📈 economics

How to Use Prices for Efficient Online Matching

This paper introduces the Sequential Equilibrium Mechanism (SEM), an online matching algorithm that achieves asymptotic efficiency, fairness, and strategy-proofness for dynamic markets with immediate matching requirements, such as placing vulnerable children in foster homes, and validates its potential to improve welfare through simulations and planned field experiments.

Original authors: Terence Highsmith

Published 2026-04-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Terence Highsmith

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are running a busy emergency shelter for homeless people, or a hospital triage room, or a foster care agency. Every day, new people arrive needing a bed, a room, or a home. You have a limited number of spots available.

The rule is strict: You must match them immediately. You can't say, "Wait here for an hour while I check if a better spot opens up tomorrow." If someone walks in, you have to give them a spot right now, or they are stuck on the street.

This is the problem Terence Highsmith II tackles in his paper. He calls it "Online Matching."

Here is the breakdown of the problem and his solution, using simple analogies.

The Problem: The "First-Come, First-Served" Trap

In many real-world situations, the current system is basically a chaotic race.

The Scenario: A caseworker gets a text about a child needing a home. They text 50 foster parents. The first one to reply gets the child.
The Flaw: This is like a "first-come, first-served" line at a bakery.
- Imagine you have 100 cupcakes (homes) and 100 customers (children).
- Customer A arrives. They are picky; they only want chocolate or vanilla.
- Customer B arrives later. They are picky too; they only want chocolate.
- If Customer A grabs the last chocolate cupcake because they were fast, Customer B gets nothing.
- But what if Customer A was actually happy with any flavor? If they had waited or been guided, Customer B could have had chocolate, and Customer A could have had vanilla. Everyone wins.

Because the system forces immediate decisions, and because people often have "indifferences" (e.g., "I'm fine with any foster home that is safe"), the current "grab what you can" method often wastes resources. It leaves the most picky people without a spot because the less picky people grabbed the good spots too early.

The Solution: The "Magic Price Tag" System

The author proposes a new system called the Sequential Equilibrium Mechanism (SEM).

Think of this not as a line, but as a virtual stock market that runs every single second.

1. The Fake Money (Token Budgets)

Imagine every person who arrives gets a stack of "fake money" (tokens).

The Twist: People who arrive earlier get more fake money than people who arrive later.
Why? This ensures that the first person to arrive can always buy their favorite item if it's available. It respects the "immediate match" rule.

2. The Magic Price Tags

In a normal market, prices are fixed (e.g., a cupcake costs $2). In SEM, the prices are randomized and constantly shifting based on what the computer predicts will happen tomorrow.

If the computer thinks a "chocolate cupcake" will be in high demand tomorrow, the price tag on it goes up today.
If a "vanilla cupcake" is likely to be left over, its price goes down.

3. The Algorithm in Action

When a new child arrives:

The system looks at the current "prices" of all available homes.
It calculates which homes are the "best deal" for that child given their fake money.
It assigns the child a lottery ticket. Instead of saying "You get Home A," it says, "You have a 70% chance of Home A and a 30% chance of Home B."
At the end of the day, the lottery is drawn, and the child gets a real home.

Why This Works (The "Crystal Ball" Effect)

The genius of SEM is that it uses prices as a crystal ball.

In the old system (First-Come, First-Served), the person arriving at 9:00 AM has no idea that a very picky person is coming at 9:05 AM. They grab the only good spot, causing a disaster later.

In the SEM system, the "prices" already know about the future.

Because the system predicts that a picky person is coming soon, the price of the "perfect spot" rises before the picky person arrives.
The early arrival sees the high price and thinks, "That spot is too expensive for my budget. I'll take this slightly less perfect spot instead."
Result: The early arrival gets a good spot, and the "perfect spot" is saved for the picky person who arrives later.

The Results: A Simulation

The author tested this with a simulation of a foster care agency.

Old Way (Random First-Come): About 82% of children found a home.
New Way (SEM): About 92% of children found a home.

That's a 10% improvement. In the real world, that means hundreds more children finding safe homes instead of staying in the system.

The Big Takeaway

The paper argues that we don't need to be perfect to be efficient. We just need to stop thinking like a "line" and start thinking like a "market."

By using fake money to give priority to early arrivals and dynamic prices to account for future demand, we can make split-second decisions that actually look ahead. It turns a chaotic scramble into a coordinated dance, ensuring that when the "picky" people arrive, the right spots are still waiting for them.

In short: SEM is a smart, automated referee that uses imaginary prices to make sure everyone gets a fair shot, even when they have to make decisions instantly.

1. Problem Definition

The paper addresses the Online Matching Problem under two specific, restrictive constraints common in critical social applications (e.g., foster care placement, emergency sheltering, hospital bed assignment):

Greedy Allocation Constraint: Agents (e.g., children, patients) arrive stochastically and must be matched immediately upon arrival. They cannot be held in a queue or discarded. If a match is possible, it must be made instantly.
Ordinal Preferences with Indifferences: Agents have ordinal preferences (rankings) rather than cardinal utilities. Crucially, agents often exhibit indifferences (e.g., two foster homes are equally suitable).

The Core Conflict: The authors demonstrate a fundamental impossibility result: It is impossible to design an online matching mechanism that is simultaneously greedy (matches immediately) and ordinally efficient (Pareto efficient with respect to stochastic dominance) in a finite market. Randomly breaking ties in a greedy mechanism often jeopardizes the welfare of future arrivals.

2. Methodology: The Sequential Equilibrium Mechanism (SEM)

To overcome the impossibility result, the authors design a novel randomized mechanism called the Sequential Equilibrium Mechanism (SEM). The approach bridges Pseudomarkets (Hylland-Zeckhauser) and Online Stochastic Matching.

A. Theoretical Framework: Large Market Equilibrium

The mechanism relies on solving a "large market" competitive equilibrium at each time step to guide immediate allocations.

Token Money & Random Prices: Agents are endowed with token money. The market designer introduces random price shocks ( $\xi$ ) to the price vector.
Demand Correspondence: Because agents have indifferences, their demand is a correspondence (a set of optimal bundles) rather than a single point. The authors use the Aumann integral to define "expected demand" over the distribution of price shocks. This integration smooths the demand correspondence, making it upper hemicontinuous and convex-valued.
Existence Proof: By applying Kakutani's Fixed Point Theorem to the expected demand, the authors prove the existence of a price equilibrium that clears the market (Expected Demand = Supply).
Asymmetric Budgets: To satisfy the greedy constraint, agents arriving earlier are given larger budgets than later arrivals. This ensures early arrivals have sufficient purchasing power to secure their most preferred objects whenever supply allows.

B. The Mechanism Algorithm

SEM operates in discrete time periods $t = 1, \dots, T$ :

State Update: Observe realized arrivals $\tilde{I}_t$ and remaining supply $S_t$ .
Equilibrium Computation: Compute a price equilibrium $p_t$ for the current fundamentals (realized past arrivals + expected future arrivals).
Lottery Allocation: Assign a lottery (probability distribution over objects) to each arriving agent based on the equilibrium demand.
Ex-Post Realization: Immediately realize the lottery into a deterministic assignment (ex-post allocation) for the current period.
Update: Reduce supply and proceed to $t+1$ .

C. Tie-Breaking Strategies

The paper analyzes two approaches to handling indifferences via price shocks:

No Tie-Breaking (NTB): Prices for indifferent goods are perfectly correlated. This guarantees ordinal efficiency but requires complex computation of multi-valued demand.
Random Tie-Breaking (RTB): Prices for indifferent goods have independent random noise. This simplifies computation (demand becomes single-valued) but requires a specific condition ( $|p_x - p_y| > 2z$ ) to ensure efficiency.

3. Key Contributions

Novel Mechanism Design: SEM is the first online matching mechanism to jointly satisfy greediness, equal-type envy-freeness (agents arriving at the same time do not envy each other's lotteries), strategyproofness (in the large), and asymptotic ordinal efficiency.
Theoretical Extension of Pseudomarkets: The authors extend the Hylland-Zeckhauser framework to dynamic, online settings with indifferences. They prove that random prices can smooth discontinuous demand correspondences, allowing for equilibrium existence where deterministic prices fail.
Asymptotic Efficiency: While finite markets cannot be perfectly efficient under greedy constraints, SEM is proven to be asymptotically efficient. As the number of arrivals per period ( $n$ ) grows, the ex-post allocation converges in probability to an ordinally efficient allocation.
Strategyproofness with Probability One: In large markets, agents have negligible price impact. The mechanism is strategyproof with probability one, meaning agents cannot gain by misreporting preferences or arrival times as $n \to \infty$ .

4. Results

Theoretical Results

Existence: Price equilibria exist under mild assumptions on price shocks and budget bounds.
Efficiency: Under specific tie-breaking conditions, the equilibrium allocation is ordinally efficient.
Convergence: The mechanism converges to the large-market equilibrium allocation as the market size increases, satisfying the greedy constraint at every step.

Empirical/Simulation Results

The authors simulate SEM against Serial Dictatorship with Random Tie-Breaking (SD-RTB), a proxy for the current "first-come, first-served" process used by a partner nonprofit (The Firm) in foster care placement.

Setup: A market with 2 host homes and 2 types of children (selective vs. non-selective) over 4 periods.
Performance:
- Placement Rates: SEM consistently outperforms SD-RTB.
- Magnitude: SEM achieves approximately 10% more matches than SD-RTB across all market sizes.
- Variance: As the arrival rate increases, the variance in performance decreases for both mechanisms, but SEM maintains a steady lead.
Interpretation: SEM's ability to "look ahead" via the large-market equilibrium allows it to internalize future demand, preventing scarce resources from being consumed by early, less critical matches in a way that blocks future high-value matches.

5. Significance and Implications

Practical Application: The paper moves beyond theoretical impossibility results to provide a deployable algorithm for high-stakes, time-sensitive matching markets (foster care, emergency housing, organ allocation).
Policy Impact: By improving placement rates by ~10% in simulations, SEM could significantly reduce the time vulnerable individuals spend in limbo, directly improving welfare outcomes.
Theoretical Advancement: It resolves the tension between "greedy" constraints and "efficiency" by utilizing randomized prices to smooth demand, offering a new toolkit for mechanism designers facing ordinal preferences and dynamic arrivals.
Future Work: The authors plan to implement a "lab-in-the-field" experiment with their partner organization to validate these theoretical and simulation results in a real-world setting.

In summary, the paper proposes that randomized pricing in a sequential equilibrium framework is the key to unlocking efficiency in online matching markets where immediate allocation is mandatory and preferences are ordinal.