Expected utility operators and coinsurance problem

Irina Georgescu

REVIEW 4 major objections 4 minor 33 references

Expected utility operators and coinsurance problem

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read An approximate formula gives the optimal coinsurance rate when risk is a fuzzy number.

desk verdict 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. read the letter →

arxiv 1908.06927 v1 pith:XCJ3MI4D submitted 2019-08-13 q-fin.MF

Irina Georgescu This is my paper · ORCID

classification q-fin.MF MSC 91B3091B1603E72

keywords expectedutilityoperatorscoinsurancefuzzynumberspossibilisticD-operatorsArrow-Prattindexriskaversiontriangular

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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*).

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What 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.

Where 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.

This 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.

Who 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.

Referee Report

4 major / 4 minor

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.

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 (4)

[Section 4, Eqs. (4.8)-(4.10) and the paragraph after (4.9)] 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 6, Example 6.5] 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 5, Proposition 5.6] 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.
[Theorem 5.3, proof] 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.

minor comments (4)

[Section 8, proof of Proposition 8.2] 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 6, before Eq. (6.7)] 'By subtotal' should be 'By subtraction'.
[Section 4, Eq. (4.4)] 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.
[General notation] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the approximate beta* formula is derived from the D-operator axioms and a first-order Taylor expansion, not fitted or assumed.

full rationale

The derivation of Corollary 5.4 is self-contained once the D-operator axioms are granted. Starting from the exact first-order condition (4.10), H'(beta*) = T(A, (x - P0)u'(g(x, beta*))) = 0, the paper applies a first-order Taylor expansion of u' around w, uses the linearity axioms (b)-(c) of Definition 3.1, the premium relation P0 = (1 + lambda)E_f(A), and the definition Var_T(A) = T(A, (x - E_f(A))^2), then solves the linearized equation algebraically. The resulting formula beta* approximately 1 - (lambda / r_u(w)) * E_f(A) / (Var_T(A) + lambda^2 E_f(A)^2) is a closed-form output depending on exogenous quantities E_f(A), Var_T(A), r_u(w), and lambda. It is not used as an input, no parameter is fitted to the quantity being predicted, and no result from the author's prior work asserts this formula; the citations to [16]-[20] supply definitions and standard operator properties, not the target formula. Theorem 5.9 uses the cited Proposition 3.5 as a second-order Taylor expansion, but it does not assume the beta* formula; it inserts the already-derived expression for 1 - beta*. The negative rates in Example 6.5 expose an uncontrolled first-order linearization that can create a spurious root, a correctness and error-bound concern rather than a circular reduction. No self-definitional, fitted-input, or self-citation-reducing step is present.

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

No free parameters are fitted to data; the formulas contain only inputs of the model (λ, f, A, u). The listed axioms are the formal framework taken from previous papers, plus standard regularity assumptions. No new entities are introduced.

assumptions (5)

domain assumption Axioms of expected utility operator: linearity, monotonicity, T(A,1_R)=E_f(A), T(A,constant)=constant
Definition 3.1, quoted from [17], [18]. The whole model uses these as the definition of T.
domain assumption D-operator axioms (D1) and (D2): interchange of T with differentiation with respect to a parameter
Definition 3.6 from [20]. Needed to compute H'(β) and derive the first-order condition. This is the key restriction that limits the scope of the results.
domain assumption Utility function u is C^2 with u'>0 and u''<0
Standard risk-averse utility assumptions used throughout Sections 4 and 5.
domain assumption Support of fuzzy number A is contained in R+ and is not a single point
Used to guarantee E_f(A)>0 and to give the coinsurance problem economic meaning.
domain assumption Possibilistic approximation formula T(A,u) ≈ u(E_f(A)) + 0.5 u''(E_f(A)) V ar_T(A)
Quoted from [16], [17] as Proposition 3.5 and used in Theorem 5.9. It is a second-order Taylor approximation with no error estimate.

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Pith. "Pith review of Expected utility operators and coinsurance problem." pith.science (2026). https://pith.science/paper/XCJ3MI4D

@misc{pith2026190806927,
  author       = {Pith},
  title        = {Pith review of: Expected utility operators and coinsurance problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XCJ3MI4D}},
  note         = {Machine review of arXiv:1908.06927}
}

abstract

The expected utility operators introduced in a previous paper, offer a framework for a general risk aversion theory, in which risk is modelled by a fuzzy number $A$. In this paper we formulate a coinsurance problem in the possibilistic setting defined by an expected utility operator $T$. Some properties of the optimal saving $T$-coinsurance rate are proved and an approximate calculation formula of this is established with respect to the Arrow-Pratt index of the utility function of the policyholder, as well as the expected value and the variance of a fuzzy number $A$. Various formulas of the optimal $T$-coinsurance rate are deduced for a few expected utility operators in case of a triangular fuzzy number and of some HARA and CRRA-type utility functions.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [19]

Georgescu, Possibilistic risk aversion and coinsurance proble m, Fuzzy Information and Engineering, 5 (2), 2013, 221-233

I. Georgescu, Possibilistic risk aversion and coinsurance proble m, Fuzzy Information and Engineering, 5 (2), 2013, 221-233

work page 2013

[2] [1]

S. S. Appadoo, A. Thavaneswaran, Possibilistic moment generat ing func- tions of fuzzy numbers with Garch applications, Advances in Fuzzy S ets and Systems, Volume 6, Issue 1, 33 - 62, June 2010 17

work page 2010

[3] [2]

K. J. Arrow, Aspects of the theory of risk bearing, Helsinki: Yri¨ o Jahnssonin S¨ a¨ ati¨ o, 1965

work page 1965

[4] [3]

Arrow, Essays in the theory of risk bearing, North–Holland, Amster- dam, 1970

K.J. Arrow, Essays in the theory of risk bearing, North–Holland, Amster- dam, 1970

work page 1970

[5] [4]

Athayde, R

G. Athayde, R. Flores, Finding a Maximum Skewness Portfolio Gene ral Solution to Three-Moments Portfolio Choice, Journal of Economic D y- namics and Control, 28 , 2004, 1335–1352

work page 2004

[6] [5]

Carlsson, R

C. Carlsson, R. Full´ er, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets Syst., 122 , 2001, 315–326

work page 2001

[7] [6]

Carlsson, R

C. Carlsson, R. Full´ er, Possibility for decision, Springer, 2011

work page 2011

[8] [7]

Carlsson, R

C. Carlsson, R. Full´ er, P. Majlender, On possibilistic correlations, Fuzzy Sets Syst., 155 , 2005, 425–445

work page 2005