Economics, Decision Theory, and Operations Research

Preferential Bayesian optimization (PBO) models a person in one sentence: there is a stable utility function, and each answer to "which do you prefer?" reads it through a little Gaussian noise. Economics has spent a century testing every part of that sentence. The probit and logit links of Section 16.5 came from its econometrics, adaptive questioning under response error is the daily business of its marketing science, and its decision theorists have asked whether a person must be able to rank every pair of options at all. No other discipline in this part is as close to the method.

The chapter moves from what a single choice reveals to how preferences are aggregated and represented, then to how practitioners elicit them, and ends with how well the underlying experiments replicate. Two findings run through it. First, several problems that PBO papers treat as new already have worked-out treatments elsewhere, collected in Section 40.13: what PBO adds is a nonparametric surrogate over a continuous design space, not adaptive questioning itself. Second, the behavioral parameters most often proposed for a preference likelihood, such as a loss aversion coefficient or a discount factor, are exactly the quantities whose magnitudes replicate worst.

40.1 Behavioral economics #

Behavioral economics documents systematic departures from the textbook model of a rational chooser. The one most often borrowed for PBO is loss aversion: a loss looms larger than a gain of the same size, and the ratio between the two weights is the loss aversion coefficient. Tversky and Kahneman estimated it at about 2.25 (Tversky and Kahneman, 1992). Losses and gains are counted from a reference point, which in the model of Kőszegi and Rabin is the outcome a person rationally expected (Kőszegi and Rabin, 2006). Applied to PBO, the argument runs: the current best design becomes the reference point, so a challenger that is worse in some respect is coded as a loss, and the likelihood near the incumbent should be asymmetric by a factor of about 2. Further proposals add present bias (overweighting what comes now against what comes later, often modeled as quasi-hyperbolic discounting, which shrinks every delayed outcome by one extra fixed factor on top of a steady discount per period) across iterations, ambiguity aversion (preferring known odds to unknown odds, as in Ellsberg's urns (Ellsberg, 1961)) to describe how users trust the posterior, and a sizable share of purely selfish people in multi-user settings.

These claims hold in direction and mostly fail in magnitude. Loss aversion and reference points are two ingredients of prospect theory, Kahneman and Tversky's account of choice under risk, in which outcomes are valued as gains and losses and probabilities are weighted unevenly, small ones counting for more than they should and large ones for less (probability weighting). Its qualitative pattern replicated across 19 countries, 4,098 participants, and 13 languages, though with smaller effects and clear differences between countries (Ruggeri et al., 2020). The size of loss aversion is contested, as Figure 40.1 shows. A meta-analysis (a statistical pooling of published estimates) of 607 estimates from 150 articles found a mean of 1.955 (95% interval 1.820 to 2.102) (Brown et al., 2024). Fitting cumulative prospect theory, the 1992 version of the theory, to individual choices between mixed gambles (gambles that can end in a gain or a loss) gave 1.31 (95% confidence interval 1.10 to 1.53), with heterogeneity between studies accounting for 91.6% of the total variation (Walasek et al., 2024). A reanalysis of 163 estimates from 84 papers (n = 149,218) found that the coefficient depends on the design of the task: it is about 2.33 when losses are smaller than gains and the items are presented in order of size, and about 1.07, not significantly different from 1, when gains and losses are symmetric and unordered (Yechiam and Zeif, 2025).

Things to try:

• At the default, 2.25 (Tversky and Kahneman's estimate), the line lies outside both pooled intervals: above 1.820 to 2.102 and far above 1.10 to 1.53.
• Move the line anywhere between 1.53 and 1.82. It lies in neither interval, and no setting lies in both: the two pooled estimates do not overlap.
• Set the line to 1. It lies below both intervals but next to the estimate of about 1.07 for symmetric, unordered presentations, the condition closest to showing two designs side by side.

The supporting mechanisms fare no better. In a real-effort experiment (one in which subjects perform an actual task for pay), effort responded to changes in expectations in a clearly non-monotonic way, against the monotonic response that models of the Kőszegi-Rabin type predict (Gneezy et al., 2017). The endowment effect, owners asking more to give up a good than buyers offer for it (Kahneman et al., 1990), showed little relation to loss aversion for risky prospects in four incentivized surveys (answers had real monetary consequences) of 4,000 American adults, according to a working paper (Chapman et al., 2023). List's field experiments with traders found that market experience reduced the endowment effect (List, 2003; List, 2004), which has been read as showing that repeated questioning makes preferences converge; but when subjects were made to trade repeatedly, the asymmetry disappeared in later tests, and the authors attribute it mainly to uncertainty about trading (Engelmann and Hollard, 2010). Experience brings learning, but the mechanism is not that preferences converge.

The other parameters are shakier still. A meta-analysis of 220 estimates from 28 articles using convex time budgets (subjects split a budget between an earlier and a later date) found present bias on average, with large heterogeneity and modest selective reporting (Imai et al., 2021), and hyperbolic patterns (much more impatience over a short delay now than over the same delay later) appear almost as strongly in tasks with no payment delay at all, so complexity-driven errors produce hyperbolic-looking choices even when the underlying discount function is exponential, discounting every period at the same rate (Enke et al., 2025). This reading of anomalies as errors in valuing complex options (Oprea, 2024) is itself disputed (Section 40.12). Ambiguity aversion replicated for gains at moderate probabilities, but with losses or low probabilities most subjects were ambiguity neutral or seeking (Kocher et al., 2018). A structural mixture model of social preferences, which sorts people into a few types with their own preference parameters, found three other-regarding types (types that also weigh what others receive) that were stable over time and no purely selfish type (Bruhin et al., 2019), although a later study did identify a "predominantly selfish" type (Fehr et al., 2026). Defaults are robust on average: across 58 studies (combined n = 73,675) the effect was Cohen's d = 0.68 (95% confidence interval 0.53 to 0.83), where d is a difference in means divided by the standard deviation, but with large variation, several null results, and two negative effects (Jachimowicz et al., 2019).

What this means for PBO. A likelihood should not fix an asymmetry of about 2 near the incumbent. PBO usually shows two designs side by side and symmetrically, which is the condition under which Yechiam and Zeif found a coefficient close to 1; if an asymmetry is modeled at all, it belongs in a per-user parameter whose prior includes 1 (inference). Quasi-hyperbolic discounting across iterations does not fit the setting, since iterations are minutes apart and nothing is consumed later; what carries over is that measured impatience partly reflects complexity, which points to noise that grows with the difficulty of the comparison (Section 27.2) (inference). For several users, stable types favor a finite mixture of user types or a hierarchical prior (each user's parameters drawn from a shared population distribution) over fully independent per-user models (inference). And the initial design of a session is a default, so it should be randomized or balanced across sessions and users (inference).

40.2 Revealed and stable preferences #

Revealed preference is the economist's method of inferring preferences from choices instead of asking about them: if a person chose $x$ when $y$ was also available, $x$ is revealed preferred to $y$. With choices from budgets (the sets of bundles a person can afford at given prices and income), some utility function explains all the choices exactly when they satisfy the Generalized Axiom of Revealed Preference (GARP): the revealed-preference relation may contain no cycle that includes a strict preference. Afriat's critical cost efficiency index (Afriat, 1972) measures how far data are from passing: it is the largest fraction $e \le 1$ such that shrinking every budget to $e$ of its size removes all violations, so an index of 0.95 means the choices become consistent once the person is allowed to waste 5% of their money. In Andreoni and Miller's experiment on altruism, 18 of 176 subjects violated at least one revealed-preference axiom and only 3 had an index below 0.95 (Andreoni and Miller, 2002). Combined with Plott's discovered preference hypothesis, that preferences are not available at first but are discovered through experience with a task (Plott, 2001), such results support a strong claim about PBO: that iterative questioning converges to a person's true underlying preference.

That reading needs three qualifications. Power indices for these tests are "better thought of as ordinal rather than cardinal measures of power" (Andreoni et al., 2013) (a working paper), that is, they rank tests by power but do not say how far apart two tests are, so a high pass rate need not mean the test was demanding. Consistency may not carry across domains: a 2025 preprint found that consistency scores from risk choices in an experiment were uncorrelated with the consistency of the same people's supermarket purchases (Chen et al., 2025b). And forced choice itself manufactures inconsistency: when subjects could pay a small cost to defer a choice, their active choices were "generally more consistent", and deferral and choice together fit a model of dominant choice with incomplete preferences (Costa-Gomes et al., 2022), the topic of Section 40.8. The discovered preference hypothesis did receive direct support. Nielsen and Rehbeck asked subjects, under incentives, which axioms they wanted their choices to satisfy; subjects often revised conflicting lottery choices to agree with those axioms (Nielsen and Rehbeck, 2022).

What this means for PBO. A session's comparison log is revealed-preference data on two-element menus, and for such data GARP reduces to a simple condition: the directed graph with an edge from each winner to its loser must have no cycle. A system can check this throughout a session with cycle detection on the comparison graph of Section 18.5, without a Gaussian process (inference), although acquisition functions concentrate queries near the incumbent, so these tests have little power late in a session and a clean log is weak evidence (inference).

import networkx as nx

def revealed_cycles(duels):
    """duels: (winner, loser) pairs from one session.
    Returns every cycle of the revealed-preference graph; an empty
    list means some utility function explains all answers."""
    g = nx.DiGraph()
    g.add_edges_from(duels)
    return list(nx.simple_cycles(g))

# The fewest answers whose reversal removes every cycle form a minimum
# feedback arc set: NP-hard in general, small for session-sized logs.

When the log contains a cycle, showing it to the user and asking which answer to revise is the Nielsen and Rehbeck procedure: preference discovery through reflection, not convergence through repetition (inference). Offering a "no clear preference" answer should reduce manufactured inconsistency, with a third outcome in the likelihood (Section 20.4) (inference).

40.3 Endogenous and changing preferences #

PBO assumes the utility it learns does not change because it is being learned. Economists call preferences endogenous when they are shaped by the very institutions, choices, or experiences being studied. Machine learning's drift models and non-stationary bandits (methods for repeated choices whose payoffs shift over time, Section 13.2) treat change as exogenous, coming from outside, while constructed preferences change endogenously. If preferences are endogenous, conventional welfare economics, which judges outcomes by the preferences people hold, loses its footing; Dold asks whether "welfare analysis and normative economics are still possible" in such a world (Dold, 2023). Infante, Lecouteux, and Sugden criticized the "preference purification" of behavioral welfare economics, the search for the preferences of an inner rational agent behind observed errors (Infante et al., 2016), which suggests that the optimum a PBO session finds might reflect a self-reinforcing construction rather than a discovery. The empirical answer is again Nielsen and Rehbeck's result, which supports a limited purification endorsed by the person, not purification by a third party.

Several welfare criteria do without stable preferences. Bernheim and Rangel proposed that one option counts as better for a person than another only if the second is never chosen over the first in any frame (way of presenting the choice) in which both appear (Bernheim and Rangel, 2009); Sugden's opportunity criterion judges institutions by the opportunities they offer rather than by stable preferences, set out in a book whose existence we verified through a review (Engelen, 2019); and with a welfare measure for people who misunderstand consequences, conventional metrics turned out to rate two financial-education interventions as equally beneficial although only one reduced the average severity of mistakes (Ambuehl et al., 2022).

What this means for PBO. When preferences change during a session, the system needs an explicit welfare criterion. The Bernheim-Rangel rule applied to the session log recommends design $A$ over design $B$ only if $B$ was never chosen over $A$ in any observed frame or at any time, and otherwise reports the set of designs that no other design unambiguously beats (inference). A check in the spirit of Ambuehl and colleagues compares the final design once more with its main alternatives in a simplified frame with labeled attributes; the disagreement between the two frames estimates how much of the result rests on misunderstanding rather than preference (inference). The Nielsen and Rehbeck endorsement step is the economic counterpart of a reflective-endorsement check at stopping time (Section 46.6) (inference), with the limit that Section 41.2 explains: when the system is what changed the preference, endorsement after the fact cannot by itself show the change was legitimate.

40.4 Stochastic choice #

Asked the same question twice, people often answer differently. Random utility models (Section 16.5) explain this by giving each option's utility a random component; the probit and logit links of Section 27.1 are random utility models with Gaussian and Gumbel noise respectively (the Gumbel is a skewed bell curve, and the difference of two independent Gumbel draws has the logistic distribution, which gives the logit). A long-standing view, usually traced to McFadden (McFadden, 1981), is that the randomness can stand for variation within one person across occasions, for differences between people, or for genuine instability of preference, and that the three readings are formally equivalent. New experiments separate them. Agranov and Ortoleva repeated the same question either far apart or back to back, telling subjects that questions would repeat, and "a large majority of subjects exhibit stochastic choice in both cases" (Agranov and Ortoleva, 2017). When a later study let subjects randomize on every row of a choice list (a table of binary choices in which one option stays fixed and the other changes from row to row), "the majority of subjects chose to randomize in the majority of questions" (Agranov and Ortoleva, 2025); an overview is in Agranov and Ortoleva (2022). Deliberate randomization is one outward sign of a person who cannot rank two options; Section 40.8 reports how often people say so directly when allowed to.

Two lines of work since 2017 bear directly on PBO. In learning-based stochastic choice, a Bayesian probit model identifies stable preferences from context-dependent data and accommodates the attraction and compromise effects (an option gains when a similar but worse option, or a more extreme one, is added beside it) that random utility models cannot (Natenzon, 2019). Response-time models with uncertain utility cast choice as optimal sequential sampling, taking noisy looks one at a time and deciding after each whether to look again, and show that "choices are more likely to be correct when the agent chooses to decide quickly, provided the agent's prior beliefs are correct", which matches the data better than the classic drift-diffusion model of Section 39.3 (Fudenberg et al., 2018).

What this means for PBO. Evaluation noise, random preference, and deliberate randomization now each have an empirical signature, and each could have its own likelihood component: a probit or logit link for evaluation noise, a utility function drawn afresh from the posterior for each judgment for random preference, and a third outcome or an explicit mixture parameter for deliberate randomization (inference). Repeating the same pair immediately and treating every disagreement as noise confounds evaluation noise with deliberate randomization; retest pairs should be spaced out, and a 50/50 split on a repeated pair should not automatically be read as indifference (inference). Letting comparison noise depend on similarity, as Natenzon's Bayesian probit model does, would let the likelihood absorb part of the context effects of Section 37.2 (inference). And the prediction of Fudenberg and colleagues that fast choices are accurate choices, when the prior is right, differs from the drift-diffusion intuition that a slow choice signals a small difference in value; which pattern holds in design comparisons should be tested before response times enter a likelihood (Section 27.2) (inference).

40.5 Rational inattention #

Rational inattention models a chooser who cannot take in everything about the options and decides how much information to acquire, paying a cost for attention. Choices come out noisy even when the person's utilities are fixed, because sometimes it is not worth looking carefully. The theory is due to Sims (Sims, 2003), and its application to discrete choice to Matějka and McKay (Matějka and McKay, 2015): with an attention cost proportional to Shannon's mutual information (Section 6.3), the optimal behavior is a generalized multinomial logit (the logit rule extended to several options and shifted by prior beliefs),

where $v_i$ is option $i$'s true value, $\lambda$ is the cost of a unit of information, and $P^0_i$ is the probability of choosing $i$ before looking, which summarizes the chooser's prior beliefs. The result has been used to argue that the noise of a preference likelihood has a normative foundation: even with a fully stable utility, an optimally attentive person answers with noise.

The foundation is narrower than that. With Bregman information costs, a class defined in that paper that contains the Shannon cost as a special case, and for a given prior, rationally inattentive choice probabilities "may take the form of any additive random utility discrete choice model", that is, any model in which each option's utility is a fixed value plus a random term (Fosgerau et al., 2020). In experiments, subjects adjusted their attention to incentives as rational inattention predicts, but the results were qualitatively inconsistent with a cost linear in Shannon entropy (the standard measure of uncertainty, so that the cost is proportional to how much the person's uncertainty shrinks on average) (Dean and Neligh, 2023). The logit foundation holds as a theorem under Shannon costs, but the empirical case for that cost is weak; the broader claim, that optimal attention produces random choice even with stable utilities, still stands.

What this means for PBO. The likelihood should be heteroscedastic: the closer two designs are in the user's perceptual metric, the noisier the comparison, and the more the user cares, the less noisy (inference). The prior acts as a utility shift, the $P^0_i$ term in Equation (40.1): a user who believes the system's incumbent is probably good will favor it even with stable preferences. That explains the anchoring on the system's proposal reported in interactive studies (Section 32.3) without any change of preference, and only designs that change what the user believes about where the options came from, such as hiding which option is the system's proposal, can tell the two apart (inference). And because rational inattention can produce any additive random utility model, it cannot decide between the probit and logit links; only fit to data can (inference).

40.6 Information economics #

In Bayesian persuasion, or information design, a sender commits in advance to how the information a receiver sees will depend on the state of the world, choosing that rule to steer the receiver's action (Kamenica and Gentzkow, 2011). In Kamenica and Gentzkow's example, a prosecutor who designs the investigation, facing a judge who convicts only when the probability of guilt is at least one half, with a prior probability of guilt of 0.3, can obtain a conviction rate of 60%. From this it has been argued that, in PBO, choosing which candidates to show is itself information design, which raises ethical questions. Closer to elicitation, a payment rule is incentive compatible if answering truthfully is in the respondent's interest, and behaviorally incentive compatible if real people, not ideal ones, then answer truthfully. The binarized scoring rule, a standard way to pay people for reporting beliefs, violates two weak conditions of behavioral incentive compatibility: explaining the incentives to subjects increased their deviations from truthful reports (Danz et al., 2022).

What this means for PBO. The information structure of PBO is the reverse of persuasion: the user holds, or is forming, the preference, and the system is trying to learn it, so choosing queries is an elicitation or screening problem, in which the side that lacks information designs the questions that draw it out. Persuasion applies to a different question, how the candidates shown change the user's beliefs about what is achievable and what is good, and it matters most when users are unsure how to evaluate the options themselves (inference). The ethical argument holds in a narrower form: a system that decides which candidates to show and also benefits from particular outcomes is a sender whose preferences differ from the user's, and the persuasion framework bounds how far it can move the user's choices (inference); Section 46.8 returns to this. If a user study pays for "accurate" answers, the payment rule should be simple and not explained in mechanical detail (inference).

40.7 Social choice and mechanism design #

Social choice theory studies how to combine many people's preferences into one collective ranking or decision, and judges each rule by axioms it satisfies or violates; mechanism design designs the rules so that participants who act in their own interest still reveal what the designer needs to know. The classic results are impossibility theorems. Arrow's theorem says no ranking rule over three or more options satisfies a short list of reasonable axioms at once (Arrow, 1950); the Gibbard-Satterthwaite theorem says every voting rule over three or more outcomes that is not a dictatorship (a rule that always follows one fixed voter) can be manipulated by some voter (Gibbard, 1973; Satterthwaite, 1975). Black's single-peaked preferences (each person's preference falls away on both sides of a favorite point along one shared axis) (Black, 1948), May's theorem on majority rule (May, 1952), Harsanyi's weighted sums of utilities (Harsanyi, 1955), and the Vickrey-Clarke-Groves mechanisms that use money to make truth-telling optimal, by charging each participant the cost their report imposes on everyone else (Vickrey, 1961; Clarke, 1971; Groves, 1973), complete the toolkit usually cited when preference learning meets social choice. Chichilnisky's topological theorem, often stated as saying that on continuous spaces of preferences no social choice function satisfies unanimity (if everyone agrees, the rule follows) and anonymity (the rule treats all people alike) together, concerns continuous aggregation maps (Chichilnisky, 1980); the common statement omits that continuity requirement on the rule.

The main development is that social choice theory is now applied directly to learning a utility from pooled comparisons, which is exactly the aggregation problem of PBO with several users (Section 20.5). Ge et al. (2024) show that the Bradley-Terry-Luce model and its broad generalizations "fail to meet basic axioms": any aggregation rule defined by a nondecreasing convex loss violates Pareto optimality (if everyone prefers $a$ to $b$, the output should too) and pairwise majority consistency (if majorities over every pair agree with one ranking, the output should be that ranking). Siththaranjan et al. (2024) show that under hidden heterogeneity, standard preference learning implicitly aggregates by Borda count, which scores each option by how many others it beats on average across people. A 2025 preprint shows that standard regularized maximum likelihood is not robust to approximate clones (adding near-duplicates of an option should not substantially change the learned reward), and that weighting by similarity restores the property (Procaccia et al., 2025).

What this means for PBO. A multi-user system that puts everyone's comparisons into one Gaussian process with a probit or Bradley-Terry likelihood inherits these axiomatic failures. The alternative is to fit a posterior per user and aggregate with an explicit rule, or to report the distribution of utilities when disagreement is large (inference). Acquisition functions generate many near-identical candidates around the incumbent, which is the clone situation: even with a single user, a cluster of near-identical comparisons can dominate the fit, and similarity weighting is a remedy (inference). Averaging cardinal utility functions is continuous, anonymous, and unanimous, so Chichilnisky's obstacle does not apply to averaging posterior means; the real difficulty is that each user's utility scale is fixed only relative to that user's own noise variance (Section 18.4), so averaging first needs an interpersonal normalization, a decision about how one user's unit of utility compares with another's, which is a normative choice in Harsanyi's sense (inference). Strategic misreporting in single-user PBO hurts only the user; with several users, three or more designs, and non-dictatorial aggregation, the Gibbard-Satterthwaite theorem applies (inference).

40.8 Decision theory #

Decision theory states axioms for rational choice and derives what follows from them, most famously that a person who satisfies Savage's axioms acts as if maximizing expected utility under some probability belief. Savage himself restricted his theory to "small worlds" whose possibilities can be listed in advance (Savage, 1954); Binmore has argued that Bayesian decision theory does not carry over to "large worlds" (Binmore, 2009), and creative design spaces look like large worlds, in which expecting a complete preference ordering is unreasonable. Debreu's representation theorems imply that a continuous utility representation requires continuous preferences, while lexicographic preferences (ranking by one attribute and using the next only to break exact ties) cannot be represented by any real-valued utility at all (Debreu, 1983), so a Gaussian process utility presupposes continuity. The classical alternative to completeness is Bewley's Knightian decision theory, which drops the completeness axiom and adds an inertia assumption: an alternative is accepted only if it is preferred to the status quo (Bewley, 2002).

The most important development of 2017 to 2026 is that incompleteness has been measured directly. Cettolin and Riedl found that about half of their participants made choices inconsistent with a model of complete preferences plus certainty independence (mixing both options, in the same proportion, with the same sure outcome does not change which one is preferred); of these, about half behaved in line with incomplete preferences, and probability weighting, choice errors, regret aversion, and intransitive indifference were all ruled out as explanations (Cettolin and Riedl, 2019). Nielsen and Rigotti elicited incompleteness directly, in a working paper (Nielsen and Rigotti, 2026). About 40% to 50% of subjects expressed incomplete preferences when allowed to, and forced choice produced more inconsistency: transitivity violations ran at 4.0% in forced choice against 1.7% among strict preferences expressed when choice was not forced. Subjects who ever reported incompleteness did so in 3.3 of 50 comparisons on average, and when the objects were made deliberately complex, 76% reported incompleteness. Decision theory under growing awareness adds a rule for options that appear later: when new possibilities are discovered, beliefs update by "reverse Bayesianism", which keeps the relative probabilities of the old possibilities and moves mass to the new ones (Karni and Vierø, 2013).

What does a forced binary likelihood do with an answer that is not a preference? In Figure 40.2 you choose between flats that trade living space against commute time, and may also answer "can't compare" or ask for a coin flip. The forced-choice model has one weight and the probit likelihood of Section 27.1, and receives every "can't compare" as the outcome of the coin you would otherwise have had to toss. The second model follows Definition 40.1: you hold a set of admissible weights, and "can't compare" means the set straddles the pair's tie point.

Things to try:

1. Answer every pair with Flat A or Flat B only. The two models then roughly agree on where your weight lies and on the shortlist. Unless your answers push the weight to one end of the axis, the set model's curve stays below even odds everywhere, because answers that never say "can't compare" point to a set so narrow that it could sit anywhere in that region.
2. Reset and answer again, choosing "Can't compare" or "Flip a coin for me" whenever the trade-off feels impossible. Each such answer narrows the forced model around a weight chosen by a coin, while the set model widens its range of admissible weights and keeps more flats on the shortlist.
3. Switch to the simulated person, press "Simulate all", and reveal the simulated weights. In the default run the forced model puts 90% of its belief on weights 0.43 to 0.57 and calls flat 4 best with probability 0.71; the set model keeps flats 3, 4, and 5, which are the flats that are best somewhere inside the person's true range. The forced model's confidence comes from the coin.

What this means for PBO. Queries should include a "can't compare" answer, and the likelihood should model it. Under Bewley's multi-utility reading, one design is better than another only when a whole set of utilities agrees; a system can return the designs that no other design beats under every posterior sample instead of a single optimum, although the spread of posterior samples measures the system's uncertainty about the person, not the person's own incompleteness, unless "can't compare" answers enter the likelihood and widen the set, as in the middle panel of Figure 40.2 (inference). The evidence also bounds the worry that forced choice is a category error: people who report incompleteness do so in a minority of comparisons, 3.3 of 50 on average, though design comparisons often are complex (inference). Reverse Bayesianism gives a rule for expanding the design space in the middle of a session without restarting, a formal answer to the "large world" objection that no PBO system implements yet (inference). Lexicographic rules and hard limits, like the veto thresholds of Section 40.10, are better represented as constraints (Section 14.4) than forced into a smooth utility (inference).

40.9 Discrete choice and conjoint analysis #

Conjoint analysis is a survey method from marketing. Respondents choose among, or rate, product profiles that vary in their attributes (price, size, brand), and a model, usually a logit, estimates the part-worth utility of each attribute level. Choice-based conjoint shows a few profiles at a time and asks for the preferred one, which makes it a close relative of PBO, with a parametric, usually linear, utility in place of a Gaussian process. Adaptive designs choose each question from the answers so far; one principle, utility balance, picks questions whose options have nearly equal predicted utility, which is what an acquisition function does when it favors pairs whose outcome is most uncertain. Before PBO existed, marketing science had polyhedral adaptive designs (Toubia et al., 2003; Toubia et al., 2004), a probabilistic version that adds response error and informative priors (Toubia et al., 2007), Bayesian designs that exploit managers' prior beliefs about the parameters (SÁndor and Wedel, 2001), designs for mixed logit models (Sándor and Wedel, 2002), and sequential Bayesian designs adapted to each respondent's earlier answers, which in simulations held up under low response accuracy where the polyhedral method did not (Yu et al., 2011).

The field also documented the known trap of adaptive questioning. Hauser and Toubia showed that adaptive, utility-balanced metric questions (paired comparisons in which respondents state how much they prefer one profile, not just which) bias part-worth estimates relative to one another, through an endogeneity like the winner's curse (the questions are chosen because of the current estimates, so their answers are not independent of those estimates). Utility-balanced questions can also be inefficient, and shrinkage estimation (pulling each respondent's estimates toward the sample average) did not remove the biases, although in their data the biases and inefficiencies were of the order of the response errors (Hauser and Toubia, 2005). Their analysis concerns these metric questions only. Since 2017, Sauré and Vielma approximated the posterior with a normal distribution, bringing response error into an approximate Bayesian method and allowing question selection by optimization (Sauré and Vielma, 2019), and Gibbard and Sadlier found that for population-level parameters the estimation accuracy of an optimal adaptive Bayesian design is "uniformly worse than the random design", so such designs suit studies focused on the specific individuals in the sample (Gibbard and Sadlier, 2025).

Environmental economics values goods that are not sold in markets, such as clean rivers, with stated preference surveys, in which respondents state what they would pay for a described change or choose among described alternatives with different costs; these methods and PBO have been described as isomorphic but developed independently. The 2017 guidelines of Johnston and colleagues (Johnston et al., 2017), which go further than those of the 1993 panel chaired by Kenneth Arrow (NOAA Panel on Contingent Valuation, 1993), say that a single binary choice is the most directly incentive-compatible format, that question order should be randomized across respondents, and that as statistical efficiency and choice complexity rise, the consistency of respondents' choices falls; a large choice experiment found robust order effects in repeated-response stated preference studies (Day et al., 2012).

What this means for PBO. Whether the winner's-curse bias of Hauser and Toubia appears in the utility magnitudes or lengthscales a Gaussian process estimates has not been tested; by its mechanism, small effects would be inflated relative to large ones, and mixing in random queries is an off-the-shelf remedy, one that studies aiming at population conclusions need anyway (inference). An acquisition function that picks the most informative query also tends to raise choice complexity and lower answer consistency, so the trade-off belongs in the acquisition design, for example as a penalty on pairs that differ on many attributes at once (inference). PBO differs from this field in its surrogate, a Gaussian process or other nonparametric model over a continuous design space; an adaptive conjoint study with such a surrogate is one of the gaps in Section 40.14.

40.10 Multi-criteria decision analysis #

Multi-criteria decision analysis (MCDA) is a family of methods from operations research for helping a decision maker rank or choose options judged on several conflicting criteria, such as cost, safety, and comfort. Compensatory methods let a gain on one criterion offset a loss on another, as a weighted sum does; non-compensatory methods do not. The outranking methods ELECTRE (Roy, 1968) and PROMETHEE (Brans and Vincke, 1985) compare options criterion by criterion with thresholds for indifference, preference, and veto (a difference on one criterion so large that no advantage elsewhere can make up for it); the analytic hierarchy process (Saaty, 1977) derives weights from pairwise ratio judgments; TOPSIS ranks options by their distance to an ideal and an anti-ideal point (Hwang and Yoon, 1981).

Closest to PBO, robust ordinal regression ranks options using every additive value function (a sum of one value function per criterion) consistent with the decision maker's pairwise statements (Greco et al., 2008). It reports two relations: $a$ is necessarily preferred to $b$ if every compatible value function ranks $a$ at least as high as $b$, and possibly preferred if at least one does. Corrente and colleagues compared the method with preference learning in machine learning (Corrente et al., 2013). Since 2017, Ciomek, Kadziński, and Tervonen compared question-selection heuristics for pairwise elicitation and validated the best of them in an experiment in which 101 participants answered pairwise questions to rank 10 mobile phone packages on four criteria (Ciomek et al., 2017); Grillo, Kotłowski, and Kadziński proved regret bounds for Bayesian methods and for regularized maximum likelihood (follow the regularized leader, which after each answer refits to all answers so far with a penalty that keeps the estimate from jumping) under linear and Bradley-Terry models (Grillo et al., 2025); and Huber, Rojas Gonzalez, and Astudillo built a Bayesian model of the decision maker's utility from pairwise comparisons, chose queries by balancing exploration and exploitation, and tested on problems with up to nine objectives (Huber et al., 2025).

What this means for PBO. The necessary and possible relations correspond to preferences with posterior probability 1 and greater than 0; reporting both would give users an interpretable summary of what their comparisons have settled (inference), and the "P(not beaten)" row of Figure 40.2 is a summary of this kind. Veto thresholds express hard limits that a smooth Gaussian process utility cannot, which supports treating some preferences as constraints (inference). We found no study that compares the question-selection heuristics of robust ordinal regression with preferential acquisition functions on the same human task; that experiment could be run today.

40.11 Interactive multi-objective optimization #

In multi-objective optimization (Section 14.5) several objectives conflict, and the answer is a Pareto front: designs that cannot be improved on one objective without worsening another. Interactive methods let a human decision maker steer the search toward the part of the front they like by stating preferences during the optimization. A related line in artificial intelligence chose queries by value of information (Chajewska et al., 2000) and by minimax regret (recommending the option whose worst-case loss, over all utilities still consistent with the answers, is smallest). A journal paper in that line proves that under both the Bayesian and the minimax-regret frameworks, an optimal recommendation set of size $k$ is also an optimal choice query of size $k$, so "there is no tradeoff to be made between good recommendations and good queries"; the analysis covers noise-free responses and constant and logistic noise, and the abstract calls the results robust to such noise (Viappiani and Boutilier, 2020). The abstract of the 2010 conference version says that for noisy responses it provides worst-case guarantees (Viappiani and Boutilier, 2010), which points to a bounded loss rather than an exact equivalence; we have not checked the theorem itself. qEUBO, which scores a set of $q$ shown candidates by the expected maximum utility among them (Section 19.4), was later proposed and analyzed as a decision-theoretic acquisition function (Astudillo et al., 2023); the equivalence is the earlier foundation of this family.

Interactive multi-objective optimization also designed interactions that resist anchoring well before PBO. The NAUTILUS method starts from the worst point and lets the decision maker improve every objective at each step without trading any off, which avoids both anchoring and loss framing (Miettinen et al., 2010). Anchoring has also been measured: with 128 participants using interactive goal programming on a two-objective problem, anchoring occurred, made decision makers miss their most preferred solution, and was stronger when they used someone else's preferences (Halstead et al., 2026). And the field has evaluation norms: a survey of how interactive methods are assessed records the type of decision maker (a utility or value function, an artificial decision maker, or a human) and the type of preference information (Afsar et al., 2021), and a reusable questionnaire and experimental design support comparisons with human decision makers (Afsar et al., 2022).

What this means for PBO. The standard interaction, incumbent against challenger, trades one attribute against another; a NAUTILUS-style path that only ever improves answers the concern about reference dependence without estimating a loss aversion parameter, and varying the initial design between sessions counteracts anchoring (inference). User studies of PBO should adopt the evaluation norms of this field: report whether the decision maker is a person or a simulated user, what preference information was collected, and how stopping was decided (Section 31.7, Section 46.7) (inference).

40.12 Replication in experimental economics #

Since several sections above lean on behavioral findings, it matters how well those findings replicate. In general, moderately well. Of 18 laboratory experiments published in two leading economics journals, 11 (61%) replicated with a significant effect in the same direction, and the replicated effects averaged 66% of the originals (Camerer et al., 2016). Of 21 social-science experiments published in Nature and Science between 2010 and 2015, 13 (62%) replicated significantly in the same direction, and replication effect sizes averaged about 50% of the originals (Camerer et al., 2018). Economic Inquiry ran a two-part symposium on reproducibility and replicability, concerned mainly with non-experimental studies (Bokhari et al., 2025a; Bokhari et al., 2025b). Table 40.1 lists the status of the specific findings that have been proposed as ingredients of preference models; works already cited in this chapter appear by author and year.

The table relies on four sources not yet introduced. On the complexity account of risk and time anomalies, Oprea answered a 2025 working paper by Banki, Simonsohn, Walatka, and Wu (Banki et al., 2025) by arguing that their reversal of his conclusion came mainly from using medians, and that with means the deviations remain highly significant (Oprea, 2025); the dispute is open. Frederick, Lee, and Baskin found that the attraction effect may be confined to stylized presentations in which every product dimension is a number; it usually disappears when consumers experience the products, or when even one attribute is presented perceptually (Frederick et al., 2014). In two preregistered replications (the analysis plan was filed before the data were collected; n = 311 and n = 316), status quo bias had strong support in three of four decision scenarios and no substantial support in the fourth (Xiao et al., 2021).

What this means for PBO. The behavioral parameters proposed for preference likelihoods rest on exactly the findings whose magnitudes are least stable; the direction of these effects is more reliable than their size, so a likelihood should not fix their magnitudes (inference). The limit on the attraction effect is especially relevant: PBO is often used for perceptual designs such as color, shape, and motion, which is where Frederick and colleagues found decoy effects rarely appear, so the worry that adding candidates reshapes the utility landscape through decoys is weakened, at least for this mechanism, although divisive normalization (Section 39.4) may still apply to unfamiliar options (inference).

40.13 Already solved elsewhere #

The sections above found, one by one, that problems PBO papers treat as new have established treatments in neighboring fields. Table 40.2 gathers them; its last column is this chapter's reading of what each treatment implies, and every entry in it is an inference.

40.14 Settled, contested, missing #

Further reading #

• Strzalecki (2025) is the reference text on stochastic choice: random utility, its alternatives, and what choice frequencies can identify.
• Agranov and Ortoleva (2022) give a short overview of the experiments showing that people sometimes randomize on purpose.
• Nielsen and Rigotti (2026) (working paper) and Cettolin and Riedl (2019) are the two most direct measurements of incomplete preferences; read them before adding a "can't compare" button.
• Johnston et al. (2017) distill decades of stated preference practice into guidance that applies almost unchanged to preferential query design.
• Hauser and Toubia (2005) is the clearest account of how adaptive questions can bias what they estimate.
• Viappiani and Boutilier (2020) prove the equivalence of recommendation sets and query sets that underlies EUBO-type acquisition functions.
• Ge et al. (2024) show, axiom by axiom, what pooling everyone's comparisons into one Bradley-Terry model does.
• Miettinen et al. (2010) and Afsar et al. (2021) describe an interaction design that avoids anchoring and the norms for evaluating such designs with people.