1 Introduction
Liquid democracy (Blum & Zuber, 2016) is an influential proposal in recent debates on democratic reforms in both Europe and the US. Several grassroots campaigns, as well as local parties, experimented with this novel type of decision making procedure. Examples include the German Piratenpartei^{1}^{1}1https://www.piratenpartei.de/ and the EU Horizon 2020 project WeGovNow (Boella et al., 2018), which have incorporated the LiquidFeedback^{2}^{2}2https://liquidfeedback.org/ platform in their decision making, as well as grassroots organizations such as the Democracy Earth Foundation^{3}^{3}3https://www.democracy.earth/. Liquid democracy is a form of proxy voting (Miller, 1969; Tullock, 1992; Alger, 2006; GreenArmytage, 2015; Cohensius et al., 2017) where, in contrast to classical proxy voting, proxies are delegable (or transitive, or transferable). Suppose we are voting on a binary issue, then each voter can either cast her vote directly, or she can delegate her vote to a proxy, who can again either vote directly or, in turn, delegate to yet another proxy, and so forth. Ultimately, the voters that decided not to delegate cast their ballots, which now carry the weight given by the number of voters who entrusted them as proxy, directly or indirectly.
Contribution
The starting point of our analysis is an often cited feature of liquid democracy: transitive delegations reduce the level of duplicated effort required by direct voting, by freeing voters from the need to invest effort in order to vote accurately. The focus of the paper is the decisionmaking problem that voters, who are interested in casting an accurate vote, face between voting directly, and thereby incurring a cost in terms of effort invested to learn about the issue at hand, or delegating to another voter in their network, thereby avoiding costs. We define a gametheoretic model, called delegation game, to represent this type of interaction. We establish pure strategy Nash equilibrium existence results for classes of delegation games, and study the quality of equilibria in terms of the average accuracy they enable for the population of voters, both analytically and through simulations. Proofs of the two main results (Theorems 1 and 2) are presented in full, while we provide proofs of the simpler secondary results as an Appendix only.
By means of simulations we also study the effects of different network structures on delegation games in terms of: performance against direct voting, average accuracy and the probability of a correct majority vote, the number and quality of voters acting as ultimate proxies (socalled gurus) and, finally, the presence of delegation cycles. To the best of our knowledge, this is the first paper providing a comprehensive study of liquid democracy from a gametheoretic angle.
Related Work
Although the idea of delegable proxy was already sketched by Dodgson (1884), only a few very recent papers have studied aspects of liquid democracy in the (computational) social choice theory (Brandt et al., 2016) literature. Kling et al. (2015) provide an analysis of election data from the main platform implementing a liquid democracy voting system (Liquid Feedback) for the German Piratenpartei. They focus on network theoretic properties emerging from the structure of delegations—with particular attention to the number of highly influential gurus or ‘supervoters’. Inspired by their experimental analysis, Gölz et al. (2018) propose and analyze a variant of the liquid democracy scheme able to restrict reliance on supervoters. Skowron et al. (2017) study an aspect of the Liquid Feedback platform concerning the order in which proposals are ranked and by which they are brought to the attention of the community. Boldi et al. (2011) investigate applications of variants of the liquid democracy voting method (called viscous democracy) to recommender systems. Brill (2018) presents some research directions in the context of liquid democracy. A general, more philosophical discussion of liquid democracy is provided by Blum & Zuber (2016).
More directly related to our investigations is the work by Christoff & Grossi (2017) and, especially, by Kahng et al. (2018). The first paper studies liquid democracy as an aggregator—a function mapping profiles of binary opinions to a collective opinion—in the judgment aggregation and binary voting tradition (Grossi & Pigozzi, 2014; Endriss, 2016). The focus of that paper are the unintended effects that transferable proxies may have due to delegation cycles, and due to the failure of rationality constraints normally satisfied by direct voting.
The second paper addresses one of the most cited selling arguments for liquid democracy: delegable proxies guarantee that better informed agents can exercise more weight on group decisions, thereby increasing their quality. Specifically, Kahng et al. (2018) study the level of accuracy that can be guaranteed by liquid democracy (based on vote delegation with weighted majority) vs. direct voting by majority. Their key result consists in showing that no ‘local’ procedure to select proxies can guarantee that liquid democracy is, at the same time, never less accurate (in large enough graphs) and sometimes strictly more accurate than direct voting. In contrast to their work, we assume that agents incur costs (effort) when voting directly, and on that basis we develop a gametheoretic model. Also, we assume agents aim at tracking their own type rather than an external ground truth, although we do assume such a restriction in our simulations to better highlight how the two models are related and to obtain insights applicable to both.
2 Preliminaries
2.1 Types, Individual Accuracy and Proximity
We are concerned with a finite set of agents (or voters, or players) having to decide whether or . For each agent one of these two outcomes is better, but the agent is not necessarily aware which one. We refer to this hidden optimal outcome as the type of agent and denote it by . Agents want to communicate their type truthfully to the voting mechanism, but they know it only imperfectly. This is captured by the accuracy of an agent : determines the likelihood that, if votes directly, she votes according to her type . We assume that an agent’s accuracy is always , i.e., at least as good as a coin toss.
We distinguish two settings depending on whether the agents’ types are deterministic or probabilistic. A deterministic type profile simply collects each agent’s type. In probabilistic type profiles
types are independent random variables drawn according to a distribution
. Given a probabilistic type profile, the likelihood that any two agents are of the same type is called the proximity where . In the probabilistic setting we assume agents know such value although, importantly, they do not know . In a deterministic type profile, we have if and otherwise. Following standard equilibrium theory, our theoretical results assume agents act as if they have access to the accuracy of each agent. More realistically, in our simulations we assume agents have access to such information only with respect to neighbors on an underlying interaction structure.2.2 Interaction Structure and Delegations
Agents are nodes in a network (directed graph) represented by a relation . For the neighborhood of in is denoted , i.e., the agents that are directly connected to . Agents have the choice of either voting themselves, thereby relying solely on their own accuracy, or delegating to an agent in their neighborhood. A delegation profile
is a vector
. Given a delegation profile we denote by the proxy selected by in . Clearly a delegation profile can be viewed as a functional graph on or, equivalently, as a map in where . When the iterated application of from reaches a fixed point we denote such fixed point as and call it ’s guru (in ). In the following, we write to denote the set of voters whose delegation does not lay on a path ending on a cycle, i.e., the set of voters for which exists. We write as a short form for .As agents may only be able to observe and interact with their direct network neighbors, structural properties of the interaction network may play a role in the model dynamics. In our simulations we will focus on undirected graphs (that is, will be assumed to be symmetric, as social ties are normally mutual) consisting of one single connected component (that is, is included in the reflexive transitive closure of ). Under these assumptions, we consider four typical network structures that are well represented in the literature on social network analysis (cf. Jackson 2008): 1) the random network, in which each pair of nodes has a given probability of being connected (Erdös & Rényi, 1959); 2) the regular network, in which all nodes have the same degree; 3) the small world network, which features a small average path length and high clustering (Watts & Strogatz, 1998); and 4) the scale free network, which exhibits a power law degree distribution (Barabási & Albert, 1999).^{4}^{4}4Although random and regular graphs are not generally applicable to realworld settings, they serve as a useful baseline to illustrate the effects of network structure on delegations.
3 A Model of Rational Delegations
3.1 Individual Accuracy under Delegable Proxy
Each agent has to choose between two options: either to vote herself with accuracy or to delegate, thereby inheriting the accuracy of another voter (unless is involved in a delegation cycle). These choices are recorded in the delegation profile and can be used to compute the individual accuracy for each agent as follows:
(1) 
In Equation (1) ’s accuracy equals the likelihood that ’s guru has the same type and votes accurately plus the likelihood that ’s guru has the opposite type and fails to vote accurately. Note that if votes directly, i.e., , then . Observe that if ’s delegation leads to a cycle (), ’s accuracy is set to . The rationale for this assumption is the following. If an agent delegates into a cycle, even though she knows her own accuracy and she actively engages with the voting mechanism by expressing a delegation, she fails to pass information about her type to the mechanism. No information is therefore available to decide about her type.
It may be worth observing that, by Equation (1), in a deterministic type profile we have that and therefore ’s accuracy reduces to: if and ; if and ; and if .
Before introducing our game theoretic analysis, we make the following observation. Agents have at their disposal an intuitive strategy to improve their accuracy: simply delegate to a more accurate neighbor. We say that a delegation profile is positive if for all either or . Furthermore, we say that a delegation from to a neighbor is locally positive if .
Proposition 1.
Let be a positive delegation profile. Further, let , , and , i.e., agent votes directly in and delegates to in . If the delegation from to is locally positive, then is positive (proof in Appendix A).
However, locally positive delegations do not necessarily correspond to optimal delegations. This can be easily seen in an example where agent is not a neighbor of a very competent agent , but would have to delegate via an intermediate agent (who delegates to ). If this intermediate agent has a lower accuracy than , then the delegation from to would not be locally positive, even though it is an optimal choice. So utilitymaximization may require information which is inherently nonlocal (accuracy of ‘far’ agents).
3.2 Delegation Games
We assume that each agent has to invest an effort to manifest her accuracy . If she delegates, she does not have to spend effort. Agents aim therefore at maximizing the tradeoff between the accuracy they can achieve (either by voting directly or through proxy) and the effort they spend. Under this assumption, the binary decision setup with delegable proxy we outlined above can be used to define a natural game—called delegation game—, with , where is the set of agents, is the (possibly degenerate) distribution from which the types of the agents in are drawn, the underlying network as defined above, is the set of strategies of agent (voting, or choosing a specific proxy), and
(2) 
is agent ’s utility function. The utility extracts from a delegation profile equals the accuracy she inherits through proxy or, if she votes, her accuracy minus the effort spent.^{5}^{5}5No utility is accrued for gaining voting power in our model. In delegation games we assume that for all . This is because if , then would prefer a random effortless choice over taking a decision with effort.
A few comments about the setup of Equation (2) are in order. First of all, as stated earlier, we assume agents to be truthful. They do not aim at maximizing the chance their type wins the vote, but rather to relay their type to the mechanism as accurately as possible.^{6}^{6}6Notice however that our modeling of agents’ utility remains applicable in this form even if agents are not truthful but the underlying voting rule makes truthfulness a dominant strategy—such as majority in the binary voting setting used here. Secondly, notice that the utility an agent extracts from a delegation profile may equal the accuracy of a random coin toss when the agent’s delegation ends up into a delegation cycle (cf. Equation (1)). If this happens the agent fails to relay information about her type, even though she acted in order to do so. This justifies the fact that is also the lowest payoff attainable in a delegation game.
The following classes of delegation games will be used in the paper: games with deterministic profiles, i.e., where is degenerate and all players are assigned a crisp type from ; homogeneous games, where all players have the same (deterministic) type;^{7}^{7}7This is the type of interaction studied, albeit not gametheoretically, by Kahng et al. (2018) and normally assumed by jury theorems (Grofman et al., 1983). and effortless voting games, where for each we have .
As an example, a homogeneous game in matrix form is given in Table 1, where , and the distribution yields the deterministic type profile . Interestingly, if we assume that with , and that^{8}^{8}8We use here the usual notation to denote ’s opponent. (i.e., the opponent’s accuracy is higher than the player’s individual accuracy minus her effort), then the game shares the ordinal preference structure of the class of anticoordination games: players need to avoid coordination on the same strategy (one should vote and the other delegate), with two coordination outcomes (both players voting or both delegating) of which the second (the delegation cycle) is worst for both players. Notice that, were the underlying network not complete (i.e., ), the matrix would be shrunk by removing the rows and columns corresponding to the delegation options no longer available.
vote  delegate (to )  

vote  
delegate (to ) 
The introduction of effort has significant consequences on the delegation behavior of voters, and we will study it in depth in the coming sections. It is worth noting immediately that the assumptions of Proposition 1 no longer apply, since agents may prefer to make delegations that are not locally positive due to the decreased utility of voting directly.
3.3 Existence of Equilibria in Delegation Games
In this section we study the existence of pure strategy Nash Equilibria (NE) in two classes of delegation games. NE describe how ideally rational voters would resolve the effort/accuracy tradeoff. Of course, such voters have common knowledge of the delegation game structure—including, therefore, common knowledge of the accuracies of ‘distant’ agents in the underlying network. Our simulations will later lift some of such epistemic assumptions built into NE.
Deterministic Types
In the following we provide a NE existence result for games with deterministic type profiles.
Theorem 1.
Delegation games with deterministic type profiles always have a (pure strategy) NE.
Proof.
First of all, observe that since the profile is deterministic, for each pair of agents and , . The proof is by construction. First, we partition the set of agents into and . We consider these two sets separately; without loss of generality let us consider . Further we consider the network . Since can be seen as a directed graph, we can partition it into Strongly Connected Components (SCCs). If we shrink each SCC into a single vertex, we obtain the condensation of this graph; note that such a graph is a directed acyclic graph (DAG). We construct a delegation profile by traversing this DAG bottom up, i.e., starting with leaf SCCs.
Let be a set of agents corresponding to a leaf SCC in the condensation DAG. We choose an agent in that has maximum . Everyone in (including ) delegates to . Now let be a set of agents corresponding to an inner node SCC in the condensation DAG and assume that we have already defined the delegation for all SCCs that can be reached from . As before, we identify an agent with maximum . Further, let be the set of all voters that can be reached from in , and for which . We distinguish two cases. (i) If , then we choose an agent with and all agents in directly or indirectly delegate to . (ii) If , all agents in delegate to . This concludes our construction (as for the analogous construction applies); let be the corresponding delegation profile.
It remains to verify that this is indeed a NE: Let be some agent in an SCC , and, without loss of generality, let . Observe that since we have a deterministic profile, if agent changes her delegation to , then ’s utility changes to if and if . First, note that for all agents , . Hence, we can immediately exclude that for agent delegating to an agent in is (strictly) beneficial, as it would yield an accuracy of at most . Towards a contradiction assume there is a beneficial deviation to an agent , i.e., there is an agent with . Let us now consider the three cases: (1) , (2) but , and (3) . In case (1), everyone in delegates to . Hence, if , a cycle would occur yielding a utility of , which is not sufficient for a beneficial deviation. If a delegation to is possible but was not chosen, then by construction and hence this deviation is not beneficial. We conclude that in case (1) such an agent cannot exist. In case (2), everyone in delegates to . Hence, if , then , a contradiction. If , the same reasoning as before applies and hence also here we obtain a contradiction. In case (3), by construction, had been chosen to maximise accuracy, hence . Since for all , , only a deviation to itself can be beneficial, i.e., . However, since was chosen because , no beneficial deviation is possible. ∎
It follows that also homogeneous games always have NE.
Effortless Voting
Effortless voting ( for all ) is applicable whenever effort is spent in advance of the decision and further accuracy improvements are not possible.
Theorem 2.
Delegation games with effortless voting always have a (pure strategy) NE.
Proof.
We prove this statement by showing that the following procedure obtains a NE: We start with a strategy profile in which all players vote directly, i.e., player ’s strategy is . Then, we iteratively allow players to choose their individual best response strategy to the current strategy profile. Players act sequentially in arbitrary order. If there are no more players that can improve their utility by changing their strategy, we have found a NE. We prove convergence of this procedure by showing that a best response that increases the player’s utility never decreases the utility of other players.
We proceed by induction. Assume that all previous best responses have not reduced any players’ utility (IH). Assume player now chooses a best response that increases her utility. Let be the delegation profile; further, let . By assumption, ’s utility started with and has not decreased since, i.e., . Since ’s best response strictly increases ’s utility, it cannot be a delegation to herself. So let a delegation to be ’s best response and consider profile . Further, let , i.e., now delegates to and by transitivity to , i.e., . Let be some player other than . We define the delegation path of as the sequence . If ’s delegation path does not contain , then ’s utility remains unchanged, i.e., . If ’s delegation path contains , then now delegates by transitivity to , i.e., we have and . By Equation (2), we have
(3)  
(4) 
We have to show that ’s utility does not decrease, i.e., , under the assumption that chooses a best response, i.e., , with:
(5)  
(6) 
In the following we will often use the fact that, for , if , then either or . By IH, since accuracies are always at least 0.5, it holds that and by Equation (5) we have and hence . Analogously, Equation (3) implies that .
Furthermore, we use the fact that
(7) 
where for . Observe that, by the definition of utility in Equation (2), the assumptions made on and , and the fact that for , if , then either or . So we have that either for , or for . We work on the first case. The other case is symmetric. Let also . From the above it follows that . Furthermore, given that , we can also conclude that . Now by substituting
in Equation (3), we obtain
(8) 
Similarly, using the appropriate instantiation of Equation (7) for with , by substituting for in Equation (4) we obtain
(9) 
Now observe that, since we have that . It remains to compare with , showing the latter is greater than the former. Observe that both expressions have a positive sign. We use the fact that implies under the assumption that . On the basis of this, and given that , we obtain that and therefore that , from which we can conclude that
It follows that the assumption (player chose a best response that increased her utility) together with Equations (8) and (9) implies that (and a fortiori that ). We have therefore shown that if some player chooses a best response, the utility of other players does not decrease. This completes the proof. ∎
Discussion
The existence of NE in general delegation games remains an interesting open problem. It should be noted that the proof strategies of both Theorems 1 and 2 do not work in the general case. Without a clear dichotomy of 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 property upon which the proof of Theorem 2 hinges (that a best response of an agent does not decrease the utility of other agents) fails in the general case due to the presence of nonzero effort. Finally, it should also be observed that Theorem 2 (as well as Proposition 1) essentially depend on the assumption that types are independent random variables. If this is not the case (e.g., because voters’ preferences are correlated), delegation chains can become undesirable.
Example 1.
Consider the following example with agents , and
. The probability distribution
is defined as , , and Consequently, , , and . Let us assume that the agents’ accuracies are , , and . A delegation from agent 1 to 2 is locally positive as . Furthermore, a delegation from 2 to 3 is locally positive as . However, the resulting delegation from to is not positive since .3.4 Quality of Equilibria in Delegation Games
In delegation games players are motivated to maximize the tradeoff between the accuracy they acquire and the effort they spend for it. A natural measure for the quality of a delegation profile is, therefore, how accurate or informed a random voter becomes as a result of the delegations in the profile, that is, the average accuracy (i.e., ) players enjoy in that profile. One can also consider the utilitarian social welfare of a delegation profile . This relates to average accuracy as follows:
It can immediately be noticed that equilibria do not necessarily maximize average accuracy. On the contrary, in the following example NE yields an average accuracy of close to , whereas an average accuracy of almost is achievable.
Example 2.
Consider an player delegation game where all players have the same type and for all and all , i.e., player 1 is a sink in and cannot delegate to anyone, but all others can delegate to everyone. Further, we have and for . The respective accuracies are and . If player does not delegate, her utility is . Hence, it is always more desirable to delegate to player (which yields a utility of for ). Consider now the profiles in which all players delegate to player (either directly or transitively). Player can only vote directly (with utility ). All such profiles are NE with average accuracy . If, however, some player chose to vote herself, all players (except ) would delegate to thereby obtaining an average accuracy of , which converges to for . This is not a NE, as could increase her utility by delegating to .
The findings of the example can be made more explicit by considering a variant of the price of anarchy for delegation games, based on the above notion of average accuracy. That is, for a given delegation game , the price of anarchy (PoA) of is given by
where denotes the set of purestrategy NE of .
Fact 1.
PoA is bounded below by and above by (see Apendix A).
An informative performance metrics for liquid democracy is the difference between the group accuracy after delegations versus the group accuracy achieved by direct voting. This measure, called gain, was introduced and studied by Kahng et al. (2018). Here we adapt it to our setting as follows: where . That is, the gain in the delegation game is the average accuracy of the worst NE minus the average accuracy of the profile in which no voter delegates. It turns out that the full performance range is possible:
Fact 2.
is bounded below by and above by (see Apendix A).
The above bounds for PoA and gain provide only a very partial picture of the performance of liquid democracy when modeled as a delegation game. The next section provides a more finegrained perspective on the effects of delegations.
4 Simulations
We simulate the delegation game described above in a variety of settings. We restrict ourselves to homogeneous games. This allows us to relate our results to those of Kahng et al. (2018). Our experiments serve to both verify and extend the theoretical results of the previous section. In particular we simulate the best response dynamics employed in the proof of Theorem 2 and show that these dynamics converge even in the setting with effort, which we could not establish analytically. In addition, we investigate the dynamics of a oneshot game scenario, in which all agents need to select their proxy simultaneously at once.
Setup
We generate graphs of size of each of the four topologies random, regular, small world, and scale free, for different average degrees, while ensuring that the graph is connected. Agents’ individual accuracy and effort are initialized randomly with and . We average results over 2500 simulations for each setting (25 randomly generated graphs 100 initializations). Agents correctly observe their own accuracy and effort, and the accuracy of their neighbors. The game is homogeneous, so proximities are .
Each agent selects from her neighborhood (which includes herself) the agent that maximizes her expected utility following Equation (2). We compare two scenarios. The iterated best response scenario follows the procedure of the proof of Theorem 2, in which agents sequentially update their proxy to bestrespond to the current profile using knowledge of their neighbors’ accuracy . In the oneshot game scenario all agents choose their proxy only once, do so simultaneously, and based only on their neighbors’ accuracy. The latter setup captures more closely the epistemic limitations that agents face in liquid democracy.
4.1 Iterated Best Response Dynamics
These experiments complement our existence theorems. They offer insights into the effects of delegations on average voter’s accuracy in equilibrium, and on the effects of different network structures on how such equilibria are achieved.
We initialize and first investigate the case in which for all (effortless voting). Across all combinations of network types and average degrees ranging from 4 to 24, we find that the best response dynamics converges, as predicted by Theorem 2, and does so optimally with for all . We observe minimal differences between network types, but see a clear inverse relation between average degree and the number of iterations required to converge (Table 2, top). Intuitively, more densely connected networks facilitate agents in identifying their optimal proxies (further details are provided in Appendix B).
We accumulate results across all network types and compare the effortless setting to the case in which effort is taken into account. When we include effort , we still observe convergence in all cases and, interestingly, the number of iterations required does not change significantly (Table 2, bottom). Although the process no longer results in an optimal equilibrium, each case still yields a single guru with (less than error) for all . In this scenario, the inclusion of effort means that a best response update of agent no longer guarantees nondecreasing accuracy and utility for all other agents, which was a key property in the proof of Theorem 2. This effect becomes stronger as the average network degree increases, and as a result higher degree networks allow a greater discrepancy between the maximal average accuracy achievable and the average accuracy obtained at stabilization (Table 3).
In lower degree graphs (e.g. degree 4) we further observe differences in convergence speed between the four different network types which coincide with differences between the average path lengths in those graphs: a shorter average distance between nodes yields a lower number of best response updates. This is intuitive, as larger distances between nodes mean longer delegation chains, but we have not yet conducted statistical tests to verify this hypothesis.
4.2 OneShot Delegation Games
Here we study oneshot interactions in a delegation game: all agents select their proxy (possibly themselves) simultaneously among their neighbors; no further response is possible. This contrasts the previous scenario in which agents could iteratively improve their choice based on the choices of others. While Kahng et al. (2018) study a probabilistic model, we instead assume that agents deterministically select as proxy the agent that maximizes their utility, as above. We compare and (the average network accuracy without and with delegation, respectively), as well as the probability of a correct majority vote under both direct democracy and liquid democracy where gurus carry as weight the number of agents for whom they act as proxy. The difference is again similar to the notion of gain (Kahng et al., 2018). In line with Condorcet’s jury theorem (see for instance Grofman et al. 1983) as , and indeed for we obtain .
First we again look at the effortless setting. Figure 1 (top) shows both metrics for the four different network types and for different average degrees.
We observe that while increases as the network degree increases (and in fact is always higher than without delegation), the probability of a correct majority outcome, , simultaneously decreases. This confirms the analysis of Kahng et al. (2018). We also observe that the number of gurus decreases exponentially as the degree increases (Figure 2, left). Simply put, giving all voting weight to a small group of gurus increases the chance of an incorrect majority vote, assuming that gurus have a less than perfect accuracy.
When we include effort (Figure 1, bottom), thereby moving away from the model of Kahng et al. (2018), we observe a drastic decrease in average network accuracy combined with a lower probability of a correct majority outcome under liquid democracy, with both decreasing as network degree increases. The main reason is the existence of delegation cycles in this case. This contrasts the best response setting above where agents could iteratively reconsider their choice of proxy and thus avoid cycles. Now, even with relatively low effort (mean 0.025), up to half of all agents are stuck in a cycle (and thereby fail to pass information about their type) when the degree increases. This confirms results on the probability of delegation cycles from Christoff & Grossi (2017) and stresses the importance of cycle resolution in concrete implementations of liquid democracy.
Figure 1 further highlights differences between the four network types. Scale free networks yield a lower probability of a correct majority outcome across all degrees, as well as a larger number of gurus with a lower average accuracy and longer delegation chains (Figure 2, right). Intuitively, this indicates that oneshot interactions in scale free networks are more likely to end up in a local optimum. In contrast, small world networks have short average distances and thus agents are more likely to be close to their optimal guru. Finally, our experiments highlight a key feature of liquid democracy: the tradeoff between a reduction in (total) effort against a loss in average voting accuracy.
5 Conclusions and Future Work
The paper introduced delegation games as a first gametheoretic model of liquid democracy. Both our theoretical and experimental results showed that voting effort is a key ingredient for understanding how delegations form and what their effects are. Our empirical findings provided further insights into the influence of interaction networks on the quality of collective decisions in liquid democracy.
The paper opens up several directions of research. A general NE existence theorem is the main open question. Our model can then be generalized in many directions, e.g.: by making agents’ utility dependent on voting outcomes; by dropping the independence assumption on agents’ types; or by assuming the voting mechanism has better than accuracy in identifying the types of agents involved in cycles.
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Appendix A Proofs of Proposition 1, Fact 1, and Fact 2
Here we provide the proofs op Proposition 1, Fact 1, and Fact 2.
Proof of Proposition 1.
First we show that . If , the statement holds since the delegation from to is locally positive. Otherwise, let . Since is positive, we know that
Now let for . Since types are independent random variables, it holds that . Using this fact, we can verify that
(10) 
We now want to prove that
(11) 
First note that a locally positive delegation from to implies that (since ); hence and . Second, observe that for any , if , then either or . Now, since and , we conclude that either or . In both cases Equation (11) holds.
Thus we obtain:
We conclude that . It remains to show that for an agent with , it still holds that . This can be shown by the same argument as above, now using the fact that the delegations from to and from to are positive. Hence is positive. ∎
Proof of Fact 1.
For the upper bound of , Example 2 shows that maximal average accuracy can be made arbitrarily close to , while the average accuracy of a worse NE can be made arbitrarily close to . For the lower bound of it suffices to consider any delegation game were (that is, no delegation is possible). ∎
Proof of Fact 2.
For the lower bound we can use again Example 2, where the average accuracy of the worse NE can be made arbitrarily close to , and the average accuracy of direct voting (which given the structure of the example is maximized by direct voting) can be made arbitrarily close to . For the upper bound, consider an effortless homogeneous game where , and for any in . Assume furthermore that is such that any agent can delegate only to . In such a game, the average accuracy of the unique NE is , while the average accuracy of direct voting can be made arbitrarily close to . We thereby obtain the desired bound of . ∎
Appendix B Additional Simulation Results
Detailed results for the “Simulations” subsection “Iterated Best Response Dynamics”. Figure 3 shows convergence speed results in the effortless setting for different network types. The larger differences between the required number of best response updates for lower degree graphs of different types (e.g. degree 4) coincide with differences between the mean distance between nodes in those graphs: a shorter average distance yields a lower number of best response updates. The latter is shown in Table 4.
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