{"id":"b3f45d01-2964-40b7-81a2-414794e453ee","arxiv_id":"1908.06927","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The optimal coinsurance rate under possibilistic risk can be approximated by one minus a risk-aversion adjusted ratio of expected loss to a variance-plus-loading term, generalizing an earlier result for a single utility operator.","lead":"This paper derives an approximate formula for the optimal coinsurance rate when the loss is modeled as a fuzzy number and utility is evaluated with an expected utility operator. The formula connects the rate to the agent's Arrow-Pratt risk aversion and to the fuzzy loss's expected value and variance.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Example 6.5 undercuts Corollary 5.4: the Taylor-based beta* is a spurious root of a linearized first-order condition; exact H' has no zero and no optimal beta* exists.","rationale":"Good faith summary: The algebraic manipulation in Theorem 5.3 is internally consistent: T(A,x-P0)=E_f(A)-P0 and T(A,(x-P0)^2)=Var_T(A)+(E_f(A)-P0)^2 follow from the axioms, and the Taylor expansion is standard. The D-operator axioms (D1)-(D2) justify differentiating under T; for the two concrete operators T1,T2 this is unproblematic. So the formal derivation is not the issue. The load-bearing weakness is that the approximation has no controlled error and is then used to solve for beta*. The paper itself presents Example 6.5 in which beta* is negative, but a direct check shows the exact problem is unbounded above: because P0=9.25 exceeds the support [4,9], selling insurance (beta->-infinity) yields utility ->+infinity, so no optimal rate exists. The negative \"optimum\" from the formula is a root of the quadratic approximation of H, not of H itself. This is a correctness failure of the central approximation claim, not merely a missing error bound. I therefore recommend REJECT rather than the reader's CONDITIONAL: the proposed approximation can fail qualitatively in a case the authors themselves treat as an application. If the authors added explicit existence conditions and an error bound, or constrained beta to [0,1], the formula might be salvageable, but as stated the central claim is not reliable.","tokens_in":16046,"tokens_out":14100,"duration_ms":148847,"concrete_test":"Recompute Example 6.5 exactly, without Taylor expansion, for both T1 and T2: evaluate H'(beta) by numerical quadrature of (3.1)/(3.2) and (4.8) on a grid beta=-12,-10,-8,-6,-4,-2,0,1, and also compute H(beta) as beta->-infinity. If H'(beta)<0 for all admissible beta and H(beta)->infinity as beta->-infinity, then no finite beta* exists; the values from (6.3)/(6.5) are not approximate optima. This settles whether the Taylor-based formula approximates the true problem.","verdict_should_be":"REJECT","load_bearing_attack":"Corollary 5.4 is derived by first-order Taylor expansion of u'(g(x,beta)) around w and then solving the linearized first-order condition. No remainder bound is supplied, so the approximate root need not be near a root of the exact condition (4.10). This is not merely a missing estimate: the paper's own Example 6.5 provides a concrete failure. Take f(t)=2t, A=(6,2,3), u(w)=ln w, w0=40, lambda=1/2. Then P0=(1+lambda)E_f(A)=37/4=9.25, which is larger than every point in supp(A)=[4,9]. For every beta in the admissible domain (g(x,beta)>0 for all x in supp(A), namely beta<6.857), we have x-P0<0 and u'(g(x,beta))>0, so the exact derivative H'(beta)=T(A,(x-P0)u'(g(x,beta))) is negative; hence H is strictly decreasing in beta and H(beta)->+infinity as beta->-infinity. The unconstrained coinsurance problem (4.7) has no maximizer for this case. Corollary 5.4 nevertheless outputs negative \"optimal\" rates (the paper prints -10.71 and -11.5; recomputation with (6.3)/(6.5) gives about -7.98 and -8.61), which are artifacts of the linearized u' becoming negative for large arguments. Thus the load-bearing step, replacing the exact first-order condition by a linearized one, can create a spurious solution where the true problem is ill-posed. Theorems 5.9, 5.7 and Proposition 5.6 inherit this uncontrolled approximation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the classical coinsurance problem to the framework of expected utility operators T defined on fuzzy numbers. After recalling the axioms of expected utility operators and D-operators, the author defines a T-premium, a T-coinsurance rate, and the resulting optimization problem max_β H(β), where H(β)=T(A,u(w0-βP0-(1-β)x)). Assuming T is a D-operator, a first-order condition is derived, and a first-order Taylor expansion of u' around w=w0-P0 leads to an approximate formula for the optimal coinsurance rate in terms of the Arrow-Pratt index, the possibilistic expectation E_f(A), and the T-variance Var_T(A). The paper also proves a Mossin-type result (β*=1 for λ=0, β*<1 for λ>0), a comparative statics statement for more risk-averse agents, a formula for convex combinations of D-operators, and an approximation of the maximal total expected utility. These results are applied to the operators T1 and T2 with triangular fuzzy numbers and HARA/CRRA utility functions, and several numerical examples are provided.","tokens_in":16412,"tokens_out":7891,"duration_ms":77572,"significance":"The axiomatic framework built on expected utility operators is elegant, and the paper usefully collects explicit formulas for the T1 and T2 operators, including triangular fuzzy number examples. If the approximate formula were rigorously justified, it would provide a convenient tool for possibilistic insurance decisions. However, the main theorem is only a Taylor heuristic without an error bound, and the paper's own Example 6.5 shows that the linearized first-order condition can produce a spurious 'optimal' rate when the exact coinsurance problem has no maximizer. The strict concavity claim used to guarantee existence of β* is not justified, and the comparative statics results are proved by comparing approximations rather than exact optimizers. These are load-bearing issues for the central claim of the paper, so the contribution is not currently established.","major_comments":[{"comment":"The claim that H is strictly concave when T is strictly increasing is not justified. From (4.9), H''(β)=T(A,u''(g(x,β))(x-P0)^2). Since u''<0, the integrand is non-positive, but it is not strictly negative pointwise: it vanishes at x=P0. The strict monotonicity axiom (d) of Definition 3.1 only yields T(A,g)<T(A,h) when g<h everywhere; it does not imply that T(A,q)<0 for every non-positive q that is negative somewhere. Thus one can only conclude H''(β)≤0, i.e., H is concave. Existence and uniqueness of β*_T in (4.7) are therefore not established, and this is a prerequisite for Theorems 5.3 and Corollary 5.4.","section":"Section 4, Eqs. (4.8)-(4.10) and the paragraph after (4.9)"},{"comment":"Example 6.5 undercuts Corollary 5.4. With f(t)=2t, A=(6,2,3), u(w)=ln w, w0=40 and λ=1/2, one computes P0=(1+λ)E_f(A)=37/4, which is larger than the upper endpoint 9 of supp(A). Hence x-P0<0 for every x in the support and u'(g(x,β))>0 whenever g(x,β)>0, so H'(β)=T(A,(x-P0)u'(g(x,β)))<0 for all admissible β. The exact coinsurance problem (4.7) has no maximizer: H is strictly decreasing on its domain. Corollary 5.4 nevertheless outputs β*_1≈-10.71 and β*_2≈-11.5, which are artifacts of the linearized u' becoming negative for large arguments. This demonstrates that replacing the exact first-order condition (4.10) by a linearized equation can create a spurious solution where the true problem is ill-posed. A valid statement of Corollary 5.4 requires additional hypotheses guaranteeing the existence of a true maximizer and an error bound showing the linearized root is close to a root of (4.10).","section":"Section 6, Example 6.5"},{"comment":"Proposition 5.6 states an exact comparative statics result, but its proof compares the approximate formulas (5.4) and (5.5). Since β*_1 and β*_2 are defined as exact maximizers of (4.7), an inequality between the approximate expressions does not imply the exact inequality β*_1 ≥ β*_2 without a remainder estimate. The same uncontrolled approximation is used in Proposition 5.7 and Theorem 5.9, where the approximate value of β* is substituted into further approximation formulas. These results therefore inherit the unquantified error of Corollary 5.4.","section":"Section 5, Proposition 5.6"},{"comment":"The first-order Taylor expansion of u'(g(x,β)) around w is used without any remainder bound. The validity of the resulting approximate root depends on (1-β)(x-P0) staying in a region where the linear approximation of u' is accurate; this is not checked, and it fails in Example 6.5. A rigorous theorem would need explicit conditions on u, A, λ and w0 under which the remainder is controlled and the linearized first-order condition has a solution near a genuine solution of (4.10). As it stands, the formula in Corollary 5.4 is an unvalidated heuristic.","section":"Theorem 5.3, proof"}],"minor_comments":[{"comment":"In the displayed inequality after 'H'(0)>0 is written as', the argument of the second u' appears as w0-x0 instead of w0-x; this is a typo.","section":"Section 8, proof of Proposition 8.2"},{"comment":"'By subtotal' should be 'By subtraction'.","section":"Section 6, before Eq. (6.7)"},{"comment":"The domain of g(x,β) is not stated explicitly; in particular, for utilities such as u(w)=ln w in Example 6.5, the admissibility condition g(x,β)>0 for all x in supp(A) restricts β, and this restriction is not discussed before the optimization problem is formulated.","section":"Section 4, Eq. (4.4)"},{"comment":"The symbol β*_1 is used both for the exact optimal rate for T1 and for the approximate value from (6.3); the distinction between exact and approximate quantities should be made explicit throughout Section 6.","section":"General notation"}],"recommendation":"reject","confidential_remarks":"The main result of the paper is an approximate formula for the optimal coinsurance rate, but the paper's own Example 6.5 shows that the linearization can produce 'optimal' rates when the exact problem has no solution. This is not a minor technical gap: it invalidates the central theorem and the results derived from it. A repair would require new conditions for existence of the exact maximizer and quantitative bounds on the Taylor remainder, which is a substantial rewrite rather than a local correction. The heavy reliance on the author's prior work is acceptable as context, but the novelty beyond [19] is incremental."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper is a clean abstract exercise in applying Taylor expansion to a coinsurance problem with fuzzy numbers and D-operators. The algebra is correct, but the main approximation formula has no error bound and no existence check, and the paper's own Example 6.5 is a concrete case where the exact first-order condition has no root while the formula outputs negative rates. That is a load-bearing flaw; it also pulls down the derived comparative-statics and convex-combination results.\n\nWhat is new: the generalization of the T1 formula from [19] to arbitrary D-operators, the T2 case, and convex combinations. This is a legitimate but modest extension. The D-operator axioms are clearly laid out, and the paper is explicit that the formula is approximate, which counts in its favor. The self-citation pattern is heavy but not circular; the target is derived from the axioms, not assumed.\n\nWhere it goes wrong: Theorem 5.3 / Corollary 5.4 replaces the exact first-order condition H'(β)=0 by a linearized version. With no remainder bound, the approximate root need not be near any true root. Example 6.5 shows this is not a pedantic worry. With f(t)=2t, A=(6,2,3), u=ln, w0=40, λ=1/2, you get P0=37/4=9.25 and supp(A)=[4,9]. So x-P0<0 and u'>0 on the whole support, and H'(β)=T(A,(x-P0)u'(g(x,β))) is strictly negative for every β in the admissible domain. The true problem has no maximizer; H increases without bound as β→-∞. The paper prints β*1≈-10.71 and β*2≈-11.5 (recomputing with (6.3)/(6.5) gives about -7.98 and -8.61). These are roots of the linearized equation, not of the original problem. So the central output of the paper is an artifact in a case the authors themselves chose.\n\nThis also sinks Proposition 5.6, which states an exact comparative-statics inequality but proves it from the same approximate formula, and it infects Theorem 5.9 and Proposition 5.7. The strict-concavity point after (4.9) is fine when T is strictly increasing, so that is not where the problem lies.\n\nWho gets value: specialists in fuzzy/possibilistic decision theory, as a framework exercise. The numerical output should not be taken at face value. The paper deserves a serious referee, because the framework is coherent and the flaw is instructive, but it needs major revision: add existence conditions, provide a remainder estimate, and repair or remove Example 6.5. My recommendation: send to review and expect heavy revision before it is usable.","headline":"A clean abstract framework with a correct Taylor derivation, but the main approximation formula is uncontrolled and the paper's own Example 6.5 produces spurious negative coinsurance rates for a problem with no true solution.","tokens_in":16912,"tokens_out":5622,"would_cite":false,"duration_ms":51323,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91B30","91B16","03E72"],"pacs":[],"model":"deepseek-v4-flash","headline":"An approximate formula gives the optimal coinsurance rate when risk is a fuzzy number.","keywords":["expected utility operators","coinsurance","fuzzy numbers","possibilistic expected utility","D-operators","Arrow-Pratt index","risk aversion","triangular fuzzy numbers"],"falsifier":"Take a concrete D-operator and utility—for example, u(x) = −$e^{{-x}}$, the triangular fuzzy number (2,4,1) with f(t) = 2t, and λ = 1—compute the exact maximizer of H(β) by solving T(A, (x−P0)u'(w−(1−β)(x−P0))) = 0 numerically, and compare with the formula in Corollary 5.4; a discrepancy that grows with λ or with the variance would mark the boundary beyond which the first-order Taylor formula stops being a usable approximation.","tokens_in":15855,"feed_emoji":"🛡️","tokens_out":6538,"duration_ms":63729,"temperature":0.7,"pith_summary":"This paper extends the coinsurance problem—choosing the fraction β of a loss to insure—to settings where the risk is modeled not by a random variable but by a fuzzy number, a set of possible values with graded membership. It works inside the abstract framework of expected utility operators T, which assign a real number to a fuzzy number and a utility function, generalizing two earlier possibilistic expected utilities. The main result is an approximate formula for the optimal coinsurance rate β*: β* ≈ 1 − (λ / r_u(w)) · E_f(A) / (Var_T(A) + λ² E_f(A)²), where λ is the insurer's loading factor, r_u is the Arrow-Pratt index of risk aversion, and E_f and Var_T are the possibilistic expectation and variance of the fuzzy loss. The paper also proves a Mossin-type theorem, a risk-aversion comparison, and explicit formulas for triangular fuzzy numbers with HARA and CRRA utilities. A sympathetic reader would care because the formula gives a workable decision rule for insurance demand under imprecise, non-probabilistic risk.","feed_headline":"Optimal coinsurance rate found for fuzzy-number risk","feed_subtitle":"Risk aversion, expected loss, and fuzzy variance determine how much insurance to buy.","key_machinery":"The central object is an expected utility operator T, a function that maps a fuzzy number A and a utility function g to a real number, satisfying natural axioms of linearity, monotonicity, and reduction to the possibilistic expectation when g is the identity. The subclass of D-operators additionally satisfies axioms (D1) and (D2), which allow the derivative of T(A, g(·,β)) with respect to β to be computed as T(A, ∂g(·,β)/∂β). This differentiation property converts the coinsurance optimization into a first-order condition, and the Arrow-Pratt index r_u(w) = −u''(w)/u'(w) enters when the Taylor expansion is written in terms of risk aversion. This mechanism carries the whole argument.","core_discovery":"The paper claims that in any T-possibilistic expected utility theory generated by a D-operator, the optimal coinsurance rate β* approximately satisfies β* ≈ 1 − (λ / r_u(w)) · E_f(A) / (Var_T(A) + λ² E_f(A)²). The derivation starts from the first-order condition H'(β) = T(A, (x − P0)u'(g(x,β))) = 0 and replaces u' by its first-order Taylor expansion around w = w0 − P0. The formula shows how insurance demand depends on risk aversion, the loading factor, and the fuzzy indicators of the loss. The paper also establishes that full insurance is optimal when the loading factor is zero, that a positive loading makes the optimal rate strictly less than one, and that a more risk-averse agent chooses a higher coinsurance rate.","pith_inferences":["The paper does not spell out, but the formula directly implies, that holding the expected loss fixed, a higher possibilistic variance Var_T(A) pushes β* upward—more uncertain losses lead to more insurance—which could be tested numerically in the triangular case.","Because the derivation is only a first-order Taylor expansion of u', adding prudence and temperance terms (u''' and u'''') would yield refined formulas involving third and fourth possibilistic moments; the paper itself lists this as an open direction.","The appearance of the same Arrow-Pratt index as in probabilistic models hints that the qualitative determinants of insurance demand—risk aversion, loading, and variance—transfer directly to imprecise-risk settings, though the numerical value of β* will differ through the fuzzy variance estimator.","One could replace the triangular fuzzy loss by a trapezoid built from a real dataset, as the paper's concluding discussion suggests, and use formula (5.3) to compare probabilistic and possibilistic optimal rates on the same data."],"forward_implications":["For a fair contract (λ = 0) the optimal choice is full insurance, β* = 1; with any positive loading the agent retains some risk, β* < 1.","If the agent's utility becomes more risk-averse in the Arrow-Pratt sense, the optimal coinsurance rate rises, all else equal.","The approximation formula applies across all D-operators, so it covers both standard possibilistic utilities T1 and T2, with explicit closed forms for triangular fuzzy numbers.","For a convex combination of two D-operators, the optimal rates combine in a reciprocal (harmonic-like) formula, β*_U ≈ 1 − 1 / (c/(1−β*_T) + (1−c)/(1−β*_S)).","The same approximation also yields a formula for the maximal attainable total utility H(β*)."],"supporting_citations":[{"why":"Supplies the probabilistic coinsurance contract, premium, and utility setup that the possibilistic model adapts, and the CRRA utility class for examples.","marker":"[12]"},{"why":"Introduces D-operators and the (D1)-(D2) axioms that allow differentiating under the expected utility operator, the key machinery for the first-order condition.","marker":"[20]"},{"why":"Earlier possibilistic coinsurance problem for operator T1 whose approximation formula is recovered as the special case of the present result.","marker":"[19]"},{"why":"Defines expected utility operators and records the approximation T(A,u) ≈ u(E_f(A)) + (1/2)u''(E_f(A))Var_T(A) used in Theorem 5.9.","marker":"[17]"},{"why":"Introduced expected utility operators and the generalized possibilistic expected utility framework on which the paper builds.","marker":"[18]"},{"why":"Mossin's theorem on full insurance at fair premiums, the probabilistic result paralleled by Proposition 5.1.","marker":"[26]"},{"why":"Provides the HARA utility form and its Arrow-Pratt index used in the worked examples.","marker":"[21]"},{"why":"Earlier possibilistic risk-aversion theory establishing the Arrow-Pratt characterization that motivates the comparison result.","marker":"[15]"}],"fun_headline_variants":["Coinsurance rate formula derived for fuzzy-number losses","Risk aversion and fuzzy variance set optimal coinsurance","Optimal coinsurance share depends on fuzzy loss distribution","Fuzzy expected utility yields coinsurance prescription","How much to insure when risk is a fuzzy number"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation only works for expected utility operators that let the derivative pass inside the operator, and it replaces the marginal utility by a first-order Taylor line with no error bound.","fun_headline_variants_meta":{"raw":{"variants":["Coinsurance rate formula derived for fuzzy-number losses","Risk aversion and fuzzy variance set optimal coinsurance","Optimal coinsurance share depends on fuzzy loss distribution","Fuzzy expected utility yields coinsurance prescription","How much to insure when risk is a fuzzy number"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000249,"raw_usage":{"total_tokens":1492,"prompt_tokens":828,"completion_tokens":664,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":444,"completion_tokens_details":{"reasoning_tokens":592}},"tokens_in":444,"tokens_out":664,"duration_ms":7157,"temperature":1.0,"reasoning_tokens":592,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:37:25.907199+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete D-operator and utility—for example, u(x) = −$e^{{-x}}$, the triangular fuzzy number (2,4,1) with f(t) = 2t, and λ = 1—compute the exact maximizer of H(β) by solving T(A, (x−P0)u'(w−(1−β)(x−P0))) = 0 numerically, and compare with the formula in Corollary 5.4; a discrepancy that grows with λ or with the variance would mark the boundary beyond which the first-order Taylor formula stops being a usable approximation.","supporting_citations":[{"cited_title":"Eeckhoudt, C","cited_arxiv_id":null,"evidence_quote":"Supplies the probabilistic coinsurance contract, premium, and utility setup that the possibilistic model adapts, and the CRRA utility class for examples."},{"cited_title":"Georgescu, L","cited_arxiv_id":null,"evidence_quote":"Introduces D-operators and the (D1)-(D2) axioms that allow differentiating under the expected utility operator, the key machinery for the first-order condition."},{"cited_title":"Georgescu, Possibilistic risk aversion and coinsurance proble m, Fuzzy Information and Engineering, 5 (2), 2013, 221-233","cited_arxiv_id":null,"evidence_quote":"Earlier possibilistic coinsurance problem for operator T1 whose approximation formula is recovered as the special case of the present result."},{"cited_title":"Georgescu, Possibility theory and the risk, Springer, 2012","cited_arxiv_id":null,"evidence_quote":"Defines expected utility operators and records the approximation T(A,u) ≈ u(E_f(A)) + (1/2)u''(E_f(A))Var_T(A) used in Theorem 5.9."},{"cited_title":"Georgescu, Expected utility operators and possibilistic risk a version, Soft Computing, 16 , 2012, 1671-1680 18","cited_arxiv_id":null,"evidence_quote":"Introduced expected utility operators and the generalized possibilistic expected utility framework on which the paper builds."},{"cited_title":"Mossin, Aspects of rational insurance purchasing, J","cited_arxiv_id":null,"evidence_quote":"Mossin's theorem on full insurance at fair premiums, the probabilistic result paralleled by Proposition 5.1."},{"cited_title":"Georgescu, Possibilistic risk aversion, Fuzzy Sets Syst., 60 , 2009, 2608– 2619","cited_arxiv_id":null,"evidence_quote":"Earlier possibilistic risk-aversion theory establishing the Arrow-Pratt characterization that motivates the comparison result."}],"review_version":1}