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REVIEW 4 major objections 5 minor 30 references

Joint Scoring Rules: Zero-Sum Competition Avoids Performative Prediction

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Zero-sum competition among multiple predictors lets a principal deterministically choose their most preferred action, restoring honest conditional predictions.

desk verdict A novel zero-sum mechanism that plausibly beats the Othman-Sandholm impossibility, but the key proofs live in an unavailable appendix. read the letter →

arxiv 2412.20732 v1 pith:MVLCGDVS submitted 2024-12-30 cs.LG

Rubi Hudson This is my paper · ORCID
classification cs.LG
keywords performativepredictionproperscoringruleszero-sumgameselicitationdecisionincentivecompatibilityAIalignmentmulti-agentlearning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [Section 3, Lemma 1 and Theorem 2] 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.
  2. [Section 3, Theorem 2 proof] 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.
  3. [Section 4, Figures 1 and 2] 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.
  4. [Section 2, 'For now, we consider the case where the ground truth q is known to all agents'] 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.
minor comments (5)
  1. [Algorithm 1, line 6] 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.
  2. [Section 3.2, Condition 1] 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.
  3. [Section 2, zero-sum scoring rule definition] 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.
  4. [Introduction] There is a duplicated word in 'it is impossible for decision maker to to deterministically identify'.
  5. [Section 3.1, Theorem 8] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the zero-sum mechanism is derived from stated definitions against an external impossibility benchmark, not from fitted values or self-citation.

full rationale

The paper's central result (Theorem 2) is a direct proof from the definition of zero-sum scoring rules and the optimistic-max decision rule, using Lemma 1, which follows from own-score maximization under a strictly proper base rule. The claimed uniqueness (Theorem 6) is presented as a theorem with proofs in an appendix, and the main text does not substitute a self-citation for the argument. The relevant impossibility baseline is Othman and Sandholm (2010), an external result, and the paper's contribution is explicitly to overcome that result with n≥2 agents. Self-citations (Hubinger et al. 2023; Oesterheld et al. 2023) are used for motivation and risk framing, not to establish the incentive or uniqueness theorems; removing them would not change the derivation. The assumption that q is known to agents is stated, and the experiments address a different setting, but that is a scope limitation, not circularity. No fitted parameter is relabeled as a prediction, and no equation reduces to its own input by construction. Score 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central theoretical results rest on the known-q assumption, strict and symmetric preferences, standard proper scoring rule theory, and Nash/strong equilibrium solution concepts. The zero-sum constant c is a free choice set to 0. No new physical or ontological entities are introduced.

free parameters (1)
  • Constant c in zero-sum scoring rule = 0
    The zero-sum scoring rule includes an additive constant c, set to 0 for all theorems; Theorem 7 explicitly relies on c=0. This value is chosen by hand and is not fitted to data.
assumptions (5)
  • domain assumption The true conditional distribution q over outcomes is known to all agents
    Section 2 states this explicitly before the theorems; all equilibrium and scoring rule definitions use q. This assumption is not met in the experiments, which train from sampled outcomes.
  • domain assumption The principal's preferences are complete, transitive, and strict, with tie-breaking, and the scoring rule is symmetric under permutations of actions, outcomes, and agents
    Section 2 defines strict preferences and symmetry modulo tie-breaking; the theorems depend on these structural conditions.
  • standard math Strictly proper scoring rules uniquely incentivize truthful reports in non-performative settings (Gneiting and Raftery 2007)
    The proofs rely on the unique maximizer property of the base scoring rule s(., q), which is the defining property of strict propriety.
  • standard math Equilibrium analysis uses Nash and strong equilibrium solution concepts
    Section 2 defines equilibrium and strong equilibrium; Theorems 2-6 use Nash equilibria, while Theorem 4 uses strong equilibria.
  • domain assumption The Independence axiom on principal preferences (for mean-max results)
    Theorem 3 and the mean-max decision rule require preferences satisfying the Independence axiom, which includes vNM and lexicographic preferences as stated in Section 3.

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Cite this review

Pith. "Pith review of Joint Scoring Rules: Zero-Sum Competition Avoids Performative Prediction." pith.science (2026). https://pith.science/paper/MVLCGDVS

@misc{pith2026241220732,
  author       = {Pith},
  title        = {Pith review of: Joint Scoring Rules: Zero-Sum Competition Avoids Performative Prediction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MVLCGDVS}},
  note         = {Machine review of arXiv:2412.20732}
}
read the original abstract

In a decision-making scenario, a principal could use conditional predictions from an expert agent to inform their choice. However, this approach would introduce a fundamental conflict of interest. An agent optimizing for predictive accuracy is incentivized to manipulate their principal towards more predictable actions, which prevents that principal from being able to deterministically select their true preference. We demonstrate that this impossibility result can be overcome through the joint evaluation of multiple agents. When agents are made to engage in zero-sum competition, their incentive to influence the action taken is eliminated, and the principal can identify and take the action they most prefer. We further prove that this zero-sum setup is unique, efficiently implementable, and applicable under stochastic choice. Experiments in a toy environment demonstrate that training on a zero-sum objective significantly enhances both predictive accuracy and principal utility, and can eliminate previously learned manipulative behavior.

Figures

Figures reproduced from arXiv: 2412.20732 by the authors.

Figure 1
Figure 1. Figure 1: (Left) In environments that incentivize performative prediction, [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Left) A zero-sum objective using two different predictions from the [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

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