CORE: Consensus Or Random Exclusion Addressing Social Choice Impossibility Through Strategy Collapse Kōmyō (Hiveism) and Claude (Anthropic) 2025-12-31 Abstract We present CORE (Consensus Or Random Exclusion), a decision-making mechanism that addresses fundamental impossibility results in social choice theory. Gibbard’s theorem establishes that any deterministic voting mechanism with three or more al- ternatives is either dictatorial or strategically manipulable. While several randomized methods (random ballot, COWPEA Lottery) escape Gibbard, they either offer no room for deliberation or leave voters with threshold decisions whose optimal resolution may depend on beliefs about others. We prove that CORE occupies a distinctive position: it provides genuine voter agency, makes honest behavior weakly dominant, and achieves outcomes that improve upon pure randomness through deliberation. Crucially, the optimal acceptance threshold in CORE is computable without modeling other voters’ strategies - a property we call strategy collapse All strategic considerations reduce to comparing proposals against a recursively- defined threat point. This threat point depends only on true preferences and genuine randomness, not on predicting others’ behavior. From each voter’s perspective, every decision reduces to a binary comparison where honest response is optimal. We further show that CORE achieves cooperation emergence : random exclusion cre- ates identity uncertainty that transforms individual self-interest into unanimous ac- ceptance of mutual gains over a fair fallback, without requiring altruism or external enforcement. Finally, we argue that CORE is an attractor of non-violent decision making - a natural endpoint for groups committed to resolving disagreements without coercion. Una- nimity ensures non-imposition; proportional randomness ensures non-domination; the mechanism ensures mutual gains emerge from self-interest alone. 1 Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part I: The Mechanism and Main Results 4 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1 The Fundamental Problem . . . . . . . . . . . . . . . . . . . . . . 4 1.2 The CORE Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Formal Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Background: Gibbard’s Theorem . . . . . . . . . . . . . . . . . . . . . . 6 3.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4. Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Lemma 4.1 (Computability of Threat Points) . . . . . . . . . . . . . . 7 Lemma 4.2 (Strategy Collapse in CORE) . . . . . . . . . . . . . . . . . 8 Lemma 4.3 (Non-Triviality of Agency) . . . . . . . . . . . . . . . . . . 10 Theorem 4.4 (CORE Existence and Characterization) . . . . . . . . . 10 4.5 Worked Example: Three Voters, Three Alternatives . . . . . . . . . . . 11 4.6 Comparison to Other Randomized Methods . . . . . . . . . . . . . . . . 13 4.6.1 Pure Lottery Methods . . . . . . . . . . . . . . . . . . . . . . . . 13 4.6.2 COWPEA Lottery . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.6.3 The Key Distinction: Deliberation . . . . . . . . . . . . . . . . . 13 4.6.4 Threshold Decisions . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.6.5 What CORE Contributes . . . . . . . . . . . . . . . . . . . . . . 14 4.6.6 Clarification: “Strategy” Vs “Manipulation” . . . . . . . . . . . . 14 5. Intuitive Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.1 The Problem with Deterministic Voting . . . . . . . . . . . . . . . 15 5.2 The Problem with Pure Randomness . . . . . . . . . . . . . . . . . 15 5.3 How CORE Resolves Both Problems . . . . . . . . . . . . . . . . . 15 5.4 The Role of Deliberation . . . . . . . . . . . . . . . . . . . . . . . 16 5.5 The Geometric Picture . . . . . . . . . . . . . . . . . . . . . . . . 16 5.6 Reduction to the Two-Candidate Case . . . . . . . . . . . . . . . . 16 5.7 The Edge of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Part II: Why It Works - Cooperation Emergence 17 6. The Survival Lottery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7. The Cooperation Premium . . . . . . . . . . . . . . . . . . . . . . . . . . 18 8. Acceptance Set Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 19 9. Interest Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 10. The Veil of Ignorance Connection . . . . . . . . . . . . . . . . . . . . . 20 11. Design Principles and Comparison . . . . . . . . . . . . . . . . . . . . . 21 11.1 Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 21 11.2 Comparison to Nash Bargaining and Kalai-Smorodinsky . . . . . 21 Part III: Foundations - Consensus as Non-Violence 22 2 12. The Precondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 13. Two Different Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 14. CORE as Attractor of Non-Violence . . . . . . . . . . . . . . . . . . . . 23 15. Layers of Agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 16. Consensus as Non-Violence, Formalized . . . . . . . . . . . . . . . . . . 24 17. What CORE Claims, and What It Doesn’t . . . . . . . . . . . . . . . . . 25 18. Connection to Recursive Alignment . . . . . . . . . . . . . . . . . . . . 25 Conclusion 26 19. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 20. Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Appendix: Recursive Computation of Threat Points . . . . . . . . . . . . . 29 3 Part I: The Mechanism and Main Results 1. Introduction 1.1 The Fundamental Problem Social choice theory seeks to aggregate individual preferences into collective decisions. A series of impossibility results - Arrow’s theorem (1951), the Gibbard- Satterthwaite theorem (1973, 1975), and related findings - demonstrate that no deterministic aggregation rule can satisfy basic fairness criteria without being either dictatorial or vulnerable to strategic manipulation. These results create an apparent dilemma: • Deterministic mechanisms allow voters to influence outcomes but create in- centives for strategic misrepresentation • Random mechanisms eliminate strategic manipulation but remove voter agency entirely Several mechanisms escape Gibbard through randomness - random ballot, propor- tional score selection, COWPEA Lottery, and others. However, they either: • Offer no room for deliberation (random ballot) • Require voters to submit preferences in advance without opportunity for delib- erative discovery (COWPEA Lottery) We propose that a further refinement is possible: a mechanism where honest behavior is weakly dominant, the optimal acceptance threshold is computable without model- ing others, full deliberative space is preserved, and outcomes can improve upon ran- dom selection when agreement exists. In the ideal case, voters would find an unanimous agreement and voting would be unnecessary. However, if unanimity fails a fallback mechanism is needed. For de- terministic methods, this mechanism is by nature predictable and introduces a non- proportional bias, for example preferring the larger group. The ability to anticipate and exploit this bias is central to the vulnerability of strategic manipulation. Purely random methods avoid this by providing a proportional, unpredictable result, but they do so on the cost of never trying. The simple idea of CORE is to first try to find a con- sensus and use randomness only as fallback. 1.2 The CORE Mechanism CORE operates as follows: 1. Deliberation: Voters discuss, propose alternatives, and seek unanimous agree- ment 2. Voluntary Exclusion: Any voter may trigger random exclusion at any time 3. Random Removal: When triggered, one voter is selected uniformly at random and excluded 4. Iteration: The process continues with remaining voters 4 5. Termination: Unanimity is guaranteed when one voter remains The mechanism creates a dynamic where voters face a choice: accept a proposed solution, or face the uncertain outcome of random exclusion. This threat point - the expected utility of not agreeing - anchors all strategic reasoning. 1.3 Main Results We prove that CORE satisfies four properties: 1. Agency: Voters can influence the outcome through their accept/reject decisions 2. Honesty Dominance: Honest behavior (accept iff utility ≥ threat point) is weakly dominant 3. Non-Triviality: Outcomes can be strictly better than pure randomness through deliberation 4. Strategy Collapse: The optimal acceptance threshold 𝑇 𝑖 (𝑆) is computable with- out modeling other voters’ strategies Deterministic mechanisms fail property 2 (Gibbard). Pure lottery methods (random ballot) satisfy 1-2 but fail 3, while 4 does not apply. Methods like COWPEA Lottery improve on random ballot but lack deliberation - voters submit preferences in advance rather than responding to proposals. The key insight is strategy collapse : every strategic consideration reduces to a single comparison against a computable threat point. This threat point depends only on the recursive structure of the mechanism and the voter’s true preferences, not on strategic modeling of other voters. Crucially, this reduces every multi-alternative decision to a binary choice from each voter’s perspective: “accept this proposal or not.” Honest response to this binary question is weakly dominant. 2. Formal Framework 2.1 Definitions Assumption (Utility Structure). Utilities are cardinal (differences are meaningful) but not interpersonally comparable (we never sum or compare utilities across voters). Each voter’s threshold is computed from their own utility function only. Definition 2.1 (Voting Mechanism). A mechanism 𝑀 is a procedure that takes a finite set of voters 𝑉 , a set of alternatives 𝐴 , and a preference profile 𝑃 = (𝑢 1 , ... , 𝑢 𝑛 ) where 𝑢 𝑖 ∶ 𝐴 → ℝ is voter 𝑖 ’s utility function, and produces an outcome 𝑎 ∈ 𝐴 (possibly probabilistically). Definition 2.2 (Deterministic Mechanism). A mechanism 𝑀 is deterministic if iden- tical inputs always produce identical outputs. 5 Definition 2.3 (Strategy). A strategy for voter 𝑖 is a function mapping the voter’s information state to an action. An honest strategy is one where actions directly reflect true preferences without misrepresentation. Definition 2.4 (Dominant Strategy). A strategy 𝑠 𝑖 is dominant for voter 𝑖 if it max- imizes 𝑖 ’s expected utility regardless of the strategies chosen by other voters. It is weakly dominant if it is never worse than any alternative and sometimes strictly bet- ter. Definition 2.5 (CORE Mechanism). Given voters 𝑉 with |𝑉 | = 𝑛 and alternatives 𝐴 : 1. Voters deliberate to reach unanimous agreement on some 𝑎 ∈ 𝐴 2. Any voter may reject the current state, triggering exclusion 3. Upon rejection, one voter is selected uniformly at random and removed 4. The process continues with 𝑉 ′ = 𝑉 ∖ { excluded voter } 5. Termination occurs upon unanimity (guaranteed when |𝑉 ′ | = 1 ) Definition 2.6 (Threat Point). For voter 𝑖 in voter set 𝑆 , let 𝑘 = |𝑆| denote the number of remaining voters. The threat point is defined recursively: 𝑇 𝑖 (𝑆) = 1 𝑘 ⋅ 𝑊 𝑖 (𝑆 ∖ {𝑖}) + 𝑘 − 1 𝑘 ⋅ 1 𝑘 − 1 ∑ 𝑗∈𝑆∖{𝑖} 𝑊 𝑖 (𝑆 ∖ {𝑗}) where the sum ranges over all voters 𝑗 in 𝑆 other than 𝑖 , and 𝑊 𝑖 (𝑆) is voter 𝑖 ’s ex- pected utility in CORE starting from voter set 𝑆 . The first term represents the case where 𝑖 is excluded (probability 1/𝑘 ); the second term represents the expected value when some other voter 𝑗 is excluded (probability (𝑘 − 1)/𝑘 , uniform over 𝑗 ≠ 𝑖 ). Base case: 𝑊 𝑖 ({𝑖}) = 𝑢 𝑖 (𝑎 ∗ 𝑖 ) where 𝑎 ∗ 𝑖 is 𝑖 ’s ideal alternative. Definition 2.7 (Acceptable Set). For voter 𝑖 in voter set 𝑆 : 𝐴 𝑖 (𝑆) = {𝑎 ∈ 𝐴 ∶ 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑆)} This is the set of alternatives 𝑖 would accept over triggering exclusion. 3. Background: Gibbard’s Theorem Theorem 3.1 (Gibbard 1973). Any deterministic voting mechanism over three or more alternatives is either: 1. Dictatorial: There exists a voter whose preference always determines the out- come, or 2. Manipulable: There exists a preference profile where some voter benefits from misrepresenting their preferences 6 3.1 Interpretation Gibbard’s theorem implies that in deterministic mechanisms, voters face strategic cal- culations: “Given the rules and others’ likely behavior, should I vote honestly or strate- gically?” This creates an infinite regress - voter 𝑖 models voter 𝑗 ’s model of voter 𝑖 ’s model, and so on. This regress is what makes deterministic mechanisms vulnerable to manipulation. CORE escapes by introducing genuine randomness that absorbs strategic uncertainty. 4. Main Results Lemma 4.1 (Computability of Threat Points) For all voters 𝑖 and voter sets 𝑆 , the threat point 𝑇 𝑖 (𝑆) is computable via bottom-up dynamic programming. Proof. The computation proceeds by increasing set size: Stage 1 (Base case): For all singleton sets {𝑖} : 𝑊 𝑖 ({𝑖}) = 𝑢 𝑖 (𝑎 ∗ 𝑖 ) This requires no input from other computations. Stage k: For all sets 𝑆 with |𝑆| = 𝑘 , compute 𝑇 𝑖 (𝑆) and 𝑊 𝑖 (𝑆) using only values from stages 1 through 𝑘 − 1 : 𝑇 𝑖 (𝑆) = 1 𝑘 (𝑊 𝑖 (𝑆 ∖ {𝑖}) + ∑ 𝑗≠𝑖 𝑊 𝑖 (𝑆 ∖ {𝑗})) All terms on the right-hand side involve sets of size 𝑘 − 1 , already computed. Agreement determination: Define the acceptable set intersection: 𝐴 ∗ (𝑆) = ⋂ 𝑗∈𝑆 𝐴 𝑗 (𝑆) = {𝑎 ∈ 𝐴 ∶ ∀𝑗 ∈ 𝑆, 𝑢 𝑗 (𝑎) ≥ 𝑇 𝑗 (𝑆)} If 𝐴 ∗ (𝑆) ≠ ∅ , let 𝑎 ∗ 𝑆 denote any element of this intersection. Agreement on 𝑎 ∗ 𝑆 yields 𝑊 𝑖 (𝑆) = 𝑢 𝑖 (𝑎 ∗ 𝑆 ) for all 𝑖 ∈ 𝑆 If 𝐴 ∗ (𝑆) = ∅ , no agreement is possible; the mechanism proceeds to exclusion and 𝑊 𝑖 (𝑆) = 𝑇 𝑖 (𝑆) Remark on tie-breaking: The choice of which 𝑎 ∗ 𝑆 ∈ 𝐴 ∗ (𝑆) to select does not affect the dominance argument. Any selection rule works because every 𝑎 ∈ 𝐴 ∗ (𝑆) satisfies 7 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑆) for all 𝑖 , making acceptance weakly dominant regardless of which element is chosen. The computation is purely arithmetic over utilities and set enumeration. No modeling of other voters’ strategic behavior is required because 𝑇 𝑖 (𝑆) depends only on: • The recursive structure of smaller games (already computed) • Voter 𝑖 ’s true utility function 𝑢 𝑖 • Other voters’ utility functions (but not their strategic choices) • Genuine randomness (uniform over voters, mechanism-defined) □ Definition 4.1 (Strategy Collapse). A mechanism exhibits strategy collapse if, for each voter, all strategic considerations reduce to a single comparison between the utility of a proposed outcome and a mechanism-defined threshold that is computable without modeling other voters’ strategies. Strategy collapse is stronger than strategy-proofness: it means not only that honesty is optimal, but that determining the optimal action requires no game-theoretic reason- ing about others. The voter need only know their own utilities and the mechanism’s structure. Lemma 4.2 (Strategy Collapse in CORE) In CORE, honest behavior - accepting proposal 𝑎 iff 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑆) - is a weakly domi- nant strategy for all voters. Moreover, CORE exhibits strategy collapse: the threshold 𝑇 𝑖 (𝑆) is computable without modeling others’ strategies. Proof. We show that honest behavior is optimal regardless of other voters’ strategies. Case 1: Another voter triggers exclusion before I act. I have no decision to make; the exclusion lottery proceeds. My strategy is irrelevant in this case. Exclusion is already triggered by them, yielding expected utility 𝑇 𝑖 (𝑆) Case 2: No other voter triggers exclusion. I face the choice: accept proposal 𝑎 , or reject (triggering exclusion). • If I accept and all others accept: outcome is 𝑎 , yielding 𝑢 𝑖 (𝑎) • If I reject: exclusion is triggered, yielding expected utility 𝑇 𝑖 (𝑆) Subcase 2a: I am pivotal (my acceptance would complete unanimity). • Accept yields 𝑢 𝑖 (𝑎) • Reject yields 𝑇 𝑖 (𝑆) • Optimal action: accept iff 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑆) - this is exactly honest behavior Subcase 2b: I am not pivotal (some other voter will reject regardless). 8 • Accept yields 𝑇 𝑖 (𝑆) (they reject, triggering exclusion) • Reject yields 𝑇 𝑖 (𝑆) • I am indifferent; any action is equally good Dominance conclusion: In Case 1: indifferent (weakly dominant satisfied trivially) In Subcase 2a: honest behavior is strictly optimal In Subcase 2b: indifferent (weakly dominant satisfied) Therefore, honest behavior is never worse than any alternative strategy, and strictly better whenever the voter is pivotal and not exactly at the indifference threshold. This is weak dominance. Why 𝑇 𝑖 (𝑆) doesn’t depend on others’ strategies: The threat point 𝑇 𝑖 (𝑆) is computed from: • Genuine randomness (uniform exclusion, mechanism-defined) • Recursive values from smaller games (computed bottom-up) • Voter 𝑖 ’s true utilities None of these involve predicting or modeling other voters’ strategic choices. The randomness “absorbs” strategic uncertainty - whatever others do, exclusion selects uniformly at random. Strategic hiding of solutions: Suppose voter 𝑖 knows a solution 𝑎 ∈ 𝐴 ∗ (𝑆) but considers hiding it, hoping for a better outcome after exclusions remove opponents. Expected utility from hiding: 𝑇 𝑖 (𝑆) (no agreement → exclusion proceeds) Expected utility from revealing: 𝑢 𝑖 (𝑎) Since 𝑎 ∈ 𝐴 ∗ (𝑆) implies 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑆) , revealing and accepting is weakly better. If 𝑢 𝑖 (𝑎) > 𝑇 𝑖 (𝑆) , revealing is strictly better. □ Remark (Three-Choice Interpretation). One might object that voters face three choices, not two: (1) accept the proposal, (2) continue deliberating, or (3) trigger exclusion. However, the threat point 𝑇 𝑖 (𝑆) already incorporates the value of continued deliberation. It is computed assuming rational play in all subgames: • If 𝐴 ∗ (𝑆) ≠ ∅ , deliberation will eventually find a mutually acceptable proposal • If 𝐴 ∗ (𝑆) = ∅ , exclusion is the eventual outcome Thus “continue deliberating” and “trigger exclusion” have the same expected value: 𝑇 𝑖 (𝑆) . The strategic question remains binary: accept this proposal (getting 𝑢 𝑖 (𝑎) ) or not (getting 𝑇 𝑖 (𝑆) in expectation). The dominant strategy is: accept iff 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑆) 9 Lemma 4.3 (Non-Triviality of Agency) CORE allows voters to achieve outcomes strictly better than random ballot in expec- tation. Proof. Random ballot (random dictatorship) gives voter 𝑖 expected utility: 𝑈 random 𝑖 = 1 𝑛 ∑ 𝑗∈𝑉 𝑢 𝑖 (𝑎 ∗ 𝑗 ) We first show 𝑇 𝑖 (𝑉 ) ≥ 𝑈 random 𝑖 CORE’s fallback (sequential random exclusion until one voter remains) preserves pro- portional chances: each voter has probability 1/𝑛 of being the final decider. This is because exclusion is uniform at each stage, and the probability of any specific voter surviving all rounds is 𝑛−1 𝑛 ⋅ 𝑛−2 𝑛−1 ⋯ 1 2 = 1 𝑛 (by symmetry). Therefore, if no agreement ever occurs at any stage: 𝑇 𝑖 (𝑉 ) = 1 𝑛 ∑ 𝑗∈𝑉 𝑢 𝑖 (𝑎 ∗ 𝑗 ) = 𝑈 random 𝑖 But CORE can only improve on this: whenever 𝐴 ∗ (𝑆) ≠ ∅ for some subset 𝑆 en- countered during the process, agreement occurs on some 𝑎 with 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑆) for all remaining voters. This weakly improves outcomes for everyone remaining. Thus 𝑇 𝑖 (𝑉 ) ≥ 𝑈 random 𝑖 , with equality only when no agreement is ever possible at any subset. Now, if ⋂ 𝑖∈𝑉 𝐴 𝑖 (𝑉 ) ≠ ∅ , immediate agreement occurs on some 𝑎 with: ∀𝑖 ∶ 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑉 ) ≥ 𝑈 random 𝑖 Therefore: 𝔼[𝑢 𝑖 ( CORE )] ≥ 𝔼[𝑢 𝑖 ( random )] with strict inequality whenever agreement is achievable. □ Theorem 4.4 (CORE Existence and Characterization) CORE is a voting mechanism satisfying: 1. Agency: Voters can influence the outcome through their choices 2. Honesty Dominance: Truthful behavior is a weakly dominant strategy for all voters 10 3. Non-Triviality: Expected outcomes can strictly improve upon pure random- ness Moreover, no deterministic mechanism and no purely random mechanism satisfies all three conditions. Proof. (I) Deterministic mechanisms fail Condition 2: By Gibbard’s theorem (Theorem 3.1), any deterministic mechanism with |𝐴| ≥ 3 is either dictatorial or manipulable. • If dictatorial: only one voter has meaningful agency, arguably failing Condition 1 (or satisfying it only trivially) • If manipulable: there exist profiles where dishonest strategy outperforms hon- esty, violating Condition 2 (II) Purely random mechanisms fail Conditions 1 and 3: If the outcome is independent of voter input: • Voters cannot influence results (fails Condition 1) • Outcomes equal random ballot by construction (fails Condition 3) (III) CORE satisfies all three conditions: • Condition 1: Voters choose to accept/reject proposals. Agreement on 𝑎 pro- duces 𝑎 ; rejection leads to exclusion and iteration. Voter choices influence out- comes. • Condition 2: By Lemma 4.2 (Strategy Collapse), honest evaluation against the threat point is a weakly dominant strategy. • Condition 3: By Lemma 4.3, CORE achieves agreement when possible, improv- ing upon random ballot. This establishes that CORE occupies the space between deterministic and random mechanisms where all three conditions can be satisfied. □ Remark. We do not claim CORE is the unique such mechanism. Variants - such as excluding multiple voters simultaneously in equal-sized random partitions - may sat- isfy the same properties. Characterizing the full equivalence class of mechanisms with these properties remains open. The essential features appear to be: (1) unanimity as the agreement criterion, (2) proportional random fallback, and (3) binary accept/reject as the only strategic input. 4.5 Worked Example: Three Voters, Three Alternatives To illustrate the computation, consider three voters {1, 2, 3} and three alternatives {A, B, C} with utilities: 11 A B C Voter 1 10 6 0 Voter 2 0 7 10 Voter 3 5 8 4 Stage 1: Singletons 𝑊 1 ({1}) = 10, 𝑊 2 ({2}) = 10, 𝑊 3 ({3}) = 8 Each voter alone gets their ideal. Stage 2: Pairs For 𝑆 = {1, 2} : 𝑇 1 ({1, 2}) = 1 2 𝑊 1 ({2}) + 1 2 𝑊 1 ({1}) = 1 2 (0) + 1 2 (10) = 5 𝑇 2 ({1, 2}) = 1 2 𝑊 2 ({1}) + 1 2 𝑊 2 ({2}) = 1 2 (0) + 1 2 (10) = 5 Note: 𝑊 1 ({2}) = 0 because if voter 2 is alone, they choose C, giving voter 1 utility 0. Acceptable sets: 𝐴 1 ({1, 2}) = {𝐴, 𝐵} (utilities 10, 6 both ≥ 5) 𝐴 2 ({1, 2}) = {𝐵, 𝐶} (utilities 7, 10 both ≥ 5) Intersection: 𝐴 ∗ ({1, 2}) = {𝐵} Agreement on B yields: 𝑊 1 ({1, 2}) = 6 , 𝑊 2 ({1, 2}) = 7 Similarly compute for {1, 3} and {2, 3} Stage 3: Full set {1, 2, 3} 𝑇 1 ({1, 2, 3}) = 1 3 (𝑊 1 ({2, 3}) + 𝑊 1 ({1, 3}) + 𝑊 1 ({1, 2})) Using computed values from Stage 2, we get each voter’s threat point. Then: 𝐴 ∗ ({1, 2, 3}) = {𝑎 ∶ 𝑢 1 (𝑎) ≥ 𝑇 1 , 𝑢 2 (𝑎) ≥ 𝑇 2 , 𝑢 3 (𝑎) ≥ 𝑇 3 } If B is in this intersection (likely, since all voters have moderate utility for B), agree- ment occurs immediately. The strategic insight: Each voter computes their threat point knowing exactly what happens in all sub-games. No voter needs to predict others’ strategies - the recursion 12 handles everything. When voter 1 asks “should I accept B?”, they compare 𝑢 1 (𝐵) = 6 against 𝑇 1 ({1, 2, 3}) . If 6 ≥ 𝑇 1 , accepting is (weakly) dominant. 4.6 Comparison to Other Randomized Methods Multiple mechanisms escape Gibbard’s theorem through randomness. CORE is not unique in this regard, but occupies a distinctive position in the space of such methods. 4.6.1 Pure Lottery Methods Random Ballot selects a voter uniformly at random and implements their preference. This is perfectly strategy-free - honesty is trivially dominant - but offers no opportu- nity for deliberation or mutual improvement. If 99 voters prefer A and 1 voter prefers B, there’s still a 1% chance of getting B. Random Top-Two Runoff picks two random ballots and runs a majority vote between them. Still essentially random, with minimal structure added. Proportional Score Selection selects outcomes with probability proportional to (nor- malized) scores. Your score only affects the lottery if selected, so honest scoring is optimal. All three escape Gibbard through the same mechanism: your input only matters in the event you’re randomly selected, and conditional on selection, honesty is trivially optimal. But none allow deliberation or improvement beyond the random baseline. 4.6.2 COWPEA Lottery COWPEA Lottery (Pereira, 2023) is a notable method for electing committees using approval ballots. The procedure works as follows: pick a random ballot and keep only candidates approved on it; pick another random ballot and eliminate candidates not approved; continue until one candidate remains. This is repeated 𝑘 times to elect 𝑘 candidates. COWPEA Lottery passes strong fairness criteria (monotonicity, Independence of Irrelevant Ballots, Universally Liked Candidate criterion) while remaining non-deterministic. It represents the state of the art in proportional approval methods. Note: COWPEA Lottery is distinct from COWPEA (without “Lottery”), which is a deterministic method that assigns candidates weighted seats in parliament based on what the lottery probabilities would be, rather than actually running the lottery. 4.6.3 The Key Distinction: Deliberation The most important difference between CORE and other randomized methods is de- liberation In COWPEA Lottery, voters submit approval ballots before the lottery runs. They can- not: - Respond to proposals from other voters - Discover creative solutions through discussion - Adjust their position based on new information 13 In CORE, deliberation is central. Voters can: - Propose any alternative at any time - Respond to specific proposals with accept/reject - Discover mutually beneficial solu- tions not initially considered This is why CORE can improve on random ballot while COWPEA Lottery’s improve- ment comes only from filtering (eliminating candidates no one in a random sequence approves). 4.6.4 Threshold Decisions Both COWPEA Lottery and CORE involve threshold-like decisions, but of different kinds: COWPEA Lottery CORE Voter decides Which candidates to approve (in advance) Whether to accept a specific proposal Timing Before the lottery During deliberation Can respond to others No Yes Threshold computation Voter chooses approval cutoff Mechanism defines 𝑇 𝑖 (𝑆) In CORE, we prove that the optimal response (accept iff 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑆) ) is computable without modeling others’ strategies. Whether COWPEA Lottery has an analogous property is a question we leave open - the mechanisms are structurally different enough that direct comparison is non-trivial. 4.6.5 What CORE Contributes The contribution of CORE is the combination: 1. No profitable misrepresentation : Honest accept/reject is weakly dominant 2. Strategy collapse : The optimal threshold 𝑇 𝑖 (𝑆) is computable without model- ing others 3. Full deliberation : Arbitrary proposals can be considered, creative solutions can emerge 4. Improvement over random : Agreement achieves outcomes better than the pro- portional lottery Random ballot satisfies 1-2 but lacks 3-4. COWPEA Lottery lacks deliberation (prop- erty 3) by design. CORE satisfies all four. 4.6.6 Clarification: “Strategy” Vs “Manipulation” CORE involves calculation - voters compute threat points and compare utilities. Is this “strategic”? The distinction: 14 • Strategic calculation : Reasoning about outcomes to make decisions • Strategic manipulation (Gibbard sense): Misrepresenting preferences to gain advantage CORE involves the former but eliminates the latter. The voter’s calculation requires no modeling of others’ strategies; it is purely arithmetic over their own utilities and the mechanism’s structure. In Gibbard-vulnerable mechanisms, voters can gain by misrepresenting preferences (e.g., voting for a less-preferred candidate who “can win”). In CORE, the only input is binary accept/reject, and honest response is weakly dominant. There is no profitable misrepresentation. We do not claim CORE is “strategy-free” - only that honest behavior is weakly domi- nant and requires no modeling of others. 5. Intuitive Explanation 5.1 The Problem with Deterministic Voting Consider a simple plurality vote with three candidates. If I prefer A > B > C, should I vote for A? Not necessarily - if A has no chance of winning, I might strategically vote for B to prevent C from winning. This strategic calculation requires modeling what others will do. But they’re modeling me. We get an infinite regress of “I think that you think that I think...” At certain configurations, this regress becomes formally undecidable: my optimal action depends on a prediction that includes my own decision process. The ideal vote no longer only reflects my honest preference but requires a strategic reasoning about misrepresenting it. Gibbard’s theorem says this is unavoidable in any non-dictatorial deterministic system. 5.2 The Problem with Pure Randomness We could eliminate strategy entirely by selecting a random voter and implementing their preference. This is perfectly fair - everyone has equal influence - and perfectly non-strategic - there’s nothing to manipulate. But it’s also perfectly useless for finding good solutions. If 99 voters prefer A and 1 voter prefers B, there’s still a 1% chance of getting B. The mechanism doesn’t use the available information that near-universal agreement exists. It represents the voter preferences proportionally, but has no way to find common ground. 5.3 How CORE Resolves Both Problems CORE asks a fundamentally different question than traditional voting. Instead of “what do you prefer?” (which can be strategically misrepresented), it asks “is this specific proposal acceptable to you?” 15 You can’t lie about this in any useful way: • If you reject something you’d actually accept, you get the random exclusion lot- tery instead - which is worse by definition (if it weren’t worse, you’d genuinely prefer to reject) • If you accept something you’d actually reject, you get an outcome worse than your threat point - directly against your interest The threat point 𝑇 𝑖 (𝑆) is your expected utility if negotiations fail. It’s calculated from genuine randomness, not from anyone’s strategy. So you don’t need to model what others are thinking - you just need to know your own preferences and do basic probability calculations. 5.4 The Role of Deliberation CORE preserves full deliberative space. Voters can propose creative solutions, dis- cover common ground, and invent alternatives nobody initially considered. This com- putation is where collective intelligence happens. The mechanism doesn’t interfere with this - it just provides a clear criterion for when to agree (proposal beats threat point) and a fair fallback when agreement fails (random exclusion, preserving proportionality). 5.5 The Geometric Picture Think of each voter as having an “acceptable region” in outcome space - all alter- natives they’d accept over their threat point. Agreement exists when these regions intersect. Deliberation expands these regions: as voters understand each other’s concerns, they may discover solutions that satisfy everyone. The threat of exclusion motivates this expansion - holding out for your ideal becomes risky when you might be excluded. The final outcome lies in the intersection of all remaining voters’ acceptable regions - by definition, everyone remaining prefers it to their alternative. 5.6 Reduction to the Two-Candidate Case A crucial insight: from each voter’s perspective, every decision in CORE reduces to a two-candidate choice. Classical voting theory establishes that with exactly two candidates, honest voting is trivially dominant - there’s no strategic advantage to voting against your preference. The impossibility results arise only with three or more candidates, where cycles and strategic possibilities emerge. CORE transforms arbitrary multi-candidate decisions into a sequence of binary choices: “Accept proposal A, or reject (triggering the exclusion lottery)?” Each such choice is a two-candidate election where honest response is obviously optimal. 16 The mechanism achieves this reduction without limiting the alternative space - any proposal can be considered. The binary structure applies to the decision , not the op- tions 5.7 The Edge of Chaos CORE occupies a distinctive position in mechanism space: • Fully deterministic mechanisms are “frozen” - their rigid rules create ex- ploitable structure. Strategic actors can compute optimal manipulations because the mapping from inputs to outputs is fixed. • Fully random mechanisms are “gaseous” - no structure exists for deliberation to act upon. Preferences are irrelevant because outcomes are disconnected from inputs. • CORE maintains a “liquid” state at the boundary. Structure exists (unanimity target, computable threat points) but isn’t rigid enough to exploit. The random- ness absorbs strategic computation while the unanimity requirement preserves deliberative agency. This boundary position may explain why CORE resolves impossibility results: it op- erates precisely where the assumptions underlying those results - deterministic ag- gregation functions - cease to apply, while retaining enough structure for meaningful collective decision-making., Part II: Why It Works - Cooperation Emergence Part I established that CORE works: honest behavior is weakly dominant, outcomes improve on random ballot, and voters retain agency. Part II asks why it works - what is the underlying mechanism that produces these properties? The answer lies in identity uncertainty . Random exclusion means each voter might not be present when the final decision is made. This uncertainty transforms how voters evaluate proposals: they accept anything better than their expected utility across all possible exclusion sequences. This expansion of acceptance sets is what creates room for agreement. Part II formalizes this insight. We show that random exclusion creates a “coopera- tion premium” - the gap between what voters would demand as guaranteed dictators versus what they accept under uncertainty. This premium is positive whenever pref- erences differ, and it is precisely this gap that enables mutual gains. The results in Part II are not additional proofs of CORE’s properties, but rather an explanation of the mechanism behind those properties. Understanding why CORE works illuminates its connection to broader ideas in game theory, bargaining, and social choice. 17 6. The Survival Lottery Lemma 6.1 (Survival Probabilities). In CORE with no agreement at any stage, each voter 𝑗 ∈ 𝑆 has equal probability 1/|𝑆| of being the final dictator. Proof. By induction on |𝑆| Base case: |𝑆| = 1 . The single voter is dictator with probability 1 = 1/1. check Inductive step: Assume true for all sets of size 𝑘 − 1 . For |𝑆| = 𝑘 : Each voter 𝑗 ∈ 𝑆 survives the first exclusion with probability (𝑘 − 1)/𝑘 Conditional on surviving, they’re in a set of size 𝑘 − 1 , where by inductive hypothesis they have probability 1/(𝑘 − 1) of being final dictator. Therefore: 𝑃 (𝑗 is final dictator ) = 𝑘−1 𝑘 ⋅ 1 𝑘−1 = 1 𝑘 check □ Corollary 6.2. Under CORE with no agreement: 𝔼[𝑢 𝑖 ] = 1 |𝑆| ∑ 𝑗∈𝑆 𝑢 𝑖 (𝑎 ∗ 𝑗 ) = 𝑃 𝑖 (𝑆) where 𝑃 𝑖 (𝑆) is voter 𝑖 ’s proportional utility - their expected utility under random dictatorship. 7. The Cooperation Premium Definition 7.1 (Dictator Utility). Voter 𝑖 ’s dictator utility is: 𝐷 𝑖 = 𝑢 𝑖 (𝑎 ∗ 𝑖 ) This is what 𝑖 receives if they are the sole decider. Definition 7.2 (Cooperation Premium). For voter 𝑖 in set 𝑆 : 𝐶 𝑖 (𝑆) = 𝐷 𝑖 − 𝑇 𝑖 (𝑆) This measures how much utility voter 𝑖 is willing to forgo (relative to their dictator utility) when accepting proposals under CORE. Lemma 7.3 (Cooperation Premium). 𝐶 𝑖 (𝑆) ≥ 0 18 with equality iff either (a) |𝑆| = 1 , or (b) all voters in 𝑆 have identical ideal alterna- tives. Proof. For |𝑆| = 1 : 𝑇 𝑖 ({𝑖}) = 𝐷 𝑖 by definition, so 𝐶 𝑖 ({𝑖}) = 0 check For |𝑆| > 1 with identical preferences: If all 𝑗 ∈ 𝑆 have 𝑎 ∗ 𝑗 = 𝑎 ∗ , then in every sub- game, the final dictator chooses 𝑎 ∗ , so 𝑊 𝑖 (𝑆 ′ ) = 𝑢 𝑖 (𝑎 ∗ ) for all 𝑆 ′ ⊆ 𝑆 . Therefore 𝑇 𝑖 (𝑆) = 𝑢 𝑖 (𝑎 ∗ ) = 𝐷 𝑖 , and 𝐶 𝑖 (𝑆) = 0 check For |𝑆| > 1 with diverse preferences: The threat point 𝑇 𝑖 (𝑆) includes: • Worlds where 𝑖 is excluded (contributing utility 𝑊 𝑖 (𝑆 ∖ {𝑖}) ) • Worlds where 𝑖 survives to smaller sets In worlds where 𝑖 is excluded, their utility is determined by others’ preferences. If some 𝑗 ≠ 𝑖 has 𝑎 ∗ 𝑗 ≠ 𝑎 ∗ 𝑖 , then with positive probability the final dictator is someone who chooses an alternative giving 𝑖 less than 𝐷 𝑖 Therefore, under preference diversity, 𝑇 𝑖 (𝑆) < 𝐷 𝑖 , so 𝐶 𝑖 (𝑆) > 0 □ Interpretation: The cooperation premium is the “price of uncertainty” - how much each voter concedes because they might not be the one deciding. This price is positive exactly when preferences differ. 8. Acceptance Set Expansion Definition 8.1 (Dictator Acceptance Set). If voter 𝑖 were guaranteed to be dictator: 𝐴 dict 𝑖 = {𝑎 ∗ 𝑖 } A guaranteed dictator accepts only their ideal. Theorem 8.2 (Acceptance Set Expansion). 𝐴 dict 𝑖 ⊆ 𝐴 𝑖 (𝑆) with strict inclusion when |𝑆| > 1 and preferences are non-trivial. Proof. 𝑎 ∗ 𝑖 ∈ 𝐴 𝑖 (𝑆) because 𝑢 𝑖 (𝑎 ∗ 𝑖 ) = 𝐷 𝑖 ≥ 𝑇 𝑖 (𝑆) For |𝑆| > 1 : Since 𝑇 𝑖 (𝑆) < 𝐷 𝑖 (Lemma 7.3 under preference diversity), alternatives 𝑎 with 𝑇 𝑖 (𝑆) ≤ 𝑢 𝑖 (𝑎) < 𝐷 𝑖 are in 𝐴 𝑖 (𝑆) but not in 𝐴 dict 𝑖 □ Interpretation: Random exclusion expands each voter’s acceptance set. They accept “good enough” alternatives, not just their ideal. This expansion is what creates room for agreement. 19 9. Interest Alignment Definition 9.1 (Proportional Improvement). Alternative 𝑎 is a proportional improve- ment for 𝑆 if: ∀𝑖 ∈ 𝑆 ∶ 𝑢 𝑖 (𝑎) ≥ 𝑃 𝑖 (𝑆) i.e., everyone prefers 𝑎 to the random dictatorship lottery. Lemma 9.2. Every group-acceptable proposal is a proportional improvement: 𝐴 ∗ (𝑆) ⊆ {𝑎 ∶ ∀𝑖, 𝑢 𝑖 (𝑎) ≥ 𝑃 𝑖 (𝑆)} Proof. For any 𝑎 ∈ 𝐴 ∗ (𝑆) : 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑆) ≥ 𝑃 𝑖 (𝑆) (by Lemma 4.3). □ Theorem 9.3 (Interest Alignment Relative to Fallback). In CORE: (i) Individual rationality: Each voter’s weakly dominant strategy is self-interested behavior (accept iff 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑆) ). (ii) Collective rationality: Agreement occurs iff 𝑎 ∈ 𝐴 ∗ (𝑆) - the group-acceptable set. (iii) Alignment: Self-interested behavior by all voters produces outcomes that Pareto-dominate the proportional fallback for all remaining voters. Proof. (i) Established in Lemma 4.2. (ii) Agreement requires unanimity. Each voter accepts iff 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑆) . So agree- ment on 𝑎 occurs iff 𝑎 ∈ ⋂ 𝑖 𝐴 𝑖 (𝑆) = 𝐴 ∗ (𝑆) (iii) By Lemma 9.2, any 𝑎 ∈ 𝐴 ∗ (𝑆) satisfies 𝑢 𝑖 (𝑎) ≥ 𝑇 𝑖 (𝑆) ≥ 𝑃 𝑖 (𝑆) for all 𝑖 . Thus agreement outcomes Pareto-dominate random dictatorship for all voters. □ 10. The Veil of Ignorance Connection Definition 10.1 (Veiled Voter). A veiled voter 𝑉 for set 𝑆 is a hypothetical agent who: • Knows all utility functions {𝑢 𝑖 } 𝑖∈𝑆 • Knows they are one of the voters in 𝑆 , but not which one • Assigns equal probability 1/|𝑆| to being each voter Definition 10.2 (Utilitarian Veiled Acceptance). A utilitarian veiled voter accepts 𝑎 iff expected utility exceeds expected threshold: 1 |𝑆| ∑ 𝑖∈𝑆 𝑢 𝑖 (𝑎) ≥ 1 |𝑆| ∑ 𝑖∈𝑆 𝑇 𝑖 (𝑆) 20