In liquid democracy, each voter either votes herself or delegates her vote to some other voter. This gives rise to what is called a delegation graph. To decide the voters who eventually votes along with the subset of voters whose votes they give, we need to resolve the cycles in the delegation graph. This gives rise to the Resolve Delegation problem where we need to find an acyclic sub-graph of the delegation graph such that the number of voters whose votes they give is bounded above by some integer λ. Putting a cap on the number of voters whose votes a voter gives enable the system designer restrict the power of any individual voter. The Resolve Delegation problem is already known to be NP-hard. In this paper we study the parameterized complexity of this problem. We show that Resolve Delegation is para-NP-hard with respect to parameters λ, number of sink nodes and the maximum degree of the delegation graph. We also show that Resolve Delegation is W[1]-hard even with respect to the treewidth of the delegation graph. We complement our negative results by exhibiting FPT algorithms with respect to some other parameters. We finally show that a related problem, which we call Resolve Fractional Delegation, is polynomial time solvable.
Liquid democracy has a natural graphical representation, the delegation graph. Consequently, the strategic aspects of liquid democracy can be studied as a game over delegation graphs, called the liquid democracy game. Our main result is that this game has bicriteria approximation guarantees, in terms of both rationality and social welfare. Specifically, we prove the price of stability for $ε$-Nash equilibria is exactly $ε$ in the liquid democracy game.
We study wisdom of the crowd effects in liquid democracy when agents are allowed to apportion weights to proxies by mixing their delegations. We show that in this setting -- unlike in the standard one where votes are always delegated in full to one proxy -- it becomes possible to identify delegation structures that optimize the truth-tracking accuracy of the group. We contrast this centralized solution with the group accuracy obtained in equilibrium when agents interact by greedily trying to maximize their own individual accuracy through mixed delegations, and study the price of anarchy of the…
Cycles in Liquid Democracy: A Game-Theoretic Justification
University of Oxford（牛津大学）；University of Glasgow（格拉斯哥大学）；University of Oslo（奥斯陆大学）；University of Groningen（格罗宁根大学）；University of Warsaw（华沙大学）；TU Eindhoven（埃因霍温理工大学）
We investigate efficient ways for the incorporation of liquid democracy into election settings in which voters submit cumulative ballots, i.e., when each voter is assigned a virtual coin that she can then distribute as she wishes among the available election options. In particular, we are interested in fine-grained liquid democracy, meaning that voters are able to designate a partial coin to a set of election options and delegate the decision on how to further split this partial coin among those election options to another voter of her choice. The fact that we wish such delegations to be trans…
University of Groningen（格罗宁根大学）；University of Amsterdam（阿姆斯特丹大学）
The paper develops a theory of power for delegable proxy voting systems. We define a power index able to measure the influence of both voters and delegators. Using this index, which we characterize axiomatically, we extend an earlier game-theoretic model by incorporating power-seeking behavior by agents. We analytically study the existence of pure strategy Nash equilibria in such a model. Finally, by means of simulations, we study the effect of relevant parameters on the emergence of power inequalities in the model.
Liquid democracy is a novel paradigm for collective decision-making that gives agents the choice between casting a direct vote or delegating their vote to another agent. We consider a generalization of the standard liquid democracy setting by allowing agents to specify multiple potential delegates, together with a preference ranking among them. This generalization increases the number of possible delegation paths and enables higher participation rates because fewer votes are lost due to delegation cycles or abstaining agents. In order to implement this generalization of liquid democracy, we ne…
Liquid democracy is a decision-making paradigm in which each agent can either vote directly for some alternative or (transitively) delegate its vote to another agent. To mitigate the issue of delegation cycles or the concentration of power, delegating agents might be allowed to specify multiple delegation options. Then, a (cycle-free) delegation is selected in which each delegating agent has exactly one representative. We study the winner determination problem for this setting, i.e., whether we can select a delegation such that a given alternative wins (or does not win). Moreover, we study the…
Power in Liquid Democracy: A Network Centrality Approach
University of Groningen（格罗宁根大学）；University of Amsterdam（阿姆斯特丹大学）；Association for Interactive Democracy（互动民主协会）；University of Warsaw（华沙大学）；University of Oxford（牛津大学）
Gran Sasso Science Institute（格兰萨索科学研究所）；Université Paris-Dauphine（巴黎第九大学）；Université PSL（巴黎文理研究大学）；CNRS（法国国家科学研究中心）；LAMSADE（LAMSADE实验室）
In this paper, we investigate the so-called ODP-problem that has been formulated by Caragiannis and Micha [10]. Here, we are in a setting with two election alternatives out of which one is assumed to be correct. In ODP, the goal is to organise the delegations in the social network in order to maximize the probability that the correct alternative, referred to as ground truth, is elected. While the problem is known to be computationally hard, we strengthen existing hardness results by providing a novel strong approximation hardness result: For any positive constant $C$, we prove that, unless $P=…
Liquid democracy under vote correlation: On the fallacies of averaging and the excluded middle
National University of Singapore（新加坡国立大学）；TU Berlin（柏林工业大学）；Dartmouth College（达特茅斯学院）；Aarhus University（奥胡斯大学）；Nanyang Technological University（南洋理工大学）
We study vote delegation with "well-behaving" and "misbehaving" agents and compare it with conventional voting. Typical examples for vote delegation are validation or governance tasks on blockchains. There is a majority of well-behaving agents, but they may abstain or delegate their vote to other agents since voting is costly. Misbehaving agents always vote. We compare conventional voting allowing for abstention with vote delegation. Preferences of voters are private information and a positive outcome is achieved if well-behaving agents win. We illustrate that vote delegation leads to quite di…
