Distributions of posterior quantiles via matching

Kolotilin, Anton; Wolitzky, Alexander Article Distributions of posterior quantiles via matching Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Kolotilin, Anton; Wolitzky, Alexander (2024) : Distributions of posterior quantiles via matching, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 19, Iss. 4, pp. 1399-1413, https://doi.org/10.3982/TE6057 This Version is available at: https://hdl.handle.net/10419/320269 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 19 (2024), 1399–1413 1555-7561/20241399 Distributions of posterior quantiles via matching Anton Kolotilin School of Economics, UNSW Business School Alexander Wolitzky Department of Economics, MIT We offer a simple analysis of the problem of choosing a statistical experiment to optimize the induced distribution of posterior medians or, more generally, qquantiles for any q∈(0, 1). We show that a single experiment—the q-quantile matching experiment—implements all implementable distributions of posterior q-quantiles, with different distributions spanned by different selections from the sets of posterior q-quantiles. A dense subset of implementable distributions of posterior q-quantiles can be uniquely implemented by perturbing the q-quantile matching experiment. A linear functional is optimized over distributions of posterior q-quantiles by taking the optimal selection from each set of posterior qquantiles. The q-quantile matching experiment is the only experiment that simultaneously implements all implementable distributions of posterior q-quantiles. Keywords. Quantiles, statistical experiments, median matching, overconfidence, gerrymandering, persuasion. JEL classification. C61, D72, D82. 1. Introduction Several problems of recent economic interest amount to characterizing the set of distributions of posterior quantiles that can be induced by some statistical experiment or to finding a distribution in this set that maximizes some objective. These problems include apparent overconfidence (Benoît and Dubra (2011)) (e.g., what distributions of medians of individuals’ beliefs about their own abilities are consistent with Bayesian updating?), partisan gerrymandering (Friedman and Holden (2008), Kolotilin and Wolitzky (2020b)) (e.g., what is the highest distribution of district median voters attained by any districting plan?), and quantile persuasion (Kolotilin and Wolitzky (2020a)) (e.g., what experiment maximizes the expected action of a receiver who minimizes the expected absolute deviation of her action from the unknown state of the world?).1 Anton Kolotilin: [email protected] Alexander Wolitzky: [email protected] 1Yang and Zentefis (2024) explore these and other applications. Kolotilin and Wolitzky (2020b)consider a more general gerrymandering model, which reduces to optimizing the distribution of posterior quantiles in a special case. Kolotilin and Wolitzky (2020a, Proposition 2’) introduce quantile persuasion as a special case of a more general persuasion model, which is further developed in Kolotilin, Corrao, and Wolitzky (forthcoming). ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE6057 1400 Kolotilin and Wolitzky Theoretical Economics 19 (2024) Our problem is as follows. There is a real-valued state θ. A statistical experiment induces a distribution over posteriors μ.Foranyq∈(0, 1), each posterior μhas at least one q-quantile. In general, a posterior can have multiple q-quantiles due to gaps in the support of μ: for example, if μputs equal weight on two states θ<θ , then the set of medians of μis the entire interval [θ,θ]. An experiment, together with a selection rule to break ties for posteriors with multiple q-quantiles, induces a distribution of posterior q-quantiles. A distribution of posterior q-quantiles is implementable if it is induced by some experiment and selection rule; it is uniquely implementable if it is induced by an experiment that almost always induces posteriors with unique q-quantiles. We ask what distributions of posterior q-quantiles are implementable or uniquely implementable, how to implement them, and how to optimize a linear functional over distributions of posterior q-quantiles. We provide a simple solution to this problem. For any q∈(0, 1), there is a single experiment—the q-quantile matching experiment—that simultaneously implements all implementable distributions of posterior q-quantiles, with different distributions spanned by different selection rules. For example, if the state is uniformly distributed on [0, 1]and the relevant quantile is the median, the q-quantile matching experiment is the median matching experiment that, whenever the true state is θ∈[0, 1/2], reveals only that the state is either θor 1/2+θ(and, hence, whenever the true state is θ∈(1/2, 1], reveals only that the state is either θor θ−1/2).2In general, the q-quantile matching experiment pools pairs of states across a q-quantile of the prior in a positively assortative manner, with weight qon the lower state in each pair. To see why the q-quantile matching experiment implements all implementable distributions of posterior q-quantiles, consider again the median matching experiment with a uniform state. When the experiment reveals that the state is θor 1/2+θwith equal probability, every value x∈[θ,1/2+θ]is a posterior median. The median matching experiment thus simultaneously implements (i) the distribution H(x)=max{0, 2x−1}, (ii) the distribution H(x)=min{2x,1 }, and (iii) every distribution Hsatisfying H≤H≤H. Conversely, simple Markov-type inequalities imply that every implementable distribution is bounded by Hand H. Moreover, the set of uniquely implementable distributions of posterior quantiles is essentially the same: any desired selection from each set of qquantiles induced by the q-quantile matching experiment can be uniquely selected by mixing each posterior under the q-quantile matching with the degenerate distribution on the desired selection with probabilities 1 −εand ε, respectively. Finally, optimizing a linear functional over distributions of posterior quantiles simply requires taking the optimal selection from each set of q-quantiles induced by the q-quantile matching experiment. See Figure 1for an illustration of our results.3 2To our knowledge, the median matching experiment first appears in Kolotilin and Wolitzky (2020a,p. 29). It is closely related to the median one-to-one matching introduced by Kremer and Maskin (1996) and further studied by Legros and Newman (2002); the title of the present paper acknowledges this connection. 3Similar figures in the literature include Figure 1 of Owen and Grofman (1988), Figure 2 of Kamenica and Gentzkow (2011), and Figure 3 of Yang and Zentefis (2024). 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE6057 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Distributions of posterior quantiles 1401 Figure 1. Implementable distributions of posterior medians. When the prior Fis uniform on [0, 1],Hand Hare the lowest and highest implementable distributions of posterior medians. A distribution His implementable if and only if H≤H≤H. Optimizing a linear functional over distributions of posterior medians requires taking the optimal selection from each horizontal dotted line. For example, the blue (red) dots are the optimal selections for an increasing (decreasing) objective function. We also show that the q-quantile matching experiment is the unique experiment that simultaneously implements all implementable distributions of posterior qquantiles. To see why, consider again a uniform state, and compare the median matching experiment with the negative assortative matching experiment that, whenever the true state is θ∈[0, 1], reveals only that the state is either θor 1 −θ.Thenegativeassortative matching experiment simultaneously implements the lowest and highest distributions of posterior medians, Hand H, but it does not implement all intermediate distributions, such as the distribution H1/2given by H1/2(x)=H(x)for x<1/4, H1/2(x)=1/2forx∈[1/4, 3/4),andH1/2(x)=H(x)for x≥3/4. Indeed, the negative assortative matching experiment induces posteriors with medians between 1/4and3/4 when the true state lies between 1/4and3/4, while H1/2assigns probability 0 to these medians. The current paper is closely related to Benoît and Dubra (2011)andYang and Zentefis (2024). Both of these papers establish results that are very similar to our Theorem 1(albeit Benoît and Dubra do so for discrete experiments with finitely many induced posteriors). Our main contribution is introducing the q-quantile matching experiment, which yields a much simpler proof of Theorem 1,aswellasnewresults(Theorems2and 3). 2. Implementable distributions of posterior quantiles This section shows that the q-quantile matching experiment implements all implementable distributions of posterior q-quantiles. Let =[θ,θ]⊂R,withθ < θ, be a compact state space, let C()be the set of continuous functions on ,let()be the set of cumulative distribution functions on , endowed with the weaktopology, and let (()) be the set of probability measures 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE6057 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1402 Kolotilin and Wolitzky Theoretical Economics 19 (2024) on (). Recall that G∈()is a nondecreasing, right-continuous function satisfying G(θ)≥0andG(θ)=1. Let δx,withx∈, denote the degenerate distribution at x,so that δx(θ)=1{θ≥x}. Fix a prior distribution F∈()and a quantile of interest q∈(0, 1). Following Kamenica and Gentzkow (2011), define an experiment as a distribution τ∈(())of posterior distributions G∈()such that Gdτ(G)=F. For each posterior G, define the set of q-quantiles of Gas X(G)=x∈:Gx−≤q≤G(x), where G(x−)denotes the left limit limθ↑xG(θ), with the convention G(θ−)=0. In addition, for each G, define its generalized inverse G−1as G−1(p)=infθ∈:G(θ)≥pfor all p∈[0, 1]. That is, G−1(p)is the smallest p-quantile of G. To define the q-quantile matching experiment, let ωbe uniformly distributed on [0, 1], and for each ω∈[0, q],letG=Gωbe the distribution that assigns probability qto F−1(ω)and assigns probability 1 −qto F−1(q+(1−q)ω/q).Theq-quantile matching experiment is defined as an experiment τsuch that for τ-almost all G,thereexists ω∈[0, q]such that G=Gω.4Formally, τis defined by τ(M)=q 0 1qδF−1(ω)+(1−q)δF−1(q+(1−q)ω/q)∈Mdω/q for all M⊂(). While all of our results hold for general Fand q, for simplicity we will provide intuition only for the uniform-median case where Fis uniform on [0, 1]and q=1/2. A distribution Hof q-quantiles is implemented by an experiment τif there exists a (measurable) selection χfrom the correspondence Xsuch that the distribution of χ(G) induced by τis H. A distribution Hof q-quantiles is uniquely implemented by an experiment τif His implemented by τand X(G)is a singleton for τ-almost all G.Let Hand Hbe the sets of implementable and uniquely implementable distributions of q-quantiles. The following theorem characterizes Hand H. Theorem 1. The following statements hold: (i) We have H={H∈():H≤H≤H},whereH(x)=max{0, (F(x)−q)/(1−q)} and H(x)=min{F(x)/q,1 }for all x∈. (ii) Every H∈His implemented by τ. 4For example, when Fis atomless, we can let ω=F(θ), so that the q-quantile matching experiment induces posteriors that assign probability qto θand assign probability 1 −qto F−1(q+(1−q)F(θ)/q)for θ∈[0, F−1(q)]. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE6057 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Distributions of posterior quantiles 1403 (iii) If Fhas a positive density on ,thenHis the closure of H. In particular, for any objective function V∈C(), we have sup H∈H V(x)dH(x)=max H∈H V(x)dH(x).(1) Figure 1illustrates the set H. The intuition for Theorem 1is straightforward. First, by simple Markov-type inequalities, any implementable Hmust satisfy H≤H≤H.For example, if the posterior median is less than xwith probability p,thenθmust be less than xwith probability at least p/2. When F(x)=x, this implies that p≤2x,sothe probability that the posterior median is less than xis at most min{2x,1 }=H(x).5 Conversely, to see that any Hsatisfying H≤H≤His implementable, consider the median matching experiment τthat induces only posteriors Gθthat assign equal probability to some θ∈[0, 1/2]and to 1/2+θ. The set of medians of such a posterior is X(Gθ)=[θ,1/2+θ]. At the same time, H≤Himplies that H−1(2θ)≥θ,andH≥Himplies that H−1(2θ)≤1/2+θ,sowehaveH−1(2θ)∈[θ,1/2+θ].Thus,χ(Gθ)=H−1(2θ) is a selection from X(Gθ). Finally, the distribution of χ(Gθ)induced by τis H, because the states that induce medians below xunder τwith selection χ(Gθ)are precisely those in [0, H(x)/2]and [1/2, 1/2+H(x)/2], and the measure of these states is H(x). As to unique implementation, for any e∈(0, 1]and any implementable and absolutely continuous distribution Hwith density h, we explicitly construct a modification of the median matching experiment τ ethat uniquely implements the distribution (1−e)H+eF of medians, by making every posterior Gθa convex combination of the median matching distribution (δθ+δ1/2+θ)/2 and the degenerate distribution δH−1(2θ) at the unique median H−1(2θ)∈[θ,1/2+θ]. Intuitively, for each θ∈[0, 1/2],τ einduces posteriors Gθand GH(θ)/2with probabilities 1 −eand e; similarly, for each θ∈(1/2, 1], τ einduces posteriors Gθ−1/2and GH(θ)/2with probabilities 1 −eand e. Then posterior medians in [x,x+dx]are induced at θ∈[H(x)/2, H(x+dx)/2]with probability 1−e,at θ∈[1/2+H(x)/2, 1/2+H(x+dx)/2]with probability 1 −e,andatθ∈[x,x+dx]with probability e. Since H(x+dx)=H(x)+h(x)dx,the density of the posterior median x multiplied by the posterior at xis equal to (1−e)h(x)(δH(x)/2+δ1/2+H(x)/2)/2+eδx,as required. To complete the proof of Theorem 1, we provide a simple argument showing that any distribution in Hcan be approximated by uniquely implementable distributions (1−e)H+eF.6 The literature contains several close antecedents of Theorem 1.Friedman and Holden (2008) study partisan gerrymandering with a finite number of legislative districts. Benoît and Dubra (2011) study testing for overconfidence in a self-ranking experiment with a finite number of bins. In our notation, Friedman and Holden and 5This argument is closely related to Kamenica and Gentzkow’s 2011 “prosecutor–judge” example. As in their example, the key observation is that if the prior probability of an event (e.g., the event that θ≤x)isx, then the maximum probability that the posterior probability of this event is at least 1/2ismin{2x,1 }. 6A complete characterization of the set Hremains an open problem. Two observations are that His apropersubsetofH(as Hand Hdo not belong to H) and that not all uniquely implementable distributions can be implemented by our modification of q-quantile matching. For example, H=δ1/2is uniquely implemented by complete pooling but not by our modification of median matching. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE6057 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1404 Kolotilin and Wolitzky Theoretical Economics 19 (2024) Benoît and Dubra consider discrete experiments with finitely many induced posteriors. Friedman and Holden show that a discrete version of His the highest implementable distribution of posterior medians. Benoît and Dubra show that the set of uniquely implementable distributions of posterior medians is a discrete version of the set {H∈():H<H<H}. In a general setting with possibly infinitely many induced posteriors in the contexts of quantile persuasion and partisan gerrymandering, respectively, Kolotilin and Wolitzky (2020a)andKolotilin and Wolitzky (2020b) show that H is the highest implementable distribution of posterior medians. Finally, in a general setting, Yang and Zentefis (2024) show that the set of implementable distributions of posterior medians is {H∈():H≤H≤H}, and also construct a dense subset of distributions that are uniquely implementable.7Relative to Benoît and Dubra and Yang and Zentefis,Theorem1shows that the q-quantile matching experiment implements every H∈Hand also yields a much simpler proof. Farther afield, Blackwell (1953), Strassen (1965), and Kolotilin (2018)characterize implementable distributions of posterior means. An interesting open question is whether a useful analogue of Theorem 1(for medians) and Strassen’s theorem (for means) exists for intermediate statistics that interpolate between the median and the mean.8 3. Optimal distributions of posterior quantiles This section uses the q-quantile matching experiment to characterize the distributions of posterior q-quantiles that maximize a continuous linear functional. Theorem 2. Let V∈C().ThenH(uniquely) maximizes V(x)dH(x)on Hif and only if H−1(p)(uniquely) maximizes Von [H−1(p),H−1(p)] for (almost) all p∈[0, 1]. Consequently, the value of the maximization problem is max H∈H V(x)dH(x)=1 0 maxV(x):x∈H−1(p),H−1(p)dp.(2) Conceptually, Theorem 2follows easily from Theorem 1. Since the median matching experiment τimplements all implementable distributions of medians, optimization just requires selecting an optimal median χ(Gθ)∈arg maxx∈[θ,1/2+θ]V(x)for each posterior Gθinduced by τ, as illustrated in Figure 1. The value of the maximization problem is, thus, 2 1/2 0maxx∈[θ,1/2+θ]V(x)dθ, and a distribution Hofmediansisoptimal if and only if H−1(2θ)∈arg maxx∈[θ,1/2+θ]V(x)for all θ∈[0, 1/2]. That is, optimal solutions can be obtained by pointwise maximization without any ironing procedure. 7To establish results similar to our Theorem 1,Yang and Zentefis characterize the extreme points of the set {H∈():H≤H≤H}. As recently emphasized by Kleiner, Moldovanu, and Strack (2021), characterizing a convex set by its extreme points can be useful for establishing some properties of the set. In contrast, we show that directly characterizing the set of implementable distributions of posterior quantiles is much easier than characterizing the extreme points of this set. 8Kolotilin, Corrao, and Wolitzky (forthcoming) study the question of characterizing optimal distributions of such intermediate statistics—the analogous problem to that of Theorem 2in the current paper. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE6057 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Distributions of posterior quantiles 1405 In general, by Theorem 1, for each H∈Hand p∈[0, 1],wehaveH−1(p)≤H−1(p)≤ H−1(p). If we consider the relaxed problem of finding a measurable function J:[0, 1]→ to maximize 1 0 VJ(p)dp subject to H−1(p)≤J(p)≤H−1(p)for all p∈[0, 1], one solution is J(p)=minargmaxV(x):x∈H−1(p),H−1(p) for all p∈[0, 1]. This function Jis monotone; moreover, the proof of Theorem 2shows that there exists H∈()such that J=H−1,soHsolves the optimization problem (2). The closest antecedent to Theorem 2is Corollary 4 of Yang and Zentefis (2024), which solves the maximization problem (2) in the special cases where Vis quasiconcave or quasi-convex. The solution follows immediately from Theorem 2.Tosee how, suppose that Vis quasi-concave with a maximum at x∈[0, 1]. For each interval [θ,1/2+θ], it is optimal to select xif x∈[θ,1/2+θ],θif x<θ,and1/2+θif x>1/2+θ. This induces the distribution of posterior medians H(x)=H(x),x<x , H(x),x≥x. Next, suppose that Vis quasi-convex with V(x)=V(1/2+x)for some x∈[0, 1/2]. Then, for each interval [θ,1/2+θ], it is optimal to select θif x>θand 1/2+θif x<θ. This induces the distribution of posterior medians H(x)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ H(x),x<x , 2x,x∈[x,1/2+x), H(x),x≥1/2+x. From the perspective of optimization, it is natural to ask whether each extreme point of His exposed, meaning that it is the unique maximizer in Hof V(x)dH(x)for some V∈C(). It turns out that some extreme points are not exposed. To see this, note that in the uniform-median case, the distribution H=(δ1/4+δ1/2)/2 is an extreme point of H, as there are no distinct H1,H2∈Hsuch that H=(H1+H2)/2. By Theorem 2, if Hmaximizes V(x)dH(x)on Hfor some V∈C(),thenV(1/4)≥V(x)for all x∈ [θ,1/2+θ]and all θ∈[0, 1/4], and, similarly, V(1/2)≥V(x)for all x∈[θ,1/2+θ]and all θ∈[0, 1/2].Thus,V(1/4)=V(1/2)≥V(x)for all x∈[0, 1]. But then the distribution δ1/2∈Halso maximizes V(x)dH(x), which shows that His not an exposed point of H.9 9The distribution Hdoes uniquely maximize V(x)dH(x)for V=2·1{x=1/4}+1{x=1/2},which is upper semi-continuous, but not continuous. An open question is whether each extreme point of H, 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE6057 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1406 Kolotilin and Wolitzky Theoretical Economics 19 (2024) 4. Unique properties of the quantile matching experiment Theorem 1shows that the q-quantile matching experiment simultaneously implements all implementable distributions of posterior q-quantiles. We now show that it is the unique experiment to do so. For simplicity, in this section we assume that Fhas a positive density on . We actually establish the stronger result that the q-quantile matching experiment is the unique experiment that simultaneously implements all optimal distributions for strictly quasi-convex objective functions. Theorem 3. The q-quantile matching experiment τis the unique experiment τthat, for each p∈[0, 1], implements the distribution Hp∈Hgiven by Hp(x)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ H(x),x<x p, p,x∈[xp,xp), H(x),x≥xp, where xp=F−1(qp )and xp=F−1(q+(1−q)p). In other words, for any experiment τ= τ,thereissomep∈[0, 1]such that τdoes not implement Hp. For example, in the uniform-median case, the negative assortative matching experiment does not implement H1/2, as noted in the Introduction. An immediate corollary of Theorem 3is that the q-quantile matching experiment is the unique experiment that minimizes the maximum regret of a designer who chooses an experiment τbefore learning her objective V, but chooses a selection χafter learning V. Formally, for each experiment τ∈(()) and each objective V∈C(), define the designer’s regret as r(τ,V)=max H∈H V(x)dH(x)−sup H∈H V(x)dH(x):His implemented by τ. Note that r(τ,V)≥0forallτand V. Say that a set of possible objective functions V⊂ C()is rich if, for all x0,x1∈, there exists a strictly quasi-convex V∈Vwith V(x0)= V(x1). We then have the following result. Corollary 1. If Vis rich, then the q-quantile matching experiment τis the unique experiment τsuch that r(τ,V)=0for all V∈V. Appendix:Proofs Proof of Theorem 1. Consider any experiment τ∈(()) and any measurable selection χ(G)from X(G).LetHbe the distribution of χ(G)induced by τ. Then, for each characterized in Theorem 1 of Yang and Zentefis (2024), is the unique maximizer of V(x)dH(x)for some upper-semicontinuous V. This is a weaker property than exposedness, as the usual theory of exposed points relies on continuity. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE6057 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Distributions of posterior quantiles 1413 Yang, Kai Hao and Alexander Zentefis (2024), “Monotone function intervals: Theory and applications.” American Economic Review, 114, 2239–2270. [1399,1400,1401,1404,1405, 1406] Co-editor Rakesh Vohra handled this manuscript. Manuscript received 26 February, 2024; final version accepted 17 June, 2024; available online 20 June, 2024. 15557561, 2024, 4, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE6057 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License