents $j$ and $k$, respectively, and these proxies also delegate all their
voting weight, the corresponding voting weight distribution would match that
produced by the delegation profile $\frac{1}{4}x_{j}+\frac{3}{4}x_{k}$ instead
of $x_{i}$. Our examples assume that agents $j$ and $k$ do not delegate a
portion of their voting weight back to agent $i$. However, in such cases, the
fraction of the voting weight is distributed proportionally among the proxies.
We formally state this observation in the following result:

######  Theorem 3.7 (Delegation property).

Consider a set $N=\\{1,...,n\\}$ of agents with delegation profiles
$x_{1},...,x_{n}\in X$, their delegation matrix $P$, and
$f\in\mathbb{R}^{n+1}$. Let $D\subset N$, such that $x_{ii}=0$ for all $i\in
D$, and suppose that there exists an agent $k\in N\setminus D$ such that
$x_{kj}>0$ for all $j\in D$, and $x_{kj}=0$ otherwise. Moreover, assume that
agent $k$ is not involved in a delegation cycle. For each $i\in D$, we define
$x^{*}_{i}\in\mathbb{R}^{n}$ as $x^{*}_{ij}:=x_{ij}$ for all $j\not=k$ and
$x^{*}_{ik}:=0$, and $x^{*}_{k}\in X$ as

(3.21) $x^{*}_{k}:=\frac{\sum_{i\in D}x_{ki}x^{*}_{i}}{1-\sum_{i\in
D}x_{ki}x_{ik}};$

We also define $P^{*}\in\mathbb{R}^{n\times n}$ as
$P^{*}:=(x_{1}|...|x_{k-1}|x^{*}_{k}|...|x_{n})$, which is the delegation
matrix obtained by replacing $x_{k}$ with $x^{*}_{k}$ in $P$. Then, we have
$V(P,f)=V(P^{*},f)$.

###### Proof.

We begin by verifying the well-definedness of (3.21), that is, $1-\sum_{i\in
D}x_{ki}x_{ik}>0$. To establish this, we assume the contrary, i.e.,
$\sum_{i\in D}x_{ki}x_{ik}=1$. Since $0\leq x_{ik},x_{ki}\leq 1$ for all $i\in
N$, and $\sum_{i\in D}x_{ki}=1$, it follows that $x_{ik}=1$ for all $i\in D$.
Subsequently, since $x_{kj}=0$ for all $j\not\in D$, it implies that
$D\cup{k}$ forms a delegation cycle, which contradicts our assumption that
agent $k$ is not part of a cycle.

To apply the procedure detailed in Section 3.5, we introduce the sets $N_{1}$,
$N_{2}$, and $N_{3}$ as specified in (3.5), (3.6), and (3.7), respectively.
Similarly, we define $N^{*}_{1}$, $N^{*}_{2}$, and $N^{*}_{3}$ using the
delegation profile $x^{*}_{k}$ instead of $x_{k}$. We assume that the agents
in $N$ are arranged in a way that agents in $N_{1}$ appear first, followed by
agents in $N_{2}$, then agents in $N_{3}$, and finally $k=\\#(N_{1}\cup
N_{2})$, for simplicity. Note that an agent $i\in N$ belongs to a cycle if and
only if $i>k$. Let us define $D^{\prime}:=D\cap(N_{1}\cup N_{2})$, and observe
that $D^{\prime}$ cannot be empty, otherwise $D\cup{k}$ would form a cycle,
which contradicts our assumption.

We now establish that $N^{*}_{3}=N_{3}$. Firstly, since agent $k$ is not part
of any cycle, changing her delegation profile by $x^{*}_{k}$ cannot resolve
any cycle. Therefore, $N_{3}\subseteq N^{*}_{3}$. On the other hand, since
$D^{\prime}\not=\emptyset$, there exists an agent $q\in D^{\prime}$ such that
$x_{qs}>0$, where $s\in(N_{1}\cup N_{2})$. By definition (3.21), it follows
that $x^{*}_{ks}>0$. Thus, agent $k$ does not belong to $N^{*}_{3}$.
Consequently, no new delegation cycle is formed when we replace $x_{k}$ with
$x^{*}_{k}$, and we can deduce that $N^{*}_{3}\subseteq N_{3}$.

The equality $N_{3}=N^{*}_{3}$ implies that $N_{1}=N^{*}_{1}$ and
$N_{2}=N^{*}_{2}$. Consequently, Algorithm 1 can be applied to $P$ and
$P^{*}$, and it can be observed that their restricted matrices, denoted by
$\tilde{P}_{r}$ and $\tilde{P}^{*}_{r}$ respectively, share the same
dimension. Specifically, we have
$\tilde{P}_{r},\tilde{P}^{*}_{r}\in\mathbb{R}^{k\times k}$. To determine
$V(P,f)$ and $V(P^{*},f)$, we need to solve the linear systems:

$(I-\tilde{P}_{r})u=f_{r},$

and

$(I-\tilde{P}^{*}_{r})u^{*}=f_{r}$

respectively, where $f_{r}$ is the restricted source term defined in Algorithm
1. It suffices to show that $u_{i}=u^{*}_{i}$ holds for all $i\in N_{1}$,
given that $V_{i}(P,f)=V_{i}(P^{*},f)=0$ for all $i\in N_{2}\cup N_{3}$. To
this end, we introduce the matrix $A:=(I-\tilde{P}_{r})$, and manipulate its
columns. By changing variables according to these operations, we define the
variable $u^{\prime}$ as follows:

(3.22) $\left\\{\begin{aligned} u^{\prime}_{i}&=u_{i}-x_{ki}u_{k}&\mbox{ for
all }i\in D^{\prime},\\\ u^{\prime}_{k}&=u_{k}(1-\sum_{i\in
D^{\prime}}x_{ki}x_{ik}),\\\ u^{\prime}_{i}&=u_{i}&\mbox{ for all }i\not\in
D^{\prime}\cup\\{k\\}.\end{aligned}\right.$

From this, it can be seen that $u^{\prime}\in\mathbb{R}^{k}$ solves the linear
system

$\Big{(}A_{1}\Big{|}...\Big{|}A_{k-1}\Big{|}\frac{A_{k}+\sum_{i\in
D^{\prime}}x_{ki}A_{i}}{(1-\sum_{i\in
D^{\prime}}x_{ki}x_{ik})}\Big{)}u^{\prime}=f_{r},$

where $A_{i}$ denotes the $i$-th column of $A$. Now, we observe that
$A_{i}=I_{i}-x_{i}$, then

(3.23) $A_{ki}-x_{ki}A_{ii}=x_{ki}-x_{ki}-x_{ki}x_{ii}=-x_{ki}x_{ii},$

and

(3.24) $A_{kk}-x_{ki}A_{kk}=1-x_{ki}x_{ik}.$

Then we have

(3.25) $A_{k}+\sum_{i\in D^{\prime}}x_{ki}A_{i}=(1-\sum_{i\in
D^{\prime}}x_{ki}x_{ik})I_{k}-\sum_{i\in D}x_{ki}x^{*}_{i},$

Based on the dependence of $\tilde{P}^{*}_{r}$ construction solely on
$x^{*}_{ki}$ values for $i\leq k$, that are determined by $x_{i}$ for $i\in
D^{\prime}\cup{k}$, we can infer that

$(I-\tilde{P}^{*}_{r})=\Big{(}A_{1}\Big{|}...\Big{|}A_{k-1}\Big{|}\frac{A_{k}+\sum_{i\in
D^{\prime}}x_{ki}A_{i}}{(1-\sum_{i\in D^{\prime}}x_{ki}x_{ik})}\Big{)}.$

In conclusion, we establish the equality $u^{\prime}=u^{*}$ and, by virtue of
the expression given in (3.22), deduce that $u_{i}=u^{\prime}_{i}=u^{*}_{i}$
holds for every $i\in N_{1}$. Thus, the theorem is demonstrated. ∎

To provide further clarification regarding Theorem 3.7 and its scope, let us
first consider the example depicted in Figure 3.1. This example showcases a
delegation scenario and its equivalent representation as per Theorem 3.7,
using $N=\\{1,...,4\\}$, $k=3$, $D=\\{2\\}$. Obtaining the equivalent
delegation setting implies replacing $x_{3}$ with $x^{*}_{3}$ based on (3.21).
The resulting configuration reveals that delegating all the voting weight to
agent $2$ is equivalent to replicating her delegation profile.

The example shown in Figure 3.2 demonstrates the outcome when an agent
delegates to another agent who, in turn, delegates a fraction of her voting
weight back to the first agent. The delegation profile in this case follows
the same parameters as the example in Figure 3.1, where $N=\\{1,...,4\\}$,
$k=3$, and $D=\\{2\\}$, according to Theorem 3.7. Using the delegation formula
in (3.21), we find that if agent $3$ delegates all her voting weight to agent
$2$, this is equivalent to copying agent $2$’s delegation profile,
disregarding the fraction delegated by $2$ to $3$, and normalizing the
resulting vector to obtain a delegation profile. Specifically, given
$x_{2}=(1/2,0,1/4,1/4)$ and $x_{3}=(0,1,0,0)$, we have
$x^{*}_{3}=(1/2,0,0,1/4)/(1-1/4)=(2/3,0,0,1/3)$.

The verification of equivalence in both examples can be easily achieved by
applying Algorithm 1. Additionally, Theorem 3.7 establishes that delegating an
agent’s entire voting weight to multiple agents who, in turn, delegate their
entire voting weight further, is equivalent to changing the delegation profile
through a convex combination of their profiles. This convex combination is
obtained by utilizing either the modified profiles as shown in Figure 3.2, or
the unmodified profiles as illustrated in Figure 3.1, as stipulated by
equation (3.21).

$1$$2$$3$$4$$1$$2$$3$$4$$1$$1/2$$1/2$$1$$1$$1$$1/2$$1/2$$1/2$$1/2$$1$ Figure
3.1. According to Theorem 3.7, the delegation configurations depicted above
are equivalent. It can be observed that the act of delegating all the voting
weight to agent $2$ is equivalent to replicating her delegation profile.
$1$$2$$3$$4$$1$$2$$3$$4$$1$$1/2$$1/4$$1/4$$1$$1$$1$$1/2$$1/4$$1/4$$2/3$$1/3$$1$
Figure 3.2. According to Theorem 3.7, the delegation setting on the left is
equivalent to the one on the right. That is, when agent $3$ delegates the
voting weight to agent $2$, the effect is the same as copying her delegation
profile and redistributing the voting weight that agent $2$ delegates back to
$3$ among her proxies in proportion to their respective weights.

The definition of the voting measure $V^{\varepsilon}$ includes a penalty on
the length of delegation chains, which causes these measures to violate the
delegation property. We observe that replicating an agent’s delegation profile
results in a configuration with shorter delegation chains, which in turn
applies less penalty and produces a different distribution of voting weights.
Nevertheless, we demonstrate that $V^{\varepsilon}$ conforms to the delegation
property within a certain margin of error that relies on the penalty
$\varepsilon$. Specifically, a reduction in the penalty leads to a decrease in
the error. We rigorously establish and verify this property, denoted as the
$\delta-$Delegation property, in the forthcoming theorem.

######  Theorem 3.8 ($\delta-$Delegation property).

Assuming the identical hypotheses as stipulated in Theorem 3.7, and
$f\in\mathbb{R}^{n+1}_{\geq 0}$. Given $\delta>0$, there exists a positive
$\varepsilon_{\delta}$ such that if $\varepsilon<\varepsilon_{\delta}$ then
$\|V^{\varepsilon}(P,f)-V^{\varepsilon}(P^{*},f)\|_{\infty}<\delta$.

###### Proof.

Define

$A:=(I-\tilde{P}^{\varepsilon}),$ $A^{*}:=(I-\tilde{P}^{*\varepsilon}),$
$u:=A^{-1}f,$

and

$u^{*}:=A^{*-1}f.$

Operating with the columns of the matrix $A$ and making the appropriate change
of variables as in the proof of Theorem 3.7, we introduce a new variable,
denoted by $u^{\prime}\in\mathbb{R}^{n+1}$, which is defined as follows:

(3.26) $\left\\{\begin{aligned}
u^{\prime}_{i}&=u_{i}-(1-\varepsilon)x_{ki}u_{k}&\mbox{ for all }i\in D,\\\
u^{\prime}_{k}&=u_{k}(1-\sum_{i\in D}(1-\varepsilon)^{2}x_{ki}x_{ik}),\\\
u^{\prime}_{i}&=u_{i}&\mbox{ for all }i\not\in
D\cup\\{k\\},\end{aligned}\right.$

and fulfills the equation $A^{\prime}u^{\prime}=f$, with

$A^{\prime}:=\Big{(}A_{1}\Big{|}...\Big{|}A_{k-1}\Big{|}\frac{A_{k}+\sum_{i\in
D^{\prime}}(1-\varepsilon)x_{ki}A_{i}}{(1-\sum_{i\in
D^{\prime}}(1-\varepsilon)^{2}x_{ki}x_{ik})}\Big{|}A_{k+1}\Big{|}...\Big{|}A_{n+1}\Big{)}.$

Let us define $\Delta:=A^{\prime}-A^{*}$. From this definition, it follows
that $A^{*}u^{\prime}=f-\Delta u^{\prime}$. By combining this expression with
the definition of $u^{*}$, we arrive at:

(3.27) $A^{*}(u^{*}-u^{\prime})=\Delta u^{\prime}.$

By invoking the definition of $A^{*}$ and following a similar approach as
carried out in equations (3.23), (3.24), and (3.25), and denoting $\Delta_{i}$
as the $i$-th column of $\Delta$, it can be established that
$\Delta_{i}=\bm{0}$ for all $i\not=k$ and

(3.28) $\Delta_{k}=\begin{pmatrix}0\\\ (1-\varepsilon)\frac{\sum_{i\in
D}x_{ki}x^{*}_{i2}}{1-\sum_{i\in
D}x_{ki}x_{ik}}-(1-\varepsilon)^{2}\frac{\sum_{i\in
D}x_{ki}x^{*}_{i2}}{1-(1-\varepsilon)^{2}\sum_{i\in D}x_{ki}x_{ik}}\\\
\vdots\\\ (1-\varepsilon)\frac{\sum_{i\in D}x_{ki}x^{*}_{in}}{1-\sum_{i\in
D}x_{ki}x_{ik}}-(1-\varepsilon)^{2}\frac{\sum_{i\in
D}x_{ki}x^{*}_{in}}{1-(1-\varepsilon)^{2}\sum_{i\in D}x_{ki}x_{ik}}\\\
(1-\varepsilon)\varepsilon\\\ \end{pmatrix},$

from we have $\Delta u^{\prime}=u^{\prime}_{k}\Delta_{k}$. Upon utilizing
Theorem 3.4 and considering the constraint that $f\in\mathbb{R}_{\geq 0}$, a
direct deduction leads to the following estimate:

(3.29) $0\leq u^{\prime}_{k}\leq\sum_{i}f_{i}.$

On the other hand, it is possible to verify that
$\Delta_{k}\in\mathbb{R}^{n}_{\geq 0}$ and $\Delta_{k}\to\bm{0}$ as
$\varepsilon\to 0$. This is evident from (3.28), where it can be observed that
$\Delta_{k1}=0$, $\Delta_{kn+1}\geq 0$, and $\Delta_{kn+1}\to 0$ as
$\varepsilon\to 0$. It can also be easily established that $\Delta_{kj}\geq 0$
for all $2\leq j\leq n+1$, by making use of (3.28) and the facts that
$\sum_{i\in D}x_{ki}x^{*}_{ij}\geq 0$ and $\sum_{i\in D}x_{ki}x_{ik}\geq 0$.
Consequently, we can derive an estimate as follows:

(3.30) $0\leq\Delta_{kj}=(1-\varepsilon)\frac{\sum_{i\in
D}x_{ki}x^{*}_{ij}}{1-\sum_{i\in
D}x_{ki}x_{ik}}-(1-\varepsilon)^{2}\frac{\sum_{i\in
D}x_{ki}x^{*}_{ij}}{1-(1-\varepsilon)^{2}\sum_{i\in D}x_{ki}x_{ik}}\leq$
$(1-\varepsilon)\frac{\sum_{i\in D}x_{ki}x^{*}_{ij}}{1-\sum_{i\in
D}x_{ki}x_{ik}}-(1-\varepsilon)^{2}\frac{\sum_{i\in
D}x_{ki}x^{*}_{ij}}{1-\sum_{i\in
D}x_{ki}x_{ik}}=\varepsilon(1-\varepsilon)\frac{\sum_{i\in
D}x_{ki}x^{*}_{ij}}{1-\sum_{i\in D}x_{ki}x_{ik}}\leq$
$\frac{\varepsilon}{1-\sum_{i\in D}x_{ki}x_{ik}}.$

Now, by applying Theorem 3.4 to (3.27) and using the estimates (3.29) and
(3.30), we obtain:

$0\leq(u^{*}-u^{\prime})_{i}\leq\sum_{1\leq j\leq
n+1}u^{\prime}_{k}\Delta_{kj}\leq\Big{(}\sum_{1\leq j\leq
n+1}f_{j}\Big{)}\Big{(}\sum_{1\leq j\leq
n+1}\Delta_{kj}\Big{)}\leq\varepsilon\frac{(n+1)\sum_{1\leq j\leq
n+1}f_{j}}{1-\sum_{j\in D}x_{kj}x_{jk}}.$

Defining the constant

(3.31) $C(P,f,n,k):=\frac{(n+1)\sum_{1\leq j\leq n+1}f_{j}}{1-\sum_{j\in
D}x_{kj}x_{jk}},$

and upon observing that (3.26) entails $u^{\prime}_{i}=u_{i}$ for all
$i\not\in D\cup\\{k\\}$, one can establish that
$\|V^{\varepsilon}(P,f)-V^{\varepsilon}(P^{*},f)\|_{\infty}<C(P,f,n,k)\varepsilon$.
Taking $\varepsilon_{\delta}=\delta/C$ we conclude the proof. ∎

######  Remark 4.

Assuming fixed parameters $n$ and $f\in\mathbb{R}^{n+1}$, the dependence of
$\varepsilon_{\delta}$ on $P$ and $k$ is apparent from (3.31). However, the
dependence on $k$ can be eliminated as follows: given $P\in\mathcal{P}^{n}$,
let $K_{P}\subset N$ be the set of agents satisfying the conditions required
for agent $k$ as stated in Theorem 3.7. We define

$C^{\prime}(P):=\max_{k\in K_{P}}C(P,f,n,k)=\frac{(n+1)\sum_{1\leq j\leq
n+1}f_{j}}{\min_{k\in K_{P}}1-\sum_{j\in D}x_{kj}x_{jk}},$

and set $\varepsilon_{\delta}=\delta/C^{\prime}$. On the other hand, the
dependence on $P$ cannot be easily disregarded since the constant $C^{\prime}$
deteriorates as $\min_{k\in K_{P}}1-\sum_{j\in D}x_{kj}x_{jk}$ approaches
zero. However, it can be deduced that, for a fixed value of $\delta$, taking
$\varepsilon<\varepsilon_{\delta}$ implies that
$\|V^{\varepsilon}(P,f)-V^{\varepsilon}(P^{*},f)\|_{\infty}<\delta$ holds for
all $P\in\\{P\in\mathcal{P}^{n}\text{ such that
}C^{\prime}(P)>\varepsilon/\delta\\}$.

##  4\. Equilibrium Analysis

###  4.1. The Delegation Game

In this paper, the decision-making process is examined as a continuous dynamic
over time, rather than as a single event, as previously stated. The actions
taken by agents, whether it be delegating voting weight or making use of it,
have an impact on other agents. As a result, feedback is received by agents,
who then adjust their delegation profiles and actions accordingly. This raises
the question of whether or not equilibrium states exist for this type of
mechanics.

To partially model this dynamic, we introduce the game
$\mathcal{G}(N,W,\varepsilon)$, wherein $N=\\{1,...,n\\}$ represents the set
of agents, and $W=\\{w_{1},...,w_{n}\\}$, with $w_{i}\in\mathbb{R}^{n+1}$
denotes the set of preferences. Here, $w_{i}\in W$ assigns to each agent a
weight, which signifies the preferences of agent $i$. Specifically, $w_{ij}$
reflects the level of satisfaction of agent $i$ with the consumption of her
voting weight by $j$ (note that $w_{i}$ may contain negative entries). The
strategy of each agent $i$ is given by its delegation profile $x_{i}\in X$.
The goal of each agent is to maximize its utility function
$U_{i}:X\to\mathbb{R}$, given by

$U_{i}(x,x_{-i}):=\sum^{n+1}_{j=1}w_{ij}V_{j}^{\varepsilon}((x|x_{-i}),\bm{\delta}_{i})=w_{i}\cdot
V^{\varepsilon}((x|x_{-i}),\bm{\delta}_{i}).$

Here $\bm{\delta}_{i}\in\mathbb{R}^{n+1}$ is such that $\bm{\delta}_{ii}=1$
and $\bm{\delta}_{ik}=0$ if $i\not=k$, and $(x|x_{-i})$ denotes the matrix
$(x_{1},...,x,...,x_{n})$ with column $x$ in the $i$-th place, where $x_{k}$
is the delegation profile of agent $k$ with $k\not=i$. We take
$0<\varepsilon<1$. As illustrated in Section 2.3, the value function
$V_{j}^{\varepsilon}(P,\bm{\delta}_{i})\in\mathbb{R}$ quantifies the amount of
voting weight consumed by agent $j$ that comes from agent $i$. Hence, $U_{i}$
can be interpreted as the sum of the voting weight delegated by agent $i$ to
other agents, weighted in accordance with her preferences.

######  Remark 5.

It is noteworthy that the strategies employed in
$\mathcal{G}(N,W,\varepsilon)$ are always pure, despite being allowed to take
values in the continuous space $X$. Hence, $\mathcal{G}$ qualifies as a
continuous game. Moreover, it should be noted that the voting weight measure
$V$ (along with $V^{\varepsilon}$) is distinct from the outcome obtained by
employing mixed strategies on simple delegations. This is evident from the
example depicted in Figure 3.2, where it is apparent that the delegation
settings are not equivalent. Further details regarding this observation are
presented in Appendix A.2.

Through conventional arguments, one can establish that the game
$\mathcal{G}(N,W,\varepsilon)$ always possesses at least one Nash equilibrium.
To achieve this, we begin by proving the following lemma.

######  Lemma 4.1.

Let $N=\\{1,...,n\\}$ be a set of agents with delegation profiles
$x_{1},...,x_{n}\in X$, $P$ their delegation matrix, and
$f\in\mathbb{R}^{n+1}_{\geq 0}$ in such a way that $f_{j}>0$ for some $j\in
N$. Then $u^{\varepsilon}_{j}>0$ for all $\varepsilon>0$.

###### Proof.

From the definition of $u^{\varepsilon}$ we have:

(4.1) $\begin{pmatrix}I-(1-\varepsilon)\tilde{P}&\mathbf{0}\\\
-\bm{\varepsilon}&1\\\ \end{pmatrix}u^{\varepsilon}=f,$

Since $f_{i}\geq 0$ for all $i$, we know from Lemma 4.1 that
$u^{\varepsilon}\geq 0$. From (4.1) we have

$u^{\varepsilon}_{j}-(1-\varepsilon)\sum_{i\not=j}P_{ji}u^{\varepsilon}_{i}=f_{j}>0,\quad\text{
if }j<n+1.$

It follows that $u^{\varepsilon}_{j}$ is greater than zero for all $j<n+1$.
The scenario where $j=n+1$ can be dealt with in a similar manner. ∎

The subsequent result demonstrates that $\mathcal{G}(N,W,\varepsilon)$
possesses at least one Nash equilibrium for all $0<\varepsilon<1$.

######  Theorem 4.1.

Let $N=\\{1,...,n\\}$ be a set of agents with preferences
$W=\\{w_{1},...,w_{n}\\}\subset\mathbb{R}^{n+1}$, and $f\in\mathbb{R}^{n+1}$.
The game $\mathcal{G}(N,W,\varepsilon)$ has a Nash equilibrium for all
$0<\varepsilon<1$.

###### Proof.

For the sake of convenience, we shall treat the elements of $W$ as row
vectors, that is to say $W\subset\mathbb{R}^{1\times n+1}$. Let
$r_{i}(x_{-i})$ denote the optimal response of agent $i$ to the strategies
employed by all other agents:

$r_{i}(x_{-i}):=\operatorname*{arg\,max}_{x_{i}}U_{i}(x_{i},x_{-i}).$

Let $x$ be an element of $\bm{X}:=X^{n}=X\times...\times X$, and let the set
valued function $r:\bm{X}\to 2^{\bm{X}}$ be defined as
$r(x):=r_{1}(x_{-1})\times...\times r_{n}(x_{-n})$. It is noteworthy that any
stationary point of $r$ constitutes a Nash equilibrium. Our aim is to
demonstrate the existence of a stationary point of $r$. To this end, we will
employ Kakutani’s fixed point theorem, which ensures the existence of a fixed
point given the fulfillment of the following conditions:

  1. (1)

$\bm{X}$ is compact, convex, and nonempty,

  2. (2)

$r(x)$ is nonempty,

  3. (3)

$r(x)$ is upper hemicontinuous,

  4. (4)

$r(x)$ is convex.

Condition 1 is easily fulfilled. Conditions 2 and 3 can be verified by
exploiting the continuous nature of the function $U_{i}$ for all $i$, and for
all $0<\varepsilon<1$, combined with Berge’s maximum theorem.

In order to establish 4, it suffices to demonstrate that $r_{i}(x_{-i})$ is
convex for all $i$, based on the definition of $r(x)$. Let $x$ and
$x^{\prime}\in r_{i}(x_{-i})$ be two strategies. Let $\eta:=x^{\prime}-x$ and
$x(t):=x+t\eta$. Our objective is to show that $x(t)\in r_{i}(x_{-i})$ for all
$t\in[0,1]$. Without loss of generality and for the sake of simplicity, we
assume that $i=1$. Referring to the definition given in (3.2), if we define
$P(t):=(x(t)|x_{-1})$, the utility function $U_{1}$ can be expressed as:

$U_{1}(x(t),x_{-1})=w_{1}D_{t}u(t),$

where
$D_{t}:=diag\big{(}(P_{11}^{\varepsilon}(t),...,P_{nn}^{\varepsilon}(t),P_{n+1n+1}^{\varepsilon}(t))\big{)}$,
and $u(t):=(I-\tilde{P}^{\varepsilon}(t))^{-1}\bm{\delta}_{1}$. We observe
that $u(t)$ satisfies the equation

$\begin{pmatrix}I-(1-\varepsilon)\tilde{P}(t)&\mathbf{0}\\\
-\bm{\varepsilon}&1\\\ \end{pmatrix}u(t)=\bm{\delta}_{1},$

with $\tilde{P}(t)=P(t)-diag\big{(}(P_{11}(t),...,P_{nn}(t)\big{)}$. Using the
definition of $x(t)$, we can rewrite the former expression as

$\left[\begin{pmatrix}I-(1-\varepsilon)\tilde{P}(0)&\mathbf{0}\\\
-\bm{\varepsilon}&1\\\
\end{pmatrix}-\begin{pmatrix}(1-\varepsilon)t\tilde{\eta}&\mathbf{0}\\\
\mathbf{0}&0\\\ \end{pmatrix}\right]u(t)=\bm{\delta}_{1},$

where $\tilde{\eta}\in\mathbb{R}^{n}$, with $\tilde{\eta}_{i}:=\eta_{i}$ for
all $i>1$ and $\tilde{\eta}_{1}:=0$. Defining

(4.2) $A_{0}:=\begin{pmatrix}I-(1-\varepsilon)\tilde{P}(0)&\mathbf{0}\\\
-\bm{\varepsilon}&1\\\ \end{pmatrix}\text{ and
}\Delta:=\begin{pmatrix}(1-\varepsilon)\tilde{\eta}&\mathbf{0}\\\
\mathbf{0}&0\\\ \end{pmatrix},$

we have $(A_{0}-t\Delta)u(t)=\bm{\delta}_{1}$, and then

(4.3) $u(t)=A^{-1}_{0}\bm{\delta}_{1}+tA^{-1}_{0}\Delta u(t).$

Also, we observe that $D_{t}=D_{0}+tD_{\Delta}$, with

(4.4) $D_{\Delta}:=\begin{pmatrix}x^{\prime}_{1}-x_{1}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}\\\ \end{pmatrix}.$

On the other hand, let us consider a constant vector $c\in\mathbb{R}^{1\times
n+1}$, with $c_{j}=C\in\mathbb{R}$ for all $j$. If we define $w^{*}=w_{1}+c$,
then it can be observed that maximizing (or minimizing) the utility function
$w_{1}D_{t}u(t)$ is equivalent to maximizing (or minimizing) the utility
function $U^{(}x(t)|x_{-1}):=w^{D}_{t}u(t)$. In fact, we have
$w^{*}D_{t}u(t)=w_{1}D_{t}u(t)+cD_{t}u(t)$, and

$cD_{t}u(t)=c\cdot
V^{\varepsilon}((x(t)|x_{-1}),\bm{\delta}_{1})=C\sum^{n+1}_{j=1}V_{j}^{\varepsilon}((x(t)|x_{-1}),\bm{\delta}_{1})=C,$

where in the last step we use Theorem 3.4. Then
$U^{*}(x(t)|x_{-1})=U(x(t)|x_{-1})+C$, and therefore

(4.5)
$\operatorname*{arg\,max}_{x}U_{1}(x,x_{-1})=\operatorname*{arg\,max}_{x}U^{*}(x,x_{-1}).$

Now, we choose $c\in\mathbb{R}^{1\times n+1}$ with $c_{j}=-w_{11}$ for all
$j$, so that $w^{*}_{1}=0$. As a consequence, from the definition of $w^{*}$
and (4.4), we have $w^{*}D_{\Delta}=\bm{0}$, thus

$U^{*}(x(t)|x_{-1})=w^{*}D_{t}u(t)=w^{*}(D_{0}+D_{\Delta})u(t)=w^{*}D_{0}u(t),$

and using (4.3) we obtain

(4.6)
$U^{*}(x(t)|x_{-1})=w^{*}D_{0}A^{-1}_{0}\bm{\delta}_{1}+tw^{*}D_{0}A^{-1}_{0}\Delta
u(t).$

From the definition of $\Delta$ (4.2) we have

$\Delta u(t)=u_{1}(t)\begin{pmatrix}\tilde{\eta}\\\ 0\\\ \end{pmatrix},$

and we can rewrite (4.6) as follow

(4.7)
$U^{*}(x(t)|x_{-1})=w^{*}D_{0}A^{-1}_{0}\bm{\delta}_{1}+tu_{1}(t)w^{*}D_{0}A^{-1}_{0}\begin{pmatrix}\tilde{\eta}\\\
0\\\ \end{pmatrix}.$

Since $x$ and $x^{\prime}$ are maximizers of $U_{1}$, from (4.5) we know that
they are also maximizers of $U^{*}$, then

$\displaystyle U^{*}(x(0)|x_{-1})$ $\displaystyle=U^{*}(x(1)|x_{-1}),$
$\displaystyle w^{*}D_{0}A^{-1}_{0}\bm{\delta}_{1}$
$\displaystyle=w^{*}D_{0}A^{-1}_{0}\bm{\delta}_{1}+u_{1}(1)w^{*}D_{0}A^{-1}_{0}\begin{pmatrix}\tilde{\eta}\\\
0\\\ \end{pmatrix},$ $\displaystyle 0$
$\displaystyle=u_{1}(1)w^{*}D_{0}A^{-1}_{0}\begin{pmatrix}\tilde{\eta}\\\ 0\\\
\end{pmatrix}.$

From Lemma 4.1, we know that $u_{1}(1)>0$. Consequently, we have

$w^{*}D_{0}A^{-1}_{0}\begin{pmatrix}\tilde{\eta}\\\ 0\\\ \end{pmatrix}=0,$

which, by using (4.7), leads to $U^{*}(x(t)|x_{-1})=U^{*}(x(0)|x_{-1})$ for
all $t\in[0,1]$. Therefore, $x(t)$ maximizes $U^{*}$, and from (4.5), we
conclude that $x(t)$ maximizes $U_{1}$ for all $t\in[0,1]$. Hence,
$r_{1}(x_{-1})$ is a convex set. Similarly, $r_{i}(x_{-i})$ is a convex set
for all $1\leq i\leq n+1$. This completes the proof of the theorem. ∎

######  Remark 6.

We note that if we restrict agents’ strategies to convex, compact, and non-
empty spaces, Theorem 4.1 can be trivially reproduced. For example, we can
consider a social network in which $\mathcal{N}_{i}\subseteq N$ denotes the
neighbors of the agent $i$, and restrict the strategies of each agent to its
neighborhood, that is $x_{i}\in X_{i}:=\\{x\in\mathbb{R}^{n}_{\geq
0}:x_{k}=0\,\,\forall k\not\in\mathcal{N}_{i},\text{ and }\sum_{k}x_{k}=1\\}$.
For this case the proof of the Theorem 4.1 can be easily replicated using
$\bm{X}:=X_{1}\times...\times X_{n}$. This setting can be seen as a
generalization of problem MINDIS, form [16][Section 3.2]. In that work, the
authors establish that a pure strategy Nash equilibrium may not exist in the
context of simple delegation, and it is even NP-complete to determine whether
one exists or not. The aforementioned problem is also discussed in [21]. In
that work, the equilibrium states with mixed strategies are examined, and the
authors establish the bicriteria approximation guarantees of the game with
respect to both rationality and social welfare. However, mixed strategies do
not serve as a good generalization of the classical model when deployed to
assess the voting weight of agents (see Appendix A.2).

###  4.2. A stable Liquid Democracy model

In the proof of Theorem 4.1, we have utilized the continuity of the utility
function, which is derived from the continuity of the voting measure
$V^{\varepsilon}$ with $\varepsilon\in(0,1)$. Although the existence of
equilibria using the voting measure $V$ is theoretically possible, proving
their existence becomes technically challenging due to the non-continuity of
$V$. However, we can demonstrate that employing the measure $V^{\varepsilon}$
in a liquid democracy system guarantees the existence of equilibria, while
ensuring that desirable properties such as the delegation property are
maintained within a certain error margin that is dependent on the penalty
parameter $\varepsilon>0$. This observation is formalized in the following
result.

######  Theorem 4.2.

Consider a system consisting of agents $N=\\{1,2,...,n\\}$, where each agent
$i\in N$ delegates their vote by using a delegation profile $x_{i}\in X$. Let
$P\in\mathcal{P}^{n}$ represent the delegation matrix formed from these
profiles. Let $f\in\mathbb{R}_{\geq 0}^{n+1}$, where $f_{i}$ represents the
inherent voting weight of agent $i$. Let
$W=\\{w_{1},w_{2},\dots,w_{n}\\}\subset\mathbb{R}^{n+1}$ denote the
preferences of these agents over others, as specified in Theorem 4.1. Given
$0<\varepsilon<1$, using the voting power measure $V^{\varepsilon}$ defined in
(3.13) the system has the following properties:

  * (I)

$\delta$-Generalization property. Given $\delta>0$ and $f=\bm{1}$, there
exists a positive $\varepsilon_{\delta}$ such that for any
$\varepsilon<\varepsilon_{\delta}$, $\max_{1\leq i\leq
n}|V^{\varepsilon}_{i}(P)-V_{i}^{c}(P)|<\delta$ for all $P\in\mathcal{A}^{n}$,
where $V^{c}$ denotes the classic LD measure as defined in (2.1).

  * (II)

$\delta$-Delegation property, as introduced in Theorem 3.8, which asserts the
existence of delegation property $(P2)$ subject to a degree of error that is
contingent upon the penalty $\varepsilon$.

  * (III)

$\delta$-Consistency. For all $i\in N$ such that $P_{ii}=0$, it holds that
$V_{i}^{\varepsilon}(P,f)=0$. Moreover, given $\delta>0$, there exists
$\varepsilon_{\delta}>0$ such that for all $\varepsilon<\varepsilon_{\delta}$
and for all $P\in\mathcal{P}^{n}$, we have $(1-\delta)f_{i}P_{ii}\leq
V_{i}^{\varepsilon}(P,f)$.

  * (IV)

Conservation of voting weight. Namely,
$\sum^{n+1}_{i=1}V^{\varepsilon}_{i}(P,f)=\sum^{n+1}_{i=1}f_{i}$ for all
$P\in\mathcal{P}^{n}$.

  * (V)

Existence of equilibrium states. The game $\mathcal{G}(N,W,\varepsilon)$ has
pure strategy Nash equilibria.

###### Proof.

The property (I) may be straightforwardly deduced from Theorem 3.6. On the
other hand, the properties (II), (III), and (IV) are readily derivable from
theorems 3.8, 3.3, and 3.4, respectively. Finally, property (V) can be
inferred from Theorem 4.1. ∎

##  5\. Conclusions

This paper presents a novel method for computing the voting power (or voting
weight) of agents in the liquid democracy paradigm. The proposed method
extends the classical model by allowing agents to fractionate and delegate
their voting power to multiple agents while retaining a portion of their vote
power for themselves. This generalization is achieved while preserving the
most reasonable properties of the classical model. The proposed voting power
measure is constructed from simpler measures that introduce a penalty factor
for the length of delegation chains. The proposed measure satisfies the
expected properties when implementing an LD system. Additionally, we
demonstrate that this voting measure coincides with the usual generalizations
as long as agents are not allowed to retain a vote fraction other than 1 or 0
for themselves.

In the second part of this paper, we explore the existence of equilibrium
states in the proposed liquid democracy system. Specifically, we define a game
in which each agent’s delegation profile constitutes their strategy, and the
goal of each agent is to maximize a utility function that quantifies their
satisfaction regarding their voting power. By leveraging the voting power
measure outlined in the first part of this paper and imposing a non-zero
penalty factor on the length of delegation chains, we prove that the game
admits pure strategy Nash equilibria. We note that this finding contrasts with
the analysis presented in [16], where the same game is examined within the
context of the classical liquid democracy model. In that study, the authors
establish that a pure strategy Nash equilibrium may not exist, and it is even
NP-complete to determine whether one exists or not. We also highlight the
results reported in [21], where the same problem is scrutinized in the
classical model framework, and the authors demonstrate that when mixed
strategies are considered, this game has bicriteria approximation guarantees
in terms of both rationality and social welfare. Unfortunately, mixed
strategies do not serve as a good generalization of the classical model when
deployed to assess the voting weight of agents, as analyzed in Appendix A.2.

Finally, we present Theorem 4.2, which summarizes the obtained results of this
work. Our results demonstrate that by introducing a penalty on the length of
delegation chains that is suitably small, it is possible to construct a liquid
democracy system that maintains the main features of the classical model
within a prescribed error margin. Furthermore, such a system allows for the
existence of pure strategy Nash equilibria. To the best of our knowledge, this
study provides the first formal proof that establishes the feasibility of a
liquid democracy system that guarantees the existence of equilibrium states.

## Appendix A

###  A.1. Standard LD generalization

We shall denote the generalized Standard LD measure as the natural extension
of the function $V^{c}$ defined in (2.1) to the set $\mathcal{P}^{n}$.
Specifically, we consider $V^{c}:\mathcal{P}^{n}\to\mathbb{R}_{\geq 0}$,

(A.1) $V^{c}(P):=\lim_{k\to\infty}(P^{k}\mathbf{1})\odot diag(P).$

As discussed in Section 2.2, the function under consideration does not
preserve consistency (property (P3)). Additionally, it is apparent that this
generalization does not fulfill the delegation property (P2). This can be
demonstrated by examining the example presented in Figure A.1. Considering
this, we have

$P_{l}=\begin{pmatrix}1/2&1/2&0\\\ 1/2&0&1\\\ 0&1/2&0\\\ \end{pmatrix},\text{
and }P_{r}=\begin{pmatrix}1/2&1/2&1\\\ 1/2&0&0\\\ 0&1/2&0\\\ \end{pmatrix}$

where the delegation matrix $P_{l}$ corresponds to the left setting, whereas
$P_{r}$ corresponds to the right setting.

$1$$2$$3$$1$$2$$3$$1/2$$1/2$$1$$1/2$$1/2$$1/2$$1/2$$1$$1/2$$1/2$ Figure A.1.
We consider two settings with $N=\\{1,2,3\\}$, where the delegation property
establishes that the voting weight distribution should be the same for both
settings. However, it can be verified that this distribution differs when
evaluated in each of the aforementioned settings using the measure $V^{c}$.

By applying (A.1), it is readily apparent that
$V^{c}(P_{l})=(\frac{9}{16},0,0)$ and $V^{c}(P_{r})=(\frac{13}{16},0,0)$.
Therefore, it can be deduced that $V^{c}$ fails to satisfy the delegation
property. It is worth noting, however, that $V(P_{l})=V(P_{r})=(3,0,0,0)$.

###  A.2. Mixed strategies as multi-agent delegation

As mentioned earlier, in [21], the authors examine a game analogous to the one
defined in Section 4 within the framework of the classic model of Liquid
Democracy. There, they establish that the game has bicriteria approximation
guarantees concerning both rationality and social welfare when mixed
strategies are taken into consideration. However, mixed strategies are not a
suitable generalization of the classical model when employed to assess the
voting power of agents.

To see this, we consider a system consisting of agents $N=\\{1,...,n\\}$,
where $x_{1},...x_{n}\in X$ represent their respective delegation profiles,
and $P\in\mathcal{P}^{n}$ is the delegation matrix derived from these
profiles. It is noteworthy that
$\mathcal{A}^{n}=\\{A\in\mathcal{P}^{n}:A_{ij}\in\\{0,1\\}\text{ for all
}1\leq i,j\leq n\\}$. We define the function
$p:\mathcal{A}^{n}\times\mathcal{P}^{n}\to[0,1]$.

$p(A,P):=\prod^{n}_{i=1}A^{t}_{i}\cdot P_{i},$

where, $A_{i}$ and $P_{i}$ correspond to the $i$-th column of $A$ and $P$,
respectively. It is easy to see that $p(\cdot,P)$ defines a point probability
function on the set $\mathcal{A}^{n}$. We can therefore define the mixed
strategy-based voting power measure $V^{MS}:\mathcal{P}^{n}\to\mathbb{R}^{n}$
as follows:

(A.2) $V^{MS}(P):=\sum_{A\in\mathcal{A}^{n}}p(A,P)V^{C}(P).$

Here, $V^{C}$ represents the voting power measure of the classic model defined
in (2.1).

We can easily observe that the voting power measure $V^{MS}$ does not satisfy
the delegation property (P2). To demonstrate this, we can consider the example
illustrated in Figure 3.2, where the delegation matrices are given by:

$P_{l}=\begin{pmatrix}1&1/2&0&0\\\ 0&0&1&0\\\ 0&1/4&0&0\\\ 0&1/4&0&1\\\
\end{pmatrix},\text{ and }P_{r}=\begin{pmatrix}1&1/2&2/3&0\\\ 0&0&0&0\\\
0&1/4&0&0\\\ 0&1/4&1/3&1\\\ \end{pmatrix}$

for the left and right settings, respectively. Formula (A.2) yields
$V^{MS}(P_{l})=(2,0,0,\frac{3}{2})$ and
$V^{MS}(P_{r})=(\frac{7}{3},0,0,\frac{5}{3})$, leading to the conclusion that
$V^{MS}$ violates the delegation property. It is noteworthy that even
normalizing the output vectors we obtain nonidentical outcomes.

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