Unanimous Vote Problem Overview

• Unanimous Vote Problem is a framework for decision protocols that ensure a unanimous outcome while preserving privacy and robustness against adversaries.
• It employs techniques like distributed bit sharing, simultaneous broadcast, and rigorous verification to guarantee correctness even under adverse conditions.
• Applications range from secure electronic voting and blockchain consensus to quantum computing protocols, balancing efficiency, fault tolerance, and strategic resilience.

The Unanimous Vote Problem encompasses a rigorous class of decision, aggregation, and protocol design tasks that require a group to reach a unanimous or near-unanimous decision, with the system guaranteeing either correctness (the output reflects unanimous input), privacy, robustness against manipulation, or efficiency under uncertainty. This concept finds formal treatment in secure multiparty computation, voting theory, consensus protocols, computational social choice, and related fields across mathematics, computer science, and economics.

1. Formal Definition and Problem Scenarios

At its core, the Unanimous Vote Problem requires a protocol or mechanism such that given $n$ agents or voters, if all (or nearly all) of their ballots or inputs agree on a decision $x$, the system outputs $x$, while preserving privacy and robustness even in adversarial environments. In distributed systems, this appears as the consensus safety condition: $\forall i, \; v_i = x \implies d_i = x$ where $v_i$ are inputs and $d_i$ final decisions. In voting, criteria such as Pareto efficiency or Arrow’s "unanimity" axiom formalize this requirement (Wood et al., 7 Sep 2024). Social choice settings may impose additional structure when ballots are preference rankings or subject to uncertainty (Bahel, 8 Jan 2024).

The problem encompasses (but is not limited to):

• Secure voting protocols with unconditional privacy and correctness even without an honest majority (0806.1931, Broadbent et al., 2010)
• Binary or multi-issue aggregation, including settings with information constraints or noisy individual signals (Chierichetti et al., 2011, Constantinescu et al., 2023)
• Anonymous or strategy-proof procedure design under variable state spaces (Bahel, 8 Jan 2024)
• Mechanisms for worst-case guarantees in adversarial environments (Moulin, 2021)
• Consensus in distributed systems under crash or Byzantine faults, with k-set or $\epsilon$-approximate agreement (Wood et al., 7 Sep 2024)
• Algorithmic coin-flipping orderings optimizing expected query cost for unanimous detection (Keles et al., 19 Oct 2025)

2. Secure Multiparty Voting Protocols

The information-theoretic voting protocols of (0806.1931) and (Broadbent et al., 2010) provide a canonical solution in settings without honest majority. These protocols implement key primitives:

• Distributed Bit Sharing: Each participant holds a random share; only the overall parity (XOR) yields meaningful information. For a distributed bit $b$, each holds $b_k$ such that $\bigoplus_k b_k = b$.
• Simultaneous Broadcast: All parties broadcast at once, eliminating adaptive message selection, with implementation relying on bit commitment for everlasting security.
• Basic Protocols and Authorities: Protocol 1 (direct interaction), Protocol 2 (introduction of authorities), Protocol 3 (utilizes cut-and-choose verification, distributed equality testing via $\oplus_{i=1}^r (a_i \oplus b_i) = 0$).
• Robustness Against Dishonest Participants: Even a coalition with all but one honest participant cannot learn individual ballots or affect the tally. Verification steps (random auditing and equality checks) revoke malformed ballots.

In the unanimous setting, if all votes are identical, the protocol’s statistical tally ($p_v$ and Chernoff bound guarantees) ensures correct output with overwhelming probability and unconditional privacy. Computational requirements remain polynomial with complexity governed by the number of candidates, voters, and a security parameter.

3. Information Aggregation and Signal-Based Voting

In decision contexts where voters receive private signals, e.g., the Condorcet Jury scenario, analysis from (Chierichetti et al., 2011) quantifies how mechanisms (plurality, cumulative, Condorcet voting) amplify correct selection:

• Plurality Voting: Each voter maps their signal to an option. To assure with probability $1-\eta$ that the correct option wins, $m = O(n^3 \epsilon^{-2} \ln(1/\eta))$ voters are required for $n$ options with minimal separation $\epsilon$ of signal distributions.
• Expressive Ballots (Cumulative, Condorcet): These require only $O(\epsilon^{-2} \ln(1/\eta))$ voters, matching the efficiency of centralized detection.
• Implications: While plurality voting is communication-minimal and widespread, its efficiency is poor for large $n$ or similar options. Richer ballot formats provide dramatically more efficient information aggregation.

4. Mechanism Design Under Uncertainty, Adversariality, and Strategy-Proofness

Mechanism design solutions incorporate worst-case guarantees, neutrality, anonymity, and strategy-proofness:

• Worst-Case Guarantees (Anonymous Mechanisms): For $n$ agents over $p$ outcomes, the uniform lottery $UNI(p)$ provides an unimprovable guarantee if $n \geq p$ (Moulin, 2021). For $p > n$, random dictator and veto mechanisms (and their combinations) form the basis of the full space of maximal guarantees, which geometrically comprises a finite union of polytopes.
• Range Unanimity and Canonical Factorization: Under uncertainty, range unanimity (the system must select an act unanimously top-ranked within its endogenous range) is the minimal efficiency axiom compatible with anonymity and strategy-proofness. Any social choice function satisfying these factors canonically decomposes into constant and binary factors, with four main types (simple, quasi-dictatorial, dyadic, iso-filtering) (Bahel, 8 Jan 2024).
• Sequential Unanimity Rules and Coalition Structures: For binary choices, sequential unanimity rules—searching predetermined voter subsets for unanimity—are equivalent to M-winning coalition rules and fully characterize neutral, strategy-proof, and fair unanimous voting (Athanasoglou et al., 20 Feb 2024).
• Algorithmic Findings: In multi-issue binary voting, as in (Constantinescu et al., 2023), finding a compromise policy (winning in the vote and close, in Hamming distance, to the majority policy) can be accomplished deterministically in polynomial time. Guaranteeing distance $\lfloor (t-1)/2 \rfloor$ from the majority policy is always possible; improving this bound is NP-hard but fixed-parameter tractable (FPT) in natural parameters.

5. Distributed and Anonymous Consensus: Computational and Blockchain Frameworks

Distributed consensus mechanisms model the Unanimous Vote Problem as a safety property for agreement protocols with adversarial tolerance:

• Arrovian Framework: Unanimity implies if all correct processes vote $x$, then $d_i = x$ for all. Relaxations (k-set agreement, $\epsilon$-approximate agreement) preserve exact unanimous decision in the unanimous input case but allow divergence when inputs differ (Wood et al., 7 Sep 2024). Topological impossibility proofs demarcate boundary conditions for agreement with crash faults.
• Byzantine and Anonymous Consensus: AVCP protocol (Cachin et al., 2019) leverages anonymous broadcast and traceable ring signatures to aggregate votes privately, achieving consensus within $O(n^3)$ message complexity. The protocol is empirically validated for group sizes up to 100 nodes with global geographical distribution.
• LLM-Driven Deliberation and Blockchain Consensus: A deliberation-based consensus framework using multi-agent LLMs (Pokharel et al., 2 Apr 2025) provides a protocol for both definite unanimous consensus (all agents converge on the same answer) and graded consensus for prioritized decisions. The protocol maintains distributed ledger properties (consistency, agreement, liveness, determinism) and addresses challenges such as LLM hallucination, malicious nodes, and scalability.

6. Quantum and Stochastic Evaluation Variants

• Quantum Majority Vote: In the quantum regime, the optimal protocol for outputting the majority state of $n$ product qubits achieves fidelity $1/2 + \Theta(1/\sqrt{n})$ in the worst case, with improved fidelity asymptotically under higher majority guarantees. The algorithm proceeds via Schur transform and a convex mixture of two extremal unitary-equivariant quantum channels, with parameters determined by linear programming (Buhrman et al., 2022).
• Stochastic Boolean Function Evaluation: The Unanimous Vote Problem as presented in (Keles et al., 19 Oct 2025) asks, given $n$ biased coins, to find an ordering minimizing the expected number of flips needed to certify unanimity or detect its failure. The problem is notable for its exact algorithm running in $O(n \log n)$ time and a tight adaptivity gap of $1.2 + o(1)$ relative to the best adaptive strategy.

7. Synthesis and Technical Implications

The Unanimous Vote Problem is multifaceted, uniting protocol design, mechanism theory, computational complexity, and practical systems engineering around a central collective action property—unanimity. State-of-the-art protocols achieve information-theoretic privacy and correctness without an honest majority, rigorous guarantees under worst-case adversaries, efficient aggregation even with severe communication constraints, and robust implementation in distributed and quantum systems.

Technical advances illuminate barriers (NP-hardness, information bottlenecks, fault tolerance limits), comparative efficiency (ballot expressivity vs voter requirements), and guarantee structures (polytope geometry, modular factorization, coalition decomposition).

Future work concerns optimally balancing practicality, privacy, fairness, and robustness across new application domains: electronic voting, decentralized ledgers, multi-agent deliberation, consensus under uncertainty, and under adversarial as well as quantum settings.