Title: Monotone Function Intervals Theory and Applications
Domain: economics

Question: In many economic models, such as voting, security design, or Bayesian persuasion with non-quadratic loss functions, outcomes are determined by ordinal comparisons rather than cardinal values. This makes quantiles of posterior distributions the key statistic of interest, as opposed to the well-studied posterior means. Given a prior distribution of an unknown state, how can one precisely characterize the complete set of all possible distributions of posterior τ-quantiles that can be generated by any conceivable information structure (i.e., any signal)? Furthermore, how does this characterization provide a unified framework for solving optimization problems in these settings, such as designing an optimal contract without relying on strong assumptions like the Monotone Likelihood Ratio Property (MLRP)?

Background: This problem is situated within the modern theory of information design, specifically Bayesian Persuasion. In this framework, a "sender" (or principal) can commit to an information disclosure rule (a "signal") to influence the actions of a "receiver" (or agent). The sender cannot lie, but can strategically control the "garbling" or "pooling" of information. Any such signal must be Bayes-plausible: the expected posterior belief across all possible signal realizations must equal the prior belief. Much of the foundational literature characterizes the set of achievable posterior mean beliefs. However, this is insufficient for many economic problems where decisions are ordinal rather than cardinal. For example, in voting, security design, or credit approval, an agent's action depends on whether their posterior belief crosses a critical threshold, not on its average value. In such cases, the relevant statistic is the posterior quantile. This query addresses a fundamental open question: what is the complete set of posterior quantile distributions that can be induced from a given prior? Solving this provides a powerful tool for designing optimal policies in settings where only relative rankings matter.

Definitions: Monotone Function Interval (MFI): For any two nondecreasing, right-continuous functions F(x) (lower bound) and F(x) (upper bound) where F(x) ≤ F(x), the MFI I(F, F) is the convex set of all nondecreasing functions H(x) such that F(x) ≤ H(x) ≤ F(x) for all x. | Extreme Points of an MFI (Theorem 1): A function H is an extreme point of I(F, F) if and only if there exists a countable collection of disjoint intervals {[x_n, x_n)} such that: | For any x not in these intervals, H(x) lies on the boundary: H(x) ∈ {F(x), F(x)}. | On each interval [x_n, x_n), H is constant. | At the endpoint of each constant interval, H must connect to a boundary, e.g., H(x_n) = F(x_n) or H(x_n⁻) = F(x_n⁻). | Left and Right Truncations of a Prior: Given a prior CDF F(x) and a quantile τ ∈ (0,1), the bounds for the posterior quantile problem are defined as: | Left Truncation (Lower Bound): F_L(x) := min{F(x)/τ, 1}. | Right Truncation (Upper Bound): F_R(x) := max{(F(x) - τ)/(1 - τ), 0}.

Methodology: Characterization via Choquet's Theorem: The proof strategy for H_τ = I(F_R, F_L) is twofold. First, the inclusion H_τ ⊆ I(F_R, F_L) is established using the martingale property of beliefs. Second, for the converse I(F_R, F_L) ⊆ H_τ, Choquet's theorem is invoked. This reduces the problem to showing that any extreme point of the MFI can be achieved as a posterior quantile distribution. This is done by explicitly constructing a signal for any given extreme point. Since any function in the MFI is a convex combination of its extreme points, and the set of achievable distributions is convex, this establishes the equality. | Optimization via Extreme Points: For problems like security design, the set of feasible contracts forms an MFI, I(0, id). The issuer's objective function (e.g., max E[x - H(x)]) is an affine functional of the contract H. A fundamental result in convex analysis states that the maximum of an affine functional over a compact convex set is always attained at an extreme point. Therefore, the search for an optimal contract can be restricted to the set of extreme points, which are contingent debt contracts.

Answer: 1. The Problem: Characterizing the Set of Posterior Quantile Distributions | In many economic models, outcomes depend on ordinal rankings, making quantiles—not means—the relevant statistic of posterior beliefs. The central question is to characterize the set of all possible distributions of posterior τ-quantiles, denoted H_τ, that can be generated from a prior distribution F via any information structure (signal) μ. A signal μ is a distribution over posteriors G, such that the average of these posteriors recovers the prior, a condition known as the Bayes-plausibility constraint: | ∫ G(x) dμ(G) = F(x) for all x. | The goal is to find the set of all possible CDFs H where H(q) = μ({G | Q_τ(G) ≤ q}), where Q_τ(G) is the τ-quantile of posterior G. | 2. The Theoretical Framework: Monotone Function Intervals (MFI) | The solution hinges on the concept of a Monotone Function Interval. Given two nondecreasing, right-continuous functions F(x) and F(x) (with F(x) ≤ F(x)), the MFI I(F, F) is the set of all nondecreasing functions H bounded by them. This set I(F, F) is convex and compact under the topology of weak convergence. | A key theoretical result characterizes the extreme points of this convex set. A function H is an extreme point of I(F, F) if and only if it has a "piecewise-constant or boundary-valued" structure. Formally, its graph consists of segments that lie either on the lower bound F, the upper bound F, or are horizontal line segments. A horizontal (constant) segment must be "attached" to one of the boundaries at an endpoint. This "bang-bang" structure is crucial for optimization. | 3. The Main Result: The Posterior Quantile Distribution Set as an MFI | The primary applied theorem provides an elegant and complete characterization: the set H_τ of all possible distributions of posterior τ-quantiles is precisely the Monotone Function Interval I(F_R, F_L). The bounds of this interval are determined solely by the prior F and the quantile τ: | Lower Bound (Left Truncation): F_L(x) = min{F(x) / τ, 1} | Upper Bound (Right Truncation): F_R(x) = max{(F(x) - τ) / (1 - τ), 0} | Any achievable posterior quantile distribution H must satisfy F_R(x) ≤ H(x) ≤ F_L(x) for all x, meaning F_R first-order stochastically dominates H, which in turn dominates F_L. | 4. Derivations and Applications | A. Information Design and Signal Construction | The proof that H_τ = I(F_R, F_L) relies on Choquet's Theorem. Since I(F_R, F_L) is a compact convex set, any function H within it can be represented as a convex combination of its extreme points. The proof strategy is to show that every extreme point of I(F_R, F_L) is an achievable posterior quantile distribution. | Detailed Construction for an Extreme Point: | Consider a simple extreme point H that follows the lower bound F_L up to a point x_1, is constant from x_1 to x_2, and follows the upper bound F_R thereafter. A signal can be constructed to generate H: | Partition the State Space: Divide the state space into three regions: X_L = (-∞, x_1), X_C = [x_1, x_2), and X_R = [x_2, ∞). | Construct Posteriors: | For each state s ∈ X_L, create a posterior G_s by pooling the atom at s with a specific mass m_s from the interval X_C. The τ-quantile of this posterior becomes exactly s. | For all states within the interval X_C, pool them together to form a single posterior G_C. The τ-quantile of this single, large posterior will be some constant value η. | For each state s ∈ X_R, a symmetric construction applies to generate posteriors G_s whose τ-quantile is s. | Resulting Distribution: The distribution of these constructed quantiles will be H. The states in X_L generate the F_L part of H, the states in X_C generate the flat part, and the states in X_R generate the F_R part. This confirms that extreme points are achievable. | A direct result is the Law of Iterated Quantiles. The set of possible τ-quantiles of posterior q-quantiles is the interval of τ-quantiles of the boundary distributions. For instance, the possible medians of posterior medians is the interquartile range of the prior: [F⁻¹(1/4), F⁻¹(3/4)]. | B. Security Design with Limited Liability | Consider an entrepreneur designing a security H(x) which pays the investor based on project profit x. The constraints are: | 0 ≤ H(x) ≤ x (Limited Liability & Physical Constraint) | H(x) is nondecreasing (Monotonicity) | The set of all feasible securities is precisely the MFI I(0, id), where id(x) = x. | The entrepreneur's optimization problem is equivalent to: | min_{H ∈ I(0, id)} ∫_0^1 H(x) dΦ(x|e*) | The objective functional L(H) = ∫ H(x)dΦ(x|e*) is affine in H. A fundamental theorem of convex analysis states that the solution must be an extreme point of the set. | Therefore, the optimal security H* must be an extreme point of I(0, id). These are contingent debt contracts, where for different ranges of profit, the investor receives either a fixed payment or the entire profit. The investor and issuer never share marginal profits. This result generalizes previous literature by showing that the optimality of debt-like structures is a fundamental consequence of the problem's convex geometry, not specific statistical properties like the MLRP.

Checklist: 1. Correctly formulate the set of all possible distributions of posterior τ-quantiles as a specific Monotone Function Interval, I(F_R, F_L), defining the bounds F_R and F_L based on the prior F.  
2. State the main characterization result (Theorem 2 from the paper), explicitly equating the set of posterior quantile distributions with this MFI.  
3. Explain the intuition by constructing a specific signal for a simple extreme point of I(F_R, F_L), demonstrating how such a distribution is achievable (as illustrated in Figure 4 of the paper).  
4. Derive the "Law of Iterated Quantiles" (Corollary 1) as a direct consequence of the main characterization, showing the range of possible medians of medians.  
5. Apply the MFI framework to the security design problem, identifying the set of feasible securities as an MFI.  
6. Explain why, in a convex optimization context, the optimal security must be an extreme point of this MFI.  
7. Characterize these extreme-point solutions as "contingent debt contracts" and explain how this generalizes previous results in the literature by relaxing the MLRP assumption.