Monotone Function Intervals: Theory and Applications

The paper studies convex sets of nondecreasing, right-continuous functions that are bounded pointwise between two monotone functions—termed monotone function intervals. The central research question is: what are the extreme points of such sets, and how can this characterization be used to analyze economic problems where monotonicity and pointwise bounds arise naturally (e.g., distributions, contracts, and payoffs)? The purpose is twofold: (i) provide a general, tractable characterization of extreme points in these intervals, and (ii) exploit this structure to obtain new results and unify existing ones in information design (posterior quantiles) and security design (limited liability). The importance stems from the ubiquity of monotone functions in economics (demand/supply curves, CDFs, limited-liability payoffs) and from two key properties of extreme points: Choquet representations for compact convex sets and optimality of extreme points in convex optimization. The paper’s main theorem (Theorem 1) gives a complete geometric description of extreme points, which then yields characterizations of distributions of posterior quantiles and identifies optimal security contracts without strong distributional assumptions like MLRP.

The work connects to classical characterizations of extreme points and orders: Hardy, Littlewood, and Pólya (1929) on majorization in finite dimensions and Ryff (1967) in infinite dimensions; Kellerer (1973) and Lakeit (1975) on extreme points of probability measures under convex and stochastic orders. Economically, Kleiner, Moldovanu, and Strack (2021) characterize extreme points for monotone functions under convex order constraints, with extensions by Candogan and Strack (2023) and Nikzad (2023) to linear constraints; this paper instead treats pointwise order between two monotone functions (equivalently, stochastic order bounds). For belief-based characterizations of signals, the results parallel Blackwell (1953) and Strassen (1965) on posterior means; Kolotilin and Wolitzky (2024) provide an alternative proof for distributions of posterior quantiles akin to this paper’s Theorem 2. Applications relate to redistricting (Owen and Grofman 1988; Friedman and Holden 2008, 2020; Gul and Pesendorfer 2010; Kolotilin and Wolitzky 2023), Bayesian persuasion with mean-based payoffs (Gentzkow and Kamenica 2016; Roesler and Szentes 2017; Dworczak and Martini 2019; Ali et al. 2022), ordinal/quantile preferences (Manski 1988; Chambers 2007; Rostek 2010; de Castro and Galvao 2021), apparent overconfidence (Benoît and Dubra 2011), and security design under moral hazard and adverse selection (Innes 1990; DeMarzo and Duffie 1999; Nachman and Noe 1994; Biais and Mariotti 2005), with the paper unifying and extending these strands via extreme points of monotone function intervals.

• Define the space F of nondecreasing, right-continuous functions and the monotone function interval I(E, F) = {H ∈ F : E ≤ H ≤ F pointwise}.
• Main characterization (Theorem 1): H is an extreme point of I(E, F) iff H coincides with a bound except on a countable collection of intervals where H is constant, and at each such interval at least one endpoint meets a bound. The proof uses step-function extremality within bounded monotone classes and a contradiction via convex decompositions if the condition fails.
• Use Choquet’s theorem: any element of a compact convex set can be represented as a convex combination (mixture) of extreme points, allowing reduction of verification/constructive tasks to extreme points.
• Quantiles application: For prior CDF F and τ ∈ (0,1), define left and right truncations F_L(x) = min{F(x)/τ, 1} and F_R(x) = max{(F(x) − τ)/(1 − τ), 0}. Show the set of distributions of posterior τ-quantiles equals I(F_R, F_L) (Theorem 2). Construct signals explicitly for extreme points and use Choquet mixtures for general H. For unique quantiles, introduce ε-perturbed bounds (F_R^ε, F_L^ε) to characterize the interior (Theorem 3).
• Applications derive corollaries (law of iterated quantiles) and economic implications (gerrymandering feasibility set, persuasion programs, misconfidence rationalizability).
• Security design: Model feasible securities as I(0, id) under limited liability and monotonicity. Use extreme-point optimality in convex programs to identify optimal contracts. Without MLRP, show that optimal securities are contingent debts (piecewise-constant with slopes 0 or 1), and under N-peaked likelihood-ratio structures, bound the number of face values via duality and sign-pattern partitions of φ_e/φ.

• Theorem 1 (Extreme points): A function H is an extreme point of I(E, F) iff H equals a bound except on countably many intervals where H is constant, and at each such interval at least one endpoint touches a bound.
• Theorem 2 (Posterior quantiles): For τ ∈ (0,1), the set of distributions of posterior τ-quantiles equals the monotone interval I(F_R, F_L) where F_L(x)=min{F(x)/τ,1} and F_R(x)=max{(F(x)−τ)/(1−τ),0}. Thus any feasible H must satisfy F_R ≤ H ≤ F_L (FOSD bounds), and conversely any such H is implementable by some signal and quantile selection rule.
• Theorem 3 (Unique quantiles): If F has full support on an interval, the set of distributions of unique posterior τ-quantiles is the interior of I(F_R, F_L), characterized by ⋃_{ε>0} I(F_R^ε, F_L^ε) ⊆ H_unique ⊆ I(F_R, F_L). Only boundary cases (e.g., F_L, F_R) are excluded by uniqueness.
• Corollary (Law of iterated quantiles): For q, r ∈ (0,1), the r-quantiles of posterior q-quantiles range exactly over [(F_L)^{-1}(r), (F_R)^{-1}(r+)], e.g., medians of posterior medians lie in the prior’s interquartile range.
• Gerrymandering (Proposition 1): The set of possible distributions of representatives’ ideal points (under districting plus median voter theorem) equals I(F_R, F_L) with τ=1/2. Hence any legislature between the “all-left” (F_L) and “all-right” (F_R) is attainable; beyond these is impossible. Enacted legislation (with median at legislature) is exactly the prior’s interquartile range.
• Quantile-based persuasion (Proposition 2, Corollary 4): When sender’s indirect payoff depends only on posterior τ-quantiles, the problem reduces to choosing H ∈ I(F_R, F_L) to maximize ∫ v_s dH. Optimal signals are given by extreme-point H’s. With pinball loss and state-independent sender utility: quasi-concave v_s leads to pooling structures H_a^L or H_a^R depending on the maximizer location; strictly quasi-convex v_s leads to a centered pooling H_{a,ā}^C; the fully revealing distribution F is never uniquely optimal under absolute loss.
• Apparent misconfidence (Corollary 5): For prior on [0,1] and partition {z_k}, a dataset of τ-quantile self-assessments (θ_k) is rationalizable iff cumulative shares satisfy Σ_{i=1}^k θ_i < F(z_k)/(1−τ) and Σ_{i=k}^K θ_i < (1−F(z_{k−1}))/(1−τ), generalizing Benoît-Dubra characterization.
• Security design with moral hazard (Proposition 3): Without assuming MLRP, an optimal security is a contingent debt contract (piecewise-constant with slopes 0/1) with at most two non-defaultable face values.
• Security design with bounded peaks (Proposition 4): If the likelihood-ratio φ_e/φ is at most N-peaked, an optimal contingent debt has at most N+1 face values (≤2 non-defaultable). Standard debt arises when there is a single crossing (MLRP case).
• Security design with adverse selection (Proposition 5): With a worst signal in FOSD, an optimal security is a contingent debt with at most one non-defaultable face value; if posteriors have full support, the solution is unique.
• Unifying insight: Limited liability + monotonicity + convexity/risk-neutrality imply optimal extreme-point securities with no equity sharing at the margin (derivative 0 or 1), separating the role of MLRP from limited-liability structure.

The characterization of extreme points provides a unifying geometric lens on problems where feasible objects are monotone and bounded pointwise. For information design, it replaces mean-based martingale constraints with FOSD bounds between truncated priors, yielding a complete, tractable description of distributions of posterior quantiles and enabling constructive signal design via extreme points. This addresses questions in political economy (limits of redistricting outcomes), persuasion with ordinal loss (pinball/absolute deviations), and behavioral data rationalization (over/underconfidence) with clear feasibility regions and policies. In security design, treating feasible contracts as a monotone function interval and exploiting extreme-point optimality in convex programs shows that contingent debts are optimal broadly, even without MLRP. The number and nature of face values follow from the sign structure of dual multipliers relative to likelihood-ratio shape, providing a transparent explanation for when standard debt suffices and when richer contingent debts are necessary. Overall, the findings unify disparate results under a common structural property (extreme points of I(E,F)), clarifying which assumptions are essential (limited liability, risk neutrality, monotonicity) and which refine the exact form (distributional shape like MLRP).

The paper characterizes extreme points of monotone function intervals and leverages this to: (i) fully describe the set of distributions of posterior quantiles via FOSD bounds between truncated priors and (ii) unify and generalize optimal security design results, establishing the optimality of contingent debts without MLRP and bounding their complexity by likelihood-ratio shape. Applications span gerrymandering feasibility, quantile-based persuasion, misconfidence rationalization, and moral hazard/adverse selection in finance. Future research directions include extending quantile-distribution characterizations to multidimensional states and other posterior statistics (e.g., moments beyond means and quantiles), exploring persuasion under alternative loss or non-expected utility models, and developing security design frameworks with risk-averse agents where convexity may fail, requiring new tools beyond extreme-point arguments.

• The analysis of posterior quantiles is restricted to one-dimensional state spaces; multidimensional extensions are open.
• While analogous to posterior means, the results give limited intuition for distributions of other posterior statistics (e.g., higher moments), and how those sets behave.
• Security design results rely on risk neutrality and convexity; extensions to risk-averse agents are challenging due to non-convexities and are not addressed.
• Some constructions rely on selection rules or non-unique quantiles; uniqueness requires interior approximations and loses boundary cases.