arXiv CSopen access

Resilient Liquid Democracy: Mitigating Voting Power Imbalances via Secure Delegation Networks

arXiv:2607.01730v1 [cs.CR] 2 Jul 2026

Resilient Liquid Democracy: Mitigating Voting Power Imbalances via Secure Delegation Networks Zhuolun Li, Evangelos Pournaras School of Computer Science, University of Leeds, UK July 3, 2026 Abstract Liquid democracy promises to improve collective decision-making by allowing voters to vote directly, delegate their voting power to trusted participants, or combine both approaches through fallback mechanisms. However, existing deployments typically rely on transparent delegation, which exposes voters to popularity-driven herding, makes coercion verifiable, and introduces systemic fragility when highly-backed delegates abstain. In this paper, we propose a secure liquid democracy mechanism that resolves the tension between informed expertise routing and systemic robustness. We introduce a sealed delegation regime using decentralized timed-release encryption, which cryptographically hides delegation choices during the formation phase to prevent herding and coercion, while restoring full public auditability for the final tally. To address delegate failures, we extend the protocol with ranked multi-delegation and personal fallback ballots. We formally prove pre-reveal secrecy and resubmission receiptfreeness for our protocol. Finally, we evaluate the mechanism on four real datasets, a municipal participatory-budgeting election with a calibration survey, twenty further participatory-budgeting elections, and 60,000 US voters with an objective competence measure. They show that whether delegation improves representational accuracy follows a recoverable-gap law; it helps only when abstention is large and systematically unrepresentative, and is otherwise neutral or harmful, with representative-style delegation safer than delegating to a competence elite. The benefit of sealed formation is primarily structural, sharply reducing power concentration rather than directly improving accuracy; and ranked multi-delegation with personal fallback ballots sharply reduces vote loss under realistic and targeted delegate failures, a result that replicates across all twenty elections.

1

Introduction

In real-world collective decision making, participants often differ in how much time, information, and confidence they have when evaluating the issue under 1

consideration. Some participants have substantial domain knowledge, some have relevant local or professional experience, and others simply trust certain people more than they trust themselves on a given decision. Liquid democracy is often proposed as a mechanism for such settings. It allows a voter to participate by voting directly, transferring their voting power to another participant, or utilizing both via fallback mechanisms. This flexibility can lower the burden of participation when a voter is busy, uncertain, or insufficiently informed, and it can also improve decision quality when voters choose delegates whom they regard as more reliable, more informed, or more aligned with their judgment. Despite this potential, existing liquid democracy systems are poorly suited for secure and reliable use in real-world deployments. Most implementations operate under a transparent delegation regime, in which delegation choices are publicly observable as they are formed or can be inferred from the system state. This creates several problems. First, visible delegations encourage herding and popularity cascades. Early or well-known delegates may attract a disproportionate share of support simply because they already appear popular. Second, transparent delegation makes influence easier to verify. If others can observe or later prove how a voter delegated, then rewards, pressure, or retaliation can be conditioned on that behavior. Third, many systems support only a single delegate. If that delegate abstains, becomes inactive, behaves maliciously, or misrepresents the voter, the delegated vote can be lost or distorted. These limitations create a basic design tension. Delegation can expand participation and improve decision quality, but naive implementations introduce new vulnerabilities that affect independence, robustness, and fairness. Concretely, we solve the problem of keeping delegation private while it is being formed, so that herding and vote-buying lose their grip, without sacrificing the public auditability that a trustworthy tally requires, and without leaving voters exposed when a delegate fails. In this paper, we address this tension by designing a liquid democracy mechanism for secure trusted delegation. Our system is based on two ideas. First, we introduce sealed delegation using timed-release cryptography to prevent participants and external observers from learning who delegated to whom while decisions are still being formed. This limits herding and removes the evidentiary basis needed for many bribery and coercion strategies. Second, we extend standard liquid democracy with ranked multi-delegation and fallback ballots. Each voter may specify an ordered list of preferred delegates, together with a personal backup ballot that is used if all delegations fail, so that votes are preserved even when delegates abstain or delegation cycles occur. This paper makes the following contributions: • We formalize liquid democracy as a mechanism for trusted delegation and introduce the notion of delegation information regimes, capturing what voters can observe about others’ delegation choices during formation and how this shapes behavior and network structure. • We design a liquid democracy protocol that combines timed-release cryp2

tography with ranked multi-delegation and fallback ballots, providing resistance to herding, bribery, and coercion while improving robustness against delegate failure. • We give a formal security analysis of the mechanism. We prove pre-reveal secrecy, meaning that no efficient adversary learns a voter’s delegation before the reveal time, and resubmission receipt-freeness, meaning that a coerced voter can comply with a demand yet still cast their true preference undetectably. • Through experiments on four real datasets, a municipal participatorybudgeting election with a calibration survey, twenty further participatorybudgeting elections, and a 60,000-voter US-election survey with an objective competence measure, we establish a recoverable-gap law characterizing when delegation improves representational accuracy. Delegation helps only when abstention is large and systematically unrepresentative; otherwise it is neutral or harmful, and representative-style delegation is safer than delegating to a competence elite. • We show that the benefit of sealed formation is primarily structural, sharply reducing the concentration of voting power and the resulting vulnerability to targeted delegate failures, and that ranked multi-delegation with fallback ballots cuts vote loss from up to 26% to about 3% under targeted delegate failures, replicating across all twenty participatorybudgeting elections. The rest of the paper is organized as follows. Section 2 reviews related work and positions our design. Section 3 formalizes liquid democracy and the notion of delegation information regimes, and Section 4 presents the sealeddelegation protocol and its algorithm. Section 5 proves pre-reveal secrecy and receipt-freeness. Sections 6 and 7 describe the experimental setup and report the evaluation across four datasets. Section 8 concludes.

2

Related Work

2.1

Delegation as a Mechanism for Expertise Aggregation

Liquid democracy builds on a broader literature studying how individuals transfer decision power to others. The paradigm originates in proposals for delegative or proxy democracy that combine direct and representative voting through freely chosen, transitive delegation [26, 30, 14]. In epistemic models of collective decision making, delegation is often motivated by differences in competence or information across participants, so that decisions can be routed toward people with higher expected decision quality. A substantial theoretical literature asks whether delegation actually improves decision accuracy. It can, but not unconditionally. Local delegation mechanisms are not guaranteed to outperform direct voting [33, 19], concentration of delegated power can degrade out3

comes [29], and only under more realistic delegation behaviour do good guarantees re-emerge [31]. Our empirical “recoverable-gap” result (Section 7.2) is a data-driven counterpart to this debate, identifying when delegation helps. Algorithmic and game-theoretic work also studies how rational agents choose between direct voting and delegation, and how the resulting delegation graph affects the final outcome [13, 24, 21]. Empirical and experimental studies show that voters do not always delegate in an idealized way [18, 28]. Participants may rely on limited information and misjudge who is a good delegate. These findings suggest that delegation mechanisms must be robust not only to strategic manipulation, but also to ordinary social influence and imperfect judgment.

2.2

Structural Risks in Delegation Networks

A key challenge in liquid democracy is that the delegation network itself can become a target of manipulation. Alouf-Heffetz et al. [6] analyze how changes to delegation edges can alter outcomes. Other work studies cycles, vote loss when delegates abstain, and instability in iterative delegation processes [22, 35]. Several proposals strengthen the delegation mechanism itself. Ranked delegation allows a voter to list backup delegates, which reduces vote loss when the first choice is unavailable [16]. Other generalized models allow more complex delegation patterns, such as splitting influence across several delegates, delegating to a group whose aggregate decision is followed, or discounting long delegation chains [11, 21]. These approaches improve reliability after delegations are formed, but they usually assume that delegation choices remain observable during formation.

2.3

Real-World Delegation Systems and Empirical Observations

Liquid democracy has been deployed in civic participation platforms and political party decision-making systems. Examples include LiquidFeedback and Liquid Friesland [3, 2], as well as party-based deployments such as the Swedish Direktdemokraterna [1]. More recently, delegation-based governance has become common in blockchain ecosystems, where decentralized autonomous organizations allow token holders to vote directly or delegate to representatives using tools such as Tally [4]. These deployments show that delegation can scale to large populations and high-stakes environments. At the same time, empirical studies repeatedly report concentration of delegation power and the emergence of highly influential delegates [34]. In blockchain ecosystems, transparent delegation in Decentralized Autonomous Organizations (DAOs) has frequently led to severe centralization controversies, with studies showing that a small subset of delegates controls the vast majority of voting power [27]. Because most deployed systems are transparent, users can often observe who is attracting support, which can reinforce popularity and centralization over time. 4

Table 1: Positioning of this work relative to prior liquid democracy systems and mechanisms. System / line of work

Formation-phase delegation visibility

Ranked delegates

Fallback ballot

Publicly auditable tally

LiquidFeedback / Liquid Friesland / Tally [3, 2, 4] Ranked delegation models [16] Anonymous delegation mechanisms [38, 52] This work

Visible

Limited

No

Yes

Visible

Yes

No

Yes

Partial (not timedrelease)

Limited

No

Limited

Hidden until timed reveal

Yes

Yes

Yes

2.4

Privacy-Preserving Voting and Delegation

A separate body of work studies how to protect voter privacy and ensure secure vote execution. Early work explored cryptographic approaches for delegation and voting to preserve privacy and anonymity [38]. More recent work studies anonymous delegation resolution mechanisms that hide who delegated to whom in the final outcome [52]. The secure e-voting literature more broadly targets receipt-freeness and coercion-resistance, ensuring that a voter cannot prove how they voted even if they wish to [10, 32], together with end-to-end verifiability, so that anyone can audit the tally [5, 20]. A common practical defense against coercion is to permit re-voting and count only a voter’s last ballot, as deployed in Estonian internet voting [50]. Our resubmission-based receipt-freeness (Section 5.3) builds on this re-voting idea but couples it with sealed formation, so that a coercer cannot even tell whether a resubmission changed the vote. This literature is important, but its focus is usually different from ours. Many privacy-preserving voting systems aim to hide ballots or identities, whereas our main concern is the timing of delegation visibility. In particular, we ask whether delegation choices should be visible while voters are still making them. That question is central to herding, coercion, and popularity effects, yet it is largely orthogonal to standard ballot secrecy. To position our design, Table 1 compares prior systems along the dimensions that matter for sealed delegation. These dimensions are whether delegation choices are visible during formation, which drives herding and contemporaneous coercion, whether a ranked list of backup delegates and a fallback ballot guard against delegate failure, and whether the final tally is publicly auditable. Our mechanism hides delegations only during formation and then discloses them at a timed reveal, so that auditability is restored once choices can no longer be influenced, which is precisely why the reveal matters. Table 1 makes the gap concrete: no prior line of work combines timed-release sealing of the formation phase with ranked delegation, a fallback ballot, and a

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publicly auditable tally. Transparent systems provide auditability but expose delegations during formation; privacy-preserving systems hide identities but neither seal the formation phase nor add ranked, fallback-protected delegation. Our design is the only row that provides all four together. Taken together, prior work and deployed systems reveal two structural vulnerabilities that any viable sealed-delegation design must address [33, 27]. First, while delegation can improve decision quality by letting less-informed voters defer to experts, transparent delegation exposes voters to popularity signals that trigger herding and degrade independent judgment. Second, delegation inherently concentrates voting power in a few delegates, and with it failure risk: without safeguards, the inactivity or incapacitation of a highly-backed delegate can disenfranchise large portions of the electorate. Our design targets both.

3

Formalization of Liquid Democracy

We formalize liquid democracy as a mechanism for trusted delegation in collective decision making, where participants may either vote directly or route their voting power toward another participant whom they regard as more reliable, more informed, or better placed to judge the decision.

3.1

Participants, Identity, and Expert Delegation Model

Let N = {1, 2, . . . , n} denote the finite set of voters, and let A = {a1 , a2 , . . . , am } denote the set of policy alternatives or candidates. Participants interact with the system through pseudonymous identities such as blockchain addresses or public keys. Ballots and delegation relationships are associated only with these pseudonymous identifiers. To support informed delegation, participants may optionally link real-world identity information or verifiable credentials to their pseudonymous identity in order to be recognized as potential delegates. Such disclosure is voluntary and may be public or selectively revealed, depending on the application context. Importantly, a delegate need not be an expert in a narrow technical sense. A voter may delegate to any participant they trust for the current decision. Each voter i ∈ N submits a ballot that can contain two components: a direct vote vi ∈ A, and a ranked list of chosen delegates. Depending on what the voter provides, this allows for: 1. Direct vote: The voter provides only a direct vote vi , which is counted directly. 2. Delegation: The voter leaves the direct vote blank and provides only a list of delegates. Their voting power is transferred. 3. Delegation with Fallback: The voter provides both a list of delegates and a direct vote vi . The direct vote acts as a fallback, which is only counted if the chosen delegates abstain, fail to vote, or form a delegation cycle that cannot be routed to a direct voter. 6

Delegation is motivated by differences across voters in knowledge, available time, confidence, and trust [14].

3.2

Delegation Graph and Vote Resolution

The set of delegation relationships induces a directed graph G = (N, E), where (i, j) ∈ E if voter i delegates to voter j. Delegation satisfies the following properties: • Transitivity. If i delegates to j and j delegates to k, then i’s vote follows the delegation path until it reaches a voter who casts a direct ballot. • Cycles. Delegation cycles may occur and must be resolved by system rules such as abstention, cycle breaking, or fallback voting. Let B ⊆ N be the set of voters whose effective choice after resolution is a direct ballot; we call such a voter g ∈ B a guru, that is, a delegate or direct voter at whom one or more delegation paths terminate. For each guru g, define its voting weight as W (g) = { i ∈ N : the delegation path of i terminates at g } . Thus W (g) is the number of voters whose voting power is ultimately routed to g; equivalently, g’s ballot is counted W (g) times in the final tally. After resolution, the resolved ballots are aggregated using a standard ballot aggregation rule that is deterministic and publicly verifiable, such as majority voting.

3.3

Information Regimes in Delegation Formation

A central concept in this work is the information regime of delegation formation. An information regime determines what a voter can observe about the partially formed delegation graph during the delegation phase. This may include, for example, delegate popularity, visible delegation chains, or aggregate statistics. While many intermediate regimes are possible, this work focuses on two canonical regimes: • Transparent delegation regime. Delegation relationships are publicly observable as they form. Voters can observe delegate popularity, which may create social influence, herding, and verifiable coercion. • Sealed delegation regime. Delegation relationships remain cryptographically hidden until a designated reveal time Tr , for instance the moment the votes are counted. During delegation formation, voters rely on their own beliefs about competence and trust, and cannot observe others’ delegation decisions.

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The information regime changes the strategic environment. In transparent regimes, a voter may condition their choice on popularity signals or social pressure. In sealed regimes, these signals are absent during formation, which reduces visibility-driven coordination and makes delegation-based bribery or coercion harder to verify. The goal of this work is to formalize and instantiate sealed delegation formation using timed-release cryptography, and to evaluate how this change affects behavior, robustness, and participation.

4

System Design Voting Phase

Reveal Phase

Decrypt

Timed-Release Encyption System

Encrypt Send Voter

...

Delegation or direct vote ballot

Aggregate

Build

Delegation graph + direct votes

Result

...

Figure 1: System workflow of the proposed sealed liquid democracy mechanism. In the voting phase, voters encrypt and submit their policy tuples to the ledger; in the reveal phase, all encrypted ballots are decrypted and the system computes the final outcome. The proposed system addresses both vulnerabilities identified in Section 2 at once. As shown in Figure 1, it separates delegation formation during the voting phase from delegation resolution during the reveal phase, enforcing secrecy during the former and transparency during the latter.

4.1

Cryptographic Primitive: Timed-Release Encryption

To enforce sealed delegation without relying on a single trusted authority, our system utilizes a decentralized Timed-Release Encryption (TRE) scheme. While traditional TRE schemes often rely on computationally intensive time-lock puzzles or timed commitments [46, 15], or on centralized time-servers, we adopt a blockchain-based “time machine” approach [36], which leverages a decentralized committee and verifiable secret sharing. This primitive has recently been applied to secure direct voting systems where the voters themselves act as the committee [37]; however, our work abstracts this layer, requiring only that the committee remains decentralized and secure, allowing us to focus on the unique complexities of delegation routing. 8

In this scheme, a conditional reveal committee is established via a smart contract. To encrypt a message for a future reveal time Tr , a voter generates a symmetric encryption key, encrypts their message with it, and uses Shamir’s Secret Sharing [49] to split the key into shares. These shares are encrypted and distributed to the committee members. The smart contract ensures that committee members are only incentivized to reveal their shares once the blockchain’s block height or timestamp surpasses Tr . Once Tr is reached, any observer can reconstruct the symmetric key from a threshold of revealed shares, allowing all sealed ballots to be decrypted. Crucially, as long as an adversary cannot compromise a majority of the committee members prior to Tr , the pre-reveal secrecy of the ciphertexts is mathematically guaranteed.

4.2

Voting Phase: Sealed Delegation Formation

Each voter constructs a policy tuple Πi = (Di , bi ), (1)

(2)

(ℓ )

where Di = [di , di , . . . , di i ] is an ordered list of ranked delegates, and bi is a direct ballot to be used if all delegations fail or if the voter chooses to vote directly. Instead of publishing delegation relationships directly, each voter encrypts Πi using the TRE scheme described above, parameterized by the reveal time Tr . The resulting ciphertext is posted to a public ledger before the submission deadline. For a voter i, the encrypted policy is written as Ci = TREncTr (Πi ; ri ), where TREncTr denotes timed-release encryption and ri is fresh randomness. If policy updates are allowed before the deadline, the voter may resubmit with a larger nonce or sequence number, and only the latest valid submission is accepted. This design should not be understood as making the election opaque. The system hides delegation decisions only during the phase in which voters are still making them. This prevents a voter from being influenced by visible delegation counts and prevents an external actor from obtaining contemporaneous proof of how a voter delegated. After Tr , however, the system reveals all accepted policies so that the tally is fully auditable. Until time Tr , delegation relationships and ballots remain computationally hidden. Although ciphertexts are publicly visible, no participant can infer who delegated to whom under standard cryptographic assumptions. As a result, voters must rely on their own judgment about competence and trust when selecting delegates. Early-mover advantages and popularity signals are therefore suppressed during formation. At time Tr , the timed-release mechanism enables decryption of all submitted policy tuples. The system then transitions from temporary secrecy to full transparency, so that delegations and fallback ballots become readable, auditable, and ready for deterministic resolution. 9

4.3

Reveal Phase: Deterministic Delegation Resolution

Once policy tuples are revealed, delegation is resolved and the ballots are aggregated deterministically. Voting power propagates along ranked delegation lists. For each voter i, the system evaluates delegates in order. If the highest-ranked delegate casts a valid vote, i’s voting power is assigned to that delegate. If that delegate abstains or becomes inactive, the system proceeds to the next ranked delegate. If all delegates fail, the direct ballot bi is used if provided. Only if no valid delegate and no direct ballot exist is the vote marked as abstention. Delegation cycles are resolved deterministically to ensure termination. Specifically, if a cycle forms, the system must break it while respecting ranked preferences of voters. The system selects one voter in the cycle to “step down” to their next highest-ranked delegate; the specific choice does not affect correctness, so we fix the lowest-indexed voter v ∗ = minv∈C v, where C is the set of voters in the cycle, to keep resolution deterministic and publicly recomputable. This iterative unrolling continues until the cycle is broken. A fallback ballot is only invoked if a selected voter has exhausted all their ranked delegates. This ensures that lower-ranked trusted proxies are prioritized over direct voting, maintaining the philosophy of expertise delegation. Because all policies are publicly visible after Tr , any observer can recompute delegation paths and verify the correctness of the final tally. The ranked delegation mechanism transforms delegation from a single-pointof-failure structure into a more resilient routing system. In single-delegate designs [3, 2, 4], failure of a popular delegate can invalidate many votes at once. By allowing multiple ranked delegates and fallback ballots, the system reroutes voting power without changing the independence of delegation formation.

4.4

Algorithm for Delegation Formation and Resolution

We now specify the full sealed delegation procedure as an explicit algorithm, presented in algorithm 1. The algorithm separates submission, reveal, cycle resolution, fallback use, and tallying steps. The algorithm is deterministic by construction. Once the reveal occurs, any observer can reconstruct the delegation graph, verify cycle handling, and recompute the final tally from the published inputs.

4.5

Correctness and Protocol-Level Guarantees

Before turning to cryptographic security, we record three guarantees that follow directly from the algorithm above. Post-reveal auditability. After Tr , all accepted policies are decrypted and published. Because resolution is deterministic, any observer can recompute the delegation graph, verify how each vote is routed, and reproduce the final tally from the revealed transcript. The mechanism therefore does not trade privacy

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Algorithm 1 Sealed ranked delegation with reveal-time resolution 1. Input: voter set N , submission deadline Ts and reveal time Tr with Ts < Tr , so that submissions close at Ts while every ciphertext stays sealed until the later reveal time Tr , and for each voter i ∈ N a policy tuple Πi = (Di , bi ), (1)

(2)

(ℓ )

where Di = [di , di , . . . , di i ] is an ordered list of ranked delegates and bi is an optional fallback ballot. 2. Submission phase: Before Ts , each voter encrypts Πi as Ci = TREncTr (Πi ; ri ), and posts Ci to the public ledger. If multiple submissions are allowed, the system retains only the latest valid submission for each voter. 3. Reveal phase: At time Tr , the system decrypts all accepted ciphertexts and reconstructs the set of revealed policies {Πi }i∈N . 4. Initialize choices: For each voter i ∈ N , set their active choice to their (1) highest-ranked valid delegate di who is not inactive and has a revealed policy. If no such delegate exists, set their active choice to their fallback ballot bi . 5. Iterative Cycle Resolution: Construct the directed delegation graph based on all current active choices. While cycles exist in the graph, do the following for each cycle: • Let C ⊆ N be the set of voters forming the cycle. Select the voter v ∗ = minv∈C v. • Advance the active choice of v ∗ to their next available ranked delegate (k+1) dv∗ who has a valid revealed policy. • If v ∗ has exhausted all ranked delegates in Dv∗ , set their active choice to their fallback ballot bv∗ . If bv∗ does not exist, mark v ∗ as abstaining. 6. Final Resolution: Once the graph is cycle-free, every voter’s path deterministically terminates at a direct voter, a fallback ballot, or an abstention. Route the voting power along these paths. 7. Tally: Aggregate all resolved ballots using the public deterministic ballot aggregation rule F . The final tally is publicly recomputable from the revealed policies and the resolution rule.

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for opacity; it delays disclosure only during formation, then reestablishes full public verifiability for tallying and audit. Deterministic termination. The resolution procedure terminates for every revealed policy profile, because ranked delegation is a finite ordered search through each voter’s delegate list, and cycles are broken by the fixed public rule of Step 5, which advances the lowest-indexed voter in the cycle. No ad hoc human intervention is needed to complete vote resolution. Robustness to delegate failure. Ranked delegates and fallback ballots reduce vote loss when delegates abstain, become inactive, or fail to submit. Because each voter may specify an ordered list of delegates, the protocol reroutes voting power to the next available delegate before resorting to the fallback ballot. This is a robustness property that single-delegate systems [3, 2, 4] lack, where the failure of one popular delegate can invalidate many votes at once; we quantify the effect empirically in Section 7.6.

5

Formal Security Analysis

Sealed delegation is only meaningful if the seal actually holds. This section proves the two security properties the design depends on, and states precisely the assumptions under which they hold. What we prove, and why it matters.

We establish two properties:

Pre-reveal secrecy (§5.2): before the reveal time Tr , no efficient adversary learns anything about how a voter delegated or voted. This is what suppresses popularity-driven herding, since no one can watch delegations accumulate while choices are still being made, and what removes the contemporaneous evidence a briber would need to verify a purchased vote during formation. Receipt-freeness via resubmission (§5.3): even a voter forced to hand over a “receipt”, that is, a ciphertext together with the randomness that produced it, can still cast a different, true policy that the coercer cannot detect before Tr . This defeats coercion of the form “prove to me how you voted.” Our strategy is to assume the cryptographic building block and reduce each protocol property to it. We do not re-prove the timed-release encryption (TRE) primitive; its security is established in the constructions we build on [36, 37]. Instead we assume TRE satisfies one precisely stated property (Definition 1) and show that breaking either protocol property would break that assumption.

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5.1

Threat Model and Assumptions

Adversary. We consider a probabilistic polynomial-time (PPT) adversary A that can (i) read the entire public ledger and observe the exact timing of every submission; (ii) post ciphertexts on behalf of voters it controls; and (iii) adaptively corrupt voters, learning their internal state, including their encryption randomness ri , and dictating their submissions. The single voter whose secrecy a theorem is about is assumed to remain honest and uncorrupted until Tr , since corrupting that voter trivially reveals its own vote, so no secrecy claim could hold for it. Trust assumptions. Our guarantees hold under two standard conditions. First, the public ledger is append-only and totally ordered, so the protocol’s “latest valid submission wins” rule is well-defined and a posted ciphertext cannot be retroactively deleted. Second, the reveal committee is honest before Tr , meaning that A controls fewer than the reconstruction threshold of committee members until Tr , so no coalition can decrypt early. These are exactly the conditions under which the timed-release primitive operates [36]. The building block and the property we rely on. The protocol uses a TRE scheme TRE = (Setup, TREnc, TRDec). The only property we need is that, before the reveal time, a ciphertext hides its plaintext. Definition 1 states this as a standard indistinguishability game. We stress a separation that the rest of the section relies on. This indistinguishability property, which formalizes what secrecy means, is a property of the primitive and is assumed, whereas the protocol-level properties we prove, namely pre-reveal secrecy and receiptfreeness, are reduced to it. Definition 1 (IND-TR-CPA security of the TRE primitive). Consider the following game between a challenger and a PPT adversary A that runs entirely before time Tr : 1. A chooses two equal-length messages m0 , m1 and sends them to the challenger. $

2. The challenger samples a secret bit b ← − {0, 1} uniformly at random and returns the challenge ciphertext c∗ ← TREncTr (mb ; r∗ ) with fresh randomness r∗ . 3. A outputs a guess b′ and wins if b′ = b. The scheme is IND-TR-CPA secure if every PPT adversary’s advantage AdvIND-TR-CPA (λ) = Pr[b′ = b] − 12 TRE,A is negligible in the security parameter λ, that is, smaller than 1/p(λ) for every polynomial p and all sufficiently large λ. Intuitively, before Tr , guessing which

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message a ciphertext hides is essentially no better than a coin flip. This is the sole cryptographic assumption used below, and it is met by the decentralized “time-machine” construction we build on [36].

5.2

Pre-Reveal Secrecy

Property. Fix an honest target voter i. Pre-reveal secrecy says that for any two policy tuples Π0 , Π1 that i might submit, the adversary’s view of the protocol before Tr is computationally independent of which one i actually submitted. This is the formal statement of “the seal holds during formation.” Theorem 1 (Pre-reveal secrecy). If TRE is IND-TR-CPA secure (Definition 1), then under the assumptions of §5.1 no PPT adversary can distinguish, before Tr , whether an honest voter i submitted Π0 or Π1 , except with negligible advantage. Proof. We give a reduction. From any adversary A that distinguishes i’s two policies with advantage ϵ, we build an adversary B that wins the IND-TR-CPA game (Definition 1) with the same advantage ϵ. Since IND-TR-CPA security forces B’s advantage to be negligible, ϵ is negligible too, which is exactly the claim. B plays the IND-TR-CPA game on the outside while simulating the whole voting protocol for A on the inside: 1. Setup. B generates the state of all voters and the ledger honestly and starts A. 2. Other voters and corruptions. Whenever A corrupts a voter or posts on behalf of a controlled voter, B answers exactly as the real protocol would; it can do so because it created all of these voters and knows their secrets. The target voter i is never corrupted, as required by the threat model. 3. Embedding the challenge. When A commits the two candidate policies Π0 , Π1 for voter i, B forwards them unchanged as its own challenge messages and receives c∗ = TREncTr (Πb ; r∗ ) from the IND-TR-CPA challenger. B posts c∗ to the simulated ledger as voter i’s submission. 4. Output. B runs A to the end of the pre-reveal phase and outputs whatever bit b′ A outputs. The crux is that the secret bit b enters A’s view at exactly one place, the challenge ciphertext c∗ , and every other part of the simulation is distributed identically to the real protocol. Hence A’s view here is identical to its view when i genuinely submits Πb , so A guesses b correctly in the simulation precisely when it would in the real run. Therefore B wins exactly when A does: AdvIND-TR-CPA (λ) = Advsecrecy (λ) = ϵ. TRE,B A IND-TR-CPA security makes the left-hand side negligible, so ϵ is negligible. 14

5.3

Receipt-Freeness via Resubmission

The coercion we defend against. Pre-reveal secrecy hides a voter’s choice from a passive observer, but a coercer can be active. It can order a voter to encrypt a dictated policy ΠA and to hand over the ciphertext together with the randomness used, as a receipt proving compliance. We show the receipt is worthless, since the voter can produce it and still vote their true preference, undetectably before Tr . The defense combines pre-reveal secrecy with one protocol rule, that only a voter’s latest valid submission is tallied. Counting only the last ballot is a known coercion mitigation; it underpins re-voting in deployed internet voting [50] and the coercion-resistance literature [32]. Our addition is that sealed formation hides whether a resubmission changed the vote at all. Theorem 2 (Resubmission receipt-freeness). Suppose the ledger is append-only and the tally uses only each voter’s latest valid submission. Then a voter coerced into producing a receipt (CA , ri ) for a dictated policy ΠA can instead cast a different true policy Πtrue such that, before Tr , the coercer cannot distinguish this voter from one who genuinely submitted only ΠA . Proof. The voter proceeds in two steps. 1. Satisfy the coercer. Encrypt the dictated policy as CA = TREncTr (ΠA ; ri ) with sequence number 1, post it, and give the coercer (CA , ri ). The coercer recomputes TREncTr (ΠA ; ri ), sees it matches the posted ciphertext, and accepts the receipt. 2. Override with the true vote. Encrypt the true policy as Ctrue = TREncTr (Πtrue ; ri′ ) with fresh randomness ri′ and sequence number 2, and post it. By the “latest valid submission” rule, Ctrue , not CA , is the ballot tallied at Tr . It remains to show the coercer cannot detect the override before Tr . The coercer sees a second ciphertext Ctrue from voter i, but posting a second ciphertext is also exactly what an obedient voter would do to refresh or re-confirm the dictated policy, so the mere existence of a resubmission is uninformative. By pre-reveal secrecy (Theorem 1), Ctrue is computationally independent of its plaintext before Tr , so the coercer cannot tell whether it re-encrypts ΠA or is a defection to Πtrue . The surrendered randomness ri certifies only the first ciphertext and reveals nothing about the second. The receipt thus loses all binding power. Scope. As is standard for this style of guarantee, the defense covers coercion that relies on a pre-reveal receipt. It does not cover a coercer who physically controls the voter at the instant of the final submission, nor one who acts after Tr once all policies are public; protecting the post-reveal phase needs orthogonal tools such as anonymous credentials, which we leave to future work.

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6

Experimental Setup and Modeling

Evaluating the behavioral impact of sealed versus transparent delegation requires observing how voters form delegation networks under different information regimes. Because no large-scale sealed delegation system is currently deployed, we evaluate our system using a data-driven experimental framework calibrated on real-world voting and demographic data. This approach allows us to isolate the effects of visibility and herding while keeping the underlying voter preferences empirically grounded. All reported results are averaged over repeated trials with independent random delegate draws, and we report 95% confidence intervals so that differences between conditions can be assessed statistically rather than read off a single run. In many real-world settings, not all voters participate equally. When decisions are complex, some individuals may abstain because they feel less informed, while others may abstain because they lack time or confidence. Classical models of political participation link abstention to decision cost [23]. In our experiments, we focus on the case where lower-confidence or less-informed agents abstain from direct voting and instead delegate their voting power.

6.1

Datasets

Most of our data comes from participatory budgeting (PB), a now widely adopted democratic process in which residents vote to allocate part of a public budget across proposed projects [17]; the design and computational analysis of PB voting rules is an active research area [7, 43]. We evaluate on four sources of real voting data, spanning participatory budgeting and general elections, chosen specifically to test whether our findings generalize beyond a single electorate. • Aarau (Stadtidee) [44]: a municipal participatory-budgeting election in which 1704 citizens voted directly, without delegation, on local projects under a fixed budget; 222 of them additionally completed demographic and political surveys. This is our primary dataset, as it uniquely pairs real ballots with a survey rich enough to calibrate a survey-based expertise measure and homophilous delegate selection based on age, district, and political self-placement. • Pabulib (20 datasets) [25]: twenty real participatory-budgeting elections from Warsaw, Amsterdam, and Budapest, with 6,000 to 96,000 voters each. These provide ballots and project costs, and the Warsaw instances include voter age and, for some, neighborhood. Lacking a survey, we proxy competence by voter engagement, that is, the number of projects a voter approved. • CES 2022 (US elections) [48]: the Cooperative Election Study common content, a nationally representative survey of 60,000 US adults. We use it to test the expertise hypothesis with an objective competence measure, a political-knowledge battery asking which party controls the U.S. House 16

and Senate, together with rich demographics. Here a “ballot” is a set of binary support/oppose positions on 34 policy items spanning guns, abortion, health care, and immigration, and the outcome is the per-item majority. • Zürich PB lab study [53]: a pre-registered behavioural experiment with 180 participants voting on a fictional Zürich PB under multiple input formats. We use it as a PB robustness check; its own finding, that citizens perceive the Method of Equal Shares as fairer than the conventional greedy rule, informs our choice of aggregation rule. The delegation model below is described for Aarau and applied to each dataset using the voter features it provides.

6.2

Data and Model Calibration

To ground our experiments empirically, we primarily use a participatory budgeting dataset from Aarau, Switzerland [44]. In this real-world election, 1704 citizens selected local projects under a fixed budget constraint. Crucially, voters in this election did not delegate; they voted directly. We use their direct ballots as the ground truth preferences. Because the fundamental goal of liquid democracy is to approximate the collective will of the electorate even when individual participation is uneven, the direct aggregation of this complete, un-delegated population serves as a robust empirical baseline. It represents the true democratic preference of the community, allowing us to accurately measure both the representational degradation caused by abstention and the degree of recovery provided by different delegation regimes. Among the electorate, 222 participants completed additional demographic and political surveys. We use these survey responses to construct a synthetic expertise proxy for each voter, detailed in Appendix A.1. We divide these 222 participants by expertise into the top 10% as high-expertise, the next 40% as mid-expertise, and the bottom 50% as low-expertise. In the experiments below, the low-expertise group acts as the delegators. Independent delegate selection. In our model, delegators tend to choose delegates who are easier for them to trust or recognize. In a real-world liquid democracy platform such as LiquidFeedback, this process is facilitated by a searchable delegate directory. Voters can browse public profiles containing self-disclosed demographic information, neighborhood affiliations, and political leanings. Rather than assuming voters magically identify the objectively best expert, we assume they use this directory to select delegates who are socially, geographically, or politically relatable. We operationalize this using similarity in age, district, and political self-placement. The detailed procedure used for the Stadtidee dataset is given in Appendix A.2.1. Delegate selection with herding. In the transparent setting, delegators can additionally observe which delegates are already attracting support. We model 17

this by increasing the probability of choosing a delegate who is already popular. This captures visibility-based reinforcement effects documented in models of information cascades and social influence [12, 8, 9, 47]. The detailed procedure is given in Appendix A.2.2. Personal vs. representative delegation. Depending on the information regime, delegates may adopt different representation styles. Delegates can have a personal representation style, where they vote entirely based on their own personal preferences. However, in certain real-world information regimes, a platform may provide delegates with anonymized, aggregated demographics of their delegators. Under these conditions, a delegate might adopt a representative style, actively trying to represent their constituency rather than voting solely for their own interests. We model this by allowing the delegate to compute an aggregate preference for their constituency, comprising the delegate and their delegators. Because the Stadtidee voting system uses binary approval ballots, we consider two natural aggregators. Under a mean rule, a project is approved if its mean approval rate across the constituency is at least 50%; under a median rule, a project is approved if a majority of the constituency approves it. We evaluate both, since the choice of aggregator turns out to materially affect the result (Section 7.4).

6.3

Outcome Accuracy Metric and Aggregation Rules

Accuracy metric. We treat the outcome produced by all 1704 voters as the ground-truth benchmark. Rather than comparing outcomes only by project overlap, we measure outcome accuracy as the structural agreement between an outcome and the benchmark, using a Condorcet-based pairwise measure [40]. For every unordered pair of projects (a, b), we determine which project wins in pairwise comparison under both the benchmark and the outcome of interest; accuracy is the fraction of project pairs on which the two outcomes agree. We use the term accuracy consistently throughout, where higher is better and 1.0 denotes perfect agreement with the full electorate. Aggregation rules. Because the winning bundle depends on how approval ballots are aggregated under the budget constraint, and because a finding that holds only for one rule would be fragile, we evaluate three aggregation rules. Our default is the Method of Equal Shares (MES), a proportional rule with strong fairness guarantees for participatory budgeting [43]. As robustness checks we also report a utilitarian greedy rule, which orders projects by approval count and adds them while the budget allows, and a cost-efficiency greedy rule, which orders projects by approvals per unit cost. Unless stated otherwise, results use MES and a fixed budget of CHF 50,000.

18

6.4

Delegate Failure Model

Delegation concentrates the responsibility for casting many ballots onto a small set of delegates, so the reliability of the outcome depends on those delegates actually voting. We therefore model delegate failure explicitly. A failed delegate submits no ballot before the tallying deadline, for example due to inactivity, abstention, technical failure, or a denial-of-service attack, and consequently both their own vote and the votes routed to them are at risk. We study two failure modes: 1. Uniform random failures: a fraction pfail of delegates, chosen uniformly at random, fail. This captures benign unreliability. 2. Targeted failures: the pfail fraction of delegates with the highest delegated voting power fail. This captures an adversary who attacks the most consequential hubs, or correlated failure of the most-relied-upon delegates. Against these we compare three designs. The single-delegate baseline uses only the first-ranked delegate. Ranked delegation lets each voter rank up to three delegates and reroutes them to their highest-ranked surviving delegate. Ranked delegation with a personal fallback ballot additionally lets a voter whose ranked delegates all fail revert to their own direct ballot. We assume the resolution itself, comprising decryption, rerouting, and tallying, is executed by the public reveal-phase protocol of Section 4.3; failure pertains only to whether a delegate submits a valid ballot, not to the integrity of the tally.

7

Experimental Evaluation

We organize the evaluation around four questions. First, how much does selective abstention degrade outcome accuracy? Second, does delegation recover that accuracy, and does the information regime, sealed versus transparent, matter? Third, how do representation style and aggregation rule affect the result? Fourth, how robust is the outcome to delegate failures? Throughout, accuracy is the agreement metric of Section 6.3, reported as a mean over 50 repetitions for the regime comparisons or 30 repetitions for the failure analysis, with 95% confidence intervals, so that differences are assessed statistically rather than from a single run.

7.1

Baseline: Abstention Degrades Outcome Accuracy

Before introducing abstention or delegation, we verify that the 222 survey respondents are broadly representative of the full electorate. Aggregating only their ballots reproduces 96% of the pairwise comparisons in the 1704-voter benchmark, indicating strong alignment between the subsample and the full population. We then consider a counterfactual in which the least-expert 50% of the 222-voter subgroup abstains. When only the remaining ballots are aggregated 19

under MES, outcome accuracy drops to 72.9%. This does not imply that all real abstention is caused by low expertise; rather, it shows that if one subgroup is systematically less likely to participate, the collective outcome can change substantially. This 72.9% is the baseline that delegation must improve upon.

7.2

Does Delegation Improve Accuracy? A Conditional, Dataset-Dependent Answer

We next ask whether delegation can recover the accuracy lost to abstention. The bottom 50% of voters delegate to experts in the top 10% using the homophilous selection mechanism (Section 6.2), and we vary the information regime (sealed vs. transparent) and the representation style (personal vs. representative). Figure 2 reports outcome accuracy with 95% confidence intervals over 50 repetitions.

Figure 2: Outcome accuracy (agreement with the 1704-voter ground truth) by delegation regime under the Method of Equal Shares, averaged over 50 repetitions with 95% confidence intervals. Every delegation regime improves on the 72.9% abstention baseline, but the sealed and transparent personal regimes are statistically indistinguishable; the median representative ballot is the strongest single regime. On Aarau, delegation clearly helps. Under MES, sealed personal delegation reaches an accuracy of 0.790 ± 0.009, a statistically significant improvement of +0.061 over the 0.729 abstention baseline (Wilcoxon signed-rank, p < 10−14 ). The effect of the information regime, however, is small. Transparent personal delegation reaches 0.797±0.011, statistically indistinguishable from sealed personal delegation (∆ = −0.007, p = 0.37). In other words, once delegate selection is grounded in real homophily data and averaged over many delegation

20

Table 2: Does delegation recover the accuracy lost to abstention? The benefit tracks the recoverable gap (1−abstention accuracy) and appears only for Aarau, whose survey-defined abstainers are strongly unrepresentative. CES uses an objective knowledge battery on 60k US voters and is our most credible test. Dataset Aarau (222) Pabulib (20 PB) CES 2022 (60k, US)

competence measure survey expertise engagement objective knowledge

abstention acc. 0.729 0.85–0.96 0.971

personal deleg. 0.790 −0.01 to −0.02 0.917

representative 0.827 (median) ≈ neutral 0.971

draws, sealing the formation phase does not by itself produce a measurable gain in outcome accuracy. An earlier single-run comparison had suggested a sealed advantage, but that advantage does not survive repeated trials. This motivates a more careful reading of the mechanism. The benefits of sealing are structural and adversarial, as shown in Sections 7.5 and 7.6, rather than a direct accuracy gain in this flat, two-level electorate. Does the recovery generalize? A cross-dataset view. The Aarau recovery is real but turns out to be conditional. We re-ran the comparison on twenty further participatory-budgeting elections from Pabulib and on 60,000 US voters from the CES 2022 study, the latter using an objective politicalknowledge measure rather than a survey or engagement proxy, as reported in Table 2. Delegation recovers accuracy in Aarau, by +0.061, but not elsewhere. Across the twenty Pabulib elections the median change is slightly negative, and on CES, our most credible test, personal delegation to experts actually reduces accuracy from 0.971 to 0.917, while representative-style delegation is neutral. The pattern has a simple explanation. Delegation can only recover accuracy in proportion to the recoverable gap 1 − (abstention accuracy). This gap is large in Aarau, at 0.271, because its survey-defined low-expertise abstainers are systematically unrepresentative, but small everywhere else, at most 0.15. When the non-participating majority already matches the electorate, there is little to recover, and importing the demographically and ideologically skewed preferences of high-competence experts via personal delegation pushes the outcome away from the electorate. The robust, dataset-independent conclusion is therefore narrower than any single number suggests. Expertise-based delegation improves representational accuracy only when abstention is large and systematically skewed; otherwise it is accuracy-neutral at best, and representative-style delegation is consistently safer than personal-expert delegation.

7.3

When Does Delegation Help? Law

A Recoverable-Gap

The cross-dataset pattern above suggests a single underlying mechanism. Delegation can only recover what abstention removed, so its benefit should scale with the recoverable gap γ = 1 − (abstention accuracy). We test this directly on the CES electorate with a controlled, semi-synthetic experiment. Rather 21

than fixing one abstaining group, we deliberately vary how ideologically skewed it is, tying the probability of abstaining to a voter’s ideology through a skew parameter, so that as skew grows the abstaining bloc becomes more one-sided and the gap γ widens. At each skew level we measure how much delegation changes accuracy for two delegate targets. Under representative delegation a voter delegates to a homophilous participant, and under expertise delegation a voter delegates to a member of the knowledge elite.

Figure 3: When delegation recovers the outcome, as a function of the recoverable gap γ = 1 − (abstention accuracy) (CES 2022, abstention skewed by ideology; mean over 5 seeds). Below a threshold gap, delegation is neutral-toharmful, delegating to the knowledge elite reduces accuracy when abstention is near-representative. Representative (homophilous) delegation recovers accuracy first and is safest at low-to-moderate skew but fades under extreme one-sided abstention; expertise delegation needs a larger gap to pay off but is stable and recovers most under extreme skew. Figure 3 reveals a clear boundary. When abstention is near-representative (γ ≈ 0), delegation does not help, and delegating to the knowledge elite actively harms accuracy (−0.05), since importing a competent but demographically and ideologically distinct bloc moves the outcome away from the electorate. As γ grows, both forms of delegation cross zero and begin to recover the outcome, with the crossover for expertise delegation near γ ≈ 0.07. The two targets trade off. Representative delegation recovers accuracy first and dominates at lowto-moderate skew, because it re-injects the abstainers’ own preferences through similar delegates, but its benefit fades under extreme one-sided abstention, when almost no similar participant remains to delegate to; expertise delegation requires a larger gap but is stable and recovers the most (+0.06) under extreme skew. This places every result in the paper on a single curve. Our real datasets 22

Table 3: Outcome accuracy by delegation regime under three aggregation rules (mean over 50 repetitions). The best regime per rule is in bold. The ranking is rule-dependent, with personal delegation strong under MES but weak under the greedy rules, where representative ballots dominate. Regime Abstention (no delegation) Sealed personal Transparent personal Sealed representative (mean) Sealed representative (median) Transparent representative (mean)

Equal Shares 0.729 0.790 0.797 0.754 0.827 0.742

Greedy 0.812 0.732 0.698 0.857 0.817 0.869

Cost-efficiency 0.790 0.739 0.744 0.865 0.838 0.852

fall where the law predicts. Pabulib and the objective-knowledge CES analysis sit at small γ (left edge), where delegation is neutral-to-harmful, while Aarau, whose survey-defined low-expertise abstainers are strongly unrepresentative (γ = 0.27), sits far to the right, where delegation recovers accuracy. The practical takeaway is a precise condition. Delegation improves representational accuracy only when abstention is sufficiently large and skewed; below that threshold it is at best neutral, representative-style delegation is the safer default, and delegating to a competence elite is justified only when abstention is severely one-sided.

7.4

Representation Style and Aggregation Rule

Representation style has a larger and more nuanced effect than the information regime. Under MES, forcing experts to cast a constituency mean ballot lowers accuracy to 0.754 ± 0.012, significantly below personal voting (∆ = +0.036, p < 10−4 ), because averaging dilutes an expert’s sharper preferences. Replacing the mean with the median (majority) rule reverses this conclusion, raising accuracy to 0.827±0.014, the best of all regimes, and significantly above the mean variant (∆ = +0.073, p < 10−9 ). The aggregator for representative ballots is therefore not a cosmetic choice. More importantly, the ranking of regimes depends on the budget aggregation rule (Table 3). Under MES, personal delegation is competitive and the median representative is best. Under the utilitarian greedy and cost-efficiency rules, the picture changes: representative ballots dominate (0.85–0.87), while personal delegation can fall below the abstention baseline, because copying a few experts’ ballots distorts the raw approval counts those rules rely on. The robust, rule-independent conclusion is therefore narrower than any single number suggests: data-grounded delegation that preserves the aggregate preference of each constituency recovers a large part of the accuracy lost to abstention, but which regime is best is sensitive to the social choice rule, underscoring the value of evaluating several.

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7.5

Structural Vulnerability: Voting Power Concentration

Outcome accuracy is only part of the story. The structure of the delegation graph determines how exposed the system is to manipulation and failure. We measure voting-power concentration using the Gini coefficient of delegated voting power. In the data-grounded two-level electorate, in which non-experts delegate to a fixed pool of experts, transparency increases concentration only modestly, because the pool of eligible delegates is small and delegation is not transitive. The Gini coefficient rises from ≈ 0.44 with no herding to ≈ 0.48 under strong herding at h = 1.5, as shown in Section 7.7. The effect becomes severe once delegation is allowed to be transitive, delegates themselves delegating to more popular delegates, as in deployed platforms. To isolate this structural mechanism, we complement the data-driven study with an abstract agent-based model in which visibility drives sequential delegate selection over multiple hops, and voting share is measured transitively along delegation chains, for example A → B → C. In this regime the transparent setting collapses into an oligarchy. The effective number of controlling delegates falls to ≈ 1 (Figure 4a) while the maximum transitive voting share of a single delegate approaches 1.0 (Figure 4b), as nearly all delegation chains funnel into one “super-delegate” (Figure 5a). Under sealing, by contrast, power remains spread across roughly 11 effective delegates with a maximum share near 0.2 (Figure 5b). This rich-get-richer dynamic matches concentration observed in early LiquidFeedback deployments [34]. The sealed regime, by removing the visibility signal during formation, prevents this coordination and keeps voting power distributed among local experts. Crucially, this structural difference exists even where the two regimes are indistinguishable in raw outcome accuracy (Section 7.2): sealing protects the health and attack-resistance of the delegation graph rather than the accuracy of a single uncontested tally, as the failure analysis below makes concrete. This concentration of power is the structural root of the systemic fragility we analyze next. The more voting power funnels into a few delegates, the more damage a small number of failures, especially targeted ones, can do.

7.6

Robustness: Outcome Accuracy Under Delegate Failures

We now turn to the question that ultimately matters for any deployment. When delegates fail, what happens to the election outcome? Using the failure model of Section 6.4, we remove a fraction pfail of delegates, uniformly at random or targeting the most-relied-upon delegates, and measure two quantities for the single-delegate, ranked, and ranked-plus-fallback designs, namely the resulting outcome accuracy and the vote-loss, or abstention, rate. Outcome accuracy. Figure 6 reports outcome accuracy as pfail grows. Under uniform random failures, accuracy is remarkably stable for all three designs, 24

a) Effective number of delegates

b) Max Voting Share

Figure 4: Voting power concentration under sealed versus transparent delegation in the abstract transitive delegation model (mean over 15 runs, 95% CI). The transparent regime collapses to a single controlling delegate (left: effective number of delegates ≈ 1 vs. ≈ 11 when sealed) that transitively amasses nearly all voting share (right: max share ≈ 1.0 vs. ≈ 0.2). In the flat two-level datagrounded model the gap is far smaller (Section 7.7).

Figure 5: Delegation graphs in the transitive structural model; node colour and size encode transitive voting weight. (a) Under the transparent regime, the rich-get-richer visibility dynamic causes massive centralization, with almost all delegation chains funnelling into a single “super-delegate.” (b) Under the sealed regime, voting power remains distributed across many local delegates. staying within roughly 0.79 to 0.81; losing a random handful of delegates does not systematically bias the outcome, because the surviving electorate still carries

25

Figure 6: Outcome accuracy as a function of the fraction of failing delegates, for uniform random failures (left) and targeted attacks on the most-relied-upon delegates (right). Shaded bands are 95% confidence intervals over 30 repetitions. Under targeted attacks, the single-delegate design loses accuracy while ranked delegation with a personal fallback ballot preserves it. Table 4: Vote loss (fraction of the electorate left unresolved) at pfail = 0.3, mean over 30 repetitions. Ranked delegation with a personal fallback ballot is the only design robust to targeted attacks. Design Single delegate Ranked delegation Ranked delegation + fallback ballot

Uniform failures 17.1% 3.3% 2.7%

Targeted failures 30.2% 7.7% 2.7%

the aggregate signal. Under targeted attacks the difference becomes clear. At pfail = 0.3 the single-delegate design degrades to 0.770 ± 0.022, whereas ranked delegation with a personal fallback ballot preserves accuracy at 0.818 ± 0.026, the highest at every failure level. Ranking provides alternative routing, and the fallback ballot guarantees that a voter whose ranked delegates are all removed still contributes their own preference instead of dropping out. Vote loss. The designs differ even more sharply in how much of the electorate they disenfranchise (Table 4). Under targeted attacks at pfail = 0.3, the single-delegate design loses 30.2% of ballots, ranked delegation loses 7.7%, and ranked delegation with fallback loses only 2.7%; under uniform failures the single-delegate loss is 17.1%. A coarse accuracy metric can therefore mask substantial disenfranchisement: it is the combination of ranked delegation and a personal fallback ballot that simultaneously minimizes vote loss and preserves outcome accuracy, precisely under the targeted, correlated failures that power concentration makes most dangerous. These results refine the trade-off inherent in delegation systems. Delega-

26

tion can improve outcomes by routing votes toward better-informed agents, but it concentrates risk. Ranked multi-delegation combined with a personal fallback ballot turns delegation from a fragile single-point-of-failure structure into a resilient routing mechanism that preserves both participation and outcome accuracy, even under targeted attacks. The resilience result replicates across 20 real elections. Because vote loss is a structural property of the routing scheme, independent of the aggregation rule and of the particular electorate, it should generalize. We confirm this directly by applying the same failure model to all twenty Pabulib datasets (Figure 7). Ranked delegation with a fallback ballot reduces the vote loss of a single-delegate system from 26.4% to 3.0% under targeted attacks at pfail = 0.3, and from 17.9% to 3.0% under uniform failures, with confidence intervals across datasets of well under one percentage point. This is the most robust of our empirical findings and, together with the structural and security results, forms the core of the case for the proposed mechanism.

Figure 7: Vote loss under delegate failures, averaged over 20 real participatorybudgeting datasets (Pabulib), with 95% confidence intervals across datasets. Ranked delegation with a personal fallback ballot keeps vote loss near 3% even when 30% of delegates are targeted, versus 26% for a single-delegate design. The effect is highly consistent across all twenty elections.

7.7

Sensitivity Analysis

Because the homophily model has several free parameters, we verify that the conclusions do not hinge on specific values, as shown in Figure 8. Sweeping the district bonus βg ∈ [1, 5], the age scale σa ∈ [5, 80], and the political scale σp ∈ [0.5, 8] leaves sealed outcome accuracy essentially flat, within overlapping confidence intervals around 0.79 to 0.81, confirming that the homophily defaults are not cherry-picked. The herding exponent h behaves as expected. Increasing h monotonically raises power concentration (Figure 8a) and, beyond h ≈ 1,

27

begins to erode outcome accuracy, consistent with strong popularity-following reducing the diversity of delegated judgment. The default h = 0.3 sits in the mild-herding regime.

Figure 8: Sensitivity analysis. (a) Increasing the herding exponent h raises power concentration (Gini of voting power) in the transparent regime. (b) Sealed outcome accuracy is flat across the homophily parameters βg , σa , and σp , indicating the defaults are not cherry-picked. Bands/bars are 95% confidence intervals over 20 repetitions.

7.8

Limitations

Only the Aarau dataset provides a survey rich enough to calibrate the full homophily model with a survey-based expertise measure; the additional datasets use weaker competence signals, namely engagement for Pabulib and the Zürich study and an objective knowledge battery for CES, and the CES analysis reframes the task from budget-constrained PB to policy-issue majorities. Competence in all cases is a proxy rather than a perfect ground truth. Homophily features also vary by dataset, and the larger datasets were subsampled for tractability. Finally, our data-grounded model is two-level, with non-experts delegating to experts, and therefore understates the transitive concentration that arises in deployed multi-level systems, which we approximate only with the abstract model of Section 7.5. Extending the study to multi-level delegation, and to settings with both real ballots and an objective competence measure, is the most important direction for strengthening external validity.

8

Conclusion and Future Work

This paper studied liquid democracy as a mechanism for trusted delegation in settings where voters differ in information, time, confidence, and trust relations. While delegation can help less-informed voters remain represented, we showed that the information regime governing delegation formation funda-

28

mentally shapes system behavior. Transparent delegation introduces visibilitydriven reinforcement dynamics that can distort delegate selection and concentrate influence. We proposed a liquid democracy mechanism with sealed delegation formation using timed-release cryptography, combined with ranked multi-delegation and fallback ballots to ensure robustness under delegate failure. Experiments across four datasets, a municipal PB election with a survey, twenty further PB elections, 60,000 US voters with an objective knowledge measure, and a PB lab study, yield a consistent, nuanced picture. Whether delegation improves representational accuracy is governed by a recoverable-gap law. Delegation recovers accuracy in proportion to how large and unrepresentative abstention is, and below a threshold it is neutral or even harmful, since delegating to a competence elite can move the outcome away from the electorate, while representative-style delegation is the safer default. The value of sealed formation is therefore not a direct accuracy gain but is structural, reducing power concentration and the resulting vulnerability to targeted delegate failures. The most robust empirical result, replicated across all twenty PB elections, is that ranked delegation with a personal fallback ballot keeps vote loss near 3% even when 30% of delegates are targeted, versus 26% for a single delegate, simultaneously minimizing disenfranchisement and preserving outcome accuracy. More broadly, the paper argues for a specific timing of transparency rather than for secrecy as a permanent principle. Delegation choices are hidden only while they are being formed, and are disclosed once the system moves to counting and verification. This makes the system less vulnerable to social pressure during formation while preserving public auditability afterward. Future work. Future research will extend this work in several directions. First, field deployments in live civic platforms or decentralized autonomous organizations (DAOs) are needed to measure the real-world behavioral impact of sealed versus transparent delegation. Second, advanced game-theoretic modeling could explore how adversaries might attempt off-chain coordination or signaling when on-chain transparency is delayed. Finally, integrating the mechanism with privacy-preserving decentralized identity systems would allow delegates to prove domain expertise or community standing using verifiable credentials, fostering informed delegation while protecting voter anonymity. Additionally, while posting individual ciphertexts to a public ledger introduces scalability challenges for large electorates, this overhead can be mitigated by off-chain aggregation. Future implementations will explore the use of zeroknowledge (ZK) rollups, where an aggregator compresses many ciphertexts and their validity proofs into a single on-chain transaction, substantially reducing the gas costs associated with sealed delegation formation [51].

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A

Stadtidee Dataset Delegation Modeling

A.1

Expertise Modeling

We construct an expertise score for each of the 222 survey participants based on self-reported political efficacy and interaction frequency with municipal actors. This score is only a proxy used in our experiments. It does not claim to measure objective expertise perfectly, but it provides a simple and interpretable way to distinguish more confident and more civically engaged participants from less confident ones. Let Ei denote the expertise score of voter i. The score is computed as a weighted linear combination of normalized responses to the following survey items: 1. “I can understand and assess important political issues on the local level well.” (weight 0.4) 2. “How strongly do you feel connected to Aarau?” (weight 0.2) 3. Frequency of interaction with: • Members of the city council (weight 0.05) • Members of the residents’ council (weight 0.05) • Members of the city administration (weight 0.05) • Other inhabitants of Aarau (weight 0.05) 4. “I have the confidence to actively participate in a conversation about local politics.” (weight 0.2) All survey responses are linearly normalized to [0, 1] before aggregation. The final expertise score is given by Ei = 0.4q1,i + 0.2q2,i + 0.2q4,i + 0.05(q3.1,i + q3.2,i + q3.3,i + q3.4,i ), where qk,i denotes the normalized response of voter i to question k. The selection of these specific survey items and their relative weights is theoretically grounded in the political science concepts of internal political efficacy and social capital. Internal political efficacy, an individual’s self-perception of their ability to understand and participate in political processes, is a wellestablished and primary predictor of political knowledge and active participation [42]. Accordingly, we assign the highest combined weight, 60%, to direct self-assessments of political understanding in Question 1 and communicative confidence in Question 4. The remaining 40% of the weight captures social capital and civic embeddedness, which are critical for local-level, municipal expertise [45]. A strong connection to the municipality in Question 2 and frequent interactions with local political actors and other inhabitants in Question 3 indicate that a voter is actively embedded in the community’s information network, granting them access to localized knowledge that an isolated voter would lack. 34

A.2

Expert Selection

A.2.1

Independent Expert Selection

To operationalize homophilous delegate selection, we extract three key demographic and ideological variables from the survey data: age, geographic location, and political orientation. Geographic proximity is determined using respondents’ self-reported residential districts, which are categorized into 18 distinct municipal neighborhoods such as Innenstadt, Zelgli, and Altstadt. Age proximity is calculated directly from self-reported age in years. Finally, ideological alignment is measured using a standard 11-point political self-placement scale, where respondents position themselves on a scale from 0 for “left” to 10 for “right”. Let Ndel ⊆ N be the subset of delegators, the low-expertise group, and Nrep ⊆ N be the subset of eligible delegates, the mid- and high-expertise groups. Each delegator i ∈ Ndel selects a delegate j ∈ Nrep with probability proportional to a selection weight wij . This weight combines three simple signals that make a delegate more recognizable or more plausible from the delegator’s point of view. The use of these three specific variables is strongly motivated by the sociological principle of homophily, the well-documented tendency of individuals to associate and bond with similar others [41]. In political and social networks, shared geographic context, generational alignment, and ideological proximity have repeatedly been shown to be primary determinants of trust and interpersonal information flow. By incorporating these factors, our model realistically captures how voters identify relatable, trustworthy delegates based on shared identity and local interests, rather than assuming they possess the ability to objectively rank all delegates by true expertise. (i) Geographic proximity. Voters residing in the same administrative district may share local knowledge and local interests. We apply a discrete bonus: ( βg if gi = gj , δij = (1) 1 otherwise, where βg > 1 is a tunable district-bonus parameter, with default value βg = 2. (ii) Age proximity. Age is used as a simple proxy for generational closeness. We apply an exponential decay over the absolute age difference:   |ai − aj | , (2) αij = exp − σa where σa > 0 controls how quickly the score decreases as age difference grows. The default is σa = 20 years. (iii) Political proximity. Voters may also prefer delegates with similar political orientation. Using self-reported left-right position pi , we apply the analogous

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decay:   |pi − pj | πij = exp − , σp

(3)

where σp > 0 controls tolerance for political distance, with default value σp = 2. Respondents who answered “don’t know” are assigned πij = 1 for all j, so this term contributes no information. The three components are multiplied to obtain the final selection weight: wij = δij αij πij .

(4)

Each delegator i then draws a delegate from Nrep according to the normalized distribution, where i → j denotes the event that i delegates to j: Pr[ i → j ] = P

wij

k∈Nrep wik

A.2.2

.

(5)

Sequential delegation with herding

In the herding variant, non-experts delegate sequentially in a uniformly random order. Each voter i, arriving at position t in the sequence, can observe the running delegation count cj (t), that is, the number of delegators who have already chosen delegate j before i’s turn. Popularity is incorporated directly into the selection weight through the scaling
h w̃ij (t) = wij 1 + cj (t) ,

(6)

where h ≥ 0 is the herding-strength parameter. The delegate is then drawn from the normalized distribution over w̃ij (t). This formulation subsumes the baseline case. At h = 0, delegation counts have no influence and the rule reduces to Equation (5). As h increases, popularity amplifies the underlying selection weight, with h = 1 corresponding to a linear multiplicative interaction between private choice and visible popularity, and h > 1 causing popularity to dominate more strongly. The use of a continuous exponent, rather than a binary switch between independent and herding behavior, reflects the observation that social influence typically grows gradually rather than appearing all at once [39].

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