⚡ electrical engineering

Welfarist Control Design -- How to fulfill the societal mandate in multi-agent control?

This paper proposes a principled framework for designing multi-agent control systems that align with societal mandates by aggregating individual preferences into objectives and utilizing feedback mechanisms within online feedback optimization, Markov decision processes, and model predictive control to ensure the ethical and efficient allocation of scarce resources.

Original authors: Sophie Hall, Kai Zhang, Ilia Shilov, Heinrich H. Nax, Saverio Bolognani

Published 2026-06-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sophie Hall, Kai Zhang, Ilia Shilov, Heinrich H. Nax, Saverio Bolognani

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 a busy highway, a power grid, or a water supply system. In all these places, there is a limited resource (like space, electricity, or water) that needs to be shared among many different people (agents).

For a long time, engineers designing the systems that control these resources have mostly followed industry rules of thumb. They ask, "How do we make this run fast and cheap?" But they often skip a crucial question: "How do we make this run fairly?"

This paper argues that as we automate more of our world, engineers need a new, principled way to decide how to share these scarce resources. They call this "Welfarist Control Design."

Here is the breakdown of their idea, using simple analogies:

1. The Problem: The "Black Box" of Fairness

Imagine a traffic light controller. It decides who gets to go and who has to wait.

Old way: The engineer sets it to move the most cars through per minute (Efficiency).
The Issue: This might mean a bus full of 50 people waits forever while a single car zips through. Or, it might mean a hospital ambulance gets stuck behind a delivery truck.
The Paper's Point: Engineers are now making "societal decisions" automatically. They need a clear recipe for what "fair" means, rather than just guessing or following old habits.

2. The Recipe: The "Welfarist" Approach

The authors propose a three-step recipe to turn the vague idea of "fairness" into a math problem a computer can solve.

Step 1: Listen to Everyone (Preferences)
First, you need to know how much each person "suffers" or "benefits" from a specific outcome.

Analogy: If you are cutting a pizza, you need to know who is starving, who is just hungry, and who is on a diet. You assign a "cost" to every slice for every person.

Step 2: Decide What You Can Compare (The "Comparability" Rule)
This is the most critical part. You have to decide how much you trust your ability to compare people's suffering. The paper outlines four levels of trust:

Level 1 (The "Who is Worst Off?" Rule): You only know who is having the worst day, but you don't know how much worse.
- Analogy: You know Person A is starving and Person B is just hungry. You don't know if Person A is "twice as hungry." So, you just try to make sure the hungriest person gets fed first. (This is called Maximin or Rawlsian fairness).
Level 2 (The "Proportional" Rule): You know people have different scales, but you can't compare them directly.
- Analogy: You know Person A loses \ $10 and Person B loses \$ 10, but maybe for Person A, $10 is a meal, and for Person B, it's a penny. You decide to cut them both by the same percentage of their budget. (This is Nash fairness).
Level 3 (The "Equal Dollar" Rule): You can compare the difference in cost, but not the total amount.
- Analogy: You know that saving Person A \ $10 is just as good as saving Person B \$ 10, even if Person A is rich and Person B is poor. You just add up all the savings and maximize the total. (This is Utilitarian fairness).
Level 4 (The "Full Knowledge" Rule): You can compare everything perfectly.
- Analogy: You know exactly how much every dollar means to everyone. You can balance efficiency and equality perfectly.

Step 3: Solve the Puzzle
Once you pick your "Level of Trust" (Step 2), you get a specific mathematical formula. You then plug in your data (Step 1) and let the computer find the best solution.

3. The Superpower: Feedback Loops

The paper highlights that control systems are special because they are dynamic. They don't just make one decision; they make decisions over and over again, watching what happens and adjusting.

The Analogy: Imagine a conductor leading an orchestra.
- Static Social Choice (Old Theory): The conductor writes the sheet music once and hopes it sounds good.
- Welfarist Control (New Theory): The conductor listens to the music as it plays. If the violins are too loud (unfair), they wave the baton to quiet them down immediately. If the drums are too soft, they boost them up.
Why this matters: Because the system keeps checking the results, it can fix mistakes. If a storm (an external factor) causes a power outage in one neighborhood, the system can automatically shift power to balance the load, ensuring no one is left in the dark for too long.

4. Real-World Examples from the Paper

The authors show how this works in three specific areas:

Internet Data (The Bandwidth Bandwidth):
- Scenario: A server sending video to three people.
- Utilitarian Approach: Give all the speed to the person with the best connection. (Fastest total video, but the others get nothing).
- Maximin Approach: Give everyone just enough speed so the person with the worst connection gets a watchable video, even if it means the others get lower quality.
- Nash Approach: Split the speed so that no one can get more without hurting someone else proportionally.
Public Transit (The Bus Schedule):
- Scenario: A city has extra buses. Should they send them to the busy city center or the quiet suburbs?
- The Paper's Insight: Depending on which "fairness rule" you pick, the answer changes.
  - If you want to help the average day, you send buses to the city.
  - If you want to ensure every single day is fair, you might send buses to the suburbs to balance out the city's crowds.
  - If you want to fix past unfairness, you might send buses to the suburb today because they were ignored yesterday.
Power Grids (The Energy Bill):
- Scenario: A neighborhood has a power limit. Who gets to use their AC?
- The Paper's Insight: If you use a "Total Cost" rule, the rich houses might get all the power because they can afford the cost. If you use a "Maximin" rule, the system might cut power to the rich houses first to ensure the poor houses stay cool.

The Bottom Line

The paper doesn't claim to have a magic wand that solves all inequality. Instead, it offers a toolkit.

It tells engineers: "Don't just guess what 'fair' means. Explicitly choose a rule (like 'help the worst off' or 'maximize total happiness'), explain why you chose it, and then use the feedback loop of your control system to make sure that rule is followed in real-time."

By doing this, we can build automated systems that don't just work efficiently, but also work in a way that society can understand, trust, and accept.

Technical Summary: Welfarist Control Design

Problem Statement

Modern socio-technical systems increasingly rely on automated control to allocate scarce resources (e.g., energy, bandwidth, water, transit capacity) among heterogeneous agents. While control engineers are tasked with making decisive assumptions about societal goals, current design choices are often driven by industry norms or ad hoc objectives rather than principled ethical frameworks. A central challenge is the "big tradeoff" between efficiency and fairness: how to aggregate individual agents' preferences into a collective decision that satisfies a societal mandate (e.g., equity, sustainability, safety) without falling into the pitfalls of undefined fairness metrics or Pareto-dominated outcomes.

The paper argues that while social choice theory has long addressed these aggregation problems, its results have remained largely abstract and disconnected from the real-time, dynamic, and uncertain nature of control systems. The core problem is bridging the gap between the axiomatic foundations of welfare economics and the operational requirements of dynamical systems control.

Methodology

The paper proposes a Welfarist Control Design framework that integrates social choice theory with control engineering. The methodology proceeds in three main stages:

1. The Welfarist Design Pipeline

The authors formalize a design procedure based on Welfarism, a stance where societal decisions depend solely on the individual well-being (costs or utilities) of agents.

Formalization: Agents $i \in \mathcal{N}$ evaluate outcomes $x$ via individual cost functions $J_i(x)$ . A Social Cost Functional (SCFL) ranks outcomes based on the cost profile $J = (J_1, \dots, J_n)$ .
Axiomatic Foundation: The design relies on three core axioms to ensure rational social ranking:
- Pareto Principle (P): If all agents prefer outcome $x$ to $y$ , society must prefer $x$ .
- Independence of Irrelevant Alternatives (IIA): The ranking of two outcomes depends only on the agents' evaluations of those two outcomes, preventing manipulation via irrelevant options.
- Pairwise Continuity (PC): Small perturbations in costs should not cause abrupt reversals in social ranking.
The Welfarism Theorem: Under these axioms, any social ranking can be represented by a continuous Social Cost Function (SCF), $C(J(x))$ , which aggregates individual costs.
Interpersonal Comparability: The specific form of the SCF is determined by the level of information available regarding cross-agent comparisons:
- Ordinal Level Comparability (OLC): Only ordinal rankings are known. Admissible SCF: Maximin (Rawlsian), $C = \max_i J_i(x)$ .
- Cardinal Non-Comparability (CNC): Costs are cardinal but scales are unknown. Admissible SCF: Nash Bargaining solution.
- Cardinal Unit Comparability (CUC): Cost differences are comparable on a common scale. Admissible SCF: Weighted Utilitarian, $C = \sum c_i J_i(x)$ .
- Cardinal Full Comparability (CFC): Both levels and differences are comparable. Admissible SCF: Utilitarian plus a function of deviations from the mean.

2. Dynamic Welfarist Control

The paper extends these static concepts to dynamic settings where decisions are repeated, outcomes are affected by exogenous disturbances, and feedback is available. Three control methodologies are adapted:

Welfarist Feedback Optimization:
- Used for systems with fast transients where the steady-state map $x = h(u, w)$ is stable.
- The controller iteratively updates decisions $u$ using projected gradient descent on the SCF $C(J(x))$ .
- Crucially, the gradient is computed via chain rule using measured outcomes $x$ and sensitivity $\nabla_u h$ , allowing the system to track the welfarist optimum without knowing the disturbance $w$ .
- Example: Data network utility maximization where different SCFs (Utilitarian vs. Nash vs. Maximin) lead to distinct power allocation strategies under fading channels.
Welfarist Control of Markov Decision Processes (MDPs):
- Addresses stochastic dynamics and uncertainty.
- The paper identifies four distinct welfare criteria based on the order of aggregation operators (Sum $\sum$ $\sum$ , Social Cost $C$ $C$ , Expectation $E$ $E$ ):
  1. SCE: Sum of expected stage costs (Ex-ante, stage-wise).
  2. SEC: Expected sum of stage costs (Ex-post, stage-wise).
  3. CES: Social cost of expected total sequence (Ex-ante, sequence-wise).
  4. ECS: Expected social cost of the total sequence (Ex-post, sequence-wise).
- Key Insight: SEC and SCE can be solved via standard Markovian policies and linear programming. ECS, which tracks realized sequences to compensate for past unfairness, requires non-Markovian policies (or state augmentation with accumulated costs) to ensure fairness over the realized trajectory.
- Example: Public transit bus allocation where different criteria yield different strategies for handling demand uncertainty between urban and suburban zones.
Welfarist Model Predictive Control (MPC):
- Incorporates constraints and predictions over a receding horizon.
- Two objective designs are proposed:
  1. Tracking a Welfarist Steady State: Minimizing the distance to a pre-computed optimal steady state $(\bar{x}, \bar{u})$ .
  2. Welfarism at Every Time: Minimizing $\sum C(J(x_t, u_t))$ , ensuring fairness at each step.
  3. Welfarism Over Time: Minimizing $C(\sum J(x_t, u_t))$ , allowing trade-offs where an agent suffers now to be compensated later.
- Example: Residential energy management where constraints (line limits) affect agents heterogeneously. The choice of SCF (e.g., Maximin vs. Utilitarian) drastically changes how the burden of constraints is distributed.

Key Contributions

Principled Design Pipeline: The paper provides a rigorous recipe for control engineers to move from "ad hoc" fairness metrics to axiomatic social cost functions, explicitly linking the choice of SCF to the available information (comparability level).
Dynamic Integration: It successfully translates static social choice theorems into dynamic control frameworks (Feedback Optimization, MDPs, MPC), demonstrating how feedback loops can be used to certify and correct allocations over time.
Distinction of Criteria: It clarifies the critical differences between ex-ante (expected) and ex-post (realized) fairness, and between stage-wise and sequence-wise aggregation, showing that these choices fundamentally alter control policies and outcomes.
Stability and Guarantees: The paper discusses theoretical guarantees for closed-loop stability (via dissipativity) and averaged performance, noting where standard control proofs hold (e.g., Utilitarian MPC) and where they remain open challenges (e.g., Maximin stability).

Results and Illustrations

The paper validates its framework through illustrative examples rather than large-scale empirical deployment:

Power Curtailment: Demonstrates how the same physical problem yields different rules (Minimum Total, Equal, or Proportional curtailment) depending on whether the designer assumes CUC, OLC, or CNC.
Data Networks: Shows that Utilitarian feedback optimization starves inefficient agents, while Nash and Maximin approaches provide different forms of fairness (proportionality vs. outcome equality) in the presence of channel fading.
Public Transit: Illustrates how different welfare criteria (SEC vs. ECS) handle demand uncertainty. SEC enforces daily equality, while ECS compensates for realized unfairness over the horizon, reducing long-term inequality in extreme scenarios.
Energy Management: Highlights that under constraints, Utilitarian objectives may concentrate burdens on specific agents, whereas Maximin objectives redistribute costs to protect the worst-off, even if it increases total system cost.

Significance and Claims

The paper claims to provide a principled foundation for "fair control" that moves beyond implicit industry conventions. Its significance lies in:

Bridging Disciplines: It connects the 100-year-old theory of social choice with modern control engineering, making abstract impossibility results concrete for applied, real-time systems.
Certifiability: By grounding decisions in axioms and specific SCFs, the framework allows for the explanation and certification of automated decisions, which is paramount for social acceptance of automation in critical infrastructure.
Dynamic Correction: Unlike static social choice, the control perspective leverages feedback to correct for exogenous disturbances and uncertainties, offering a mechanism to achieve fairness "over time" rather than just in a single snapshot.

The authors are modest in their claims, acknowledging that while the framework provides a rigorous starting point, significant open challenges remain. These include proving stability for non-utilitarian SCFs (like Maximin), handling risk-averse social planners, developing efficient learning algorithms for unknown environments, and extending the theory to non-time-separable objectives. The paper positions itself as a "leap" toward a future where social choice is operationally integrated into the control of socio-technical systems.