On Rational Delegations in Liquid Democracy

The Thirty-Third AAAI Conference on Artiﬁcial Intelligence (AAAI-19)

Daan Bloembergen Davide Grossi Martin Lackner Centrum Wiskunde & Informatica Bernoulli Institute TU Wien Amsterdam, The Netherlands University of Groningen Vienna, Austria Groningen, The Netherlands

Abstract accurate vote, face between voting directly, and thereby in- curring a cost in terms of effort invested to learn about the Liquid democracy is a proxy voting method where proxies are issue at hand, or delegating to another voter in their network, delegable. We propose and study a game-theoretic model of liquid democracy to address the following question: when is thereby avoiding costs. We define a game-theoretic model, it rational for a voter to delegate her vote? We study the exis- called delegation game, to represent this type of interaction. tence of pure-strategy Nash equilibria in this model, and how We establish pure strategy Nash equilibrium existence re- group accuracy is affected by them. We complement these sults for classes of delegation games, and study the quality theoretical results by means of agent-based simulations to of equilibria in terms of the average accuracy they enable for study the effects of delegations on group’s accuracy on vari- the population of voters, both analytically and through sim- ously structured social networks. ulations. Proofs of the two main results (Theorems 1 and 2) are presented in full, while we provide proofs of the simpler 4 Introduction secondary results only as supplementary material. By means of simulations we also study the effects of dif- Liquid democracy (Blum and Zuber 2016) is an influen- ferent network structures on delegation games in terms of: tial proposal in recent debates on democratic reforms in performance against direct voting, average accuracy and the both Europe and the US. Several grassroots campaigns, as probability of a correct majority vote, the number and qual- well as local parties, experimented with this novel type of ity of voters acting as ultimate proxies (so-called gurus) and, decision making procedure. Examples include the German 1 finally, the presence of delegation cycles. To the best of our Piratenpartei and the EU Horizon 2020 project WeGov- knowledge, this is the first paper providing a comprehensive Now (Boella et al. 2018), which have incorporated the Liq- study of liquid democracy from a game-theoretic angle. uidFeedback2 platform in their decision making, as well as grass-roots organizations such as the Democracy Earth Related Work Although the idea of delegable proxy was Foundation3. Liquid democracy is a form of proxy voting already sketched by Dodgson (1884), only a few very re- (Miller 1969; Tullock 1992; Alger 2006; Green-Armytage cent papers have studied aspects of liquid democracy in the 2015; Cohensius et al. 2017) where, in contrast to classical (computational) social choice theory (Brandt et al. 2016) proxy voting, proxies are delegable (or transitive, or trans- literature. Kling et al. (2015) provide an analysis of elec- ferable). Suppose we are voting on a binary issue, then each tion data from the main platform implementing a liquid voter can either cast her vote directly, or she can delegate her democracy voting system (Liquid Feedback) for the Ger- vote to a proxy, who can again either vote directly or, in turn, man Piratenpartei. They focus on network theoretic prop- delegate to yet another proxy, and so forth. Ultimately, the erties emerging from the structure of delegations—with par- voters that decided not to delegate cast their ballots, which ticular attention to the number of highly influential gurus now carry the weight given by the number of voters who or ‘super-voters’. Inspired by their experimental analysis, entrusted them as proxy, directly or indirectly. Golz¨ et al. (2018) propose and analyze a variant of the liquid democracy scheme able to restrict reliance on super-voters. Contribution The starting point of our analysis is an of- Skowron et al. (2017) study an aspect of the Liquid Feed- ten cited feature of liquid democracy: transitive delegations back platform concerning the order in which proposals are reduce the level of duplicated effort required by direct vot- ranked and by which they are brought to the attention of ing, by freeing voters from the need to invest effort in order the community. Boldi et al. (2011) investigate applications to vote accurately. The focus of the paper is the decision- of variants of the liquid democracy voting method (called making problem that voters, who are interested in casting an viscous democracy) to recommender systems. Brill (2018) Copyright c 2019, Association for the Advancement of Artificial presents some research directions in the context of liquid Intelligence (www.aaai.org). All rights reserved. democracy. A general, more philosophical discussion of liq- 1https://www.piratenpartei.de/ uid democracy is provided by Blum and Zuber (2016). 2https://liquidfeedback.org/ 3https://www.democracy.earth/ 4https://arxiv.org/abs/1802.08020

1796 More directly related to our investigations is the work neighbors on an underlying interaction structure. by Christoff and Grossi (2017) and, especially, by Kahng, Mackenzie, and Procaccia (2018). The first paper studies liq- Interaction Structure and Delegations uid democracy as an aggregator—a function mapping pro- Agents are nodes in a network (directed graph) represented files of binary opinions to a collective opinion—in the judg- by a relation R N 2. For i N the neighborhood of i in ment aggregation and binary voting tradition (Grossi and N,R is denoted⊆R(i), i.e., the∈ agents that are directly con- Pigozzi 2014; Endriss 2016). The focus of that paper are the hnectedi to i. Agents have the choice of either voting them- unintended effects that transferable proxies may have due to selves, thereby relying solely on their own accuracy, or del- delegation cycles, and due to the failure of rationality con- egating to an agent in their neighborhood. A delegation pro- n straints normally satisfied by direct voting. file is a vector d = d1, . . . , d N . Given a delega- h ni ∈ The second paper addresses one of the most cited sell- tion profile d we denote by di the proxy selected by i in d. ing arguments for liquid democracy: delegable proxies guar- Clearly a delegation profile can be viewed as a functional antee that better informed agents can exercise more weight graph on N or, equivalently, as a map in d : N N → on group decisions, thereby increasing their quality. Specif- where d(i) = di. When the iterated application of d from ically, Kahng, Mackenzie, and Procaccia (2018) study the i reaches a fixed point we denote such fixed point as di∗ level of accuracy that can be guaranteed by liquid democ- and call it i’s guru (in d). In the following, we write N ∗ racy (based on vote delegation with weighted majority) vs. to denote the set of voters whose delegation does not lay direct voting by majority. Their key result consists in show- on a path ending on a cycle, i.e., the set of voters i for ing that no ‘local’ procedure to select proxies can guarantee which di∗ exists. We write d0 = (d i, j) as a short form − that liquid democracy is, at the same time, never less accu- for d0 = d1, . . . , di 1, j, di+1, . . . , dn . rate (in large enough graphs) and sometimes strictly more As agentsh may only− be able to observei and interact with accurate than direct voting. In contrast to their work, we as- their direct network neighbors, structural properties of the sume that agents incur costs (effort) when voting directly, interaction network may play a role in the model dynamics. and on that basis we develop a game-theoretic model. Also, In our simulations we will focus on undirected graphs (that we assume agents aim at tracking their own type rather than is, R will be assumed to be symmetric, as social ties are nor- an external ground truth, although we do assume such a re- mally mutual) consisting of one single connected component striction in our simulations to better highlight how the two (that is, N 2 is included in the reflexive transitive closure of models are related and to obtain insights applicable to both. R). Under these assumptions, we consider four typical network structures that are well represented in the literature on Preliminaries social network analysis (cf. Jackson 2008): 1) the random network, in which each pair of nodes has a given probability Types, Individual Accuracy and Proximity of being connected (Erdos¨ and Renyi´ 1959); 2) the regular We are concerned with a finite set of agents (or voters, or network, in which all nodes have the same degree; 3) the players) N = 1, . . . , n having to decide whether x = 1 or small world network, which features a small average path x = 0. For each{ agent one} of these two outcomes is better, length and high clustering (Watts and Strogatz 1998); and 4) but the agent is not necessarily aware which one. We refer the scale free network, which exhibits a power law degree to this hidden optimal outcome as the type of agent i and distribution (Barabasi´ and Albert 1999).5 denote it by τi 0, 1 . Agents want to communicate their type truthfully∈ to { the voting} mechanism, but they know it A Model of Rational Delegations accuracy q only imperfectly. This is captured by the i of an Individual Accuracy under Delegable Proxy agent i: qi determines the likelihood that, if i votes directly, Each agent i has to choose between two options: either to she votes according to her type τi. We assume that an agent’s accuracy is always 0.5, i.e., at least as good as a coin toss. vote herself with accuracy qi or to delegate, thereby inher- We distinguish two≥ settings depending on whether the iting the accuracy of another voter (unless i is involved in agents’ types are deterministic or probabilistic. A determin- a delegation cycle). These choices are recorded in the delegation profile d and can be used to compute the individual istic type profile T = τ1, . . . , τn simply collects each h i accuracy for each agent i N ∗ as follows: agent’s type. In probabilistic type profiles types are indepen- ∈ dent random variables drawn according to a distribution P. ∗ ∗ ∗ ∗ qdi pi,di + (1 qdi ) (1 pi,di ) if i N ∗ Given a probabilistic type profile, the likelihood that any two qi∗(d) = · − · − ∈ agents i, j N are of the same type is called the proximity 0.5 if i / N ∗ ∈ ∈ pi,j where pi,j = P(τ(i) = τ(j)) = P(τ(i) = 1) P(τ(j) = (1) · 1)+(1 P(τ(i) = 1)) (1 P(τ(j) = 1)). In the probabilistic − · − In (1) i’s accuracy equals the likelihood that i’s guru has setting we assume agents know such value although, impor- the same type and votes accurately plus the likelihood that tantly, they do not know P. In a deterministic type profile, i’s guru has the opposite type and fails to vote accurately. we have pi,j = 1 if τi = τj and pi,j = 0 otherwise. Fol- Note that if i votes directly, i.e., d = i, then q (d) = q . lowing standard equilibrium theory, our theoretical results i i∗ i assume agents act as if they have access to the accuracy of 5Although random and regular graphs are not generally appli- each agent. More realistically, in our simulations we assume cable to real-world settings, they serve as a useful baseline to illus- agents have access to such information only with respect to trate the effects of network structure on delegations.

1797 Observe that if i’s delegation leads to a cycle (i / N ∗), i’s vote delegate (to 1) ∈ accuracy is set to 0.5. The rationale for this assumption is vote q1 e1, q2 e2 q1 e1, q1 − − − the following. If an agent delegates into a cycle, even though delegate (to 2) q2, q2 e2 0.5, 0.5 she knows her own accuracy and she actively engages with − the voting mechanism by expressing a delegation, she fails Table 1: A two-player delegation game. The row player is to pass information about her type to the mechanism. No agent 1 and the column player is agent 2. information is therefore available to decide about her type. It may be worth observing that, by (1), in a deterministic type profile we have that p 0, 1 and therefore i’s ac- A few comments about the setup of (2) are in order. First i,j ∈ { } curacy reduces to: qd∗ if i N ∗ and τ(i) = τ(d∗); 1 q ∗( ) of all, as stated earlier, we assume agents to be truthful. They i ∈ i − d i if i N ∗ and τ(i) = τ(di∗); and 0.5 if i / N ∗. do not aim at maximizing the chance their type wins the Before∈ introducing6 our game theoretic∈ analysis, we make vote, but rather to relay their type to the mechanism as accu- the following observation. Agents have at their disposal an rately as possible.7 Secondly, notice that the utility an agent intuitive strategy to improve their accuracy: simply delegate extracts from a delegation profile may equal the accuracy of to a more accurate neighbor. We say that a delegation profile a random coin toss when the agent’s delegation ends up into d is positive if for all j N either dj = j or qj∗(d) > qj. a delegation cycle (cf. (1)). If this happens the agent fails Furthermore, we say that∈ a delegation from i to a neighbor j to relay information about her type, even though she acted is locally positive if qj pi,j + (1 qj) (1 pi,j) > qi. in order to do so. This justifies the fact that 0.5 is also the · − · − lowest payoff attainable in a delegation game. Let d be a positive delegation profile. Fur- Proposition 1. The following classes of delegation games will be used ther, let s, t N, d = s, and d = (d , t), i.e., agent s s 0 s in the paper: games with deterministic profiles, i.e., where votes directly∈ in d and delegates to t in d−. If the delegation 0 is degenerate and all players are assigned a crisp type from s to t is locally positive, then d is positive. P 0 from 0, 1 ; homogeneous games, where all players have However, locally positive delegations do not necessarily the same{ (deterministic)} type;8 and effortless voting games, correspond to optimal delegations. This can be easily seen in where for each i N we have ei = 0. an example where agent i is not a neighbor of a very compe- As an example,∈ a homogeneous game in matrix form is tent agent j, but would have to delegate via an intermediate given in Table 1, where N = 1, 2 , R = N 2 and the distri- agent k (who delegates to j). If this intermediate agent k bution yields the deterministic{ type} profile T = 1, 1 . Inter- h i has a lower accuracy than i, then the delegation from i to estingly, if we assume that qi ei > 0.5 with i 1, 2 , and 9 − ∈ { } k would not be locally positive, even though it is an opti- that q i > qi ei (i.e., the opponent’s accuracy is higher mal choice. So utility-maximization may require informa- than the− player’s− individual accuracy minus her effort), then tion which is inherently non-local (accuracy of ‘far’ agents). the game shares the ordinal preference structure of the class of anti-coordination games: players need to avoid coordina- Delegation Games tion on the same strategy (one should vote and the other delegate), with two coordination outcomes (both players voting We assume that each agent i has to invest an effort ei to man- or both delegating) of which the second (the delegation cy- ifest her accuracy qi. If she delegates, she does not have to spend effort. Agents aim therefore at maximizing the trade- cle) is worst for both players. Notice that, were the underlying network not complete (i.e., R N 2), the matrix would off between the accuracy they can achieve (either by voting ⊂ directly or through proxy) and the effort they spend. Un- be shrunk by removing the rows and columns corresponding der this assumption, the binary decision set-up with dele- to the delegation options no longer available. gable proxy we outlined above can be used to define a natu- The introduction of effort has significant consequences on the delegation behavior of voters, and we will study it in ral game—called delegation game—G = N, P,R, Σi, ui , h i with i N, where N is the set of agents, P is the (possibly depth in the coming sections. It is worth noting immediately degenerate)∈ distribution from which the types of the agents that the assumptions of Proposition 1 no longer apply, since in N are drawn, R the underlying network as defined above, agents may prefer to make delegations that are not locally positive due to the decreased utility of voting directly. Σi N is the set of strategies of agent i (voting, or choosing a specific∈ proxy), and Existence of Equilibria in Delegation Games q∗(d) if d = i u (d) = i i 6 (2) In this section we study the existence of pure strategy Nash i q e if d = i i − i i Equilibria (NE) in two classes of delegation games. NE is agent i’s utility function. The utility i extracts from a dele- 7Notice however that our modeling of agents’ utility remains gation profile equals the accuracy she inherits through proxy applicable in this form even if agents are not truthful but the under- 6 or, if she votes, her accuracy minus the effort spent. In del- lying voting rule makes truthfulness a dominant strategy—such as egation games we assume that qi ei 0.5 for all i N. majority in the binary voting setting used here. − ≥ ∈ 8 This is because if qi ei < 0.5, then i would prefer a random This is the type of interaction studied, albeit not game- effortless choice over− taking a decision with effort. theoretically, by Kahng, Mackenzie, and Procaccia (2018) and normally assumed by jury theorems (Grofman, Owen, and Feld 1983). 6No utility is accrued for gaining voting power in our model. 9We use here the usual notation −i to denote i’s opponent.

1798 describe how ideally rational voters would resolve the ef- Hence, if j S, then dj∗ = di∗, a contradiction. If j / S, fort/accuracy trade-off. Of course, such voters have common the same reasoning∈ as before applies and hence also here∈ we knowledge of the delegation game structure—including, obtain a contradiction. In case (3), by construction, di∗ / S therefore, common knowledge of the accuracies of ‘distant’ had been chosen to maximise accuracy, hence j S. Since∈ ∈ agents in the underlying network. Our simulations will later for all k S, d∗ = d∗, only a deviation to i itself can be ∈ k i lift some of such epistemic assumptions built into NE. beneficial, i.e., j = i. However, since di∗ was chosen because q∗∗ > qi ei, no beneficial deviation is possible. Deterministic Types In the following we provide a NE d (i) − existence result for games with deterministic type profiles. It follows that also homogeneous games always have NE. Theorem 1. Delegation games with deterministic type profiles always have a (pure strategy) NE. Effortless Voting Effortless voting (ei = 0 for all i N) is applicable whenever effort is spent in advance of the∈ de- Proof. First of all, observe that since the profile is determin- cision and further accuracy improvements are not possible. istic, for each pair of agents i and j, p 0, 1 . The proof i,j ∈ { } Theorem 2. Delegation games with effortless voting always is by construction. First, we partition the set of agents N into have a (pure strategy) NE. N1 = i N τ(i) = 1 and N0 = i N τ(i) = 0 . { ∈ | } { ∈ | } We consider these two sets separately; without loss of gen- Proof. We prove this statement by showing that the follow- N erality let us consider 1. Further we consider the network ing procedure obtains a NE: We start with a strategy profile R1 = (i, j) N1 N1 :(i, j) R . Since (N1,R1) can { ∈ × ∈ } in which all players vote directly, i.e., player i’s strategy is be seen as a directed graph, we can partition it into Strongly i. Then, we iteratively allow players to choose their indi- Connected Components (SCCs). If we shrink each SCC into vidual best response strategy to the current strategy profile. a single vertex, we obtain the condensation of this graph; Players act sequentially in arbitrary order. If there are no note that such a graph is a directed acyclic graph (DAG). more players that can improve their utility by changing their d We construct a delegation profile by traversing this DAG strategy, we have found a NE. We prove convergence of this bottom up, i.e., starting with leaf SCCs. procedure by showing that a best response that increases the Let S N1 be a set of agents corresponding to a leaf SCC player’s utility never decreases the utility of other players. ⊆ i S in the condensation DAG. We choose an agent in that has We proceed by induction. Assume that all previous best q e S i maximum i i. Everyone in (including ) delegates to responses have not reduced any players’ utility (IH). Assume i S − N . Now let 1 be a set of agents corresponding to an player i now chooses a best response that increases her util- inner node SCC⊆ in the condensation DAG and assume that ity. Let d be the delegation profile; further, let di∗ = s. By we have already defined the delegation for all SCCs that can assumption, i’s utility started with q e = q and has not S i S i i i be reached from . As before, we identify an agent decreased since, i.e., u (d) q . Since− i’s best response q e T N S ∈ i i with maximum i i. Further, let 1 be the set strictly increases i’s utility, it≥ cannot be a delegation to her- j − ⊆S (\N ,R ) of all voters that can be reached from in 1 1 , and self. So let a delegation to j = i be i’s best response and for which qj∗ > qi ei. We distinguish two cases. (i) If T = 6 − 6 consider profile d0 = (d i, j). Further, let dj∗ = t, i.e., i k T q = max q − , then we choose an agent with k∗ j T j∗ now delegates to j and by transitivity to t, i.e., d = d = t. ∅ S ∈ k∈ i0∗ j0∗ and all agents in directly or indirectly delegate to . (ii) Let k be some player other than i. We define the delegation T = S i If , all agents in delegate to . This concludes our path of k as the sequence (d(k), d(d(k)), d(d(d(k))),... ). ∅ N construction (as for 0 the analogous construction applies); If k’s delegation path does not contain i, then k’s utility re- let d be the corresponding delegation profile. mains unchanged, i.e., uk(d0) uk(d). If k’s delegation It remains to verify that this is indeed a NE: Let i be some path contains i, then k now delegates≥ by transitivity to t, i.e., agent in an SCC S, and, without loss of generality, let i ∈ we have dk∗ = s and dk0∗ = t. By Equation (2), we have N1. Observe that since we have a deterministic profile, if agent i changes her delegation to j, then i’s utility changes u (d) = q p + (1 q ) (1 p ) and (3) k s · k,s − s · − k,s to qj∗(d) if i N1 and 1 qj∗(d) if i N0. First, note ∈ − ∈ uk(d0) = qt pk,t + (1 qt) (1 pk,t). (4) that for all agents k N, q∗(d) q e 0.5. Hence, · − · − ∈ k ≥ k − k ≥ we can immediately exclude that for agent i delegating to an We have to show that k’s utility does not decrease, i.e., agent in j N0 is (strictly) beneficial, as it would yield an u (d0) u (d), under the assumption that i chooses a best ∈ k ≥ k accuracy of at most 1 qj∗ 0.5. Towards a contradiction response, i.e., u (d ) > u (d), with: − ≤ i 0 i assume there is a beneficial deviation to an agent j N1, ∈ i.e., there is an agent j R(i) N1 with q∗(d) > q∗(d). Let ui(d) = qs pi,s + (1 qs) (1 pi,s) and (5) ∈ ∩ j i · − · − us now consider the three cases: (1) d = i, (2) d∗ S but i i ∈ ui(d0) = qt pi,t + (1 qt) (1 pi,t). (6) di = i, and (3) di∗ / S. In case (1), everyone in S delegates · − · − to i6 . Hence, if j ∈S, a cycle would occur yielding a utility In the following we will often use the fact that, for a, b of 0.5, which is∈ not sufficient for a beneficial deviation. If [0, 1], if ab + (1 a)(1 b) 0.5, then either a, b [0, 0.5]∈ a delegation to j / S is possible but was not chosen, then or a, b [0.5, 1]−. By IH,− since≥ accuracies are always∈ at least ∈ ∈ by construction qj∗ qi ei and hence this deviation is 0.5, it holds that ui(d) qi 0.5 and by Equation (5) we not beneficial. We conclude≤ − that in case (1) such an agent have q p +(1 q ) (1≥ p ≥) 0.5 and hence p 0.5. s · i,s − s · − i,s ≥ i,s ≥ j cannot exist. In case (2), everyone in S delegates to d∗. Analogously, Equation (3) implies that p 0.5. i k,s ≥

1799 Furthermore, we use the fact that type it is not possible to assign delegations for all agents in an SCC (as we do in the proof of Theorem 1). And the key pk,i = pk,sps,i + (1 pk,s)(1 ps,i) − − (7) property upon which the proof of Theorem 2 hinges (that + ( 2(2x 1) (2x 1) (x 1)x ) − k − · i − · s − s a best response of an agent does not decrease the utility of other agents) fails in the general case due to the presence where x = (τ(j) = 1) for j k, i, s . Observe that, j P of non-zero effort. Finally, it should also be observed that by the definition of utility in (2), the∈ assumptions { } made on d Theorem 2 (as well as Proposition 1) essentially depend on and d , and the fact that for a, b [0, 1], if ab + (1 a)(1 0 the assumption that types are independent random variables. b) 0.5, then either a, b [0∈, 0.5] or a, b [0−.5, 1]. So− If this is not the case (e.g., because voters’ preferences are we≥ have that either x 0.∈5 for j k, i, s ∈, or x 0.5 j j correlated), delegation chains can become undesirable. for j k, i, s . We work≥ on the first∈ { case. The} other≤ case is symmetric.∈ { Let} also γ = 2(2x 1) (2x 1) Example 1. Consider the following example with agents k,s,i − k − · i − · (xs 1)xs. From the above it follows that 0.5 γk,i,s 1, 2 and 3. The probability distribution P is defined as 0. Furthermore,− given that p = p 0.5, we≥ can also≥ P(τ(1) = 1 τ(2) = 1 τ(3) = 0) = 0.45, P(τ(1) = i,s s,i ∧ ∧ conclude that p 0.5. Now by substituting≥ 0 τ(2) = 1 τ(3) = 1) = 0.45, and P(τ(1) = k,i ≥ ∧ ∧ 1 τ(2) = 1 τ(3) = 1) = 0.1. Consequently, p1,2 = p = p p + (1 p )(1 p ) ∧ ∧ k,s k,i i,s − k,i − i,s 0.55, p2,3 = 0.55, and p1,3 = 0.1. Let us assume that + ( 2(2x 1) (2x 1) (x 1)x ) the agents’ accuracies are q1 = 0.5001, q2 = 0.51, and − k − · s − · i − i | {z } q3 = 0.61. A delegation from agent 1 to 2 is locally pos- γk,i,s itive as q2 p1 2 + (1 q2) (1 p1 2) = 0.501 > q1. · , − · − , in Equation (3), we obtain Furthermore, a delegation from 2 to 3 is locally positive as q3 p2,3 + (1 q3) (1 p2,3) = 0.511 > q2. How- ui(d) ever, the· resulting− delegation· − from 1 to 3 is not positive since z }| { u (d) = (2p 1)(2q p q p + 1) (8) q3 p1,3 + (1 q3) (1 p1,3) = 0.412. k k,i − s i,s − s − i,s · − · − + 1 pk,i + γk,i,s(2qs 1). − − Quality of Equilibria in Delegation Games Similarly, using the appropriate instantiation of (7) for xj In delegation games players are motivated to maximize with j k, i, t , by substituting p p + (1 p )(1 ∈ { } k,i · i,t − k,i − the tradeoff between the accuracy they acquire and the ef- pi,t) + γk,i,t for pk,t in Equation (4) we obtain fort they spend for it. A natural measure for the quality 0 ui(d ) of a delegation profile is, therefore, how accurate or in- z }| { formed a random voter becomes as a result of the dele- uk(d0) = (2pk,i 1) (2qtpi,t qt pi,t + 1) (9) gations in the profile, that is, the average accuracy (i.e., − · − − 1 P + 1 pk,i + γk,i,t(2qt 1). q¯∗(d) = n i N qi∗(d)) players enjoy in that profile. One − − ∈ can also consider the utilitarian social welfare SW(d) = Now observe that, since pk,i 0.5 we have that (2pk,i P ≥ − i N ui(d) of a delegation profile d. This relates to av- 1) 0. It remains to compare γk,i,s(2qs 1) with ∈ SW(d)+P ≥ − i|d(i)=i ei γk,i,t(2qt 1), showing the latter is greater than the former. erage accuracy as follows: q¯∗(d) = n . Observe that− both expressions have a positive sign. We use It can immediately be noticed that equilibria do not nec- the fact that ab+(1 a)(1 b) < cd+(1 c)(1 d) implies essarily maximize average accuracy. On the contrary, in the cd > ab under the− assumption− that a, b,− c, d −[0.5, 1]. On following example NE yields an average accuracy of close to ∈ the basis of this, and given that ui(d0) > ui(d), we obtain 0.5, whereas an average accuracy of almost 1 is achievable. that q p < q p and therefore that q x < q x , s · i,s t · i,t s · s t · t Example 2. Consider an n-player delegation game where from which we can conclude that all players have the same type and (i, j) R for all j and ∈ γk,i,s all i > 1, i.e., player 1 is a sink in R and cannot delegate z }| { to anyone, but all others can delegate to everyone. Further, ( 2(2xk 1) (2xs 1) (xi 1)xi) (2qs 1) − − · − · − · − we have e1 = 0 and ei = 0.5 for i 2. The respective < ( 2(2x 1) (2x 1) (x 1)x ) (2q 1). − ≥ k t i i t accuracies are q1 = 0.5+2 and q = 1. If player i 2 does |− − · {z − · − } · − i ≥ γk,i,t not delegate, her utility is 0.5 + . Hence, it is always more desirable to delegate to player 1 (which yields a utility of It follows that the assumption ui(d0) > ui(d) (player i 0.5+2 for i). Consider now the profiles in which all players chose a best response that increased her utility) together with delegate to player 1 (either directly or transitively). Player 1 Equations (8) and (9) implies that uk(d0) > uk(d) (and a can only vote directly (with utility 0.5+2). All such profiles fortiori that u (d0) u (d)). We have therefore shown that k ≥ k are NE with average accuracy 0.5 + 2. If, however, some if some player chooses a best response, the utility of other player j 2 chose to vote herself, all players (except 1) players does not decrease. This completes the proof. would delegate≥ to j thereby obtaining an average accuracy 0.5 2 of 1 − , which converges to 1 for n . This is not Discussion The existence of NE in general delegation − n → ∞ games remains an interesting open problem. It should be a NE, as j could increase her utility by delegating to 1. noted that the proof strategies of both Theorems 1 and 2 do The findings of the example can be made more explicit by not work in the general case. Without a clear dichotomy of considering a variant of the price of anarchy for delegation

1800 games, based on the above notion of average accuracy. That Degree 4 8 12 16 20 24 is, for a given delegation game G, the price of anarchy (PoA) BR updates 298.1 261.7 254.2 251.6 250.6 250.0 ∗ maxd∈Nn q¯ (d) (effortless) (18.2) (11.1) (6.9) (4.5) (3.3) (2.6) of G is given by PoA(G) = min ¯∗(d) , where NE(G) d∈NE(G) q Full passes 3.6 3.0 2.9 2.8 2.7 2.5 denotes the set of pure-strategy NE of G. (effortless) (0.5) (0.1) (0.2) (0.4) (0.5) (0.5) Fact 1. PoA is bounded below by 1 and above by 2. BR updates 294.7 259.4 252.9 250.9 250.2 249.9 An informative performance metrics for liquid democracy (with effort) (18.4) (10.6) (6.6) (4.8) (4.9) (4.3) is the difference between the group accuracy after delega- Full passes 3.6 3.0 2.8 2.6 2.4 2.4 tions versus the group accuracy achieved by direct voting. (with effort) (0.5) (0.3) (0.6) (0.8) (0.9) (1.0) This measure, called gain, was introduced and studied by Kahng, Mackenzie, and Procaccia (2018). Here we adapt it Table 2: Total number of best response updates by individ- to our setting as follows: G(G) = mind NE(G) q¯∗(d) q¯ ∈ − ual agents and corresponding full passes over the network where q¯ =q ¯∗( 1, . . . , n ). That is, the gain in the delegation h i required for convergence. Reported are the mean (std.dev.) game G is the average accuracy of the worst NE minus the over all network types. Note: not all agents update their dele- average accuracy of the profile in which no voter delegates. gation at each full pass, but any single update requires an ad- It turns out that the full performance range is possible: ditional pass to check whether the best response still holds. Fact 2. G is bounded below by 0.5 and above by 0.5. − The above bounds for PoA and gain provide only a very Degree 4 8 12 16 20 24 partial picture of the performance of liquid democracy when maxj qj 0.8908 0.8908 0.8904 0.8909 0.8904 0.8910 modeled as a delegation game. The next section provides a q¯∗(d) 0.8906 0.8903 0.8897 0.8899 0.8890 0.8893 more fine-grained perspective on the effects of delegations. Table 3: Comparing the maximum accuracy across all agents Simulations and the mean accuracy under delegation d for different net- We simulate the delegation game described above in a vari- work degrees, averaged across network types. The differ- ety of settings. We restrict ourselves to homogeneous games. ences are statistically significant (paired t-test, p = 0.05). This allows us to relate our results to those of Kahng, Mackenzie, and Procaccia (2018). Our experiments serve to voter’s accuracy in equilibrium, and on the effects of differ- both verify and extend the theoretical results of the previous ent network structures on how such equilibria are achieved. section. In particular we simulate the best response dynam- We initialize q (0.75, 0.05) and first investigate the ics employed in the proof of Theorem 2 and show that these i case in which e =∼ 0 Nfor all i (effortless voting). Across all dynamics converge even in the setting with effort, which i combinations of network types and average degrees ranging we could not establish analytically. In addition, we inves- from 4 to 24, we find that the best response dynamics con- tigate the dynamics of a one-shot game scenario, in which verges, as predicted by Theorem 2, and does so optimally all agents need to select their proxy simultaneously at once. with di∗ = arg maxj qj for all i. We observe minimal dif- Setup We generate graphs of size N = 250 of each of the ferences between network types, but see a clear inverse re- four topologies random, regular, small world, and scale free, lation between average degree and the number of iterations for different average degrees, while ensuring that the graph required to converge (Table 2, top). Intuitively, more densely is connected. Agents’ accuracy and effort are initialized ran- connected networks facilitate agents in identifying their op- domly with qi [0.5, 1] and qi ei 0.5. We average timal proxies. results over 2500∈ simulations for each− setting≥ (25 randomly We accumulate results across all network types and com- generated graphs 100 initializations). Agents correctly ob- pare the effortless setting to the case in which effort is taken × serve their own accuracy and effort, and the accuracy of their into account. When we include effort ei (0.025, 0.01), neighbors. The game is homogeneous, so proximities are 1. we still observe convergence in all cases∼ and, N interestingly, Each agent i selects from her neighborhood R(i) (which the number of iterations required does not change signifi- includes i herself) the agent j that maximizes her expected cantly (Table 2, bottom). Although the process no longer re- utility following Equation 2. We compare two scenarios. The sults in an optimal equilibrium, each case still yields a sin- iterated best response scenario follows the procedure of the gle guru j with qj maxk qk (less than 1% error) for all proof of Theorem 2, in which agents sequentially update k N. In this scenario,≈ the inclusion of effort means that their proxy to best-respond to the current profile d using a best∈ response update of agent i no longer guarantees non- knowledge of their neighbors’ accuracy qi∗(d). In the one- decreasing accuracy and utility for all other agents, which shot game scenario all agents choose their proxy only once, was a key property in the proof of Theorem 2. This effect do so simultaneously, and based only on their neighbors’ ac- becomes stronger as the average network degree increases, curacy. The latter setup captures more closely the epistemic and as a result higher degree networks allow a greater dis- limitations that agents face in liquid democracy. crepancy between the maximal average accuracy achievable and the average accuracy obtained at stabilization (Table 3). Iterated Best Response Dynamics In lower degree graphs (e.g. degree 4) we further observe These experiments complement our existence theorems. differences in convergence speed between the four differ- They offer insights into the effects of delegations on average ent network types which coincide with differences between

1801 0.95 1.05 thetance average between path nodes lengths yields in those a lower graphs: number a ofshorter best response average distanceupdates.9 betweenThis is intuitive, nodes yields as larger a lower distances number between of best nodes re- 0.9 1 0.85 sponsemean longerupdates. delegation This is intuitive, chains, as but larger we distanceshave not between yet con- 0.95 nodes mean longer delegation chains, but we have not yet 0.8 ducted statistical tests to verify this hypothesis. 0.9 conducted statistical tests to verify this hypothesis. 0.75 0.85 One-Shot Delegation Games 0.7 Prob. correct majority One-ShotHere we study Delegation one-shot Games interaction in a delegation game: Mean network accuracy 0.65 0.8 Hereall agents we study select one-shot their proxy interactions (possibly in themselves) a delegation simulta- game: 0.6 0.75 4 8 12 16 20 24 4 8 12 16 20 24 all agents select their proxy (possibly themselves) simulta- Degree Degree neously among their neighbors; no further response is pos- 0.95 1.05 neouslysible. This among contrasts their neighbors; the previous no scenario further response in which is agents pos- 0.95 1.05 0.9 Random Regular Small1 World Scale Free sible.could This iteratively contrasts improve the previous their choice scenario based in on which the choices agents 0.9 1 0.85 0.85 0.95 couldof others. iteratively While improve Kahng, theirMackenzie, choice andbased Procaccia on the choices (2018) 0.95 0.8 0.8 ofstudy others. a probabilistic While Kahng, model, Mackenzie, we instead and assume Procaccia that (2018) agents 0.9 0.9 0.75 studydeterministically a probabilistic select model, as proxy we instead the agent assumej thatR( agentsi) that 0.75 0.85 2 0.7 0.85 deterministicallymaximizes their utility, select as above. proxy Wethe compareagent j q¯ andR(iq¯)⇤ that(the 0.7

∈ Prob. correct majority

Mean network accuracy 0.8 maximizes their utility, as above. We compare q¯ and q¯∗ (the 0.65 Prob. correct majority average network accuracy without and with delegation, re- Mean network accuracy 0.65 0.8 averagespectively), network as well accuracy as the without probability and of with a correct delegation, majority re- 0.6 0.75 0.6 4 8 12 16 20 24 0.75 4 8 12 16 20 24 4 8 12 16 20 24 4 8 12 16 20 24 spectively),vote under both as well direct as thedemocracy probabilityPD and of a liquid correct democracy majority Degree Degree Degree Degree votePL where under both gurus direct carry democracy as weightP theD and number liquid of democracy agents for PwhomL where they gurus act as carry proxy. as The weight difference the numberP ofP agentsis the forno- Random Regular Small World Scale Free L D Random Regular Small World Scale Free whomtion of theygain act(Kahng, as proxy. Mackenzie, The difference and ProcacciaPL PD 2018).is again In − Figure 1: Top: Without effort. Bottom: With effort ei similarline with tothe Condorcet’s notion of jurygain theorem(Kahng, (see Mackenzie, for instance and Grof- Pro- Figure 1: Top: Without effort. Bottom: With effort ei (0.025, 0.01). Left: average network accuracy under liq-⇠∼ cacciaman, Owen, 2018). and In lineFeld with 1983) Condorcet’sPD 1 as juryN theorem, and (see in- (0.025, 0.01). Left: mean accuracy under liquid democ- ! !1 NuidN democracy. The solid (dashed) line shows the mean (std. fordeed instance for N = Grofman, 250 we obtain Owen,P andD Feld1. 1983) PD 1 as racy, q¯∗(d). The solid (dashed) line shows the mean (std. ⇡ → dev.) of q; the dotted line shows maxi qi. Right: probability N First we, and again indeed look for atN the= effortless 250 we obtainsetting.P FigureD 1. 1 (top) dev.) of the initial accuracy q; the dotted line shows max q . → ∞ ≈ of a correct majority vote under liquid democracy. i i showsFirst weboth again metrics look for at the the foureffortless different setting. network Figure types 1 (top) and Right: probability of a correct majority vote under d. showsfor different both metrics average for degrees. the four We different observe network that while typesq¯⇤ and(d) 0.3 2 0.3 2 forincreases different as average the network degrees. degree We increases observe that(and while in factq¯∗ is(d al-) 0.25 1.8 0.25 increasesways higher as the than networkq¯ without degree delegation), increases the(and probability in fact is al- of 1.8 0.2 waysa correct higher majority than q¯ outcome,without delegation),PL, simultaneously the probability decreases. of 0.2 1.6 1.6 aThis correct conﬁrms majority the outcome, analysis ofPL Kahng,, simultaneously Mackenzie, decreases. and Pro- 0.15 0.15 1.4 Thiscaccia conﬁrms (2018). the We analysis also observe of Kahng, that Mackenzie, the number and of gurus Pro- 0.1 1.4 cacciadecreases (2018). exponentially We also observe as the degree that the increases number (Figure of gurus 2, 0.1

Percentage of gurus 1.2 0.05 Mean distance to guru

Percentage of gurus 1.2 0.05 decreasesleft). Simply exponentially put, giving as all the voting degree weight increases to a small (Figure group 2, Mean distance to guru 0 1 left).of gurus Simply increases put, giving the chance all voting of an weight incorrect to a majority small group vote, 0 4 8 12 16 20 24 1 4 8 12 16 20 24 4 8 12Degree 16 20 24 4 8 12Degree 16 20 24 ofassuming gurus increases that gurus the have chance a less of than an incorrect perfect accuracy. majority vote, Degree Degree Random Regular Small World Scale Free assumingWhen wethat include gurus have effort a (Figureless than 1, perfect bottom), accuracy. thereby mov- Random Regular Small World Scale Free ingWhen away we from include the model effort of (Figure Kahng, 1, Mackenzie, bottom), thereby and Procac- mov- Figure 2: Percentage of guru nodes under d (left) and mean ingcia away(2018), from we observe the model a drastic of Kahng, decrease Mackenzie, in average and network Pro- Figuredistance 2: between Percentage (non-guru) of guru nodes andunder theird (left) gurus and (right). mean cacciaaccuracy (2018), combined we observe with a a lower drastic probability decrease of in a average correct net- ma- distance between (non-guru) nodes and their gurus (right). workjority accuracy outcome combined under liquid with democracy, a lower probability with both of decreas- a cor- ing as network degree increases. The main reason is the ex- rect majority outcome under liquid democracy, with both de- longer delegation chains (Figure 2, right). Intuitively, this istence of delegation cycles in this case. This contrasts the creasing as network degree increases. The main reason is moreindicates likely that to one-shot end up in interaction a local optimum. in scale In free contrast, networks small is best response setting above where agents could iteratively the existence of delegation cycles in this case. This con- worldmore likelynetworks to end have up short in a averagelocal optimum. distances In and contrast, thus agents small reconsider their choice of proxy and thus avoid cycles. Now, trasts the best response setting above where agents could areworld more networks likely to have be close short toaverage their optimal distances guru. and Finally, thus agents our even with relatively low effort (mean 0.025), up to half of all iteratively reconsider their choice of proxy and thus avoid experimentsare more likely highlight to be close a key to feature their optimal of liquid guru. democracy: the agents are stuck in a cycle (and thereby fail to pass informa- cycles. Now, even with relatively low effort (mean 0.025), trade-off between a reduction in (total) effort against a loss tion about their type) when degree increases. This confirms up to half of all agents are stuck in a cycle (and thereby in averageConclusions voting accuracy. and Future Work failresults to pass on the information probability about of delegation their type) cycles when from the Christoff degree increases.and Grossi This (2017) confirms and stresses resultsthe on the importance probability of cycle of delega- reso- The paper introduced delegation games as a first game- tionlution cycles in concrete from Christoff implementations and Grossi of (2017) liquid democracyand stresses such the theoretic modelConclusions of liquid anddemocracy. Future Both Work our theoretical importanceas Liquid Feedback. of cycle resolution in concrete implementations Theand experimental paper introduced results delegation showed that games voting as effort a first is game- a key of liquidFinally, democracy. Figure 1 highlights differences between the four theoreticingredient model for understanding of liquid democracy. how delegations Both our form theoretical and what networkFigure types. 1 further Scale highlights free networks differences yield a between lower probability the four andtheir experimental effects are. Our results empirical showed findings that voting provided effort further is a key in- networkof a correct types. majority Scale free outcome networks across yield all a degrees, lower probability as well as ingredientsights into for the understanding influence of interaction how delegations networks form on andthe qual- what ofa largera correct number majority of gurus outcome with across a lower all average degrees, accuracy as well and as theirity of effects collective are. decisions Our empirical in liquid findings democracy. provided further in- The paper opens up several directions of research. A gen- a larger9 number of gurus with a lower average accuracy and sights into the influence of interaction networks on the qual- longerMore delegation detailed results chains supporting (Figure 2, this right). finding Intuitively, are presented this in ityeral of NE collective existence decisions theorem in is liquid the main democracy. open question. The indicatesthe supplementary that one-shot material, interactions Appendix in B. scale free networks are frameworkThe paper can opens then up be several generalized directions along of natural research. lines, A e.g.:gen-

1802 eral NE existence theorem is the main open question. Our of the 16th Conference on Autonomous Agents and Multi- model can then be generalized in many directions, e.g.: by Agent Systems, 858–866. International Foundation for Au- making agents’ utility dependent on voting outcomes; by tonomous Agents and Multiagent Systems. dropping the independence assumption on agents’ types; or Dodgson, C. L. 1884. The Principles of Parliamentary Rep- by assuming the voting mechanism has better than 0.5 accu- resentation. Harrison and Sons. racy in identifying the types of agents involved in cycles. Endriss, U. 2016. Judgment aggregation. In Brandt, F.; Conitzer, V.; Endriss, U.; Lang, J.; and Procaccia, A. D., Acknowledgements eds., Handbook of Computational Social Choice. Cam- We are indebted to the anonymous reviewers of IJ- bridge University Press. CAI/ECAI’18 and AAAI’19 for many helpful comments on Erdos,¨ P., and Renyi,´ A. 1959. On random graphs, I. Publi- earlier versions of this paper. We are also grateful to the cationes Mathematicae (Debrecen) 6:290–297. participants of the LAMSADE seminar at Paris Dauphine Golz,¨ P.; Kahng, A.; Mackenzie, S.; and Procaccia, A. D. University, and the THEMA seminar at University Cergy- 2018. The fluid mechanics of liquid democracy. In Pontoise where this work was presented, for many helpful WADE’18, arXiv:1808.01906. comments and suggestions. Daan Bloembergen has received Green-Armytage, J. 2015. Direct voting and proxy voting. funding in the framework of the joint programming initia- Constitutional Political Economy 26(2):190–220. tive ERA-Net Smart Energy Systems’ focus initiative Smart Grids Plus, with support from the European Union’s Horizon Grofman, B.; Owen, G.; and Feld, S. 1983. Thirteen theo- 2020 research and innovation programme under grant agree- rems in search of truth. Theory and Decision 15:261–278. ment No 646039. Davide Grossi was partially supported by Grossi, D., and Pigozzi, G. 2014. Judgment Aggregation: EPSRC under grant EP/M015815/1. Martin Lackner was A Primer. Synthesis Lectures on Artificial Intelligence and supported by the European Research Council (ERC) under Machine Learning. Morgan & Claypool Publishers. grant number 639945 (ACCORD) and by the Austrian Sci- Jackson, M. O. 2008. Social and Economic Networks. ence Foundation FWF, grant P25518 and Y698. Princeton University Press. Kahng, A.; Mackenzie, S.; and Procaccia, A. 2018. Liquid References democracy: An algorithmic perspective. In Proc. 32nd AAAI Alger, D. 2006. Voting by proxy. Public Choice 126(1- Conference on Artificial Intelligence (AAAI’18). 2):1–26. Kling, C. C.; Kunegis, J.; Hartmann, H.; Strohmaier, M.; Barabasi,´ A.-L., and Albert, R. 1999. Emergence of scaling and Staab, S. 2015. Voting behaviour and power in online in random networks. science 286(5439):509–512. democracy: A study of liquidfeedback in germany’s pirate party. In Ninth International AAAI Conference on Web and Blum, C., and Zuber, C. I. 2016. Liquid democracy: Poten- Social Media. tials, problems, and perspectives. Journal of Political Phi- Miller, J. C. 1969. A program for direct and proxy voting in losophy 24(2):162–182. the legislative process. Public choice 7(1):107–113. Boella, G.; Francis, L.; Grassi, E.; Kistner, A.; Nitsche, A.; Skowron, P.; Lackner, M.; Brill, M.; Peters, D.; and Elkind, Noskov, A.; Sanasi, L.; Savoca, A.; Schifanella, C.; and E. 2017. Proportional rankings. In Proceedings of the Tsampoulatidis, I. 2018. WeGovNow: A map based plat- Twenty-Sixth International Joint Conference on Artificial In- WWW ’18 Com- form to engage the local civic society. In telligence, IJCAI 2017, Melbourne, Australia, August 19-25, panion: The 2018 Web Conference Companion, 1215–1219. 2017, 409–415. Lyon, France: International World Wide Web Conferences Steering Committee. Tullock, G. 1992. Computerizing politics. Mathematical and Computer Modelling 16(8-9):59–65. Boldi, P.; Bonchi, F.; Castillo, C.; and Vigna, S. 2011. Vis- cous democracy for social networks. Communications of the Watts, D. J., and Strogatz, S. H. 1998. Collective dynamics ACM 54(6):129–137. of ‘small-world’ networks. Nature 393(6684):440. Brandt, F.; Conitzer, V.; Endriss, U.; Lang, J.; and Procac- cia, A. D., eds. 2016. Handbook of Computational Social Choice. Cambridge University Press. Brill, M. 2018. Interactive democracy. In Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems, 1183–1187. International Foundation for Autonomous Agents and Multiagent Systems. Christoff, Z., and Grossi, D. 2017. Binary voting with delegable proxy: An analysis of liquid democracy. In Proceed- ings of TARK’17, volume 251, 134–150. EPTCS. Cohensius, G.; Mannor, S.; Meir, R.; Meirom, E.; and Orda, A. 2017. Proxy voting for better outcomes. In Proceedings

1803