Maximum Utility Split Method for Utility Preference Elicitation

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.

• The paper introduces the maximum utility split method, which places pairwise lottery queries where the plausible utility functions differ most and halves that local utility range.
• The paper proves Hausdorff convergence to the true utility under error-free VNM preferences and provides an explicit finite-sample bound, while reducing query-point computation to linear programs.
• The paper reports faster ambiguity reduction and better portfolio recommendations than random and polyhedral baselines, although MUS requires more computation and assumes noiseless responses.

Overview and motivation

This paper proposes the maximum utility split (MUS) scheme, an adaptive questionnaire-generation method for eliciting a decision maker's (DM) Von Neumann–Morgenstern (VNM) utility function through pairwise lottery comparisons. The method operates within the preference robust optimization (PRO) framework of Armbruster and Delage, in which elicited preference information defines an ambiguity set $\mathcal{U}_m$ of plausible utility functions rather than a single fitted approximation. The central deficiency the authors identify in existing schemes—random utility split (RUS), random relative utility split (RRUS), and polyhedral cut methods—is the absence of a theoretical guarantee that repeated questioning shrinks the ambiguity set to a singleton, together with inefficient query placement. RUS selects the deterministic outcome $r_2^{m+1}$ uniformly at random from $[0,1]$, which wastes queries at points where the range of $\mathcal{U}_m$ is narrow.

MUS corrects this by placing the deterministic outcome at the point where the gap between the largest and smallest utility values attainable over $\mathcal{U}_m$ is maximal, and by setting the probability $p^{m+1}$ so that this range is halved at that point. The paper's main claims are: (i) MUS converges to the true utility function under the Hausdorff distance induced by the Kolmogorov norm, with an explicit finite-sample bound; (ii) the key optimization problem admits a semi-closed form solvable via linear programs; and (iii) when elicitation terminates early, a nominal utility function can be identified as the midpoint between Kantorovich-metric extremes of the ambiguity set.

The MUS scheme and convergence theory

The initial ambiguity set is $\mathcal{U}_{cv}$: normalized ($u(0)=0$, $u(1)=1$), monotonically increasing, concave, Lipschitz continuous functions with modulus bounded by $L$. Each query presents a binary lottery with outcomes fixed at 0 and 1 against a deterministic outcome $r_2^{m+1}$0, and the DM's response adds a half-space constraint to the set. The distinguishing step solves

$r_2^{m+1}$1

with $r_2^{m+1}$2 set to the midpoint of the range at $r_2^{m+1}$3.

The main theoretical result establishes four properties: compactness of $r_2^{m+1}$4 under the Kolmogorov norm; equality of the Kantorovich distance between two utility functions with the area between their curves; Hausdorff convergence of $r_2^{m+1}$5 to the singleton $r_2^{m+1}$6; and an explicit complexity bound. The convergence proof exploits the fact that each MUS query halves the maximum range $r_2^{m+1}$7, while Lipschitz continuity controls how much the range can change between nearby query points. Part (iv) gives a concrete guarantee: for any $r_2^{m+1}$8, after $r_2^{m+1}$9 queries, every utility function in $[0,1]$0 lies within $[0,1]$1 of $[0,1]$2 in the uniform norm. This is a stronger statement than anything available for RUS or RRUS, for which no such guarantee is known—a point the authors state explicitly.

Two assumptions underpin these results and should be noted plainly: the DM's preferences follow VNM expected utility theory, and there is no error in responses, observation, or data. The authors acknowledge that error-free elicitation "may be undesirable in some practical applications" and defer probabilistic convergence under random errors to future work.

Tractable computation of the query point

The computational core of MUS is solving the non-concave, non-parametric min-max problem above. The authors derive semi-closed forms for the lower bound function $[0,1]$3 and upper bound function $[0,1]$4:

• Lower bound: connecting the minimum utility values $[0,1]$5 at the elicited breakpoints yields a piecewise linear concave function that itself belongs to $[0,1]$6 and equals $[0,1]$7 everywhere.
• Upper bound: $[0,1]$8 is piecewise linear with one or two pieces per interval between adjacent elicited points, depending on whether the maximum right derivative $[0,1]$9 exceeds the minimum left derivative $\mathcal{U}_m$0; unlike $\mathcal{U}_m$1, it need not belong to $\mathcal{U}_m$2 globally.

The quantities $\mathcal{U}_m$3, $\mathcal{U}_m$4, $\mathcal{U}_m$5, and $\mathcal{U}_m$6 are each computed by solving linear programs over piecewise-linear approximations of the utility class, using a lemma showing membership in $\mathcal{U}_m$7 reduces to feasibility checks at the elicited points. Because $\mathcal{U}_m$8 has at most two linear pieces per interval, the maximizer lies in a finite candidate set, giving an interval-based algorithm whose cost scales with $\mathcal{U}_m$9 subproblems. An improved variant prunes intervals using lower and upper bounds on $\mathcal{U}_m$0 before computing derivatives, reducing the number of LPs solved in practice. The authors note that the semi-closed-form bounds also apply to related functionally robust problems such as pricing with unknown demand functions.

Nominal utility identification

Since practical elicitation terminates before $\mathcal{U}_m$1 becomes a singleton, the paper develops a procedure for selecting a nominal utility function. The smallest function $\mathcal{U}_m$2 is exactly $\mathcal{U}_m$3, shown to be the unique minimizer of the Kantorovich distance $\mathcal{U}_m$4 over $\mathcal{U}_m$5. The largest function $\mathcal{U}_m$6 requires more care because $\mathcal{U}_m$7 generally; it is obtained by maximizing $\mathcal{U}_m$8 over $\mathcal{U}_m$9, reformulated equivalently over tangent-line-based piecewise-linear upper approximations and ultimately as a second-order cone program. A closed-form expression for the Kantorovich distance between two piecewise-linear functions—derived via Lagrangian duality of a QCQP—supports both the construction and the final selection.

The nominal function is $p^{m+1}$0, which belongs to $p^{m+1}$1 and solves $p^{m+1}$2. Notably, the containment guarantee is $p^{m+1}$3 with $p^{m+1}$4, not radius $p^{m+1}$5: the authors exhibit a concrete example where some member of $p^{m+1}$6 sits at exactly distance $p^{m+1}$7 from $p^{m+1}$8, so the factor of three is tight. This is an honest limitation of the nominal-function approach relative to what one might hope for.

Extensions beyond concavity

The framework extends to general increasing Lipschitz utilities without concavity, where both bound functions become two-piece structures with slopes either zero or $p^{m+1}$9. For S-shaped utilities—convex on losses $\mathcal{U}_{cv}$0 and concave on gains $\mathcal{U}_{cv}$1, consistent with prospect theory—the domain is extended to $\mathcal{U}_{cv}$2 with lotteries spanning $\mathcal{U}_{cv}$3 to $\mathcal{U}_{cv}$4. A reflection mapping reduces the convex-loss segment to the concave-gain case, yielding semi-closed forms for both bounds. One structural observation here carries economic content: because the minimum of concave functions remains concave but the maximum does not, worst-case consensus among risk-averse DMs preserves risk aversion while best-case attitudes may diverge. The reference point is assumed known at $\mathcal{U}_{cv}$5; the authors concede that identifying it through pairwise comparisons "falls outside the scope of the current paper."

Numerical results

The academic example uses a true exponential-type utility with $\mathcal{U}_{cv}$6 and compares MUS against RUS, RRUS, and two configurations of the modified polyhedral method over 50 queries. MUS achieves the fastest reduction in both the Kantorovich and Kolmogorov distances between the largest and smallest elicited functions, and its distances to the true utility are consistently smallest across all four measured quantities. Two diagnostic findings are worth highlighting. First, RRUS fails to improve the largest utility function for its first 21 queries and Poly₁ for its first 31, because randomly drawn outcomes frequently leave the extreme function unchallenged; MUS-generated queries never fall into these degenerate cases. Second, polyhedral performance depends heavily on initial breakpoint placement—Poly₂, seeded with the breakpoint $\mathcal{U}_{cv}$7, converges far faster than Poly₁ despite having fewer breakpoints.

Computational cost favors the random schemes, as expected:

Queries	MUS	RUS	RRUS	Poly₁	Poly₂
10	2.67 s	0.32 s	0.56 s	17.97 s	15.57 s
20	23.65 s	1.10 s	2.15 s	104.64 s	80.13 s
40	249.82 s	4.11 s	8.37 s	6469.62 s	1058.87 s

MUS is roughly an order of magnitude cheaper than Poly₁ at 40 queries but substantially slower than RUS/RRUS—an inherent price of adaptive optimization.

The robo-advisor experiment embeds MUS into a portfolio recommendation loop over five U.S. assets (XLE, XLK, GLD, IEF, USDU plus cash) with weekly returns from 2020–2023, rebalancing every four weeks and asking one query per cycle with response rate $\mathcal{U}_{cv}$8. With full response ($\mathcal{U}_{cv}$9, 44 answered queries), MUS attains an average relative deviation of 0.2613 and MSE of 3.47E-05 against the true-optimal benchmark, versus 0.3860/1.53E-04 for RUS, 0.7611/3.16E-04 for RRUS, and 0.4527/1.10E-04 for Poly₂. At $u(0)=0$0 the gap widens sharply (MUS ARD 0.3295 versus 0.9346 for RUS), indicating that MUS extracts more value per answered query—precisely the regime where elicitation efficiency matters most. These results are preliminary in scale (a single simulated user, one asset universe, no transaction costs), which the authors do not dispute.

Limitations and open questions

Three limitations are conceded explicitly. First, the error-free elicitation assumption excludes measurement noise and inconsistent responses; extending to probabilistic convergence under random errors is left open. Second, the S-shaped extension presumes a known reference point at zero, whereas in prospect-theoretic applications the turning point may be unknown or dynamically adapting; incorporating reference-point identification into MUS is stated as necessary future research. Third, the numerical validation is limited to single-user simulations on one academic utility and one historical dataset, so the reported performance advantages have not been tested against heterogeneous user populations or out-of-sample market regimes. Additionally, the tightness of the factor-3 Kantorovich ball around the nominal utility means downstream PRO decisions based on $u(0)=0$1 inherit a quantified but nontrivial residual ambiguity.

Conclusion

The paper contributes a provably convergent, computationally tractable adaptive elicitation scheme that improves on RUS by targeting queries at the point of maximal ambiguity-set range. Its theoretical contributions—Hausdorff convergence with explicit sample complexity, LP-based semi-closed forms for the bound functions, a closed-form Kantorovich distance between piecewise-linear utilities, and a tight nominal-utility construction—are matched by numerical evidence of faster ambiguity reduction and better robo-advisor portfolio fidelity than existing benchmarks. The method's dependence on error-free responses and a known reference point, and the modest scale of the empirical evaluation, define the boundaries within which these results should be interpreted.