CORE and Strategic Manipulation An Addendum Kōmyō (Hiveism) and Claude (Anthropic) 2026-03-10 1. The Question The CORE paper proves that honest behavior — accepting a proposal if and only if it exceeds your threat point — is weakly dominant for the acceptance decision. Follow- ing publication, a substantive objection arose in discussion (in particular from cdsmith on r/EndFPTP): Gibbard’s theorem (1973, extended to lotteries in 1978) establishes that any mechanism with more than two alternatives is either dictatorial or strategically manipulable. CORE is non-dictatorial and has more than two alternatives. Therefore CORE must be strategic in ways the paper doesn’t address. This objection has valid form, and it points to something the original paper left un- derspecified. The original paper’s claims about weak dominance are correct for the acceptance decision, but the paper does not fully address strategic behavior in the deliberation phase that precedes acceptance. This note fills that gap. The short answer: CORE is not strategy-proof in Gibbard’s technical sense. But it achieves something different — what we call strategy-resilience . Any attempt at strate- gic manipulation can be counteracted by other participants, and the expected outcome cannot be pushed below proportional fairness at any level of the decision hierarchy. This is a property Gibbard’s theorem does not rule out. 1 Contents 1. The Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. The CORE Mechanism (Recap) . . . . . . . . . . . . . . . . . . . . . . . 3 3. What Gibbard’s Theorem Says . . . . . . . . . . . . . . . . . . . . . . . . 3 4. Why CORE Appears to Violate the Theorem . . . . . . . . . . . . . . . . 3 5. Why the Theorem Doesn’t Directly Apply . . . . . . . . . . . . . . . . . 4 6. The Subtle Point About Deliberation . . . . . . . . . . . . . . . . . . . . 4 7. Strategy-Resilience vs Strategy-Proofness . . . . . . . . . . . . . . . . . 5 7.1 Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7.2 Why CORE Satisfies Strategy-Resilience . . . . . . . . . . . . . . 6 7.3 The Role of Convexity . . . . . . . . . . . . . . . . . . . . . . . . 7 7.4 The Core Claim . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 8. The Acceptance Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 9. What CORE Actually Guarantees . . . . . . . . . . . . . . . . . . . . . . 8 10. The Bargaining Interpretation . . . . . . . . . . . . . . . . . . . . . . . 10 11. Relationship to the Original Paper . . . . . . . . . . . . . . . . . . . . . 10 12. Responding to the Critique . . . . . . . . . . . . . . . . . . . . . . . . . 11 13. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 2. The CORE Mechanism (Recap) For readers unfamiliar with the original paper, CORE (Consensus Or Random Exclu- sion) works as follows: 1. A group of voters deliberates freely, proposing and discussing alternatives 2. At any time, any voter may trigger random exclusion 3. Upon triggering, one voter is selected uniformly at random and removed from the group 4. The reduced group continues deliberating 5. The process terminates when unanimous agreement is reached (guaranteed when one voter remains) The threat point 𝑇 𝑖 (𝑆) is the expected utility for voter 𝑖 when the group 𝑆 proceeds to random exclusion without reaching agreement. The original paper proves that this threat point is computable via bottom-up dynamic programming over subsets, without modeling other voters’ strategies. Honest behavior — accepting a proposal 𝑃 if and only if 𝑢 𝑖 (𝑃 ) ≥ 𝑇 𝑖 (𝑆) — is weakly dominant for the acceptance decision. When no agreement is ever reached at any stage, sequential random exclusion to a single remaining voter is equivalent to random ballot: each voter has equal probability 1/𝑛 of being the final decider. CORE can only improve on this baseline when mutually acceptable proposals exist. For the full formal treatment, see the original paper. 3. What Gibbard’s Theorem Says Gibbard (1977) proved that any strategy-proof social decision scheme - a mechanism mapping preference profiles to probability distributions over outcomes - must be a convex combination of two types of restricted schemes: • Unilateral schemes: One voter determines the outcome; others’ votes are irrel- evant • Duple schemes: The outcome is restricted to a fixed pair of alternatives The key word is convex combination with fixed probabilities . A strategy-proof mecha- nism must be decomposable into a predetermined weighted mixture of unilateral and duple components, where the weights are fixed independently of the votes cast. A corollary (credited to Sonnenschein) establishes that the only strategy-proof mech- anism satisfying both anonymity and ex post Pareto efficiency is random dictatorship - selecting a voter uniformly at random and implementing their top choice. 4. Why CORE Appears to Violate the Theorem CORE seems to satisfy properties that Gibbard’s theorem says cannot coexist: • It improves over random ballot when unanimous solutions exist • It has more than two possible outcomes (the full proposal space) • It is non-dictatorial (no single voter determines the result) • The original paper claims honest behavior is weakly dominant 3 If CORE were a mechanism in Gibbard’s sense, the theorem would imply it must be strategic. The critic’s syllogism is valid - the question is whether the premises apply. 5. Why the Theorem Doesn’t Directly Apply CORE is not a fixed probability mixture in Gibbard’s sense. The mechanism has two modes: 1. Unanimous agreement: Everyone accepts a proposal; that outcome is imple- mented 2. Random fallback: No unanimity is reached; the process iterates with random exclusion, converging to proportional randomness The critical difference from Gibbard’s framework: which mode activates is not deter- mined by fixed weights assigned in advance. It depends on whether deliberation pro- duces a proposal everyone prefers to the fallback. The “probability” of each mode is not a parameter of the mechanism but an emergent property of the preference profile and the deliberation process. Gibbard’s theorem requires that a strategy-proof mecha- nism be decomposable into a predetermined weighted mixture of unilateral and duple components with fixed weights. CORE’s weights are endogenous — they arise from the interaction between preferences and deliberation. An analogy may help, though the technical point above is what matters. CORE is like a system at a critical point between phases. Water at the triple point is not “a fixed mixture of solid, liquid, and gas” — it’s poised to fall into whichever phase the local conditions favor. Similarly, CORE manifests as different mechanism types depending on whether unanimity emerges. Gibbard’s theorem classifies mechanisms that are already in one of these phases. It doesn’t address systems that can transition between them based on outcomes. 6. The Subtle Point About Deliberation The critic’s deeper concern is that finding unanimous agreement involves strategy. If Alice knows others will accept proposal P, she might hold out for something bet- ter. If Bob anticipates Alice’s holding out, he might adjust his behavior. Doesn’t this strategic complexity in deliberation “bleed back” into the mechanism? This concern points to something real, but the structure of CORE addresses it in a specific way. Consider what “unanimous agreement” means. For a proposal to be unanimously accepted, every voter must genuinely prefer it to the random fallback. This is not a voting threshold that can be gamed - it’s the definition of Pareto improvement over the outside option. Now consider the strategic possibilities: Strategic acceptance: Alice accepts a proposal she doesn’t actually prefer to the fall- back. But this is irrational, not strategic. She could guarantee herself the fallback 4 value T_i(S) by rejecting. Accepting something worse is simply a mistake. Strategic rejection: Alice rejects a proposal she does prefer to the fallback, hoping to extract something even better. This is genuine strategic behavior. But it’s observable — others see that unanimity fails. And anyone can respond by pulling the random exclusion trigger themselves. (The strength of this observability claim depends on the communication protocol. In face-to-face deliberation, rejection is immediately apparent. In more structured settings, the protocol should be designed so that failed unanimity is common knowledge. The key requirement is that participants can distin- guish “no proposal on the table yet” from “a proposal was made and someone rejected it.”) The critical insight: if agreement is strategic in the sense of misrepresenting prefer- ences, then it’s not unanimous. If agreement is unanimous, then it’s honest - everyone really does prefer the outcome to the fallback. You cannot fake unanimity because the fallback is always available. Strategic rejection doesn’t break the mechanism; it simply activates the random fall- back. The fallback is the honest answer when genuine agreement doesn’t exist. This reveals a structural point about how CORE relates to Gibbard’s impossibility. Gibbard shows that deterministic aggregation of conflicting preferences must be ei- ther dictatorial or strategic. But CORE conditionally avoids the domain where this conclusion applies. When genuine unanimity exists, the outcome is determined by real agreement — there is nothing to game, because everyone prefers the result to the alternative. When unanimity does not exist, the resolution is random — there is noth- ing to game, because the outcome is determined by uniform chance. The strategic middle ground that Gibbard identifies — where a deterministic rule must aggregate conflicting preferences and therefore creates manipulation opportunities — simply never activates. CORE is deterministic only when preferences don’t conflict, and ran- dom only when they do. 7. Strategy-Resilience vs Strategy-Proofness The preceding analysis points to a distinction that existing social choice vocabulary does not capture. We propose the following refinement. Strategy-proofness (Gibbard’s sense): For every voter, truthful reporting of prefer- ences is a weakly dominant strategy. No individual deviation from honest input can improve a voter’s expected outcome, regardless of what others do. Strategy-resilience (what CORE achieves): Any attempt at strategic manipulation can be counteracted by other participants, and the expected outcome cannot be pushed below proportional fairness. Gains above the baseline require genuine unanimous consent, and these guarantees hold recursively at every level of the decision hierarchy. Strategy-proofness makes honesty individually optimal regardless of others’ behavior. Strategy-resilience makes fairness the guaranteed outcome because any manipulation attempt can be counteracted — there is no fixed deadline forcing voters to commit to 5 preferences that can then be exploited. The distinction matters because Gibbard’s theorem shows strategy-proofness is unachievable for non-dictatorial mechanisms with more than two alternatives. Strategy-resilience is a different kind of property — one that Gibbard does not rule out. 7.1 Formal Definition Let 𝐵 𝑖 = 1 𝑛 ∑ 𝑗∈𝑉 𝑢 𝑖 (𝑎 ∗ 𝑗 ) denote voter 𝑖 ’s expected utility under random ballot (pro- portional randomness). This is the baseline — what each voter receives when no agreement exists and the mechanism reduces to random dictatorship. Definition (Strategy-resilience). A mechanism 𝑀 with proportional baseline 𝐵 is strategy-resilient if, for every subgame-perfect equilibrium 𝜎 of the extensive-form game induced by 𝑀 : (i) Floor. 𝐸 𝜎 [𝑢 𝑖 ] ≥ 𝐵 𝑖 for all voters 𝑖 No equilibrium can push any voter below their proportional share. (ii) Pareto surplus. If unanimous agreement is reached on outcome 𝑃 , then 𝑢 𝑖 (𝑃 ) ≥ 𝑇 𝑖 (𝑆) for all voters 𝑖 in the agreeing group 𝑆 , where 𝑇 𝑖 (𝑆) ≥ 𝐵 𝑖 Any outcome above the baseline requires genuine consent from all participants. Gains cannot be extracted from unwilling voters. (iii) Recursive fairness. Let 𝑄 ⊆ 𝐴 be any subset of achievable outcomes that arises as a sub-decision within 𝑀 — for instance, choosing among multiple Pareto improve- ments. The induced mechanism on 𝑄 is itself strategy-resilient with respect to the proportional baseline over 𝑄 The same guarantees hold at every level of the decision hierarchy: choosing among outcomes, choosing among Pareto improvements, choosing among distributions over Pareto improvements, and so on. 7.2 Why CORE Satisfies Strategy-Resilience Floor guarantee. At any point in the process, any voter 𝑖 can unilaterally trigger ran- dom exclusion. Under sequential uniform random exclusion to completion, each voter survives to be the sole decider with probability 1/𝑛 (by symmetry — the sequential removal is equivalent to a uniform random permutation). Therefore triggering exclu- sion guarantees 𝑖 an expected utility of 𝐵 𝑖 . Since this deviation is always available, any subgame-perfect equilibrium must give 𝑖 at least 𝐵 𝑖 Pareto surplus. The only way to terminate with an outcome different from the random-exclusion baseline is unanimous agreement on some proposal 𝑃 by the current group 𝑆 Unanimity means every voter in 𝑆 has accepted, which — by the weak dominance result of Section 8 — implies 𝑢 𝑗 (𝑃 ) ≥ 𝑇 𝑗 (𝑆) for all 𝑗 ∈ 𝑆 Voters excluded before agreement was reached were removed by uniform random selection; no strategy controlled who was removed. Their ex ante expected utility is determined by the symmetry of random exclusion, giving them 𝐵 𝑖 regardless of 6 others’ strategies. So excluded voters receive their baseline, remaining voters receive at least their baseline, and any surplus above baseline is genuinely consensual. Recursive fairness. Suppose multiple Pareto improvements 𝑎, 𝑏, 𝑐 exist and voters disagree about which to implement. This disagreement is itself a decision problem. If voter 1 strategically obstructs 𝑏 and 𝑐 to force acceptance of 𝑎 , voters 2 and 3 can re- spond by obstructing 𝑎 . When no Pareto improvement is agreed upon, the mechanism falls back to random exclusion on the original problem. The key structural fact: random exclusion is equally likely to remove any voter, in- cluding the obstructor. Coordinated obstruction of particular alternatives does not systematically shift which alternative is eventually selected, because the composition of the group after exclusions is determined by uniform random sampling. Averaging over all possible exclusion sequences (weighted by probability), each voter’s influence on the sub-decision is proportional — the same guarantee as at the top level, now ap- plied within the space of Pareto improvements. 7.3 The Role of Convexity Why does the proportional fixed point exist at every meta-level? Because the space of achievable outcomes, once mixed strategies (lotteries) are permitted, is convex. If 𝐿 1 and 𝐿 2 are both achievable lotteries, so is any mixture 𝛼𝐿 1 + (1 − 𝛼)𝐿 2 . The set of Pareto improvements over any baseline is also convex (by linearity of expected utility). Within any convex set of contested alternatives, the proportional mixture — giving each voter’s preferred alternative its fair share — is a well-defined interior point. Open-ended deliberation is what allows the mechanism to access these meta-levels. When disagreement arises at any level, participants can propose mixed strategies over the contested alternatives. There is no fixed protocol that must resolve the dis- agreement in a particular way, and therefore no protocol that can be exploited. The proportional baseline at each level is always available as a fallback. 7.4 The Core Claim Strategy-resilience can be summarized in one sentence: at every level of the decision hierarchy, the worst case under strategic play is the proportional baseline for that level. Strategic behavior in CORE can cause delay — obstruction, failed proposals, repeated exclusions. But it cannot shift the expected outcome below proportional fairness. Not because strategy is impossible, but because the only unilateral action available (trig- gering exclusion) is symmetric in its effects, and the only way to achieve an outcome different from the baseline (unanimous agreement) requires everyone’s genuine con- sent. This is what distinguishes CORE from mechanisms where Gibbard’s theorem bites hardest. In standard mechanisms, a voter’s report can asymmetrically affect the out- come — my ballot can shift the result toward my preferred candidate at your expense. 7 In CORE, my only unilateral power move (trigger exclusion) affects everyone symmet- rically, and my only asymmetric influence (persuasion in deliberation) requires your consent to change anything. Symmetric individual action combined with unanimous collective action forecloses any path to asymmetric advantage. CORE is strategic in Gibbard’s technical sense — the full game admits strategic behav- ior. Strategy-resilience is the claim that this strategicness is bounded and symmetric: it cannot produce unfair outcomes, only delay fair ones. 8. The Acceptance Decision The CORE paper’s claim about weak dominance of honest behavior concerns a spe- cific decision: accepting or rejecting a proposal on the table. Given a proposal P, should voter i accept? The paper proves: accepting iff u_i(P) ≥ T_i(S) is weakly dominant for this decision. This is correct. The decision is binary. The alternatives are: (a) accept, contributing to possible unanimity, or (b) reject, triggering random exclusion. If everyone else accepts, you get P by accepting and T_i(S) in expectation by rejecting. If anyone else rejects, you get T_i(S) either way. So accepting when u_i(P) ≥ T_i(S) is weakly dominant. A crucial structural point: T_i(S) is determined entirely by the mechanism’s recursive structure and voter utilities. It does not depend on what happens during deliberation — not on which proposals are made, in what order, or with what strategic intent. What- ever deliberation dynamics produce or fail to produce, the threshold against which any proposal is evaluated remains fixed. This is why the acceptance decision is clean even when the deliberation that precedes it is not: strategic behavior during deliber- ation may change which proposals reach the table, but cannot change the standard against which they are judged The critic’s point is that there’s more to the process than this single decision. Before reaching a proposal P, there’s deliberation. During deliberation, strategic behavior is possible. This is true, but it’s a feature rather than a bug. CORE doesn’t eliminate strategic considerations from deliberation — it channels them. Strategic holding-out in deliber- ation either produces a better proposal (if others accommodate) or produces deadlock (if they don’t). Deadlock activates the fair fallback. As established in Section 7, the mechanism provides a floor at every level of the decision hierarchy that no strategy can breach. 9. What CORE Actually Guarantees In the language of Section 7, CORE is strategy-resilient. This can be unpacked con- cretely. Guaranteed (the three conditions of strategy-resilience): 8 The floor: no voter can be pushed below their proportional share in expectation. 𝑇 𝑖 (𝑆) ≥ 𝐵 𝑖 is an inviolable lower bound, available to any voter by unilaterally trig- gering exclusion. Pareto surplus: improvement over random ballot is possible when Pareto improve- ments exist, but only through genuine unanimous consent. Any outcome above the baseline reflects the real preferences of all participants in the agreeing group. Recursive fairness: when multiple Pareto improvements exist and voters disagree about which to implement, the same guarantees apply to this sub-decision. No voter can exploit the choice among improvements to extract disproportionate surplus. The proportional baseline over the contested improvements is the fallback for this level too. What about surplus distribution? A natural concern: even if no voter falls below the floor, a strategic voter might influence which Pareto improvement is selected, captur- ing more of the surplus from agreement. This is the real strategic prize, and Section 7’s recursive fairness condition addresses it directly. Strategic obstruction of partic- ular improvements is countered by others’ ability to obstruct in turn, and deadlock at this level falls back to proportional randomness over the improvements — or, if no agreement is reached at all, to the original baseline. The surplus distribution that emerges from deliberation may not be the theoretically optimal one, but no voter can systematically tilt it in their favor at equilibrium. Not guaranteed: That the best Pareto improvement is found. Deliberation might miss it, settle on a sub- optimal one, or fail to reach agreement at all. Strategy-resilience guarantees fairness, not optimality. That strategic play doesn’t cause delays or inefficiency. Obstruction wastes time. The mechanism guarantees that delay doesn’t shift expected outcomes, but the time cost is real. That deliberation terminates in any particular timeframe. CORE is a process, not a function. If external constraints impose a genuine deadline (a building is on fire, a legal filing date), that is a fact about the situation, not a feature of the mechanism. Ar- tificially imposing a time limit would itself be a pre-decision that advantages whoever benefits from the default when time expires — reintroducing exactly the kind of fixed structure that creates strategic leverage. Not claimed: That CORE is strategy-proof in Gibbard’s sense. It is not. The full game admits strate- gic behavior. That deliberation is free of strategic considerations. It is not. Deliberation is where strategic complexity lives. That voters have no reason to model others’ behavior. They do. What they cannot do is profit from such modeling beyond their proportional share. 9 10. The Bargaining Interpretation CORE is better understood as bargaining with a fair outside option than as preference aggregation. Traditional voting mechanisms ask: given conflicting preferences, how do we select a single outcome? This is the domain Gibbard’s theorem addresses. The theorem shows that any such aggregation must be either dictatorial or manipulable. CORE asks a different question: given a fair fallback (proportional randomness), can we find something everyone prefers? If yes, implement it. If no, the fallback is the answer. The fallback — random ballot — is symmetric by construction. It treats all voters equally in expectation. The mechanism doesn’t impose a theory of “fair division of surplus” when agreement exists. Which Pareto improvement is found, and how sur- plus is distributed, emerges from deliberation. This is actually appropriate, and the recursive structure of strategy-resilience explains why. Any fixed rule for distributing surplus would itself be contestable — and choos- ing such a rule is itself a decision problem to which CORE applies. If voters disagree about how to divide the gains from agreement, that disagreement is handled at the meta-level with the same proportional fallback. The mechanism doesn’t need to pre- scribe a surplus-division rule because it provides the tools to negotiate one, with fair- ness guaranteed at every level of the negotiation. In this sense, CORE is not a decision mechanism in the classical sense. It does not ag- gregate conflicting preferences into an outcome. It uncovers agreement when agree- ment exists and defaults to fair randomness when it does not. The process itself does not discriminate between options — only unanimous agreement does, and unanimity is non-strategic by definition. 11. Relationship to the Original Paper Does this addendum invalidate the CORE paper’s claims? We think not. The paper claims honest acceptance is weakly dominant. This is correct for the ac- ceptance decision, and the invariance of 𝑇 𝑖 (𝑆) to deliberation dynamics (Section 8) is what makes this hold regardless of strategic behavior in deliberation. The paper claims the optimal threshold 𝑇 𝑖 (𝑆) is computable without modeling others’ strategies. This is correct — 𝑇 𝑖 (𝑆) depends only on utilities and the mechanism’s structure. The paper doesn’t claim that deliberation is strategy-free. It claims that strategic con- siderations collapse to a single threshold comparison. This is also correct: whatever happens in deliberation, the final decision point is binary, and the optimal response at that point is computable. One point of clarification: the original paper characterizes strategy collapse as “stronger than strategy-proofness.” This should be understood as applying to the ac- 10 ceptance decision specifically — for that decision, the voter needs no game-theoretic reasoning at all, not even the knowledge that honesty is dominant. For the full deliberation game, however, CORE is not strategy-proof in Gibbard’s sense. The appropriate property for the full game is strategy-resilience, not strategy collapse. Strategy collapse describes what happens at the decision point; strategy-resilience describes what happens across the entire process. What the paper doesn’t fully articulate is the property we have here called strategy- resilience — the formal guarantee that strategic play in deliberation is bounded by proportional fairness at every level of the decision hierarchy, and that the recursive structure of the mechanism ensures this bound holds not only for the top-level deci- sion but for all sub-decisions about how to distribute surplus from agreement. This note fills that gap. 12. Responding to the Critique The critic is right that CORE is not strategy-proof in Gibbard’s technical sense. Strate- gic behavior exists in deliberation. We concede this fully. But the critic’s conclusion — that CORE is “manifestly strategic” and therefore flawed — does not follow. The implicit assumption is that a mechanism is either strategy- proof or defective. Strategy-resilience is a third possibility that Gibbard’s framework does not consider. Strategic behavior in CORE is observable: others can see when unanimity fails and respond accordingly. It is counterable: no strategic move goes unanswered, because any voter can trigger the fair fallback. It is bounded: the floor guarantee ensures no voter falls below their proportional share at any level of the decision hierarchy. And it is symmetric in its worst case: the fallback treats all voters equally. The mechanism isn’t a fixed probability mixture of unilateral and duple schemes. It is a dynamic process where the only unilateral action (triggering exclusion) is symmet- ric and the only way to move beyond the baseline (unanimous agreement) requires genuine consent. This combination — symmetric individual power, unanimous col- lective power — is what forecloses asymmetric advantage. The deeper point is that Gibbard’s theorem tells us some strategy profile involves manipulation. It does not tell us whether manipulation is profitable at equilibrium, or whether it can shift outcomes below a fair baseline. Strategy-resilience addresses exactly these questions, and the answers are: manipulation is not profitable beyond one’s proportional share, and the fair baseline holds at every level of the decision hierarchy. There is also a subtler way in which CORE transcends Gibbard’s framework. Gibbard classifies mechanisms as either deterministic (and therefore dictatorial or strategic) or random (and therefore not really deciding anything). CORE is neither in a fixed sense. Whether any given stage of the process resolves deterministically (through unanimous agreement) or randomly (through exclusion) is not a parameter of the mechanism — it is an emergent property of the preference landscape. Predicting 11 which mode will activate at which point would require knowing the outcome of de- liberation in advance, which is akin to solving the halting problem for the group’s collective reasoning. In the absence of such prediction, the mechanism occupies a third category — neither deterministic nor random but responsive to what the actual preferences allow. This is not itself random; it is conditioned on whether genuine agreement exists. 13. Conclusion CORE is not strategy-proof. Gibbard was right that you cannot have strategy- proofness plus efficiency plus non-dictatorship in a mechanism with fixed structure over more than two alternatives. CORE achieves strategy-resilience instead: a fair floor that no manipulation can breach, surplus that requires genuine unanimous consent, and recursive fairness that extends these guarantees to every level of the decision hierarchy — including the choice among competing Pareto improvements. Strategic play is possible but bounded and symmetric. It cannot produce unfair outcomes, only delay fair ones. The improvement over random ballot comes from finding unanimous solutions — gen- uine Pareto improvements where everyone prefers the outcome to the fallback. When preferences genuinely conflict and no Pareto improvement exists, fair randomness is the honest answer. CORE delivers exactly that. This is a different solution concept from Gibbard’s framework. Not incentive compati- bility, but equilibrium fairness with a neutral outside option at every level. The mech- anism does not resolve fundamental disagreement through some clever aggregation scheme. It surfaces agreement when agreement exists, and defaults to proportional fairness when it doesn’t — all the way down. 12