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

Abstract: At the core of most socio-technical systems lies a scarce resource that is allocated among agents: highway lanes, public transit, road space, water rights, energy access, grid capacity, user attention, pollution rights, etc. With further automation of the underlying allocation processes, control engineers are increasingly tasked to make decisive assumptions regarding what society wants. In practice to date, design choices are largely driven by industry norms and conventions rather than a result of conscientiously responsible and ethical design. In this paper, we look at tools available to control engineers to design systems in a more principled manner in order to match the societal mandate. We consider three control design paradigms: online feedback optimization, control of Markov decision processes, and model predictive control. Beginning with aggregating individual agents' preferences into control design objectives, subsequently ensuring and certifying the fulfillment of those specifications, we argue that the feedback nature of control systems enables appropriate allocation of the shared resources in ways hitherto unparalleled.

• The paper introduces welfarist control, a framework that combines social-choice axioms, interpersonal comparability assumptions, and closed-loop optimization to align multi-agent resource allocation with explicit societal objectives.
• The paper shows that utilitarian, Nash, and maximin objectives produce different feedback policies and welfare outcomes, including distinct approaches to fairness in photovoltaic curtailment, wireless streaming, transit, and energy management.
• The paper develops welfare criteria and optimization methods for feedback control, MDPs, and MPC while identifying unresolved challenges in transient guarantees, model uncertainty, Bellman recursions, trajectory-wise fairness, and non-utilitarian stability certification.

Motivation and problem statement

Socio-technical systems—power grids, transit networks, communication infrastructure, water distribution—allocate scarce resources among heterogeneous agents in real time. As these allocation processes become automated, control engineers are implicitly making normative decisions about what society wants: who gets curtailed, who waits longer, whose service degrades. The authors of "Welfarist Control Design" (2606.23931) observe that, in practice, such design choices are driven by industry norms and ad hoc objectives rather than by a principled account of the societal mandate. Their proposal is to import the axiomatic machinery of welfarist social choice theory into control design, producing what they call welfarist control: taking welfarist decisions repeatedly, under noise and uncertainty, from online measurements, while certifiably maximizing social welfare over the long run.

The paper's central claim is that feedback is the distinguishing asset of control. Whereas classical social choice concerns one-shot allocations decided by an authority, control problems feature dynamics, uncertainty, and disturbances—and hence the possibility of compensating exogenous factors and correcting realized inequalities. The appropriate object of certification is therefore not each individual decision but the closed-loop behavior of the system.

The welfarist tutorial: axioms, representation, and comparability

The foundational layer is a tutorial on welfarism [(2606.23931); see also shilov2025welfare]. Agents $i \in \mathcal{N}$ evaluate outcomes $x \in \mathcal{X}$ through individual cost functions $J_i$, and a Social Cost Functional (SCFL) maps cost profiles to complete, transitive social rankings. Three axioms—Pareto Principle (P), Independence of Irrelevant Alternatives (IIA), and Pairwise Continuity (PC), optionally with Anonymity (A)—yield the Welfarism Theorem: any SCFL satisfying them admits a numerical representation by a continuous social cost function $C(J_1(x),\dots,J_n(x))$, so allocation reduces to optimizing $C$ over $\mathcal{X}$.

Two aspects of this tutorial deserve emphasis for a control audience:

• Inequality metrics can violate Pareto. The Gini index prefers a Pareto-dominated outcome $(3,3)$ over $(1,2)$ because it is more equal; the same failure arises with Jain, coefficient of variation, and Hoover indices. This is a pointed critique of much of the "fair control" literature that optimizes such indices directly.
• Non-IIA rules are manipulable. Range-normalized aggregation rules break ties between two alternatives when a third, irrelevant alternative is added—a concrete vulnerability in repeated settings.

The axioms pin down representability but not the form of $C$; that is determined by the assumed level of interpersonal comparability, formalized as invariance of the social ranking under families of transformations $\Phi$. Four classes yield distinct canonical SCFs:

Class	Admissible SCF	Interpretation
OLC	$x \in \mathcal{X}$0 (Rawlsian maximin)	Only ordinal cross-agent comparisons
CNC	Nash product with benchmark $x \in \mathcal{X}$1	Cardinal per-agent scales, no cross-agent comparison; requires relaxing IIA to Partial Independence
CUC	Weighted utilitarian sum	Cost differences comparable across agents
CFC	Mean plus homogeneous deviation term	Levels and differences fully comparable

An impossibility result under full CNC with unrestricted IIA motivates the benchmark-dependent Nash form. A running example on photovoltaic curtailment shows how the same physical problem yields minimum-total-curtailment (utilitarian/CUC), equal-curtailment (maximin/OLC), or proportional-curtailment (Nash/CNC) rules depending on which welfare comparisons the operator can justify—an explicit, defensible design specification rather than an implicit convention.

Welfarist feedback optimization

For stable plants with fast transients, $x \in \mathcal{X}$2, the steady-state welfarist program minimizes $x \in \mathcal{X}$3 subject to $x \in \mathcal{X}$4 and input constraints. Feedback optimization tracks its solution via projected gradient steps using measured outcomes $x \in \mathcal{X}$5 and model sensitivities $x \in \mathcal{X}$6, without measuring the disturbance $x \in \mathcal{X}$7. The structure of the update differs sharply across comparability levels:

• Utilitarian: resources flow to agents with the highest marginal utility sensitivity—but this starves agents needing more resources for equal quality, and it is fragile to sensitivity modeling errors.
• Nash: the log-transformed gradient is invariant to individual rescaling of costs; in the wireless video-streaming example it converges to equal power allocation regardless of measured fading, embodying equality of allocation rather than of outcome.
• Maximin: subgradient descent transfers resources to the currently worst-off agent based on measured outcomes, implementing a closed-loop form of distributive justice that compensates exogenous factors.

Three limitations are stated plainly. First, welfarist guarantees hold only at steady state; individual iterates need not satisfy any welfarist specification, so time-varying disturbances can affect agents "unfairly" during transients. Second, robustness to mismatch in plant sensitivities must be certified per agent, not merely in aggregate. Third, under CNC the benchmark outcome $x \in \mathcal{X}$8 (an attainable worst case) must be determined online under time-varying disturbances, which is itself a nontrivial problem given only steady-state sensitivities.

Welfarist Markov decision processes

For stochastic dynamics modeled as finite-horizon MDPs, the key structural observation is that three aggregation operators—summation over time, expectation over uncertainty, and the SCF across agents—are generally non-commutative. Their orderings define four welfare criteria: SCE (welfare of average stage outcomes, ex-ante), SEC (average stage welfare, ex-post), CES (welfare of the average sequence, ex-ante), and ECS (average welfare of realized sequences, ex-post). A two-stage task-allocation example demonstrates that these criteria produce genuinely different policy orderings: e.g., stage-wise randomization is optimal under SCE but strictly suboptimal under SEC and ECS, while sequence-level randomization is optimal under both CES and ECS.

Computationally, all four criteria admit convex programs over occupancy measures: SEC reduces to a linear program for any SCF since $x \in \mathcal{X}$9 is precomputable; SCE and CES require reformulations (LPs for utilitarian and Rawlsian objectives, a convex program via logarithms for Nash). A notable structural result concerns ECS, which depends on the joint distribution of trajectories rather than occupancy measures alone and thus requires non-Markovian policies—but state augmentation with accumulated per-agent costs $J_i$0 restores a Markovian formulation solvable by LP or dynamic programming. In contrast, SCE and CES generally violate stage-additivity and do not admit standard Bellman recursions, though backward induction remains possible for SCE by joint optimization over states at each stage.

A public-transit spare-bus example illustrates the practical divergence: under maximin, SEC enforces daily equality between city and suburb; CES achieves equality only in expectation over typical demand realizations and accumulates large inequality in extreme scenarios; ECS compensates along the realized sequence, substantially reducing end-of-horizon inequality even in all-high-demand weeks. The ex-ante/ex-post distinction thus maps directly onto feedforward versus feedback compensation.

Open challenges here include replacing risk-neutral expectation with risk measures such as CVaR (the relation between ECS and explicit risk-awareness is unknown), accommodating heterogeneous temporal preferences and risk tolerances across agents, and reinforcement learning: Q-learning does not generalize to criteria lacking Bellman recursions, and scalable learning-based welfarist control remains open.

Welfarist model predictive control

MPC accommodates constraints and predictions, and the paper identifies three ways to embed welfarism in the receding-horizon program: tracking a welfarist steady state computed offline; applying $J_i$1 at every time step ($J_i$2); or aggregating trajectory costs before applying $J_i$3 ("welfarism over time"). The first two preserve time separability and inherit economic-MPC proof techniques; the third does not.

On guarantees, the picture is incomplete and honestly reported. For the time-separable formulations, asymptotic stability of the welfarist steady state follows from strict dissipativity with supply rate built from $J_i$4—but dissipativity results are established essentially only in the utilitarian case. Quadratic tracking objectives extend certification to maximin (via min-max MPC techniques) and plausibly to Nash (via log-barrier MPC), while dissipativity for maximin with general costs is, to the authors' knowledge, an open problem. Averaged-performance results from economic MPC carry over to certify that closed-loop behavior matches or beats the welfarist steady state in the limit.

Two empirical findings in the energy-management example (household batteries on a CIGRE feeder) carry weight. First, the "welfarism over time" maximin solution delivers no energy to one consumer early in the horizon and compensates later; applied receding-horizon, this risks that consumer never receiving energy—open-loop welfarism does not imply closed-loop welfarism without certificates. Second, when line-capacity constraints bind heterogeneously, the choice of SCF strongly redistributes burden: under maximin, affected consumers are compensated through lower energy prices paid by unaffected consumers, whereas under utilitarianism affected consumers bear nearly all of the cost while others benefit. Whether constrained closed-loop trajectories actually minimize the chosen SCF, and how Lagrangian dual variables interact with SCF gradients, remain open questions, as does the entire non-time-separable case where dissipativity and turnpike arguments fail.

Limitations and open questions

The paper is a tutorial-position piece, and several boundaries should be noted. The welfarist pipeline presumes that individual cost functions $J_i$5 are available in functional form together with a justified comparability level; how to learn these models online while controlling, and how axiomatic guarantees survive stochasticity and model mismatch, are explicitly left open. The steady-state-only guarantees of feedback optimization, the absence of Bellman recursions for SCE/CES, the missing dissipativity theory for non-utilitarian MPC objectives, and the unresolved status of trajectory-wise ("welfarism over time") optimization are all flagged by the authors rather than resolved. The framework also assumes a fixed, well-defined set of agents with limited-scope decisions; whether it extends to changing participation or multi-generation horizons is outside its stated scope. Finally, the claim that welfarist certification will improve social acceptance of automation rests on procedural-fairness evidence from social psychology, not on validation within the proposed control loops.

Conclusion

This paper articulates a coherent pipeline—axioms, comparability assumptions, social cost function, certified closed-loop implementation—for embedding welfarist social choice into feedback optimization, MDP control, and MPC. Its distinctive contributions are the demonstration that comparability assumptions uniquely determine admissible allocation rules in engineering terms, the taxonomy of ex-ante versus ex-post welfare criteria for stochastic control, and a candid map of where existing stability and learning theory falls short of certifying welfarist closed-loop behavior. The agenda it leaves open—ex-ante fairness as output regulation, learning welfarist policies, and Bellman principles for social welfare over trajectories—defines the technical work required before axiomatic guarantees can be certified in operational socio-technical systems.