为流动民主辩护_InDefenseofFluidDemocracy资源-CSDN文库

版权申诉

92 浏览量 2022-01-20 17:26:48 上传评论收藏 355KB PDF 举报

资源详情

资源评论

arXiv:2107.11868v1 [cs.DM] 25 Jul 2021

In Defense of Fluid Democracy

Daniel Halpern

Harvard University

dhalpern@g.harvard.edu

Joseph Y. Halpern

∗

Cornell University

halpern@cs.cornell.edu

Ali Jadbabaie

†

MIT

jadbabai@mit.edu

Elchanan Mossel

‡

MIT

elmos@mit.edu

Ariel D. Procaccia

Harvard University

arielpro@seas.harvard.edu

Manon Revel

MIT

mrevel@mit.edu

Abstract

Fluid democracy is a voting paradigm that allows vo ters to choose between dir ectly voting

and transitively delegating their votes to other voters. While ﬂuid democracy has been viewed

as a system that can combine the best aspects of direct and representative democracy, it can

also result in situations where few voters amass a large amount of inﬂuence. To analyze the

impact of this shortcoming, we consider what has been called an epistemic setting, where voters

decide on a binary issue for which there is a ground truth. Previous work has shown that

under certain assumptions on the delegation mechanism, the concentration of power is so severe

that ﬂuid democracy is less likely to identify the gr ound truth than direct voting. We examine

diﬀerent, arguably more realistic, classes of mechanisms, and prove they behave well by ensuring

that (with high probability) there is a limit on concentration of power. Our proofs demonstrate

that delegations can be treated as stochastic processes and that they can be compared to well-

known processes from the literature — such as preferential attachment and multi-types br anching

process — that are suﬃciently bounded for our purposes. Our results sug gest that the concerns

raised about ﬂuid democracy can be overcome, thereby bolstering the case for this emerging

paradigm.

∗

Supported in p art by the NSF grants IIS-178108 and IIS-1703846, a grant from the Open Philanthropy Foundation,

the ARO grant W911NF-17-1-0592, and the MURI grant W911NF-19-1-0217.

†

Supported in part by the ARO MURI grant W911NF-19-1-0217; and th e ONR 2016 Vannevar Bush Faculty

Fellowship.

‡

Supported in p art by the 2020 ONR Vann evar Bush Faculty Fellowship, the NSF grant CCF-1918421, the ARO

MURI grant W911NF-19-1-0217, and the Simons Investigator Award 622132.

Supported in part by the NSF grants CCF-2007080, CCF-1733556 and IIS-2024287, and the ONR grant N00014-

20-1-2488.

To whom correspondence may be addressed (mrevel@mit.edu). Supported in part by the Michael Hammer

Fellowship and the ONR 2016 Van nevar Bush Faculty Fellowship.

1 Introduction

Fluid democracy (also known as liquid democracy) is a voting paradigm that is conceptually situated

between direct democracy, in which voters have direct inﬂuence over decisions, and repre se ntative

democracy, where voters choose delegates who represent them for a period of time. Under ﬂuid

democracy, voters have a choice: they can either vote directly on an issue similar to direct d emoc-

racy, or they can delegate their vote to another voter, entrusting them to vote on their behalf. The

deﬁning feature of ﬂuid democracy is that these delegations are transitive: if voter 1 delegates to

voter 2 and voter 2 delegates to voter 3, then voter 3 votes (or delegates) on behalf of all three

voters.

In recent years, ﬂuid democracy has gained prominence around the world. The most imp ressive

example is that of the German Pirate Party, which adopted the LiquidFeedback platform in 2010 [

22].

Other political parties, such as the Net Party in Argentina and Flux in Austr alia, have run on the

wily promise that, once elected, their representatives would be essentially controlled by voters

through a ﬂuid democracy platform. Compan ies are also exploring the use of ﬂuid demo cr acy for

corporate governance; Google, for example, has run a proof-of-concept experiment [17].

Practitioners, however, recognize that there is a potential ﬂaw in ﬂuid democracy, namely, the

possibility of concentration of power, in the sens e that certain voters amass an enormous number

of delegations, giving them pivotal inﬂuence over the ﬁnal decision. This scenario seems inherently

undemocratic — and it is not a mere thought experiment. Indeed, in the LiquidFeedback platform

of the German Pirate Party, a linguistics professor at the University of Bamberg received so many

delegations that, as noted by Der Spiegel,

his “vote was like a decree.”

Kahng et al. [

21] examine ﬂuid democracy’s concentration-of-power phenomenon from a th e-

oretical viewpoint, and established a troubling imp ossibility result in what has been called an

epistemic setting, that is, one where there is a ground truth.

Informally, they demonstrate that,

even under the strong assumption that voters only delegate to more “competent” voters, any “local

mechanism” satisfying minimal conditions will, in certain instances, fall victim to a concentration

of power, leading to r elatively low accuracy.

More speciﬁcally, Kahng et al. model the problem as a decision problem where voters decide

on an issue with two outcomes, {0, 1}, where 1 is correct (the ground truth) and 0 is incorrect.

Each of the voters i ∈ {1, . . . , n } is characterized by a competence p

∈ [0, 1]. The binary vote

of each voter i is draw n independently f rom a Bernoulli distribution, that is, each voter votes

correctly with p robability p

. Under direct democracy, the outcome of the election is determined

by a majority vote: the correct outcome is selected if and only if more than half vote for the

correct outcome. Under ﬂuid democracy, there exists a set of weights, weight

for each i ∈ [n],

which represent the number of votes that voter i gathered transitively after delegation. If voter i

delegates, then weight

= 0. The outcome of the election is then determined by a weighted majority;

it is correct if and only if

i=1

weight

≥ n/2.

Kahng et al. also introduce the concept of a

deleg ation mechanism, which determines whether voters delegate and if so to whom they delegate.

They are especially interested in local mechanisms, where the delegation decision of a voter only

depends on their local neighborhood according to an underlying social network. They assume that

voters only d elegate to those with strictly higher competence, which excludes the possibility of

http://www.spiegel.de/international/germany/liquid-democracy-web-platform-makes-professor-most-powerful-

pirate-a-818683.html

The use of the term “epistemic” in this context is well-established in the social choice literature [

23, 28].

cyclic delegations.

In order to evaluate ﬂuid democracy, Kahng et al. [

21] test the intuition that society makes

more informed decisions un der ﬂuid democracy than under direct d emocracy (especially given the

foregoing assumption about upward delegation). To that end, they deﬁne the gain of a delegation

mechanism to be the diﬀerence between the probability the correct outcome is selected under ﬂuid

democracy and the probability the correct outcome is selected under direct democracy. A delegation

mechanism satisﬁes positive gain if its gain is s trictly positive in some cases, and it satisﬁes do no

harm if its loss (negative gain) is at most ε for all suﬃ ciently large instances. Th e main result of

Kahng et al. [

21] is that local mechanisms can never satisfy these two requirements. Caragiannis

and Mich a [

7] further strengthen this negative result by showing that there are instances where local

mechanisms perform much worse than either direct democracy or dictatorship (the most extreme

concentration of power).

These theoretical critiques undermine the case for ﬂuid democracy: the beneﬁts of delegation

appear to be reversed by concentration of power. However, the negative conclusion relies heavily on

modeling assumptions, and h as not been borne out by experiments [

2]. In this paper, we provide a

rebuttal by introducing an arguably more realistic model, in which ﬂuid d emocracy is able to avoid

extreme concentration of power, th er eby satisfying both do no harm and positive gain (for suitably

deﬁned extensions).

1.1 Our Contributions and Techniques

Our point of departure from the existing literature is the way we model delegation in ﬂuid democr acy.

Our delegation mechanisms are deﬁned by M = (q, ϕ), wh er e q : [0, 1] → [0, 1] is a function that

maps a voter’s competence to the probab ility they delegate and ϕ : [0, 1]

→ R

≥0

maps a pair

of competencies to a weight. In this mo del, each voter i votes directly with probability 1 − q(p

)

and, conditioned on delegating with probability q(p

), delegates to voter j 6= i with pr ob ab ility

proportional to ϕ(p

, p

). Crucially, a voter does not need to “know” the competence of another voter

in order to decide whether to delegate; rather, the delegation probabilities are m er ely inﬂuenced by

competence in an abstract way captured by ϕ. Also note that delegation cycles are possible, and

we take a worst-case approach to d ealing with them: If the delegations form a cycle th en all voters

in the cycle are assumed to be incorrect (vote 0).

The most signiﬁcant diﬀerence between our model of delegation and that of Kahng et al. [21]

is that in our model each voter has a chance of delegating to any other voter, whereas in their

model an under lying s ocial network restricts delegation options. Our model captures a connected

world where, in particular, voters may have heard of experts on various issues even if they do not

know them pers on ally. Although our model eschews an explicit social network, it can be seen as

embedded into the delegation process, where the probability that i delegates to j takes into account

the probability that i is familiar with j in the ﬁrst place.

Another diﬀerence between our model and that of Kahng et al. [

21] is that we model the com-

petencies p

, . . . , p

as being sampled independently from a distrib ution D. While this assumption

is m ad e mostly for ease of exposition, it allows us to avoid edge cases and obtain robust results.

Our goal is to identify delegation mechanisms that satisfy (probabilistic versions of) positive

gain and do no harm. Our ﬁrst technical contribution, in Section 3, is the formulation of general

conditions on the mechanism and competence distribution that are suﬃcient for these properties

In LiquidFeedback delegation cycles are, in fact, ignored.

to hold (Lemma 1). In particular, to achieve the more diﬃcult do no h arm property, we present

conditions that gu arantee the maximum weight max-weight(G

) accumulated by any voter be su b-

linear with high probability and that the expected increase in competence post-delegation be at

least a positive constant times the population size. These conditions intuitively prevent extreme

concentration of power, and ensure that the representatives post-delegation are suﬃciently better

than the entire population to compensate for any concentration of power that does happen.

It then suﬃces to identify mechanisms and distribution classes that verify th ese conditions. A

delegation mechanism M and a competence distribution D induce a distribution over delegation

instances that generates random graphs in ways that relate to well-known graph processes, w hich

we leverage to analyze our mechanisms. Speciﬁcally, we introduce three m echanisms, all shown to

satisfy do no harm and positive gain under any continuous d istribution over competence levels. The

ﬁrst two mechanisms, upward delegation an d conﬁdence-based delegation, can be seen as interesting

but somewhat restricted case studies, whereas the ge ne ral continuous delegation mechanism is, as

the name suggests, quite general an d arguab ly realistic. Despite the simplicity of the ﬁrst two

mechanisms, the three mechanisms, taken together, reveal the robustness of our approach.

Upward Delegation. In Section

4, we consider a model according to which the probability of

delegating p is exogenous and constant across competencies and delegation can only occur towards

voters with strictly higher competence. That is, the probability that any voter i delegates is

q(p

) = p and th e weight that any voter i puts on an other voter j is φ(p

, p

) = I

−p

>0}

. This

mechanism captures the fact that there might be some reluctance to delegate regardless of the

voter’s competence, but does assume that voters act in the interest of society in only delegating to

voters that are more competent than they are.

To generate a random graph induced by such a mechanism, one can add a single voter at a

time in ord er of decreasing competence, and allow th e voter to either not delegate and create their

own disconnected component, or delegate to the creator of any other compon ent with probability

proportional to p times the size of the component. This works because delegating to any voter

in the previous components is possible (since they have strictly higher competence) and would

result in the votes being concentrated in the originator of that component by transitivity. Such a

process is exactly the one that generates a preferential attachment graph with positive probability

of not attaching to the existing components [

30]. We can then show that, w ith high probability, no

component grows too large, so long as p < 1. Further, there needs to be a constant improvement

by continuity of the competence distribution, which ensu res that a positive fraction of voters below

a certain competence delegate to a positive fraction of voters with strictly higher competencies.

Conﬁdence-Based Delegation. In Section

5, we consider a model in which voters delegate with

probability decreasing in their competencies and choose someone at random when they delegate.

That is, the probability q(p

) that any voter i delegates is decreasing in p

and the weight that any

voter i gives to any voter j is φ(p

, p

) = 1. In other words, in this model competence does not

aﬀect the probability of receiving d elegations, only the probability of delegating.

To generate a random graph induced by such a mechanism, one can begin from a random vertex

and study the delegation tree that starts at that vertex. A delegation tree is deﬁned as a branching

process, where a node i’s “children” are the nodes that delegated to node i. In contrast to classical

branching process, the probability for a child to be born increases as the nu mber of people who

already r eceived a d elegation decreases. Nevertheless, we prove that, with high probability, as long

as a delegation tree is no larger than O(log n), our heterogeneous branching process is dominated

by a sub-critical graph branching process [

1]. We can then conclude that no component has size

larger than O(log n) with high probability. Next, we show that the expected competence among

the voters that do not delegate is strictly higher than the average competence. Finally, given that

no voter has weight larger than O(log n), we prove that a sm all number of voters end up in cycles

with high probability. We can therefore show that the conditions of Lemma 1 are satisﬁed.

General Continuous Delegation. Finally, we consider a general model in Section 6 where the

likelihood of delegating is ﬁxed and the weight assigned to each voter when delegating is in cr easing

in their competence. That is, each voter i delegates with probability q(p

) = p and the weight that

voter i places on voter j is ϕ(p

, p

), where ϕ is continuous and increases in its second coordin ate.

Thus, in this model, the delegation distr ibution is slightly skewed towards more competent voters.

To generate a random grap h in duced by such a mechanism, we again consider a branching

process, but now voters j and k place diﬀerent weights on i per ϕ. Therefore, voters h ave a type

that governs their delegation behavior; this allow us to deﬁne a multi-type br an ching process with

types that are continuous in [0, 1]. The major part of the analysis is a proof that, with high

probability, as long as the delegation tree is no larger than O(log n), our heterogeneous branching

process is dominated by a sub-critical Poisson multi-ty pe branching process. To do so, we group

the competencies into buckets that partition the segment [0, 1] into small enough pieces. We deﬁne

a new ϕ

′

that outputs, for any pair of competencies p

, p

, th e maximum weight a voter from

i’s bucket could place on a voter from j’s bucket. We can show that such a discrete multi-ty pe

branching process is sub-critical and conclude that no component has size larger than O(log n ) with

high probability. In a similar fashion to Conﬁdence-Based Delegation, we also show that there is

an expected increase in competence post-delegation.

1.2 Related work

Our work is most closely related to that of Kahng et al. [

21], which was discussed in detail above.

It is worth noting, though, that they complement their main negative result with a positive one:

when the mechanism can restrict the maximum number of delegations (transitively) received by

any voter to o(

√

log n), do no harm and positive gain are satisﬁed. Imposing such a restriction

would r equire a central planner that monitors and controls delegations. G¨olz et al. [

14] build on

this idea: they study ﬂuid demo cr acy s y stems wher e voters may nominate multiple delegates, and

a central planner chooses a single delegate for each delegator in order to minimize the maximum

weight of any voter.

Similarly, Brill and Talmon [6] propose allowing voters to specify ordinal preferences over dele-

gation options, and possibly restricting or modifying delegations in a centralized way. Caragiannis

and Micha [7], and then Becker et al. [2] also consider central planners; they show that, for given

competencies, the problem of choosing among d elegation options to maximize the probability of

a correct decision is hard to approximate. In any case, imp lementing these proposals would re-

quire a fundamental rethinking of the practice of ﬂuid democracy. By contrast, our positive results

show that decentralized delegation mechanisms are inherently s elf-regulatory, which supports the

eﬀectiveness of the current practice of ﬂuid democracy.

More generally, there has been a signiﬁcant amount of theoretical research on ﬂuid democracy

in recent years. To give a few examples: Green-Armytage [

15] studies whether it is rational for

voters to delegate their vote from a utilitarian viewpoint; Christoﬀ and Grossi [

8] examine a similar

剩余31页未读，继续阅读

评论收藏

内容反馈

版权申诉

为流动民主辩护_In Defense of Fluid Democracy

评论0

最新资源

为流动民主辩护_In Defense of Fluid Democracy

评论0

最新资源

相关推荐

The_Finite_Volume_Method_in_Computational_Fluid_Dynamics2016

A_Gallery_of_Fluid_Motion.pdf

fct-vof.rar_FCT VOF_FCT-VOF_fct_vof_volume of fluid

Fundamentals_of_Computational_Fluid_Dynamics

流体图册1-An_Album_Of_Fluid_Motion-1.pdf

ANSYS Fluent_workshop01_Fluid Flow and Heat Transfer.pdf

An_introduction_to_computational_fluid_dynamics.pdf

fluidsD3D9.rar_cuda_cufft_fluid_fluid simulation

An_Introduction to Computational Fluid Dynamics 2nd_Edition

A_mathematical_intro_to_fluid_mech_chorin.pdf

Computational Fluid Mechanics.zip_CFD_Fluid mechanics_fluid CFD

An_introduction_to_Computational_Fluid_Dynamics

Computational_Fluid_Dynamics-master.zip_CFD_computational fluid_

fluid_FEM.rar_FEM_Matlab程序 流体_fluid_fluid FEM_计算流体

fluid_mechanics.rar_C++ mechanics_Fluid mechanics

LBM_WSS_Poiseuille.rar_LBM_Poiseuille_Poiseuille LBM_fluid simul

meshfile.rar_Mesh_computational fluid_download meshfile_dynamic

Foundations of Fluid Mechanics with Applications: Problem Solving Using mma

Fundamentals of Fluid Mechanics, 6th Edition By Munson.流体力学基础

Cobalt Strike下载

计算机系统-笔记-HUN2021级

北京邮电大学计算机考研复试笔试资料

cs1.6老版本供下载

合成孔径雷达的经典成像算法cs(matlab)仿真代码（吐血整理，内容全，注释全）

港大CS（MSC）面试整理

合成孔径雷达RD CS OmegaK算法点目标仿真.rar

计算机科学导论原书第二版答案.zip

Cobalt-Strike-4.5

cobaltstrike4.3.zip

fluid_FEM.rar_FEM_Matlab程序流体_fluid_fluid FEM_计算流体