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Research Paper AAMAS 2020, May 9–13, Auckland, New Zealand

Incentivising Participation in Liquid with Breadth-First Delegation∗

Grammateia Kotsialou Luke Riley - King’s College London Informatics - King’s College London, Quant Network London, United Kingdom London, United Kingdom [email protected] luke.riley@{kcl.ac.uk,quant.network}

ABSTRACT Party in Argentina and Partido de Internet in Spain have been ex- allows members of an electorate to either directly perimenting with liquid democracy implementations. Additionally, vote over alternatives, or delegate their rights to someone the local of the London Southwark borough and the they trust. Most of the liquid democracy literature and implementa- Italian cities Turino and San Dona di Piave are working on integrat- tions allow each voter to nominate only one delegate per . ing liquid democracy for community engagement processes (Boella However, if that delegate abstains, the voting rights assigned to et al. [2018]). In the technology domain, the online platform Liq- her are left unused. To minimise the number of unused delegations, uidFeedback uses a liquid democracy system where a user selects it has been suggested that each voter should declare a personal a single guru for different topics (Behrens et al. [2014]; Kling etal. ranking over voters she trusts. In this paper, we show that even if [2015]). Another prominent online example is GoogleVotes (Hardt personal rankings over voters are declared, the standard delegation and Lopes [2015]), where each user wishing to delegate can select method of liquid democracy remains problematic. More specifically, a ranking over other voters. we show that when personal rankings over voters are declared, it Regardless of the increasing interest in liquid democracy, there could be undesirable to receive delegated voting rights, which is exists outstanding theoretical issues. This work focuses on liquid contrary to what liquid democracy fundamentally relies on. To solve democracy systems where each individual wishing to delegate can this issue, we propose a new method to delegate voting rights in an select a ranking over other voters. In such systems, given the com- election, called breadth-first delegation. Additionally, the proposed mon interpretation that delegations of voting rights are multi-step 1 method prioritises assigning voting rights to individuals closely and transitive , we observe that: searching for a guru follows a connected to the voters who delegate. depth-first search in a graph that illustrates all delegation pref- erences within an electorate, e.g. nodes represent the voters and KEYWORDS directed edges the delegation choices for each voter. For this rea- son, we name this standard approach of delegating voting rights liquid democracy, , economic mechanism design as depth-first delegation. Despite its common acceptance, we came ACM Reference Format: across an important disadvantage even for the rule with Grammateia Kotsialou and Luke Riley. 2020. Incentivising Participation binary issues. In particurlar, we show that when depth-first dele- in Liquid Democracy with Breadth-First Delegation. In Proc. of the 19th gation is used, it could be undesirable to receive the voting rights International Conference on Autonomous Agents and Multiagent Systems of someone else. At this point, we emphasize that disincentivising (AAMAS 2020), Auckland, New Zealand, May 9–13, 2020, IFAAMAS, 7 pages. voters to participate as gurus is in contrast to the of liquid democracy due to the following. How can a liquid democracy sys- 1 INTRODUCTION tem flourish if voters may not be incentivised to receive delegated votes? Motivated by this, we propose a simple idea solution to this Liquid democracy is a middle ground between direct and repre- issue: a new rule for delegating voting rights, called the breadth- sentative democracy, as it allows each member of the electorate to first delegation, which guarantees that casting voters (those who do directly vote on a topic, or temporarily choose a representative by not delegate or abstain) weakly prefer to receive delegated voting delegating her voting rights to another voter. Therefore individuals rights, i.e. to participate as gurus. For these reasons, we consider who are either apathetic for an election, or trust the knowledge this work of a high societal importance and immediate applications. of another voter more than their own, can still have an impact on We outline this paper as follows: In the introduction we discuss the election result (through delegating). An individual who casts the latest applied and theoretical developments in liquid democracy a vote for themselves and for others is known as a guru (Christoff and give the preliminaries of our model. In the next two sections, and Grossi [2017]). Liquid democracy has recently started gaining we define delegation graphs, delegation rules and two types ofpar- attention in a few domains which we discuss to show an overview ticipation. Afterwards, we formally introduce a new delegation rule of the general societal interest. In the political domain, local parties and compare this rule with the standard one. Finally, we conclude such as the in , in , the Net this work with future research goals.

∗This work is supported by the EPSRC grant EP/P031811/1.

Proc. of the 19th International Conference on Autonomous Agents and Multiagent Systems 1 (AAMAS 2020), B. An, N. Yorke-Smith, A. El Fallah Seghrouchni, G. Sukthankar (eds.), May We assume the following interpretation for the meaning of transitive delegations of 9–13, 2020, Auckland, New Zealand. © 2020 International Foundation for Autonomous voting rights: If a voter i delegates to a voter j, then i transfers to j the voting rights Agents and Multiagent Systems (www.ifaamas.org). All rights reserved. of herself and all the others that had been delegated to i.

638 Research Paper AAMAS 2020, May 9–13, Auckland, New Zealand

1.1 Related work x over alternative y.A preference relation over voters for voter i ∈ V ≻V V ∈ V There currently exists a lack of theoretical analysis on liquid democ- is denoted by i and is a binary relation on , i.e.: for i, x,y 2 ≻V racy. However, we summarise the main differences of our work to with i , x,y and x , y, the expression x i y indicates that voter the main undertaking so far. i strictly prefers to delegate her voting rights to voter x instead of As outlined by Brill ([2018]), one of the main ongoing issues in voter y. For both preference relations, we allow an index to identify liquid democracy is how to handle personal rankings over voters. ranking positions e.g. for any i ∈ Vd , her m-th preferred voter is ≻V His work discusses possible solutions around this issue without denoted by i,m. giving a formalised model, which this paper does. For two election ≻W ≻W For a set W , consider a binary relation i . Then, i is: alternatives where one is the ground truth, Kahng et al. ([2018]) ( ) ∈ ≻W ≻W find that: (a) there is no decentralised liquid democracy delegation a complete iff for every pair x,y W either x i y or y i x rule that is guaranteed to outperform and (b) holds, ( ) ∈ ≻W there is a centralised liquid democracy delegation rule that is guar- b antisymmetric iff for every pair x,y W , if x i y then ≻W anteed to outperform direct democracy as long as voters are not y i x does not hold, and ( ) ∈ ≻W ≻W completely misinformed or perfectly informed about the ground c transitive iff for all x,y, z W , if x i y and y i z, then truth. In comparison, our model can be used in a wider variety ≻W x i z. of , as it allows for multiple alternatives and no ground Both preference relations over alternatives and preference relations truth. Additionally our delegation rules can be used in a central or over voters are antisymmetric and transitive but not complete (we decentralised manner, thus the negative result (a) does not apply to do not enforce voters to rank every other member of the electorate our paper. The work of Christoff and Grossi ([2017]) focuses on the as we consider this an unrealistic scenario for large electorates). existence of delegation cycles and inconsistencies that can occur The set of all possible preference relations ≻A and ≻V , for any when there are several binary issues to be voted on with a different i i i ∈ V, are denoted by R A and RV , respectively. A preference profile guru assigned for each issue. In comparison, we avoid delegation A RA A cycles by stating that a delegation chain (a path from a delegating over alternatives is a function P : E(V) → 2 , where P (N ) voter to their assigned guru) cannot include the same voter more returns a set of preference relations over alternatives (maximum than once. Furthermore, individual rationality issues between mul- one for each voter in N ). For example, given an electorate N = { } A( ) {( ≻A) ( ≻A)} tiple elections is out of scope for this work. Last, Brill and Talmon i, j,k , a preference profile P N could return i, i , j, j , ([2018]) introduce a special case of Christoff and Grossi’s model, meaning that agent i and j have been assigned a preference relation which allows a single voter to be assigned several gurus. However, over alternatives but k has not (as k either delegates or abstains). our model assigns one guru per voter. Therefore Brill and Talmon Similarly, we define as a preference profile over voters a function V ([2018]) does allow more fine-grained delegations than our model, P V : E(V) → 2R , where P V (N ) returns a set of preference i.e. they allow for different pairs of alternatives to be ranked by relations over voters (maximum one for each voter in N ). Given different gurus. But the delegating voter cannot choose a preference profiles P A and P V , voters are assigned to the V a, V c and V d ( ≻A) ∈ A( ) relation over voters for any pair of alternatives, which is what this electorates as follows. If i, i P N , we infer that voter i paper investigates how to do. ≻A casts a vote according to i and therefore becomes a member Similar to our work, GoogleVotes (Hardt and Lopes [2015]) allows A A V of the casting electorate V c . If (i, ≻ ) < P (N ) and (i, ≻ ) ∈ a user to select a ranking over other voters and uses, what the i i P V (N ), then i becomes a member of the delegating electorate V d . authors describe as, a back-track breadth first search to assign a guru If (i, ≻A) P A(N ) and (i, ≻V ) P V (N ), i becomes a member of to a voter. We cannot complete a more comprehensive comparison i < i < the abstaining electorate V a. to GoogleVotes as they have published only a general description of Given an electorate N , adding or removing a preference relation their system (without a formal model). However, we know that their over alternatives (or over voters) from a preference profile over delegation rule is different to our proposed breadth-first delegation P A(N ) P V (N ) rule as in the Tally/Coverage section of their video example (from alternatives (or over voters ), is denoted as follows. (i, ≻A) ∈ P A(N ) j ∈ V \ N j minute 32 of Hardt [2014]), their rule assigns guru C to voter F, For a tuple i , a voter and ’s assigned ≻A ∈ A while our rule would assign guru A to voter F. preference relation over alternatives j R : A( ) A( )\{( ≻A)} 1.2 Preliminaries P−i N := P N i, i , V P A (N ) := P A(N ) ∪ {(j, ≻A)}. Consider a set of voters and a set of alternatives or outcomes +(j, ≻A ) j A. The set of possible electorates is given by E(V) = 2V \{∅}, i.e. j non-empty subsets of V. In our model, for every election there are Similarly, for a tuple (i, ≻V ) ∈ P V (N ), a voter j ∈ V \ N and j’s a c d i j i three sets of electorates V ,V ,V ∈ E(V) such that V ∩ V = ∅ ≻V ∈ V assigned preference relation over voters j R : for i , j ∈ {a,c,d} and V a ∪ V c ∪ V d = V, where sets V a, V c , V d consist of those who abstain, cast a vote and delegate their voting V V V P−i (N ) := P (N )\{(i, ≻i )}, rights, respectively. V ( ) V ( ) ∪ {( ≻V )} ∈ V P V N := P N j, j . A preference relation over alternatives for a voter i is denoted +(j, ≻j ) ≻A A ∈ by i and is a binary relation on , i.e.: for x,y A with x , y, the A ≻ 2 expression x i y indicates that voter i strictly prefers alternative A voter cannot include herself in her preference relation over voters.

639 Research Paper AAMAS 2020, May 9–13, Auckland, New Zealand

To simplify the above cases notation, we will sometimes be using 3 CAST AND GURU PARTICIPATION A V A A V V P , P , P−i , P+j , P−i and P+j , accordingly. The key property that we investigate is participation. The partici- pation property holds if a voter, by joining an electorate, is at least 2 DELEGATION GRAPH AND DELEGATION as satisfied as before joining. This property has been defined only RULES in the context of vote casting (Fishburn and Brams [1983]; Moulin To model all possible delegation choices between voters, we use a [1988]). Due to the addition of delegations in our model, we es- weighted directed graph defined as follows. tablish two separate definitions of participation to reflect this new functionality3. Definition 1. A delegation graph is a weighted directed graph For both of the following definitions, note that for an electorate G = (V, E,w), where N ∈ E(V), the set of all preference profiles over alternatives is •V is the set of nodes representing the agents registered as given by P A,N , while the set of all preference profiles over dele- voters; gates is given by P V,N . • E is the set of directed edges representing delegations between A voting rule f satisfies the cast participation property when voters; and every voter i ∈ V weakly prefers joining any possible voting • 7→ w is the weight function w : E N that assigns a value to an electorateV c compared to abstaining and regardless of who is in the ( , ) edge i j . delegating electorate V d . For the next two definitions, we add the To decide which values the weight function w has assigned to the notation ⪰i of weakly preferring as in these two cases the resulting outgoing edges of each vertex, we introduce the following function outcomes might be identical. For non-identical outcomes, a voter i д that generates a delegation graph. Recall that we allow an index has a strict preference (as indicated in the model preliminaries). to identify ranking positions of trustees e.g. for any i ∈ Vd , her Definition 4. The Cast Participation property holds for a voting m ≻V -th preferred voter is denoted by i,m. rule f iff: Define as д the which A V A A V Definition 2. delegation graph function f (d(P ,д(P ))) ⪰ f (d(P ,д(P ))), takes as input a preference profile over voters P V and returns the re- i −i lated delegation graph G = (V, E,w) for which the following property for every possible disjoint casting and delegating electorates V c ,V d ∈ holds. E(V), where i ∈ V c , and every possible preference profile for these A A,V c V V,V d ∈ V ( ≻V ) ∈ electorates P ∈ P and P ∈ P . Property 1. For every i, j and i , j, if there exists i, i V ≻V ( ) ∈ c d P with i,x = j, then there exists i, j E such that For any casting and delegating electorates V and V , a voting w((i, j)) = x. rule f satisfies the guru participation property when any voter i ∈ V c weakly benefits from receiving additional voting rights of We evaluate a delegation graph through the following. any voter j ∈ V. Definition 3. A delegation rule function d takes as input a Definition 5. The Guru Participation property holds for a voting preference profile over alternatives P A together with a delegation rule f iff: graph G, and returns another preference profile over alternatives PˆA A V A A V that resolves delegations as follows, f (d(P ,д(P ))) ⪰i f (d(P ,д(P−j ))), (1) A • ( ≻ ) ∈ ˆA c d if i, i P , then i casts her vote, for every possible disjoint casting and delegating electorates V ,V ∈ A c d A A,V c • if (i, ≻ ) ∈ PˆA for a voter j , i, then j becomes i’s final E(V), where i ∈ V , j ∈ V , and every possible profile P ∈ P j d delegate, i.e. her guru, and P V ∈ P V,V that assign j’s vote to guru i, i.e. • if (i, ≻A) PˆA for every k ∈ V, then i abstains. A A V k < (j, ≻i ) ∈ d(P ,д(P )). For each voter i ∈ V, a delegation rule analyses the subtree of Let F be the set of all voting rules. It is known that only a subset the delegation graph rooted at node i and decides whether i casts, F¯ ⊂ F satisfy (cast) participation: for example, Fishburn and Brams delegates or abstains. If voter i is found to delegate, the chosen ([1983]) show that does not satisfy (cast) delegation rule function will traverse i’s subtree to find i’s guru. participation, while Moulin ([1988]) shows there is no Condorcet- To get the outcome of an election, we use a voting rule function consistent voting rule satisfying this property given 25 or more f . In our model, f takes as input the modified preference profile voters. over alternatives PˆA (which incorporates delegations) and returns We are interested in exploring guru participation for voting a single winner or a ranking over alternatives (depending on the rules in the subset F¯ and this paper focuses on the voting rule used). In Section 5, we show that the output of the with binary issues. Our results build on Observation 1, which we voting rule depends on the chosen delegation rule, meaning that intuitively describe as follows. First, recall that if a voting rule we could get different election results when only the delegation satisfies cast participation, then any voter weakly prefers casting a rule function is different, i.e. vote instead of abstaining. When delegations of votes are allowed in A V ′ A V f (d(P ,д(P ))) , f (d (P ,д(P ))), 3There could be other interesting participation properties for liquid democracy, such ′ as incentivising deviation from delegating to casting. But this is out of the paper’s when d , d . scope, as we focus on finding delegation rules that weakly benefit casting voters who become gurus.

640 Research Paper AAMAS 2020, May 9–13, Auckland, New Zealand

the election and there is only one guru who gains an additional vote, Observe that the variable x in the expression Ci,x also indicates then this guru weakly prefers receiving this additional vote (and how deep the voter Ci,x is in the delegation graph subtree rooted therefore casting it, since cast participation is satisfied). Therefore with vertex i. Thus sometimes we refer to x as the depth of Ci,x in guru participation is satisfied. Formally, we write the above as Ci . The function w takes as input a delegation chain and returns a follows (see preliminaries for full form notation and definitions of list of the weights assigned to each edge among voters in Ci , that the profiles used). is,

Observation 1. Let f ∈ F¯ be the majority rule. Consider voters w(Ci ) = [w(Ci )1,...,w(Ci )x ,...,w(Ci )n−1], ′ i, j ∈ V, a profile PˆA returned by d(P A,д(P V )) and a profile Pˆ A where w(Ci )x is the weight of edge (Ci,x ,Ci,x+1) and n = |Ci |. ( A ( V )) returned by d P ,д P+j , where i has been assigned as j’s guru, i.e. Delegation chains can be used as a tool to find a guru for a voter ′ ( ≻V ) ∈ ˆ A ∈ d j, i P . Then guru i weakly benefits after j delegates if for i V . The standard interpretation of liquid democracy delegations prioritises all possible delegation chains involving i and i’s most every k ∈ V: ≻V preferred voter i,1 before all possible delegation chains involving i (a) k’s vote is assigned to guru l ∈ V by both returned preference V ′ and i’s second preferred voter ≻ and so on. Note that this priority profiles, i.e. (k, ≻V ) ∈ PˆA ∩ Pˆ A, or i,2 l rule hold for the deeper levels of the delegation graph subtree rooted (b) k’s vote is assigned to guru i after j joins the delegating elec- at i. In other words, we observe that the standard way to select i’ ′ ( ≻A) ∈ ˆ A guru is to choose the first casting voter found through a depth first torate, i.e. k, i P . search in i’s subtree, which motivates the next definition. A depth-first delegation rule dD assigns guru j to i iff: (a) there is a delegation chain Ci that can be formed from i to j, and (b) there 4 INTRODUCING BREADTH-FIRST ′ is no other delegation chain Ci leading to a different guru k that DELEGATION has a smaller weight at the earliest depth after the root, compared Recall that liquid democracy allows for multi-step delegations. to Ci . d c Therefore, the guru of any i ∈ V could be any voter j ∈ V Definition 7. For i, j,k ∈ V, a depth-first delegation rule dD who is in the sub tree of the delegation graph with root i. Further- returns a profile PˆA with (i, ≻A) ∈ PˆA iff (a) and (b) hold: more, the assigned guru j may not be included in i’s preference j V V (a) delegation chain C with C = j, ≻ š ≻ i i, |Ci | relation i , i.e. it could be that x such that i,x = j. In this case, ∃ there is at least one intermediate delegator between voter i and the ′ (b) š any delegation chain Ci such that: assigned guru j. To find who exactly are intermediate delegators, . ′ = b1 Ci, |C′ | k for k , j, we introduce the delegation chains as follows. i ′ b2. ∗ y: w(Ci )y < w(Ci )y , and A delegation chain for a voter i ∈ V d starts with i, then lists the ∃ ′ ∗ w(C )x ≤ w(Ci )x for all 0 < x < y. intermediate voters in V d who have further delegated i’s voting i rights and ends with i’s assigned guru j ∈ V c . These chains (see Example 4.1. Consider the delegation graph in Figure 1 (a). There Definition 6) must satisfy the following conditions: (a) no voter 4 are two delegation chains available for voter p ∈ V: Cp = ⟨p,r⟩ occurs more than once in the chain (to avoid infinite delegation ′ ′ and C = ⟨p,q,s⟩ with weights w(Cp ) = [2] and w(C ) = [1, 2], cycles that could otherwise occur) and (b) each member of the p p D ˆA chain must be linked to the next member through an edge in the respectively. The d rule returns profile P that assigns s as the A ˆA ′ guru of p, i.e. (p, ≻s ) ∈ P , due to inequality w(Cp )1 < w(Cp )1. delegation graph, which is generated from the given preference ′ profile over voters. Note that Cp satisfies Definition 7.

Definition 6. Given profiles P A and P V , a voter i ∈ V d and her In Example 4.1, p’s voting right is assigned to guru s, but why guru j ∈ V c , we define a delegation chain for i to be an ordered tuple should s (who is the second preference of q) outrank agent p’s explicit second preference r? This question gains even more impor- Ci = ⟨i,..., j⟩ such that: tance the longer the depth-first delegation chain is. Given this issue, ( ) ∈ [ | |] a for an integer x 1, Ci , notation Ci,x indicates the voter we define a novel delegation rule that prioritises a voter’s explicit at the x-th position in Ci , preferences as follows. A breadth-first delegation rule dB assigns (b) for every pair of integers x,y ∈ [1, |Ci |] with x , y, guru j to i iff: (a) there is a delegation chain Ci that can be formed ′ from i to j; and (b) there is no other delegation chain Ci leading to Ci,x , Ci,y , a different guru k with: either a shorter length or, an equal length and a smaller weight at the earliest depth after the root, compared (c) for every integer z ∈ [1, |Ci | − 1], there exists an edge to Ci . V (Ci,z ,Ci,z+1) ∈ E ∈ д(P ), Definition 8. For i, j,k ∈ V, a breadth-first delegation rule dB ( ) ∈ [ | | − ] ˆA ( ≻A) ∈ ˆA ( ) ( ) d for every integer z 1, Ci 1 , the voter positioned at Ci,z returns a profile P with i, j P iff a and b hold: belongs in the V d electorate , and (a) delegation chain Ci with Ci, |C | = j, ∃ i ( ) c e the voter positioned at Ci, |Ci | belongs in the V electorate. 4Recall that ⟨p, q, t ⟩ does not satisfy the delegation chain definition as t < V c .

641 Research Paper AAMAS 2020, May 9–13, Auckland, New Zealand

delegation graph delegation rule Yes No Figure 1 (a) dD 1 3 Figure 1 (b) dD 3 2 Figure 1 (a) dB 2 2 Figure 1 (b) dB 3 2 Table 1: Election results for Figure 1 when using either the depth-first or the breadth-first delegation rule.

B Using rule d instead, voter p is assigned guru r through Cp (see (a)(b) Example 4.2), while q’s guru remains the same. Therefore dB returns another preference profile over alternatives: Figure 1: (a) A delegation graph with electorates V a = A A A A {(p, ≻r ), (q, ≻s ), (s, ≻s ), (r, ≻r )}. {t}, V c = {s,r } and V d = {p,q}, meaning that t abstains, s,r cast, and p,q delegate. The preference relations over In the next example we focus on the case where the previously A A abstaining voter t decides to delegate and show that the election alternatives are: “No" ≻s “Yes" and “Yes" ≻r “No". (b) A D delegation graph with electorates V a = {}, V c = {s,r } and result is inversed only when d is used (see Table 1). V d = {p,q, t}. The preference relations over alternatives Example 5.2. Consider the delegation graph in Figure 1(b) and remain the same. The only difference to (a) is that voter t all possible delegation chains available to each voter in V d with decides to delegate with a preference relation over voters their respective edge weights: p ⪰V r. t delegation chain edge weights

Cp = ⟨p,r⟩ w(Cp ) = [2], ′ ′ (b) š any delegation chain C such that C = k, for k , j, C′ = ⟨p,q,s⟩ w(C′ ) = [1, 2], i i, |Ci | p p and ′′ ′′ Cp = ⟨p,q, t,r⟩ w(Cp ) = [1, 1, 2], ′ b1. |Ci | < |Ci |, or Cq = ⟨q,s⟩ w(Cq ) = [2], ′ b2. ∗ |Ci | = |Ci | and ′ ⟨ ⟩ ( ′ ) [ ] ′ Cq = q, t,r w Cq = 1, 2 , ∗ y: w(Ci )y < w(Ci )y and ′′ ′′ ∃ ′ Cq = ⟨q, t,p,r⟩ w(Cq ) = [1, 1, 2], ∗ w(Ci )x ≤ w(Ci )x for all 0 < x < y. Ct = ⟨t,r⟩ w(Ct ) = [2], Example 4.2. Consider the delegation graph in Figure 1 (a). There ′ ′ Ct = ⟨t,p,r⟩ w(Ct ) = [1, 2], are two delegation chains available for voter p ∈ V: Cp = ⟨p,r⟩ ′ ′ C′′ = ⟨t,p,q,s⟩ w(C′′) = [1, 1, 2]. and Cp = ⟨p,q,s⟩ with weights w(Cp ) = [2] and w(Cp ) = [1, 2], t t respectively. The dB rule returns profile PˆA that assigns r as the A ˆA ′ D guru of p, i.e. (p, ≻r ) ∈ P , due to inequality |Cp | < |Cp |. Note Using rule d , observe that voter p is assigned guru r through ′′ ′′ ′′ ′ that Cp satisfies Definition 8. the chain Cp due to w(Cp )1 < w(Cp )1, w(Cp )1 = w(Cp )1 and ′′ ′ ′′ w(Cp )2 < w(Cp )2. Voter q is also assigned guru r through chain Cq ′′ ′′ ′ ′′ ′ since w(Cq )1 < w(Cq )1, w(Cq )1 = w(Cq )1 and w(Cq )2 < w(Cq )2. ′′ ( ′′) 5 DEPTH-FIRST VERSUS BREADTH-FIRST Last, voter t is assigned guru s through chain Ct because w Ct 1 < ( ) ( ′′) ( ′) ( ′′) < ( ′) DELEGATION w Ct 1, w Ct 1 = w Ct 1 and w Ct 2 w Ct 2. Therefore rule dD returns the preference profile over alternatives Through the next two examples, we show that different delegation A A A A A rules can have different properties. More specifically, we present {(p, ≻r ), (q, ≻r ), (s, ≻s ), (r, ≻r ), (t, ≻s )}. an instance where the depth-first delegation rule cannot guarantee Using rule dB instead, voter p is assigned guru r through the chain ′ ′′ guru participation, while the breadth-first delegation rule does. Cp due to inequalities |Cp | < |Cp | < |Cp |. Voter q is assigned guru ′ ′′ s through Cq due to |Cq | < |Cq | < |Cq | and voter t is assigned Example 5.1. Consider the delegation graph in Figure 1(a) and all ′ ′′ d guru r through Ct because of |Ct | < |C | < |C |. Therefore, rule possible delegation chains available to each voter in V : Cp = ⟨p,r⟩, t t ′ dB returns the profile over alternatives Cp = ⟨p,q,s⟩ and Cq = ⟨q,s⟩. {(p, ≻A), (q, ≻A), (s, ≻A), (r, ≻A), (t, ≻A)}. D ′ r s s r r Using rule d , voter p is assigned guru s through chain Cp (see Example 4.1), while voter q is also assigned guru s through chain D Examples 5.1 and 5.2 show that guru participation may not hold Cq . Therefore d returns the preference profile over alternatives for depth-first delegation when a cycle exists in the delegation A A A A {(p, ≻s ), (q, ≻s ), (s, ≻s ), (r, ≻r )}. graph. Due to this cycle, when t joins the election, both r and s

642 Research Paper AAMAS 2020, May 9–13, Auckland, New Zealand

receive new delegated voting rights, thus Observation 1 does not from i, voter l also increases the times she becomes a guru. Next we occur5. We summarise the above for the majority rule f ∈ F¯. describe that, when dD is used, this case can only arise through the following circumstance. Let guru i be assigned to voter j through Theorem 5.3. Given the majority rule f ∈ F¯, guru participation delegation chain C = ⟨j,...,i⟩ and guru l be assigned to voter k is not guaranteed to hold when using the depth-first delegation rule j through delegation chain C = ⟨k,..., j,...,l⟩. Chain C must pass dD . k k through j because all chains without j are available before j dele- Proof. Consider the preference profile over alternatives and the gates. Note that even if both chains pass through voter j, they end preference profile over voters of Figure 1(b), at different gurus. For dD , this only occurs if there exists a voter h A A A with h , i, j,l, such that P = {(r, ≻r ), (s, ≻s )} V V V V Cj = ⟨j,...,h,...,i⟩ and (3) P = {(p, ≻p ), (q, ≻q ), (t, ≻t )}, Ck = ⟨k,...,h,..., j,...,l⟩. (4) r s ≻A where the preferences over alternatives for and are: “Yes” r The reason for the above is that k’s delegation goes through h to ≻A , , “No", “No" s “Yes” and the preferences over voters for p q t are: reach j, but then the preferred delegation from j passes through h q ≻V r, t ≻V s and p ≻V r. We prove this theorem using Exam- p q t (see chain Cj ). As k’s delegation already includes h before j, and ples 5.1 and 5.2. In Example 5.1, where voter t abstains, rule dD an intermediator voter cannot be repeated (definition 3), k uses returns profile another route to guru l (through a less preferred option of j). From D A V (3) and (4), observe that there exists a cycle in the graph, i.e. the d (P ,д(P−t )) = cycle ⟨h,..., j,...,h⟩, which contradicts our assumption and proves A A A A {(p, ≻s ), (q, ≻s ), (s, ≻s ), (r, ≻r )}, the lemma. □ which gives three votes (via s) for alternative “No" and one vote Theorem 5.5. Given the majority rule f ∈ F¯ and a delegation (via r) for alternative “Yes" (see also Table 1). From Example 5.2 D graph with no cycles, guru participation is guaranteed to hold when where voter t delegates, rule d returns profile using the depth-first delegation rule dD . dD (P A,д(P V )) = Proof. We prove this using Lemma 5.4 and Observation 1. □ {(p, ≻A), (q, ≻A), (s, ≻A), (r, ≻A), (t, ≻A)}, r r s r s We have previously shown that depth-first delegation does not which gives three votes for “Yes" and two votes for “No". Ob- guarantee guru participation when the delegation graph contains serve that the election result changes from “No" to “Yes" despite cycles. The next theorem states that breadth-first delegation always A the fact that t votes for “No" through her guru s, i.e. (t, ≻s ) ∈ guarantees guru participation. To show this, we first introduce the D A V A d (P ,д(P )). Note that due to the preference “No"≻s “Yes”, we following observation and lemma. get Observation 2. Consider two voters j and k in a delegating elec- D A V A D A V B f (d (P ,д(P ))) ≺s f (d (P ,д(P−t ))), (2) torate. Using the breadth-first delegation rule d , if k is assigned guru where f is a voting rule satisfying cast participation. However, l through a delegation chain Ck with j < Ck , then k is assigned guru the preference expressed by (2) implies that guru s becomes worst l even when j abstains. off after t delegates to her, which violates the definition of guru B The above observation occurs because rule d has used Ck ahead participation (1), proving this theorem. □ of any possible delegation chain that includes j. Therefore chain B We highlight that if a delegation graph has no cycle then guru Ck will still be used by d when j is in the abstaining electorate participation is guaranteed to hold for the depth-first delegation and no possible delegation chain that includes j can be formed. rule, which show through Lemma 5.4 and Theorem 5.5. Lemma 5.6. Consider two voters j and k in a delegating electorate. B Lemma 5.4. When using depth-first delegation rule dD , if there is Using the breadth-first delegation rule d , if voter k is assigned her no cycle in the delegation graph then Observation 1 holds. guru through a delegation chain Ck with j ∈ Ck , then k is assigned the same guru as j. Proof. Assume there exists a delegation graph with no cycles B where Observation 1 does not hold. We show that the only case Proof. Assume that, using d , voter j is assigned guru i through where Observation 1 does not hold is when a cycle exists. delegation chain Cj = ⟨j ...,i⟩ and k is assigned a different guru l Recall that (by Observation 1) guru participation is guaranteed through a delegation chain that includes j, i.e. to hold if whenever a voter j joins the delegating electorate, there Ck = ⟨k,..., j,...,l⟩. exists only one voter, say i, in the casting electorate who increases Then either (a) or (b) occurs: the number of times she becomes a guru. Consider a voter k who ( ) B ′ = ⟨ ,..., ⟩ changes her assigned guru to another voter l after j joins the del- a rule d should use chain Cj j l , which contradicts egating electorate, where l , i and k , j. This means that, apart the assumption that Cj is used, (b) there exists a shared intermediate voter e such that 5Observation 1 states how guru participation can be satisfied when a voting rule satisfying cast participation is used: when a voter joins the delegating electorate, if Cj = ⟨j,..., e,д,...,i⟩ and only one voter increase the number of times assigned as a guru, then this voter is = ⟨ ,..., , ,..., ⟩, weakly better off. Ck k f j l

643 Research Paper AAMAS 2020, May 9–13, Auckland, New Zealand

where e ∈ ⟨k,..., f ⟩. Recall that dB prioritises shorter favours keeping delegated voting rights close to their origin, could length delegation chains (see definition 8). We show that this issue be resolved by using this rule? Other interesting future voter k has a shorter delegation chain available that does directions are investigating guru participation with voting rules ′ | ′ | < | | not include j, i.e. there exists a Ck such that Ck Ck and that do not satisfy cast participation, relaxing the assumption of ′ ′ ⟨ ,..., , ,..., ⟩ B strict personal rankings over voters, and analysing other types of j < Ck . Let Ck = k e д i . According to d , the delegation chain used to assign j’s guru, ⟨j ..., e,д,...,i⟩, participation. is shorter or equal in length to any other alternative, thus |⟨j,..., e,д,...,i⟩| ≤ |⟨j,...,l⟩|. Observe that 7 ACKNOWLEDGEMENTS |⟨д,...,i⟩| < |⟨j,..., e,д,...,i⟩| ≤ |⟨j,...,l⟩| ⇒ We would like to thank the EPSRC grant EP/P031811/1 for the |⟨k,..., e⟩| + |⟨д,...,i⟩| < |⟨k,..., e⟩| + |⟨j,...,l⟩|. research (and travel) funds for this work as well as the anonymous reviewers for their valuable comments for improvements which Since e ∈ ⟨k,..., f ⟩, we can rewrite the previous as helped us better shape this paper’s idea, resulting in this publication. |⟨k,..., e⟩| + |⟨д,...,i⟩| < |⟨k,..., f ⟩| + |⟨j,...,l⟩|. B ′ REFERENCES Therefore, rule d should useCk to assign k’s guru. However, since j < C′ , the assumption is contradicted. [1] Jan Behrens, Axel Kistner, Andreas Nitsche, and Bjorn Swierczek. 2014. The k Principles of LiquidFeedback. Interaktive Demokratie. The contradictions of both (a) and (b) prove this lemma. □ [2] Guido Boella, Louise Francis, Elena Grassi, Axel Kistner, Andreas Nitsche, Alexey Noskov, Luigi Sanasi, Adriano Savoca, Claudio Schifanella, and Ioannis Tsam- Theorem 5.7. Given the majority rule f ∈ F¯, guru participation poulatidis. 2018. WeGovNow: A Map Based Platform to Engage the Local Civic B Society. In Companion Proceedings of the The Web Conference, WWW 2018, Lyon , is guaranteed to hold when using the breadth-first delegation rule d . France, April 23-27, 2018. 1215–1219. https://doi.org/10.1145/3184558.3191560 [3] Markus Brill. 2018. Interactive Democracy. In Proceedings of the 17th Interna- Proof. 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[8] Steve Hardt. 2014. Liquid Democracy with Google Votes. https://www.youtube. gation rule proving that only the latter has the desirable property com/watch?v=F4lkCECSBFw. (2014). Accessed: 2018-07-17. that every guru weakly prefers receiving delegated voting rights [9] Steve Hardt and Lia C. R. Lopes. 2015. Google Votes: A Liquid Democracy Ex- periment on a Corporate Social Network. Technical Report. Technical Disclosure under the majority rule with two alternatives. The immediate future Commons, Google. questions that arise are to investigate what holds for more than two [10] Anson Kahng, Simon Mackenzie, and Ariel D. Procaccia. 2018. Liquid Democracy: alternatives in this model or for other voting rules that satisfy cast An Algorithmic Perspective. In Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, New Orleans, Louisiana, USA, February 2-7,. 2018 https: participation such as (Felsenthal [2010]). 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