{"id":"3544b789-8aa5-4f2b-89f0-1a6e0c7374bd","arxiv_id":"2412.20732","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Joint zero-sum scoring of multiple conditional predictors lets a principal deterministically take their most preferred action without performative manipulation.","lead":"This paper proposes a zero-sum competition between two or more prediction agents that removes their incentive to manipulate a decision maker, allowing the decision maker to reliably choose their most preferred action. If correct, it offers a practical route to safer use of AI predictions in decisions, and a uniqueness theorem says only zero-sum competition can do this.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"No significant objection identified.","tokens_in":10736,"tokens_out":151,"duration_ms":2311,"concrete_test":"No significant objection identified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the problem of a principal who wants to use conditional predictions from expert agents to deterministically choose their most preferred action, in a setting where predictions can influence which action is taken. The authors claim that while a single expert cannot be incentivized to report honestly under a symmetric scoring rule (the Othman and Sandholm impossibility), this becomes possible with two or more agents if the agents are evaluated with a zero-sum joint scoring rule. The main theoretical result (Theorem 2) states that, for n >= 2, an optimistic-max decision rule combined with a zero-sum scoring rule is quasi-strictly proper and selects the principal's preferred action in all equilibria. The paper also claims a uniqueness theorem (only zero-sum rules work), an efficient binary-search procedure over actions, extensions to stochastic choice, and toy experiments showing that zero-sum training avoids and even removes performative prediction behavior.","tokens_in":10658,"tokens_out":10839,"duration_ms":113997,"significance":"If the main theorems are correct, the paper would provide a genuinely useful way around a well-known impossibility result: honest conditional predictions can be elicited without specifying the principal's full utility function, and the principal can deterministically take their preferred action. The uniqueness result (Theorem 6) and the efficient search and stochastic-choice extensions are also valuable because they narrow the design space and address practical obstacles. The experimental results, despite the toy-scale environment, are a useful sanity check. However, the paper currently lacks proofs for several load-bearing claims and contains an experimental baseline that is not a genuine zero-sum implementation, so the significance can be fully assessed only after those issues are resolved.","major_comments":[{"comment":"Lemma 1 is stated with the note 'The proofs for this and all theoretical results are provided in the technical appendix,' but no technical appendix is included in the manuscript. Since the proof of Theorem 2 relies directly on Lemma 1, and since Theorems 3-12 have no proofs in the main text, the central soundness of the paper cannot currently be verified. Please include the appendix or, at minimum, a complete proof of Lemma 1 and of the uniqueness claim in Theorem 6.","section":"Section 3, Lemma 1 and Theorem 2"},{"comment":"In the proof of Theorem 2, the sentence 'there exists some agent k != j who is either reporting honestly or can change their prediction to p_{k,a'} = q_{a'} without affecting the action taken' is asserted without justification. This step is load-bearing because the contradiction depends on an agent being able to deviate to a truthful report for the chosen action while keeping the decision rule's output fixed. The proof should spell out why such an agent exists under the optimistic-max rule, including the case where the chosen action is a* itself and the case with exactly two agents.","section":"Section 3, Theorem 2 proof"},{"comment":"The 'exact' zero-sum baseline, described as pitting the agent against a detached version of itself that makes identical predictions, does not implement the zero-sum mechanism analyzed in Section 3. When the opponent's prediction is identical and detached, the zero-sum loss is identically zero (or, if gradients are taken through only one term, reduces to standard proper-scoring training), so this baseline cannot provide evidence about zero-sum competition. The paper should either specify precisely what is computed in the 'exact' condition and explain how it constitutes a zero-sum objective, or remove this baseline and interpret Figures 1 and 2 using only the dropout-based variant.","section":"Section 4, Figures 1 and 2"},{"comment":"All theoretical results, including Theorems 2-6, are proved under the assumption that the true conditional distribution q is known to all agents. The experiments, however, train models without known q (the ground truth is a randomly initialized neural network, and the models must learn predictions). The empirical results are therefore not covered by the theory. The paper should state this as an explicit limitation and either prove an extension in which q is unknown and scoring is based on realized outcomes, or clearly separate the experimental claims from the theoretical guarantees.","section":"Section 2, 'For now, we consider the case where the ground truth q is known to all agents'"}],"minor_comments":[{"comment":"The line 'A <- D((pA2, pA2))' appears to be a typo for D((pA1, pA2)); as written, the binary search never compares the two halves of the action set.","section":"Algorithm 1, line 6"},{"comment":"Condition 1 reads 'If p'_{i,a} > p'_{i,a} forall a in A', which should presumably be 'p'_{i,a} > p_{i,a}'; the current statement is vacuous.","section":"Section 3.2, Condition 1"},{"comment":"The notation s(pi,a,qa) is used for the expected score, while S(a,p,o) is the realized score; please clarify the relationship, for example by writing s(pi,a,qa) = E_{o ~ qa}[s(pi,a,o)], so that the expected-score formula is unambiguous.","section":"Section 2, zero-sum scoring rule definition"},{"comment":"There is a duplicated word in 'it is impossible for decision maker to to deterministically identify'.","section":"Introduction"},{"comment":"The claimed O(1)-comparison procedure begins by eliciting a non-conditional, non-zero-sum prediction about which action will be chosen, which appears to reintroduce the performative incentive the paper aims to avoid; the statement needs a clearer explanation of why that initial prediction can be trusted.","section":"Section 3.1, Theorem 8"}],"recommendation":"major_revision","confidential_remarks":"The paper's core idea is interesting and, if fully proved, would be a solid contribution to the decision-rule and performative-prediction literature. The main blockers are the absent technical appendix and the flawed 'exact' baseline. I would encourage the editor to request a revision in which the missing proofs are supplied and the experiments are either corrected or reinterpreted. No concerns about the citation pattern beyond the usual need to cite the companion technical appendix once it is available."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea here is genuinely new: have two or more predictors compete in a zero-sum scoring game, and the incentive to manipulate the principal's action disappears. Theorems 2 and 6 are the heart of the paper, and if they hold, this is a real step past Othman and Sandholm's single-agent impossibility. The paper also gives a uniqueness result claiming only zero-sum rules work, which is the sort of strong claim that makes the contribution worth engaging with seriously.\n\nWhat the paper does well: it identifies the right benchmark (Othman-Sandholm), states a clean mechanism, and is transparent about many of its own limitations. The stochastic-choice extension is a nice addition, and the toy experiments support the intuition, even if they are not the main event. The author also engages honestly with the q-known assumption, flagging it and listing it as future work.\n\nThe soft spots are real but not fatal. The biggest issue is that Lemma 1, which Theorem 2's proof leans on, is deferred to a technical appendix that is not in the arXiv version. The uniqueness theorem (Theorem 6) is also appendix-only. That means a referee cannot actually verify the central claims from the text. I would not desk-reject over this, but the author needs to supply those proofs before acceptance. Second, the experiments train agents without known q, so the theory does not directly cover the empirical setting; the paper acknowledges this indirectly but should state it more explicitly. Third, the \"exact\" zero-sum baseline is confusing: if the detached model makes identical predictions, the score difference is zero, and calling it \"exact\" while saying it behaves like a non-performative environment does not clarify the setup. There are also no error bars or replication details, which matters for a toy experiment meant to demonstrate a qualitative effect.\n\nOverall, the argument is plausible and the contribution is meaningful. The missing appendix is the main obstacle, not the idea itself. I would send this to peer review, with the condition that the author provides the technical appendix and cleans up the experimental reporting.","headline":"A novel zero-sum mechanism that plausibly beats the Othman-Sandholm impossibility, but the key proofs live in an unavailable appendix.","tokens_in":11192,"tokens_out":1998,"would_cite":true,"duration_ms":22861,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Zero-sum competition among multiple predictors lets a principal deterministically choose their most preferred action, restoring honest conditional predictions.","keywords":["performative prediction","proper scoring rules","zero-sum games","prediction elicitation","decision rules","incentive compatibility","AI alignment","multi-agent learning"],"falsifier":"Enumerate all small finite action/outcome spaces, all symmetric strictly proper scoring rules, and all strict preference orders, and check every combination of a zero-sum rule with the optimistic-max decision rule: if any equilibrium chooses an action other than the principal's most preferred one, Theorem 2 is false. Alternatively, in the paper's toy setup let two agents train on separate noisy samples of the same environment; if either agent's equilibrium report for the preferred action systematically differs from its own posterior, the known-$q$ assumption is load-bearing in a way the theory does not cover.","tokens_in":10499,"feed_emoji":"⚖️","tokens_out":7443,"duration_ms":66886,"temperature":0.7,"pith_summary":"This paper claims that a principal can deterministically select their most preferred action using honest conditional predictions from two or more agents, despite each agent having an incentive to steer the decision. The trick is to score the agents jointly in a zero-sum way: each agent receives their own proper-score minus the average of the others' scores. With an optimistic-max decision rule, no agent can profit from distorting the action taken, and every equilibrium forces honest reporting of the preferred action. A companion theorem shows that any symmetric scoring rule achieving this must be zero-sum, so the construction is not arbitrary. Toy experiments suggest zero-sum training prevents and even reverses performative behavior.","feed_headline":"Zero-sum scoring lets principals pick their true best action","feed_subtitle":"Pitting two predictors against each other removes their incentive to steer your decision, restoring honest forecasts.","key_machinery":"The zero-sum joint scoring rule $S_i(a,p,q)=s(p_{i,a},q_a)-\\frac{1}{n-1}\\sum_{j\\neq i}s(p_{j,a},q_a)$, where $s$ is a symmetric strictly proper single-agent scoring rule, together with the optimistic-max decision rule that selects an action iff some agent's prediction for it weakly dominates all other predictions for every action. This pair makes honesty for the preferred action an equilibrium and guarantees that in every equilibrium the principal's most preferred action is chosen and honestly reported. Quasi-strict propriety is the formal target: at least one equilibrium exists, all equilibria select $a^*$ with honest reports for it, and honesty on untaken actions is weakly incentivized.","core_discovery":"The central result is Theorem 2: for $n\\ge 2$ agents, the optimistic-max decision rule paired with a zero-sum scoring rule is quasi-strictly proper. In every equilibrium each agent reports the true conditional distribution for the principal's favorite action $a^*$, and the max decision rule applied to any agent's report selects $a^*$. The proof isolates each agent's incentives: holding other reports fixed, the agent faces a strictly proper rule, while the zero-sum structure exactly cancels any gain from changing which action the principal takes. Theorem 6 completes the picture by showing that any symmetric scoring rule/decision-rule pair that is quasi-strictly proper must be zero-sum, meaning the design space collapses to this family.","pith_inferences":["The theoretical results assume agents know the true conditional distribution $q$; the experiments train without a known $q$, so extending the uniqueness and incentive guarantees to the estimated-$q$ setting is an open question the paper does not fully answer.","If the mechanism works with two dropout-generated views of the same model, it could become a drop-in training objective for large models, since no separate agent population is needed.","The uniqueness theorem suggests that any training pipeline that successfully avoids performative prediction must locally approximate a zero-sum objective; checking for that structure could help audit existing systems for hidden manipulation incentives.","The $O(1)$ search result relies on an initial non-conditional prediction, and that first prediction would itself be subject to performative pressure, so the practical gain depends on the honesty of that upstream step."],"forward_implications":["A principal can deterministically take their most preferred action without eliciting a utility function over outcomes, using only symmetric proper scores and multiple agents' conditional predictions.","Because non-zero-sum symmetric rules cannot achieve this (Theorem 6), honest conditional prediction must be explicitly engineered with a zero-sum objective; it will not arise by default.","Large action spaces can be narrowed to the optimal action in $O(\\log |A|)$ comparisons, or even $O(1)$ comparisons when a non-conditional prediction is available (Theorems 7 and 8).","With stochastic decision rules satisfying Conditions 1–3, honest predictions are strictly incentivized for untaken actions that have positive probability, while the principal randomizes away from $a^*$ only with arbitrarily small probability (Theorems 10–12).","In the paper's toy environment, zero-sum training improves both predictive accuracy and principal utility and removes previously learned performative behavior faster than simply removing the performative gradient."],"supporting_citations":[{"why":"Proves the single-agent impossibility result for symmetric scoring rules that this paper overcomes with multiple agents.","marker":"Othman and Sandholm (2010)"},{"why":"Defines strictly proper scoring rules and the uniqueness of honest reporting, which supply the base score s.","marker":"Gneiting and Raftery (2007)"},{"why":"Introduces performative prediction, the phenomenon the zero-sum mechanism is designed to avoid.","marker":"Perdomo et al. (2020)"},{"why":"Shows that single-agent proper scoring rules incentivize influencing the distribution, motivating the joint-evaluation approach.","marker":"Oesterheld et al. (2023)"},{"why":"Provides the full-support randomization baseline for eliciting honest conditional predictions, which the stochastic-choice results extend with partial support.","marker":"Chen et al. (2011)"}],"fun_headline_variants":["Zero-sum scoring eliminates incentive to manipulate decisions","Competing agents neutralize performative prediction bias","Zero-sum rules uniquely guarantee honest conditional forecasts","Pit two predictors to get the principal's true best action","Zero-sum scoring: manipulation canceled, honest reports emerge"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorems assume every agent knows the true conditional distribution $q$ over outcomes; when $q$ must be estimated from data, the incentive and uniqueness results are not directly proven to survive.","fun_headline_variants_meta":{"raw":{"variants":["Zero-sum scoring eliminates incentive to manipulate decisions","Competing agents neutralize performative prediction bias","Zero-sum rules uniquely guarantee honest conditional forecasts","Pit two predictors to get the principal's true best action","Zero-sum scoring: manipulation canceled, honest reports emerge"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0002,"raw_usage":{"total_tokens":1315,"prompt_tokens":826,"completion_tokens":489,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":417}},"tokens_in":442,"tokens_out":489,"duration_ms":4937,"temperature":1.0,"reasoning_tokens":417,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:13:21.549679+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all small finite action/outcome spaces, all symmetric strictly proper scoring rules, and all strict preference orders, and check every combination of a zero-sum rule with the optimistic-max decision rule: if any equilibrium chooses an action other than the principal's most preferred one, Theorem 2 is false. Alternatively, in the paper's toy setup let two agents train on separate noisy samples of the same environment; if either agent's equilibrium report for the preferred action systematically differs from its own posterior, the known-$q$ assumption is load-bearing in a way the theory does not cover.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the single-agent impossibility result for symmetric scoring rules that this paper overcomes with multiple agents."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces performative prediction, the phenomenon the zero-sum mechanism is designed to avoid."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that single-agent proper scoring rules incentivize influencing the distribution, motivating the joint-evaluation approach."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the full-support randomization baseline for eliciting honest conditional predictions, which the stochastic-choice results extend with partial support."}],"review_version":1}