Holarchic Automatic Market Maker

• Holarchic AMM is a multi-layered market mechanism that propagates scarcity signals across nested hierarchical layers to achieve robust and fair pricing.
• It employs sequential and parallel compositions to integrate local markets into a coherent pricing system, ensuring differentiability and well-posed market equilibrium.
• Simulations in digital electricity and decentralized finance demonstrate its potential for enhanced fairness, lower costs, and scalable real-time resource allocation.

A Holarchic Automatic Market Maker (AMM) is an architectural and mathematical generalization of conventional automated market-making mechanisms, designed to operate across multiple nested hierarchical layers. It supports continuous, event-driven market operations by propagating bounded scarcity and pricing signals through a multilevel structure—termed a holarchy—where each layer corresponds to a physically and/or logically distinct aggregation (e.g., node, cluster, zone, region, system). This design ensures robust, stable, and fair market outcomes, with applications in domains such as digital electricity markets and decentralized finance. The holarchic framework is both compositional and extensible, permitting the construction of arbitrarily complex market networks exhibiting provable stability, closure, and optimality properties (Sweeney et al., 15 Dec 2025, Engel et al., 2021).

1. Mathematical Foundations and Formalism

The holarchic AMM framework is grounded in the compositional theory of automated market makers. An $n$-asset AMM is specified by a twice-differentiable, strictly convex function $A:\mathbb{R}_{>0}^n \to \mathbb{R}$, with the state manifold

$$\mathcal{S}(A) = \{ x \in \mathbb{R}_{>0}^n \mid A(x) = 0 \},$$

satisfying smoothness, monotonicity ($\partial_{x_i} A(x) > 0$), and strict convexity requirements. Marginal prices at a given state are given by the gradient:

$$p_{i \to j}(x) = -\frac{\partial_{x_i}A(x)}{\partial_{x_j}A(x)}.$$

These properties ensure continuous price curves, absence of arbitrage-dominant states, and well-posedness during trading operations (Engel et al., 2021).

Holarchic AMM networks are constructed recursively using:

• Sequential composition ($\otimes$): Connects two AMMs along a shared asset, where the output flow of one serves as input for the next, yielding a higher-level AMM whose pricing is implicitly defined via nested constraint satisfaction.
• Parallel composition ($\|$): Aggregates multiple AMMs servicing the same asset pair, enabling input splitting across liquidity pools and forming a total output curve by maximizing aggregate returns.

This compositional closure is crucial: any network (tree) of sequential and parallel AMM compositions is itself an AMM of the corresponding form, preserving differentiability, monotonicity, and unique market equilibrium. For higher-dimensional assets, virtualization steps (convex combinations) are used to reduce multi-asset overlap to tractable cases (Engel et al., 2021).

2. Holarchic Architecture and Scarcity Propagation

Holarchic AMMs instantiate their compositional structure in real-world systems by mapping AMM nodes to layers of aggregation. For example, in a digital electricity market, the holarchy may include System (national) $\rightarrow$ Region $\rightarrow$ Zone/Cluster $\rightarrow$ Node $A:\mathbb{R}_{>0}^n \to \mathbb{R}$0 Device hierarchies. At each layer $A:\mathbb{R}_{>0}^n \to \mathbb{R}$1, the local scarcity index

$A:\mathbb{R}_{>0}^n \to \mathbb{R}$2

is computed, with $A:\mathbb{R}_{>0}^n \to \mathbb{R}$3 representing the total supply and demand at aggregation level $A:\mathbb{R}_{>0}^n \to \mathbb{R}$4 and time $A:\mathbb{R}_{>0}^n \to \mathbb{R}$5. Scarcity signals $A:\mathbb{R}_{>0}^n \to \mathbb{R}$6 are recursively aggregated up the holarchy to determine the tightest constraint $A:\mathbb{R}_{>0}^n \to \mathbb{R}$7 (Sweeney et al., 15 Dec 2025).

Prices, allocations, and all exogenous control signals (e.g., buy/sell pairs $A:\mathbb{R}_{>0}^n \to \mathbb{R}$8) are generated using bounded monotonic functions of the tightest binding scarcity, ensuring the system-wide propagation of the most acute constraint—generalizing and subsuming classical nodal and zonal pricing as special cases. Prices are capped and explainable, being strictly coupled to physical and operational limitations, as:

$A:\mathbb{R}_{>0}^n \to \mathbb{R}$9

where $$\mathcal{S}(A) = \{ x \in \mathbb{R}_{>0}^n \mid A(x) = 0 \},$$0 are monotone, bounded-slope functions and $$\mathcal{S}(A) = \{ x \in \mathbb{R}_{>0}^n \mid A(x) = 0 \},$$1 is the minimum scarcity index over all layers (Sweeney et al., 15 Dec 2025).

3. Market Clearing, Dispatch, and Fairness Mechanisms

Clearing in the holarchic AMM is continuous and event-driven, replacing batch auctions with a real-time dynamic resource allocation protocol:

• Measurement: $$\mathcal{S}(A) = \{ x \in \mathbb{R}_{>0}^n \mid A(x) = 0 \},$$2 (supply), $$\mathcal{S}(A) = \{ x \in \mathbb{R}_{>0}^n \mid A(x) = 0 \},$$3 (demand), and local deficit $$\mathcal{S}(A) = \{ x \in \mathbb{R}_{>0}^n \mid A(x) = 0 \},$$4 at each node.
• Propagation: Calculate and propagate $$\mathcal{S}(A) = \{ x \in \mathbb{R}_{>0}^n \mid A(x) = 0 \},$$5.
• Pricing: Determine $$\mathcal{S}(A) = \{ x \in \mathbb{R}_{>0}^n \mid A(x) = 0 \},$$6 per node.
• Prioritization: Allocate essential demand first; flexible requests are queued and cleared using a fairness algorithm ("FairPlayAllocation").
• Dispatch: Supply offers are ranked by ascending $$\mathcal{S}(A) = \{ x \in \mathbb{R}_{>0}^n \mid A(x) = 0 \},$$7 (pay-as-bid, merit order); accepted offers clear demand, each generator paid its bid price (Sweeney et al., 15 Dec 2025).

Non-fuel costs are assigned via a multi-layered Shapley value decomposition. Generators' cost shares are allocated according to their marginal contributions to energy, flexibility, adequacy, congestion relief, and stability, using symmetrized clusters to enable scalable, equitable settlements. Under symmetry, nested Shapley clustering exactly recovers the full cooperative solution (Sweeney et al., 15 Dec 2025).

4. Stability, Performance, and Simulation Evidence

Holarchic AMMs exhibit bounded-input, bounded-output (BIBO) stability; for any input perturbation in demand $$\mathcal{S}(A) = \{ x \in \mathbb{R}_{>0}^n \mid A(x) = 0 \},$$8, the aggregate price response is bounded, precluding runaway price spikes:

$$\mathcal{S}(A) = \{ x \in \mathbb{R}_{>0}^n \mid A(x) = 0 \},$$9

A Lyapunov-style argument, with $\partial_{x_i} A(x) > 0$0, establishes that the allocation-feedback mechanism rapidly dissipates surpluses (outside shocks), driving the market to equilibrium.

Large-scale simulations at national grid scale (UK) demonstrate:

• Zero structural waste (98% renewables utilization).
• Distributional improvements: Gini coefficient of household bills reduced from 0.48 to 0.43, maximum generator rent share reduced from 25% to 10%.
• Procurement costs bounded between £20–30bn/year, undercutting classical locational marginal pricing (LMP) solutions (£33bn/year) (Sweeney et al., 15 Dec 2025).

5. Compositional Theory and Holarchic Networks

The underlying mathematical theory ensures that sequential and parallel compositions of $\partial_{x_i} A(x) > 0$1, strictly convex AMMs yield new AMMs of the same form, preserving stability and well-posedness. This permits the construction of arbitrarily deep holarchic (tree-structured) networks:

• Each node in the tree, whether a leaf (local market) or an internal aggregate (cluster, zone, region), is a valid AMM.
• Markets composed in this fashion collectively act as one coherent AMM at the root, exposing a single pricing surface and equilibrium for external participants.
• Multi-asset overlaps and cycles in the asset graph require normalization via asset virtualization steps to preserve consistency (Engel et al., 2021).

This holarchic compositional approach forms the mathematical and algorithmic foundation of scalable, modular digital marketplace architectures, both for programmable digital asset trading and for physical resource allocation in sectors like electricity.

6. Transition, Implementation, and Governance Considerations

Implementing a holarchic AMM architecture entails a phased rollout strategy:

1. Behind-the-meter pilots deploying local holarchic logic with quality-of-service contracts.
2. Integration at distribution and system operator layers.
3. Full deployment with revenue allocation advanced by generator-level Shapley value settlements.

Essential tooling for deployment includes an API gateway, real-time data stores, embedded fairness allocation engines, Shapley-based revenue allocation modules, and regulatory dashboards. The approach institutionalizes "regulation as code," ensuring algorithmic compliance and the possibility of audit, while holarchic dashboards provide real-time stress and fairness telemetry at every aggregation level (Sweeney et al., 15 Dec 2025).

7. Applications, Limitations, and Research Directions

Holarchic AMMs generalize and unify disparate market designs, enabling bounded, explainable, and equitable outcomes in settings with physical constraints and stochastic supply/demand shocks. They subsume nodal/zonal frameworks, mitigate the need for administrative ex post corrections, and facilitate decentralized digital regulation.

Limitations include requirements for consistent stake valuation during virtualization, explicit loop-avoidance in sequentially chained compositions, and numerical stability in deeply recursive compositional architectures. Ongoing research addresses transition challenges, optimization of cost allocation rules, and incorporation of complex network effects in both digital and physical resource markets (Sweeney et al., 15 Dec 2025, Engel et al., 2021).