{"id":"aa48bf0d-757c-484e-8c98-686afcf6528a","arxiv_id":"2505.22468","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Under a cone-positivity condition, the competitive spectral radius is 1-Lipschitz in the matrix sets and can be approximated to arbitrary precision by solving a discretized nonlinear eigenproblem.","lead":"This paper studies a two-player game where players alternately pick matrices and the product's growth rate is the payoff. It shows that this growth rate varies smoothly with the allowed matrix sets under a positivity condition, and gives a method to compute it to any desired accuracy, demonstrated on a population model.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Approximability is claimed under the same assumptions as continuity, but it requires the strictly stronger Small Cone Assumption IV.1, which is not verified for the population example and fails for simple positive matrix families.","rationale":"The reader's identification of Assumption IV.1 as the weakest structural point is correct and is the most load-bearing concern. The continuity theorem (Corollary III.2) is solid, and the approximation bounds in Theorems IV.6 and IV.8 appear to follow from the stated assumptions once the Small Cone hypothesis is in place. My independent check of the proof of Theorem IV.6 shows that the lower-bound step requires identifying the original game value with the restricted game on X=K∩Δ; this identification is plausible via the independence of the escape rate from the initial state and the forward invariance of K, though it is not spelled out in the paper. The central problem is that the abstract and introduction claim approximability under the same 'strict positivity' condition as continuity, while the actual algorithm requires the strictly stronger Assumption IV.1. This matters practically: there are simple positive matrix families in a common part (e.g., diagonal matrices with distinct positive diagonal entries) that satisfy the continuity hypotheses but fail the Small Cone assumption, so the approximation scheme does not apply to them. Moreover, the paper's own numerical example does not verify IV.1, so the illustrative results are not backed by the theorem's hypotheses. These are scope and presentation issues rather than mathematical errors, so the reader's CONDITIONAL verdict is appropriate and I do not propose changing it.","tokens_in":14129,"tokens_out":26611,"duration_ms":280295,"concrete_test":"For the population dynamics example, check Assumption IV.1 by solving the feasibility of a common invariant cone for the nine Leslie matrices L(α,β) with α∈A, β∈B. For instance, parametrize K as the conical hull of a polytope or as a Hilbert ball around a candidate ray, and test whether L(α,β)(K)⊂K for all 9 pairs with K∩Δ⊂relint Δ. If the test fails or is inconclusive, the numerical results in Table I do not demonstrate the theorem; separately, run the algorithm on the diagonal family {diag(2,1), diag(1,2)} to show that no invariant subcone exists and the algorithm cannot even be set up, confirming that approximability does not hold under the continuity hypothesis alone.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline contribution is that, under the 'strict positivity' condition ensuring continuity (Corollary III.2), the competitive spectral radius can also be approximated (Abstract and Section I-B: 'under the same assumptions'). This is not what the theorems prove. The discretization and the RVI-KM error bounds (Theorems IV.6 and IV.8) depend critically on Assumption IV.1: there must exist a closed cone K⊂C with Tab(K)⊂K for all actions and X=K∩Δ compactly contained in relint Δ. This assumption is strictly stronger than the common-part/positivity condition. For instance, the family {diag(2,1), diag(1,2)} preserves the positive orthant and lies in a common part of End(Int R_+^2), but one can check directly that no nondegenerate subcone K is invariant under both maps; hence the algorithm is not applicable to it. The paper does not give a constructive criterion for IV.1, and Section V does not verify it for the Leslie matrices in the population dynamics example. Consequently, the central approximation claim is only conditional on an extra hypothesis that is neither stated in the front matter nor validated in the example. This is a scope mismatch, not an internal inconsistency of the theorems, but it is the most load-bearing weakness in the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the competitive spectral radius, defined as the value of a two-player matrix multiplication game, for families of linear operators that preserve the interior of a closed convex pointed cone. It proves a continuity theorem: for compact matrix sets contained in a fixed part of End(Int C), the competitive spectral radius is 1-Lipschitz with respect to the Hausdorff distance induced by the Thompson metric (Theorem III.1 and Corollary III.2). It then gives an approximation scheme under an additional small-cone assumption (Assumption IV.1): a cross-section of an invariant subcone is discretized, a finite-dimensional nonlinear eigenproblem for a discretized Shapley operator is solved by relative value iteration with Krasnoselskii-Mann damping, and two-sided error bounds are obtained (Theorems IV.6 and IV.8). The algorithm is illustrated on a three-age population dynamics model with Leslie matrices.","tokens_in":14282,"tokens_out":14643,"duration_ms":169932,"significance":"The continuity result is clean and quantitative: Theorem III.1 gives an explicit Lipschitz bound with no tuned parameters, and the proof is self-contained. The approximation scheme is a genuine step toward computable values of non-rectangular matrix multiplication games, with an explicit O(1/h^2) iteration bound and total complexity O(1/h^{2d}). The use of McShane-Whitney interpolation to obtain two-sided bounds is elegant. However, the advertised scope is larger than what is proved: the paper's front matter says approximation holds under the same assumptions as continuity, whereas the algorithmic theorems require the strictly stronger small-cone Assumption IV.1, which is not verified in the numerical example. This gap does not invalidate the theorems, but it substantially qualifies the central claim.","major_comments":[{"comment":"The Abstract and Section I-B claim that, under the same assumptions that give continuity (the common-part / strict-positivity condition), the competitive spectral radius can be approximated up to arbitrary accuracy. The approximation results in Section IV all depend on Assumption IV.1 (Small Cone), which is strictly stronger. For example, take A = {diag(2,1), diag(1,2)} and B = {I} in R_+^2; this is a compact set contained in a single part of End(Int R_+^2), so Corollary III.2 applies, but no closed cone K ⊂ R_+^2 with K ∩ Δ ⊂ relint Δ is invariant under both diagonal matrices: the orbit of any positive ray under products of these maps accumulates on a boundary ray, so any closed invariant cone must intersect the boundary. Thus Theorems IV.6 and IV.8 are not available for this family. The front matter should either state Assumption IV.1 explicitly or weaken the approximation claim to be conditional on it.","section":"Abstract and §I-B, Assumption IV.1"},{"comment":"The numerical application to age-structured population dynamics does not verify Assumption IV.1. The text only checks that the Leslie matrices lie in a common part of End(Int C), which is sufficient for continuity by Corollary III.2 but not for the discretization error bounds. Consequently, the entry in Table I and the reported numerical value (for example 1.3147 at 8646 points) are not certified by Theorems IV.6 and IV.8 unless the small-cone condition is established for this family. The authors should either prove IV.1 for the Leslie example or explicitly label the numerical results as heuristic.","section":"§V, Assumption IV.1"},{"comment":"Theorem IV.7 as stated is formally incorrect: since v_k belongs to L_xbar and \\hat F_hn v takes value 0 at xbar, the equation \\hat F_hn v = λ + v forces λ = 0. The intended statement is either \\hat F_hn v = v for the normalized operator or \\hat F_h^+ v = λ + v for the unnormalized operator; the proof of Theorem IV.8 uses an eigenpair of \\hat F_h^+. This is a load-bearing statement in the algorithm section and should be corrected.","section":"Theorem IV.7 and following text"},{"comment":"Corollary IV.9 only discretizes the action spaces, using Corollary III.2; it does not by itself provide an approximation of the competitive spectral radius. The state-space discretization, which is the core of the algorithm, still requires Assumption IV.1. The sentence introducing Corollary IV.9 ('Corollary III.2 allows us to reduce to this case, after discretization of the action spaces') could be read as removing the small-cone requirement, which it does not. The presentation should separate the two discretizations and state clearly that the small-cone assumption remains necessary.","section":"Corollary IV.9"}],"minor_comments":[{"comment":"The abstract contains the typo 'competive spectral radius'.","section":"Abstract"},{"comment":"The notation 'C = Rn≥0' is typeset incorrectly; it should be R_{≥0}^n.","section":"Example II.1"},{"comment":"In the sentence 'For β ∈ A and α ∈ B' the roles of α and β appear to be reversed relative to the preceding definition of L(α, β).","section":"Section V"},{"comment":"The runtime for the 13041-point row (1584.62 s) is two orders of magnitude larger than the 8646-point row; if this is not a typographical error, the authors should explain the superlinear growth.","section":"Section V, Table I"},{"comment":"The same symbol h is used for the state-space mesh size and for the termination threshold in Algorithm 1; this double use is confusing and should be clarified by renaming one of them.","section":"Section IV"}],"recommendation":"major_revision","confidential_remarks":"The core theorems appear mathematically sound, but the paper's advertised scope is substantially broader than the hypotheses actually used by the approximation results. In revision, the authors should align the abstract and introduction with Assumption IV.1 and either verify the small-cone condition for the population-dynamics example or recast the numerical section as heuristic. The explicit diagonal-matrix counterexample may be useful to include in the revision to delimit the applicability of the algorithm."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the continuity result is solid and the discretization framework is a real step forward, but the approximation claim overreaches. The algorithm requires the Small Cone Assumption IV.1, which is strictly stronger than the part condition used for continuity, and the paper never verifies it for its own population example.\n\nWhat is genuinely new: Theorem III.1 and Corollary III.2 give a clean 1-Lipschitz continuity result for the competitive spectral radius as a function of compact matrix sets in a common part, under a positivity-type assumption. That extends the rectangular/entropy-game cases and the authors' earlier escape-rate framework, and it is proved directly from the Shapley operator formulation. The discretized eigenproblem in Section IV — interpolation operators, two-sided bounds in Theorem IV.6, and the RVI-KM termination guarantee in Theorem IV.8 — is also new and nontrivial. The complexity estimate O(1/h^{2d}) is plausible given the state discretization.\n\nThe soft spots are in the framing and in the verification. The abstract and Section I-B say approximation works 'under the same assumptions' as continuity, meaning the common-part positivity condition. Theorems IV.6 and IV.8 actually need Assumption IV.1: a closed cone K preserved by all operators and compactly contained in the interior of the cross-section. As the stress test notes, positive diagonal matrices in a common part need not admit such a K. The paper does not give a constructive criterion or even check IV.1 for the Leslie matrix example in Section V. That is a real scope mismatch, not a hidden contradiction, but it changes what the headline result is. Also, Table I lists grid sizes but not the mesh parameter h, so the stated error bounds cannot be checked against the numbers. Corollary IV.9 introduces a second discretization of the action spaces without a combined error analysis; that is a smaller issue but worth flagging.\n\nFor a reader working on joint spectral radii, entropy games, or cone-preserving dynamics, the continuity theorem and the discretization idea are worth the read. The paper deserves serious peer review — I would send it out — but it needs a revision that (1) restates the approximation theorem with IV.1 in the hypothesis, (2) verifies IV.1 for the example or drops the example, and (3) reports h in the numerics.","headline":"The continuity theorem is real, but the approximation claim needs Assumption IV.1, which is stronger than advertised; fix the framing and verify the example.","tokens_in":14914,"tokens_out":3356,"would_cite":true,"duration_ms":36410,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A50","15A60","47H09","65K15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for cone-preserving families satisfying strict positivity, the competitive spectral radius is 1-Lipschitz in the matrix sets and can be approximated to arbitrary accuracy by a discretized eigenproblem.","keywords":["competitive spectral radius","matrix multiplication games","joint spectral radius","cone-preserving operators","Thompson metric","Shapley operator","relative value iteration","RVI-KM algorithm"],"falsifier":"Take any two compact sets of positive matrices in the same part with Thompson-Hausdorff distance $\\varepsilon$ and compute their competitive spectral radii by an independent high-accuracy method; if $|\\rho(A,B)-\\rho(A',B')|$ exceeds $\\varepsilon$, the 1-Lipschitz claim fails. Alternatively, run Algorithm 1 on a family satisfying continuity but not the Small Cone assumption, such as a set of diagonal positive matrices, and check whether the computed eigenvalue converges to the true value with error shrinking like $h$; failure would show Assumption IV.1 is genuinely necessary for Theorem IV.6.","tokens_in":13836,"feed_emoji":"🎲","tokens_out":9177,"duration_ms":101489,"temperature":0.7,"pith_summary":"The paper studies the competitive spectral radius, a two-player generalization of the joint spectral radius: Min and Max alternately choose matrices from compact sets, and the growth rate of the resulting product is the payoff. The central claim is that for cone-preserving families satisfying a strict-positivity condition, the competitive spectral radius is a 1-Lipschitz function of the matrix sets with respect to the Thompson metric. Under the additional Small Cone assumption, the paper shows the value can be approximated to arbitrary accuracy by solving a discretized nonlinear eigenproblem on a finite grid. The discretized eigenvalue brackets the true value within the mesh size, and the proposed RVI-KM algorithm terminates in $O(1/h^2)$ iterations, for a total complexity of $O(1/h^{2d})$ for a $d$-dimensional cone. This makes non-rectangular matrix multiplication games with positive matrices approximable, a case not covered by the earlier rectangular/entropy-game theory.","feed_headline":"Positive matrix games are approximable to any precision","feed_subtitle":"A discretized eigenvalue problem brackets the two-player growth rate within mesh size $h$ in $O(1/h^2)$ iterations.","key_machinery":"The load-bearing object is the Shapley operator of the escape-rate game, restricted to a cross-section of the cone: $F v(x) = \\inf_a \\sup_b [\\log\\langle T_{ab}x,e^*\\rangle + v(T_{ab}x/\\langle T_{ab}x,e^*\\rangle)]$. Its unique additive eigenvalue equals the competitive spectral radius. The approximation scheme discretizes this operator: a finite $h$-net $X_h$ in the Hilbert metric, together with an interpolation operator $I_h^+ v(x)=\\min_y [v(y)+\\operatorname{Funk}(x,y)]$, produces a finite-dimensional Shapley operator whose eigenvalue is computable and whose error is controlled by $h$. The RVI-KM algorithm is the iterative engine: relative value iteration with averaging, whose convergence and error estimate come from nonexpansive iteration theory.","core_discovery":"The paper establishes the competitive spectral radius $(A,B) \\mapsto \\rho(A,B)$ is 1-Lipschitz with respect to the Hausdorff distance induced by the Thompson metric on compact subsets of a common part of $\\mathrm{End}(\\operatorname{Int} C)$. It then proves that under the Small Cone assumption, the additive eigenvalue $\\lambda$ of the discretized Shapley operator on a finite grid $X_h$ satisfies $-h+\\lambda \\le \\rho \\le \\lambda$. The RVI-KM algorithm computes an approximate eigenvalue $\\lambda_k$ and stops in $O(1/h^2)$ iterations with the guaranteed enclosure $\\rho \\in [\\lambda_k-3h, \\lambda_k+2h]$. The authors present this as answering the approximability question for non-rectangular families of positive matrices.","pith_inferences":["Editorial inference: the Small Cone assumption is the real bottleneck. The paper neither gives a constructive criterion for it nor verifies it for the Leslie-matrix example, so the numerical table there is not covered by the proven error bounds unless such a cone is exhibited.","Editorial inference: the continuity theorem is stated for general hemi-metric spaces, so the 1-Lipschitz stability should transfer to parameterized families of nonexpansive maps beyond linear cone-preserving operators, for instance switched systems in a Funk-metric setting.","Editorial inference: the total cost $O(1/h^{2d})$ makes the method sensitive to dimension; replacing the full interpolation operator $I_h^+$ by a local or adaptive interpolation, which the paper names as a bottleneck, could lower the effective exponent.","Editorial inference: the same discretization could be used to approximate worst-case versus best-case growth in population or network models where two players have opposing objectives, providing a computable two-player analogue of the lower spectral radius for positive matrices."],"forward_implications":["Non-rectangular matrix multiplication games with positive matrices can be approximated to any prescribed accuracy with explicit error bounds, answering in the positive case a question raised for general matrix multiplication games.","If the allowed matrix sets are perturbed by at most $\\varepsilon$ in Thompson-Hausdorff distance, the competitive spectral radius changes by at most $\\varepsilon$.","The RVI-KM algorithm gives two-sided bounds on the value, not just a heuristic estimate: at precision $h$ it returns $\\lambda_k$ with $\\rho \\in [\\lambda_k-3h, \\lambda_k+2h]$.","The same continuity and approximation machinery applies to other cone-preserving operator families, such as congruence operators on positive semidefinite matrices, so two-player versions of singular-value growth rates are covered.","In the population-dynamics illustration, the computed optimal play enters a periodic cycle and the state converges to a projective fixed point; the paper reports this turnpike behavior as a numerical observation."],"supporting_citations":[{"why":"Defines competitive spectral radius and escape rate games, and supplies the characterizations used throughout.","marker":"[3]"},{"why":"Introduced matrix multiplication games and the rectangular/entropy-game framework that this paper goes beyond.","marker":"[1]"},{"why":"Gives the upper semicontinuity of the lower spectral radius and the embedded-cone condition that motivates the positivity assumption.","marker":"[5]"},{"why":"Establishes continuity of the lower spectral radius under domination conditions, the one-player analogue the paper extends.","marker":"[4]"},{"why":"Provides the equivalence of topologies on parts of a cone that underlies the discretization and Hausdorff-distance arguments.","marker":"[22]"},{"why":"Shows that for the positive orthant the Small Cone assumption is equivalent to the tubular condition, connecting to prior work.","marker":"[17]"},{"why":"Supplies the relative Krasnoselskii-Mann iteration for mean-payoff and entropy games on which the RVI-KM algorithm is based.","marker":"[19]"},{"why":"Gives the convergence theorem for averaged nonexpansive iterations used in the termination proof.","marker":"[34]"},{"why":"Provides the asymptotic regularity estimate used to derive the $O(1/h^2)$ iteration bound.","marker":"[35]"}],"fun_headline_variants":["Competitive spectral radius is Lipschitz and approximable","Two-player matrix growth: continuous and computable","Discretized eigenproblem brackets competitive spectral radius","Positive matrix games: precise approximation via discretization","Competitive spectral radii: continuity and arbitrary accuracy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All the quantitative approximation guarantees depend on the Small Cone assumption: the players' matrices must all map some smaller cone inside the main cone back into itself, so the finite grid stays strictly inside the simplex; if no such invariant subcone exists, the discretized bounds are not proven.","fun_headline_variants_meta":{"raw":{"variants":["Competitive spectral radius is Lipschitz and approximable","Two-player matrix growth: continuous and computable","Discretized eigenproblem brackets competitive spectral radius","Positive matrix games: precise approximation via discretization","Competitive spectral radii: continuity and arbitrary accuracy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000188,"raw_usage":{"total_tokens":1276,"prompt_tokens":830,"completion_tokens":446,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":446,"completion_tokens_details":{"reasoning_tokens":373}},"tokens_in":446,"tokens_out":446,"duration_ms":5060,"temperature":1.0,"reasoning_tokens":373,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:09:10.494399+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any two compact sets of positive matrices in the same part with Thompson-Hausdorff distance $\\varepsilon$ and compute their competitive spectral radii by an independent high-accuracy method; if $|\\rho(A,B)-\\rho(A',B')|$ exceeds $\\varepsilon$, the 1-Lipschitz claim fails. Alternatively, run Algorithm 1 on a family satisfying continuity but not the Small Cone assumption, such as a set of diagonal positive matrices, and check whether the computed eigenvalue converges to the true value with error shrinking like $h$; failure would show Assumption IV.1 is genuinely necessary for Theorem IV.6.","supporting_citations":[{"cited_title":"Optimal rates of asymptotic regularity for averaged nonexpansive mappings","cited_arxiv_id":null,"evidence_quote":"Provides the asymptotic regularity estimate used to derive the $O(1/h^2)$ iteration bound."},{"cited_title":"Entropy Games and Matrix Multiplication Games","cited_arxiv_id":null,"evidence_quote":"Introduced matrix multiplication games and the rectangular/entropy-game framework that this paper goes beyond."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the upper semicontinuity of the lower spectral radius and the embedded-cone condition that motivates the positivity assumption."},{"cited_title":"Continuity properties of the lower spectral radius","cited_arxiv_id":null,"evidence_quote":"Establishes continuity of the lower spectral radius under domination conditions, the one-player analogue the paper extends."},{"cited_title":"Lemmens and R","cited_arxiv_id":null,"evidence_quote":"Provides the equivalence of topologies on parts of a cone that underlies the discretization and Hausdorff-distance arguments."},{"cited_title":"Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture","cited_arxiv_id":null,"evidence_quote":"Shows that for the positive orthant the Small Cone assumption is equivalent to the tubular condition, connecting to prior work."},{"cited_title":"Solv- ing Irreducible Stochastic Mean-Payoff Games and Entropy Games by Relative Krasnoselskii-Mann Iteration","cited_arxiv_id":null,"evidence_quote":"Supplies the relative Krasnoselskii-Mann iteration for mean-payoff and entropy games on which the RVI-KM algorithm is based."},{"cited_title":"Fixed Points and Iteration of a Nonexpansive Mapping in a Banach Space","cited_arxiv_id":null,"evidence_quote":"Gives the convergence theorem for averaged nonexpansive iterations used in the termination proof."}],"review_version":1}