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. Following publication, a substantive objection arose in discussion (in particular from cdsmith on r/EndFPTP): Gibbard’s theorems establish that any non-dictatorial mechanism with more than two alternatives admits strategic behavior. CORE is non-dictatorial and has more than two alternatives. Therefore CORE must be strategic. This objection is correct. The original paper’s claims about weak dominance hold 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. CORE is not strategy-proof in Gibbard’s technical sense, and we do not claim oth- erwise. What CORE achieves is something different — a property we call strategy- resilience , defined and developed in Section 6. Contents 1. The Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. The CORE Mechanism (Recap) . . . . . . . . . . . . . . . . . . . . . . . 2 3. Gibbard’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4. CORE is Strategic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5. What Gibbard’s Theorems Do and Do Not Show . . . . . . . . . . . . . . 3 6. Strategy-Resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6.1 The Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6.2 Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6.3 Why CORE Satisfies Strategy-Resilience . . . . . . . . . . . . . . 5 6.4 The Role of Convexity . . . . . . . . . . . . . . . . . . . . . . . . 5 6.5 The Structural Reason . . . . . . . . . . . . . . . . . . . . . . . . . 6 7. The Acceptance Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 8. What CORE Actually Guarantees . . . . . . . . . . . . . . . . . . . . . . 7 1 9. CORE as Process, Not Aggregation . . . . . . . . . . . . . . . . . . . . . 8 10. Relationship to the Original Paper . . . . . . . . . . . . . . . . . . . . . 9 11. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 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. Gibbard’s Theorems There are three relevant results by Gibbard, progressively more general. Gibbard 1973 applies to deterministic game forms — any process where an outcome depends on individual actions. The theorem states: if a deterministic game form has at least three possible outcomes and is not dictatorial, then it is not straightforward. That is, there exist situations where a player’s optimal action depends on what others do. This applies to any process of collective decision, not just voting rules. Gibbard 1977 applies to social decision schemes — mechanisms that map preference profiles to probability distributions over outcomes. Any strategy-proof social decision scheme must be a convex combination of unilateral schemes (one voter determines the outcome) and duple schemes (the outcome is restricted to a fixed pair of alternatives). A corollary credited to Sonnenschein establishes that the only strategy-proof scheme satisfying anonymity and ex post Pareto efficiency is random dictatorship. Gibbard 1978 extends this to game forms with lotteries as outcomes. Gibbard defines a game form as “any system which makes an outcome depend on individual actions of some kind, called strategies,” and defines it to be straightforward if every player, for every utility function, has a dominant strategy — a complete contingency plan that is 2 optimal regardless of what others do. The theorem proves that any straightforward game form (with finitely many alternatives, players, and strategies) is, on an adequate domain, a probability mixture of unilateral and duple game forms. The 1978 result is the most relevant to CORE, since CORE is a process where partic- ipants take actions (propose, accept, reject, trigger exclusion), not merely a function from preference reports to outcomes. 4. CORE is Strategic We concede fully: CORE is not straightforward in Gibbard’s sense. The 1978 theorem requires that every player always have a dominant strategy — a complete plan for the entire game that is optimal no matter what anyone else does. In CORE, the “entire game” includes open-ended deliberation: what to propose, when to propose it, how to respond to others’ proposals, when to trigger exclusion. No complete contingency plan for this full game is guaranteed to be dominant. The CORE paper’s claim of weak dominance applies to the specific binary ac- cept/reject decision, not to the full strategy space. For the accept/reject decision, the voter faces a simple comparison: is this proposal better than my threat point? That comparison requires no modeling of others. The full game — including deliberation — is a different matter, and it is there that the interesting questions arise. Not whether strategic behavior exists, but what can be said about its character . This is what the rest of the note addresses. 5. What Gibbard’s Theorems Do and Do Not Show Gibbard’s theorems establish that CORE admits strategic behavior. They do not es- tablish that this behavior can produce unfair outcomes, push any voter below their proportional share, or be exploited asymmetrically. These are separate questions, and they are the questions that matter for mechanism design in practice. There is also a deeper point about the scope of these results. Gibbard’s 1978 theorem assumes finitely many alternatives, finitely many players, and finitely many strategies for each player (a limitation Gibbard himself notes). A “strategy” in this framework is a complete contingency plan — a predetermined specification of what to do in every possible state of the game. This is appropriate for mechanisms where voters submit ballots and an algorithm produces a result. CORE is a different kind of object. Its deliberation is open-ended: participants can generate novel proposals, discover solutions nobody initially considered, and always respond to what others do. A “complete contingency plan” for an open-ended de- liberative process is not a finite object. It would need to specify responses to every possible proposal (an unbounded space), in every possible sequence of exclusions, at every possible stage of deliberation. Any formal model that represents this as a finite strategy set must either be incomplete (not covering all possible moves) or must ar- tificially constrain the deliberative process. This is not a technicality — it reflects a 3 genuine feature of CORE. The mechanism works precisely because participants can al- ways transcend the current situation by finding a new solution. This open-endedness is what prevents any fixed strategic structure from being exploited asymmetrically. We do not claim this places CORE “outside” Gibbard’s theorems. If one models CORE as a game form with a fixed (if large) strategy set, the theorem applies and the con- clusion (not straightforward) holds. Rather, the formal framework captures some but not all of what makes CORE work. What it misses is the dynamic by which strategic exploitation is counteracted — the subject of the next section. 6. Strategy-Resilience 6.1 The Concept Gibbard’s framework draws a binary distinction: a mechanism is either strategy-proof (straightforward) or it admits strategic manipulation. Existing social choice vocabu- lary does not distinguish between mechanisms where strategic behavior can produce unfair outcomes and mechanisms where it cannot. We propose the following refine- ment. 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 preferences that can then be exploited. 6.2 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 4 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. 6.3 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 7 — 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 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. 6.4 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 5 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. 6.5 The Structural Reason Why can’t strategic behavior in CORE produce asymmetric advantage? Because the only unilateral action available to any voter — triggering exclusion — is symmetric in its effects. And the only way to achieve an outcome different from the symmetric baseline — unanimous agreement — requires everyone’s genuine consent. When a voter attempts strategic manipulation, other voters can respond. In mech- anisms with fixed structure (submit ballots, algorithm produces result), there is no opportunity to respond — the manipulation succeeds or fails based on the fixed rules. In CORE, there is always a response available: counter-propose, refuse to agree, or trigger the fair fallback. This back-and-forth of strategy and counter-strategy cannot produce asymmetric advantage because both sides hold the same symmetric threat: cooperate, or the proportional baseline. The mechanism forces participants to oper- ate at the highest level of equal influence — genuine cooperation or fair randomness. There is no breaking down into competing factions with different structural leverage. 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. 7. 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 — 6 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 — de- liberation precedes it, and deliberation is strategic. This is true. But CORE channels strategic deliberation rather than eliminating it: holding out either produces a better proposal or produces deadlock, and deadlock activates the fair fallback. The floor established in Section 6 holds at every level. 8. What CORE Actually Guarantees In the language of Section 6, CORE is strategy-resilient. This can be unpacked con- cretely. Guaranteed (the three conditions of strategy-resilience): 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 6’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. 7 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, that deliberation is free of strategic considerations, or that voters have no reason to model others’ behavior. What voters cannot do is profit from strategic modeling beyond their proportional share. 9. CORE as Process, Not Aggregation 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 theorems address. The theorems show that any such aggregation — any function from preferences to outcomes — 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. And the recursive structure of strategy- resilience explains why this is appropriate: any fixed rule for distributing surplus would itself be contestable — and choosing such a rule is itself a decision problem to which CORE applies, with the same proportional fallback at every level. CORE is not a decision mechanism in the classical sense. It does not aggregate con- flicting preferences into an outcome. It uncovers agreement when agreement exists and defaults to fair randomness when it does not. The process itself does not dis- criminate between options — only unanimous agreement does. When there is gen- uine unanimity, the outcome is determined by real agreement and there is nothing to game. When there is not, the resolution is random and there is nothing to game. The strategic middle ground that Gibbard identifies — where a deterministic rule must aggregate conflicting preferences and therefore creates manipulation opportunities — never activates in pure form. CORE is deterministic only when preferences don’t conflict, and random only when they do. 8 10. 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 7) 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- 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. 11. Conclusion CORE is strategic in Gibbard’s sense. Any non-dictatorial mechanism with more than two alternatives admits strategic behavior, and CORE is no exception. What CORE achieves is strategy-resilience: 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. Strategic play can- not produce unfair outcomes, only delay fair ones. Gibbard’s theorems tell us that some strategy profile involves manipulation. They do 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 answer in both cases is no. The deeper reason is that CORE’s open-ended deliberation prevents any fixed strate- gic structure from being exploited asymmetrically. Any manipulation attempt can be 9 met with a novel response. When the back-and-forth reaches an impasse, the fall- back is proportional fairness — the unique resolution that preserves symmetry. The mechanism surfaces agreement when agreement exists, and defaults to proportional fairness when it doesn’t — all the way down. 10