Optimal Market Composition In Monopoly Screening

Imagine a bustling marketplace where a single, powerful shopkeeper (the Monopolist) sells custom-tailored suits. The shopkeeper doesn't know exactly how much money each customer has, but they know there are some "bargain hunters" and some "wealthy tycoons." To make the most profit, the shopkeeper creates a menu: a cheap, low-quality suit for the budget-conscious, and an expensive, high-quality suit for the rich. This is screening: trying to get different people to reveal their true value by offering them different deals.

Now, imagine there is a Gatekeeper standing at the entrance of this market. This Gatekeeper (an Upstream Actor, like a social media platform, a government regulator, or a broker) decides who gets to walk into the shop. They don't tell the shopkeeper what prices to charge; they just decide the mix of customers the shopkeeper will see.

This paper asks a fascinating question: What mix of customers should the Gatekeeper choose to create the best outcome?

The Two Goals: Profit vs. Happiness

The Gatekeeper has a specific goal in mind, which is a balance between two things:

The Shopkeeper's Profit: How much money the seller makes.
The Customers' Happiness (Surplus): How much value the buyers get for their money.

The paper explores what happens when the Gatekeeper changes their mind about which goal is more important.

The Big Discovery: The "Top-Heavy" vs. The "Balanced" Market

Scenario A: The Gatekeeper Loves the Shopkeeper (or doesn't care about customers)

If the Gatekeeper cares mostly about the shopkeeper's profit (or is indifferent), they will do something very strange: They will only let the wealthiest tycoons into the shop.

The Analogy: Imagine the Gatekeeper closes the doors to everyone except the top 1% of earners.
The Result: The shopkeeper sees only rich people. Since everyone is rich, the shopkeeper doesn't need to offer cheap suits to trick people. They just sell the best, most expensive suit to everyone. The shopkeeper makes a fortune, but the customers get no "extra" happiness because they are paying exactly what their suit is worth. The market is tiny, but super efficient for the seller.

Scenario B: The Gatekeeper Loves the Customers

If the Gatekeeper cares more about the customers' happiness (specifically, if they value it more than half as much as the seller's profit), the strategy flips completely.

The Analogy: The Gatekeeper opens the doors to a wide variety of people: the middle class, the working class, and the rich.
The Result:
1. No One is Left Out: The shopkeeper must serve everyone who walks in. They can't just ignore the poor.
2. No "Bunching": In a normal monopoly, the shopkeeper might group similar people together and give them the same bad deal. Here, the Gatekeeper forces the shopkeeper to treat almost everyone slightly differently. The "middle" of the market becomes very crowded and diverse.
3. The "Premium" Corner: Even though the market is diverse, the Gatekeeper keeps a small, exclusive corner for the absolute richest tycoons. These people get the perfect, top-tier suit at the perfect price.
4. The Trade-off: The shopkeeper has to work harder to separate the customers. They have to offer lower-quality suits to the middle class to make them feel like they got a deal. This creates "information rents" (extra happiness) for the customers.

The "Sweet Spot" (The Efficient Frontier)

The paper draws a map of all possible outcomes. It shows that you can't have your cake and eat it too.

If you want maximum profit, you get a tiny, rich-only market.
If you want maximum customer happiness, you get a huge, diverse market where the shopkeeper makes less money.
The Frontier: The paper proves that by simply changing who is allowed into the market, the Gatekeeper can slide smoothly along a line between these two extremes. You can't get more customer happiness without hurting the seller, and you can't get more seller profit without hurting the customers. This line is the "Pareto Frontier"—the absolute limit of what is possible.

Why Does This Matter? (Real World Examples)

This isn't just about suits; it's about how we design our digital and economic world.

Social Media & Ads: Imagine Facebook (the Gatekeeper) showing ads to a company (the Shopkeeper). If Facebook wants to maximize the company's profit, it might only show the ad to people who are guaranteed to buy expensive things. But if Facebook wants to help the users (the customers), it might show the ad to a broader, more diverse group. This forces the company to offer better deals or lower prices to attract the less wealthy users, increasing user happiness but lowering the company's profit margin.
Regulators: A government regulator might decide which banks can lend money. If they only let banks lend to the safest, richest clients, banks make huge profits. If they force banks to lend to a mix of risky and safe borrowers, banks have to offer better rates to compete, helping the borrowers but making the banks work harder.

The "Cost" of Quality

The paper also looks at what happens if making high-quality goods becomes cheaper or more expensive.

If it becomes cheaper to make good products, the market actually becomes more polarized. The Gatekeeper lets in even more poor people (because the suits are cheap) but also keeps a huge chunk of rich people (because the top suits are now even better).
This is different from just caring more about customers. Caring about customers makes the market less top-heavy. Cheaper technology makes the market more top-heavy but also broader.

The Bottom Line

The main takeaway is simple: You don't need to force a shopkeeper to change their prices to help customers.

Instead, you can change the crowd they face. By curating a more diverse, less "top-heavy" audience, you naturally force the seller to offer better deals and generate more happiness for the buyers, even if the seller tries to do their best to maximize profit. The Gatekeeper holds the power to reshape the entire market just by deciding who gets to walk through the door.

1. Problem Statement

The paper investigates a two-stage game where an upstream actor (e.g., a platform, regulator, or marketplace) chooses the distribution of buyer valuations (market composition) that a downstream monopolist will face. The monopolist then observes this distribution and offers an optimal quality-price menu (nonlinear pricing/screening) to maximize profit.

Objective: The upstream actor maximizes a weighted sum of Consumer Surplus ($CS$) and Seller Profit ( $\Pi$ ): $W_k(G) = k \cdot CS(G) + (1-k) \cdot \Pi(G)$ , where $k \in [0, 1]$ represents the weight on consumer welfare.
Core Question: What is the optimal market composition $G$ (distribution of valuations) that the upstream actor should induce to achieve the best trade-off between consumer welfare and seller profit, given that the seller will subsequently screen optimally?
Context: Unlike standard screening models where the demand distribution is primitive, here the demand environment is endogenous. This models scenarios where platforms determine which users see a seller, or regulators set eligibility criteria.

2. Methodology

The paper employs a standard Mussa-Rosen (1978) screening framework with convex costs of quality provision, $c(q)$ .

Quantile Space Reformulation: The author reformulates the problem in quantile space ( $u \in [0, 1]$ $u \in [0, 1]$ ). Instead of choosing a distribution $G$ $G$ directly, the upstream actor effectively chooses the ironed virtual value function $\phi(u)$ $ϕ (u)$ .
- The seller's optimal quality schedule is pointwise determined by the virtual value: $q(u) = q(\phi(u))$ , where $q(\cdot)$ is the inverse of the marginal cost function.
- The actual valuation $Q(u)$ is derived from the virtual value via the revenue curve: $Q(u) = \frac{1}{1-u} \int_u^1 \phi(t) dt$ .
Variational Calculus: The upstream actor's problem is transformed into maximizing a functional $J_k(\phi)$ over the space of non-decreasing, right-continuous functions. The author derives necessary first-order conditions using variational inequalities and the Euler-Lagrange equation.
Free-Boundary Problem: The solution is characterized as a free-boundary value problem where the optimal virtual value schedule hits a boundary (either 0 or 1) at specific cutoffs.
Assumptions:
- Assumption 1: Standard convex cost function ( $c(0)=0, c'(0)=0, c''>0$ ).
- Assumption 2 (Curvature Condition): A mild condition on the third derivative of the cost function ( $-q c'''/c'' < 2$ ) to ensure uniqueness and strict monotonicity of the solution.

3. Key Results

A. Structural Characterization of the Optimal Market

The optimal market composition depends critically on the weight $k$ placed on consumer surplus:

Profit-Biased Regime ( $k \leq 1/2$ ):
- The optimal market collapses to a degenerate distribution at the highest valuation ( $G^* = \delta_1$ ).
- The monopolist serves only the top type at the first-best quality level.
- Consumer surplus is zero; the monopolist extracts the entire surplus.
Consumer-Oriented Regime ( $k > 1/2$ ):
- No Exclusion: All consumer types in the support are served (no mass at zero quality).
- No Interior Bunching: The optimal virtual value schedule $\phi^*(u)$ is strictly increasing on the interior of the support. There are no flat regions where different types are bunched together.
- Premium Top Segment: There exists a cutoff $b \in (0, 1)$ such that for all $u \in [b, 1]$ , $\phi^*(u) = 1$ . This corresponds to a strictly positive mass (atom) at the highest valuation $v=1$ , served efficiently.
- Interior Region: For $u \in (0, b)$ , $\phi^*(u)$ is strictly increasing and satisfies a specific Euler-Lagrange differential equation. This region generates informational rents for consumers.

B. Comparative Statics

As the weight on consumer surplus ( $k$ ) increases (for $k > 1/2$ ):

Market Heterogeneity: The market becomes more heterogeneous. The support of the distribution expands downward (the lowest valuation $v$ decreases).
Top Segment Shrinkage: The size of the atom at the top valuation ( $1-b$ ) decreases.
Quality Distortion: The virtual value profile $\phi(u)$ and the resulting quality schedule $q(u)$ decrease pointwise for all interior types. The monopolist faces a "less favorable" screening environment.
Welfare Trade-offs:
- Consumer Surplus ($CS$) increases.
- Seller Profit ( $\Pi$ ) decreases.
- Total Surplus ($TS$) decreases. The gain to consumers comes at the cost of efficiency losses (increased distortion) and a smaller total pie.

C. The Pareto Frontier

Theorem 2: The set of implementable $(CS, \Pi)$ pairs forms a convex set. The supported frontier (solutions to the weighted maximization problem) coincides exactly with the Pareto frontier.
This implies that by varying $k$ , the upstream actor can trace the entire efficient trade-off curve between consumer surplus and profit across all possible market compositions.
Relation to Information Design: The paper shows that this frontier is the envelope of all fixed-prior information-design problems (e.g., Bergemann et al., 2015; Roesler and Szentes, 2017). When the market composition itself is endogenous, segmentation within a fixed market does not expand the Pareto frontier beyond what is achievable by choosing the optimal aggregate distribution.

D. Linear Cost Benchmark

In the special case of constant marginal cost ($c(q) = Mq$), the screening problem collapses to posted pricing.

The optimal market for $k > 1/2$ is a shifted equal-revenue distribution with a top atom.
This recovers and generalizes the results of Condorelli and Szentes (2020).

4. Significance and Contributions

Shift from Menu to Environment: The paper shifts the design focus from the seller's contract (the menu) to the market environment (the distribution of types). It demonstrates that shaping the market composition is a powerful instrument for welfare redistribution.
No Bunching Result: A counter-intuitive finding is that an upstream actor seeking to help consumers does not create bunching (collapsing types) to reduce discrimination. Instead, the optimal strategy is to create a more heterogeneous, strictly separating interior where the monopolist must screen against a wider, less favorable range of virtual values, thereby generating more rents for consumers.
Policy Implications: For platforms and regulators, the results suggest that to improve consumer outcomes, one should not ban differentiation or make buyers look identical. Instead, the goal should be to reshape the market to be less top-heavy and more heterogeneous, preserving an efficient premium segment while expanding the active middle class of buyers.
Robustness: The structural results hold even under restrictions (e.g., fixed mean valuation) and in the presence of "hold-up" costs (where creating a consumer-friendly market is costly), provided the costs are not too severe.

5. Conclusion

Kyriazis provides a rigorous characterization of how upstream institutions can optimally design market composition to influence monopoly screening outcomes. The paper establishes that the most effective way to redistribute surplus from a monopolist to consumers is not to suppress market differentiation, but to engineer a market composition that forces the monopolist to screen a broader, more diverse set of types, thereby reducing the efficiency of price discrimination and increasing consumer rents, albeit at the cost of total surplus.