Academic Paper Outline¶

Title: Backpressure Economics: Capacity-Constrained Monetary Flow Control for Agent Economies

Target: ACM CCS / IEEE S&P / FC (Financial Cryptography) - theoretical contribution + systems paper

Section 1: Introduction (2 pages)¶

Motivating example: three-stage AI agent pipeline - transcription → summarization → report generation - where the summarizer hits GPU limits and payment continues flowing into a bottleneck
Problem statement: streaming payment protocols lack flow control; money cannot be "dropped" like data packets - requires new routing primitives
Contribution summary: (1) formal model, (2) throughput optimality proof, (3) protocol design, (4) simulation

2.1 Backpressure Routing¶

Tassiulas-Ephremides (1992): max-weight scheduling, throughput optimality, Lyapunov drift
Neely (2010): utility-optimal extensions, V-parameter tradeoff
Multi-commodity flow formulations

2.2 Network Pricing¶

Kelly proportional fairness (1998): shadow prices, dual decomposition
Srikant (2004): congestion control as optimization
Bridge: Kelly's "price" = Lagrange multiplier on capacity constraints = our capacity signal

2.3 Token Engineering¶

Zhang-Zargham (2020): dynamical systems for token economies
Superfluid Protocol: CFA/GDA streaming primitives
Position: BPE adds flow-control layer above streaming primitives

2.4 Demurrage Economics¶

Gesell (1916), Fisher (1933): stock-based velocity mechanisms
Position: BPE is flow-based allocation - orthogonal to demurrage's velocity

2.5 Sender-Side Capacity Awareness¶

AMMs as sender-side routing (Uniswap, Curve)
BPE as receiver-side routing: distinct contribution

2.6 AI Agent Payment Protocols¶

Google AP2, Coinbase x402, OpenAI-Stripe ACP, Visa TAP (2024-2026)
Gap: all handle authorization/trust, none handle flow control

Section 3: Model (4 pages)¶

3.1 Network Graph¶

G = (V, E) where V = Sources ∪ Sinks, E = payment flow edges
C(K, t, τ): capacity of sink K at time t for task type τ
Q(K, t, τ): virtual queue backlog of unprocessed payment at sink K
F(e, t): payment flow rate on edge e at time t

3.2 Backpressure Payment Routing¶

Differential backlog: W(K, t) = Q(source, t) - Q(K, t) (simplified single-hop)
Max-weight scheduling: allocate to sink K* = argmax_K W(K, t) · C(K, t)
Money-specific constraint: no packet drops → overflow buffer B(t)

3.3 Multi-Commodity Extension¶

Task types as commodities: τ ∈ T
Separate virtual queues per (sink, task type)
Capacity allocation across task types at each sink

3.4 EWMA Smoothing¶

C_smooth(K, t) = α·C_raw(K, t) + (1-α)·C_smooth(K, t-1)
Stability analysis: α tradeoff between responsiveness and oscillation

Section 4: Throughput Optimality (3 pages)¶

4.1 Lyapunov Function¶

L(t) = Σ_K Q(K,t)² - sum of squared virtual queue backlogs
Drift: Δ(t) = E[L(t+1) - L(t) | Q(t)]

4.2 Main Theorem¶

Show backpressure payment routing achieves throughput optimality for payment flows within the capacity region
Proof sketch via Lyapunov drift minimization
Key modification from Tassiulas: overflow buffer B(t) absorbs drops; show B(t) is bounded under sufficient capacity

4.3 Stability Conditions¶

Sufficient capacity condition (total sink capacity > total source flow rate)
Buffer bound theorem

Section 5: Protocol Design (3 pages)¶

Map model to Superfluid GDA implementation
CapacityRegistry, BackpressurePool, StakeManager, EscrowBuffer
Pipeline composition for multi-stage

Section 6: Security Analysis (2 pages)¶

Sybil resistance (min stake + concave cap)
Capacity truthfulness (incentive-compatible under slashing)
MEV resistance (commit-reveal)

Section 7: Simulation & Evaluation (3 pages)¶

7.1 Setup¶

Python agent-based model, 50 sinks, 10 sources, 3 task types

7.2 Experiments¶

Convergence: time to steady-state allocation vs round-robin
Shock response: sink failure → rebalance speed
Sybil resistance: attack cost vs gain under min-stake
EWMA α sweep: stability vs responsiveness

Section 8: Discussion & Future Work (1.5 pages)¶

Pricing (v0.2), multi-hop routing, cross-chain, TEE verification
Limitations: oracle trust, on-chain latency, task type coverage