pith. sign in

arxiv: 2606.12868 · v1 · pith:YFSVG6BRnew · submitted 2026-06-11 · 🧮 math.OC

Maximum Utility Split Method for Utility Preference Elicitation

Bo Chen , Jia Liu , Huifu Xu This is my paper

Pith reviewed 2026-06-27 06:23 UTC · model grok-4.3

classification 🧮 math.OC
keywords utility elicitationmaximum utility splitambiguity setlottery questionnairelinear programmingKantorovich metricnominal utilitypreference elicitation
0
0 comments X

The pith

The maximum utility split method designs lottery questions that halve the range of possible utility functions at each step until the true utility is isolated.

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

The paper introduces the maximum utility split scheme to elicit a person's utility function by asking a sequence of lottery questions. One lottery uses two fixed outcomes with adjustable probability while the second offers a sure outcome placed exactly where the current range of possible utilities is largest, with the probability chosen to cut that range in half. An interval-based procedure locates the right sure outcome by solving a series of linear programs. Under moderate conditions the repeated splits shrink the ambiguity set until it contains only the true utility. When questioning stops early, a nominal utility is recovered by solving optimization problems that find the smallest and largest utilities in the set under the Kantorovich metric.

Core claim

The MUS scheme improves on random utility split by fixing one lottery's outcomes and selecting the deterministic outcome at the point of maximum utility range, then setting the probability so the ambiguity set range halves. This process can be continued indefinitely and converges to the true utility function. The key computational step is an interval-based algorithm that identifies each deterministic lottery by solving linear programs; when the process ends before the set collapses to a singleton, the nominal utility is obtained as the midpoint between the Kantorovich-minimal and Kantorovich-maximal utilities inside the set. The method is shown to work for concave utilities and extends to ge

What carries the argument

The interval-based algorithm that locates the deterministic-outcome lottery by solving a sequence of linear programs to find the point of largest utility range inside the current ambiguity set.

If this is right

  • Each MUS questionnaire halves the range of the ambiguity set.
  • Repeated application converges to the true utility function as the number of questions grows.
  • The interval-based linear-program sequence efficiently finds each split point.
  • A nominal utility can always be recovered from a nonempty ambiguity set by Kantorovich optimization even if elicitation stops early.
  • The same splitting logic extends directly to non-concave utility functions including S-shaped ones.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Fewer questions may be needed in practice to reach a usable utility estimate than with non-adaptive methods.
  • The linear-program structure suggests the scheme can be embedded inside larger optimization models for portfolio choice or insurance design.
  • The nominal-utility recovery step supplies a concrete way to proceed when real decision makers stop answering before the set collapses to one function.
  • Extension to S-shaped utilities opens direct use in settings where loss aversion or reference dependence is expected.

Load-bearing premise

The ambiguity set of utility functions has a structure that lets the maximum utility range point be identified by solving a sequence of linear programs.

What would settle it

A concrete ambiguity set built from preference data for which the interval-based algorithm returns a point that does not halve the utility range when the corresponding MUS question is posed.

read the original abstract

In this paper, we propose a new approach, called maximum utility split (MUS) scheme, which is built on random utility split (RUS) scheme but with a notable difference: one lottery is designed with two fixed outcomes but with varying probability, and the other has a deterministic outcome specifically chosen at the point where the range between the largest and smallest possible utility values is maximized. Consequently, the probability of random lottery is set such that the range of the ambiguity set of utility functions is reduced by half at the point. Under moderate conditions, we show that MUS can successively generate a sequence of such questionnaires and effectively reduce the ambiguity set, eventually converging to the true utility function as the number of questionnaires increases. The main challenge is to effectively identify the point with the largest utility range for a given ambiguity set constructed from preference information. Based on the structure of the ambiguity set, we propose an interval-based algorithm which identifies each certain-outcome lottery by solving a sequence of linear programs. Moreover, to deal with the case where elicitation terminates before the ambiguity set reduces to a singleton, we demonstrate how to figure out a nominal utility function by solving optimization programs. These identify the smallest and largest utility functions under the Kantorovich metric within the ambiguity set, after which we identify a nominal utility function located in the middle of them. Finally, numerical results demonstrate the efficiency of the MUS method and the performance of a robo-advisor system based on MUS-type queries and the nominal utility elicitation. While the main discussions focus on concave utility functions, we also demonstrate how the MUS approach can be extended to accommodate general non-concave utility functions, particularly S-shaped ones.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper proposes the Maximum Utility Split (MUS) method for utility preference elicitation. Building on random utility split, MUS constructs questionnaires consisting of one lottery with two fixed outcomes and varying probability, paired with a deterministic outcome chosen at the point of maximum utility range in the current ambiguity set; the probability is chosen to halve that range. Under moderate conditions the procedure is claimed to generate a sequence of such queries that successively shrinks the ambiguity set until convergence to the true utility function. For concave utilities an interval-based algorithm locates the required split point by solving a sequence of linear programs; the approach is extended to S-shaped utilities. When elicitation stops short of a singleton set, a nominal utility is recovered by solving optimization problems that identify the Kantorovich-metric extremal utilities and selecting the midpoint. Numerical experiments illustrate efficiency on robo-advisor instances.

Significance. If the convergence result and the correctness of the LP-based identification hold, the work supplies a concrete, query-efficient procedure for reducing ambiguity sets of utility functions that is directly usable in automated decision systems. The explicit use of linear programs to locate the maximum-range point and the Kantorovich-metric nominal selection are technically attractive features that could be reproduced or extended.

major comments (1)
  1. [Abstract / algorithm description] The abstract states that the interval-based algorithm identifies the split point 'by solving a sequence of linear programs' whose correctness rests on the structure of the ambiguity set (concave utilities, Kantorovich metric). Because the full derivation of why each LP correctly locates the required point is not visible in the provided material, the load-bearing algorithmic claim cannot yet be verified; a concrete counter-example or missing hypothesis on the ambiguity set would undermine the halving guarantee.
minor comments (2)
  1. [Abstract] The abstract mentions 'moderate conditions' for convergence but does not list them; adding an explicit statement of the required assumptions (e.g., compactness of the ambiguity set, continuity of the utility class) would improve readability.
  2. [Numerical results] Numerical results are summarized only at the level of 'demonstrate the efficiency'; reporting the precise number of queries needed to reach a given ambiguity-set diameter, together with the dimension of the utility space used in the experiments, would allow direct comparison with prior elicitation schemes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and positive assessment of the significance of the MUS method. We address the major comment below and will revise the manuscript accordingly to improve verifiability.

read point-by-point responses

  1. Referee: [Abstract / algorithm description] The abstract states that the interval-based algorithm identifies the split point 'by solving a sequence of linear programs' whose correctness rests on the structure of the ambiguity set (concave utilities, Kantorovich metric). Because the full derivation of why each LP correctly locates the required point is not visible in the provided material, the load-bearing algorithmic claim cannot yet be verified; a concrete counter-example or missing hypothesis on the ambiguity set would undermine the halving guarantee.

    Authors: We agree that explicit verification of the LP sequence is essential for the halving guarantee. The current manuscript describes the interval-based algorithm in Section 3 and states that it relies on concavity and the Kantorovich metric to locate the maximum-range point, but the step-by-step justification for why successive LPs correctly identify that point (without counterexamples under the maintained assumptions) is not fully expanded. In the revision we will add a dedicated subsection containing the complete derivation, including the key structural properties of the ambiguity set that ensure each LP solves the required subproblem and that the overall procedure halves the range. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines the MUS procedure explicitly through a lottery construction (one random lottery with fixed outcomes and varying probability, one deterministic lottery at the maximum utility-range point) and an interval-based algorithm that locates the split via a sequence of linear programs whose correctness is tied directly to the assumed structure of the ambiguity set (concave utilities, Kantorovich metric). The convergence claim is stated as a theorem under moderate conditions on that structure; no step reduces by construction to a fitted parameter, a self-citation chain, or a renaming of an input. The nominal-utility recovery step is likewise an explicit optimization program inside the same ambiguity set. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that utility functions are concave (with extension discussed for non-concave) and that the ambiguity set admits an interval structure solvable by linear programs; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Utility functions under consideration are concave (main case).
    Stated as the focus of main discussions, with extension to general non-concave shown separately.
  • domain assumption The ambiguity set constructed from preference information has sufficient structure for the maximum utility range point to be found via linear programs.
    Invoked to justify the interval-based algorithm.

pith-pipeline@v0.9.1-grok · 5830 in / 1423 out tokens · 16126 ms · 2026-06-27T06:23:50.468110+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

40 extracted references · 3 canonical work pages

  1. [1]

    Management science46(11), 1497–1512 (2000)

    Abdellaoui, M.: Parameter-free elicitation of utility and probability weighting functions. Management science46(11), 1497–1512 (2000)

    2000

  2. [2]

    Journal of Financial Econometrics19(2), 369–392 (2021)

    Alsabah, H., Capponi, A., Ruiz Lacedelli, O., Stern, M.: Robo-advising: Learning investors’ risk preferences via portfolio choices. Journal of Financial Econometrics19(2), 369–392 (2021)

    2021

  3. [3]

    Management science61(1), 111–128 (2015)

    Armbruster, B., Delage, E.: Decision making under uncertainty when preference informa- tion is incomplete. Management science61(1), 111–128 (2015)

    2015

  4. [4]

    Management science46(11), 1485–1496 (2000)

    Bleichrodt, H., Pinto, J.L.: A parameter-free elicitation of the probability weighting function in medical decision analysis. Management science46(11), 1485–1496 (2000)

    2000

  5. [5]

    Springer, Boston, MA (1993)

    Brown, R.F.: A Topological Introduction to Nonlinear Analysis. Springer, Boston, MA (1993)

    1993

  6. [6]

    IMA Journal of Management Mathematics (2026) https://doi.org/10.1093/imaman/dpag006

    Chen, B., Liu, J.: A robo-advisor system: Von neumann-morgenstern expected utility modeling via pairwise comparisons. IMA Journal of Management Mathematics (2026) https://doi.org/10.1093/imaman/dpag006

    work page doi:10.1093/imaman/dpag006 2026

  7. [7]

    Journal of the Operations Research Society of China 10(3), 529–558 (2022)

    Cui, X.-Y., Li, D., Qiao, X., Strub, M.S.: Risk and potential: An asset allocation framework with applications to robo-advising. Journal of the Operations Research Society of China 10(3), 529–558 (2022)

    2022

  8. [8]

    Cengage Learning, Mason, OH (2013)

    Clemen, R.T., Reilly, T.: Making Hard Decisions with DecisionTools. Cengage Learning, Mason, OH (2013)

    2013

  9. [9]

    Management Science64(1), 327–344 (2018)

    Delage, E., Li, J.Y.-M.: Minimizing risk exposure when the choice of a risk measure is ambiguous. Management Science64(1), 327–344 (2018)

    2018

  10. [10]

    In: Homburg, C., Klarmann, M., Vomberg, A

    Eggers, F., Sattler, H., Teichert, T., V¨ olckner, F.: Choice-based conjoint analysis. In: Homburg, C., Klarmann, M., Vomberg, A. (eds.) Handbook of Market Research, pp. 781–

  11. [11]

    Springer, Cham (2021)

    2021

  12. [12]

    Management science30(11), 1283–1300 (1984)

    Farquhar, P.H.: State of the art—utility assessment methods. Management science30(11), 1283–1300 (1984)

    1984

  13. [13]

    The Journal of Wealth Manage- ment20(1), 30 (2017)

    Faloon, M., Scherer, B.: Individualization of robo-advice. The Journal of Wealth Manage- ment20(1), 30 (2017)

    2017

  14. [14]

    Annals of Operations Research, 1–39 (2024)

    Gomez, A.A., Consigli, G., Liu, J.: Multi-period portfolio selection with interval-based conditional value-at-risk. Annals of Operations Research, 1–39 (2024)

    2024

  15. [15]

    International statistical review70(3), 419–435 (2002)

    Gibbs, A.L., Su, F.E.: On choosing and bounding probability metrics. International statistical review70(3), 419–435 (2002)

    2002

  16. [16]

    Operations Research72(5), 2241–2261 (2024)

    Guo, S., Xu, H., Zhang, S.: Utility preference robust optimization with moment-type information structure. Operations Research72(5), 2241–2261 (2024)

    2024

  17. [17]

    European Journal of Operational Research254(1), 214–225 (2016)

    Haskell, W.B., Fu, L., Dessouky, M.: Ambiguity in risk preferences in robust stochastic optimization. European Journal of Operational Research254(1), 214–225 (2016)

    2016

  18. [18]

    Academic Press, New York (1963) 58

    Hildebrandt, T.H.: Introduction to the Theory of Integration. Academic Press, New York (1963) 58

    1963

  19. [19]

    Operations Research 67(6), 1564–1585 (2019)

    Hu, J., Li, J., Mehrotra, S.: A data-driven functionally robust approach for simultaneous pricing and order quantity decisions with unknown demand function. Operations Research 67(6), 1564–1585 (2019)

    2019

  20. [20]

    IIE Transactions47(4), 358–372 (2015)

    Hu, J., Mehrotra, S.: Robust decision making over a set of random targets or risk-averse utilities with an application to portfolio optimization. IIE Transactions47(4), 358–372 (2015)

    2015

  21. [21]

    SIAM Journal on Optimization27(4), 2230–2257 (2017)

    Hu, J., Stepanyan, G.: Optimization with reference-based robust preference constraints. SIAM Journal on Optimization27(4), 2230–2257 (2017)

    2017

  22. [22]

    SIAM Journal on Optimization32(2), 1446–1470 (2022)

    Haskell, W.B., Xu, H., Huang, W.: Preference robust optimization for choice functions on the space of cdfs. SIAM Journal on Optimization32(2), 1446–1470 (2022)

    2022

  23. [23]

    Mathematical Programming, 1–47 (2024)

    Hu, J., Zhang, D., Xu, H., Zhang, S.: Distributional utility preference robust optimization models in multi-attribute decision making. Mathematical Programming, 1–47 (2024)

    2024

  24. [24]

    Mathematical Programming, 1–93 (2025)

    Liu, J., Chen, Z., Xu, H.: Preference ambiguity and robustness in multistage decision making. Mathematical Programming, 1–93 (2025)

    2025

  25. [25]

    Management Science67(11), 7113–7141 (2021)

    Li, J.Y.-M.: Inverse optimization of convex risk functions. Management Science67(11), 7113–7141 (2021)

    2021

  26. [26]

    Economic Theory19(4), 823–831 (2002)

    Maccheroni, F.: Maxmin under risk. Economic Theory19(4), 823–831 (2002)

    2002

  27. [27]

    In: Zarembka, P

    McFadden, D.: Conditional logit analysis of qualitative choice behavior. In: Zarembka, P. (ed.) Frontiers in Econometrics, pp. 105–142. Academic Press, New York (1974)

    1974

  28. [28]

    CRC Press, Taylor & Francis Group, Boca Raton (2011)

    Raghavarao, D., Wiley, J.B., Chitturi, P.: Choice-Based Conjoint Analysis: Models and Designs. CRC Press, Taylor & Francis Group, Boca Raton (2011)

    2011

  29. [29]

    Operations Research63(4), 789–806 (2015)

    Shi, Y., Cui, X., Yao, J., Li, D.: Dynamic Trading with Reference Point Adaptation and Loss Aversion. Operations Research63(4), 789–806 (2015)

    2015

  30. [30]

    Operations Research67(2), 315–338 (2019)

    Saur´ e, D., Vielma, J.P.: Ellipsoidal methods for adaptive choice-based conjoint analysis. Operations Research67(2), 315–338 (2019)

    2019

  31. [31]

    Marketing Science26(5), 596–610 (2007)

    Toubia, O., Hauser, J., Garcia, R.: Probabilistic polyhedral methods for adaptive choice- based conjoint analysis: Theory and application. Marketing Science26(5), 596–610 (2007)

    2007

  32. [32]

    Journal of Marketing Research41(1), 116–131 (2004)

    Toubia, O., Hauser, J.R., Simester, D.I.: Polyhedral methods for adaptive choice-based conjoint analysis. Journal of Marketing Research41(1), 116–131 (2004)

    2004

  33. [33]

    Management Science59(3), 613–640 (2013)

    Toubia, O., Johnson, E., Evgeniou, T., Delqui´ e, P.: Dynamic experiments for estimat- ing preferences: An adaptive method of eliciting time and risk parameters. Management Science59(3), 613–640 (2013)

    2013

  34. [34]

    Econometrica47(2), 263–292 (1979)

    Tversky, A., Kahneman, D.: Prospect theory: An analysis of decision under risk. Econometrica47(2), 263–292 (1979)

    1979

  35. [35]

    Journal of Risk and uncertainty5, 297–323 (1992)

    Tversky, A., Kahneman, D.: Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and uncertainty5, 297–323 (1992)

    1992

  36. [36]

    Marketing Science22(3), 273–303 (2003) 59

    Toubia, O., Simester, D.I., Hauser, J.R., Dahan, E.: Fast polyhedral adaptive conjoint estimation. Marketing Science22(3), 273–303 (2003) 59

    2003

  37. [37]

    Princeton University Press, Princeton (1947)

    Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior, 2nd Rev. Princeton University Press, Princeton (1947)

    1947

  38. [38]

    arXiv preprint arXiv:2003.01899 , year=

    Vayanos, P., Ye, Y., McElfresh, D., Dickerson, J., Rice, E.: Robust active preference elicitation. arXiv preprint arXiv:2003.01899 (2020)

    work page arXiv 2003

  39. [39]

    Research in International Business and Finance64, 101879 (2023)

    Yu, S., Wang, H., Dong, C.: Learning risk preferences from investment portfolios using inverse optimization. Research in International Business and Finance64, 101879 (2023)

    2023

  40. [40]

    arXiv preprint arXiv:2503.23269 , year=

    Zhang, S., Guo, S., Sim, M., Xu, H.: Modified polyhedral method for elicitation of shape-free utility and conservatism reduction in robust optimization. arXiv preprint arXiv:2503.23269 (2025) 60

    work page arXiv 2025