The Speed of Trust
From Velocity of Money to Velocity of Commitments
Inspired by how Dirac expanded the wave function into multiple dimensions, this article introduces a framework for understanding polycentric networks of commitments and agreements. Through an expansion of the classical MV=PY Exchange Equation into a matrix form, we explore how to investigate flows (or see current) of commitments for resources through polycentric networks of resource pools.
*N.B. This article is for heterodox economists, physicists and mathematicians. But might be interesting for anyone interested in measuring trust in networks - with a little understanding of matrix equations and linear algebra. I will leave out full derivations as exercises for the adventurous reader.
Why Classical Economics Falls Short
In classical economics, the equation: MV=PY provides a foundational understanding of how money flows through an economy. Here:
M represents the money supply,
V the velocity of money (how often money circulates),
P the price level,
Y the real output or GDP.
While useful for centralized monetary systems, this scalar equation simplifies the complex interactions of decentralized economies.
In systems like Rotational Labor Associations (ROLAs) or mutual aid networks, money is considered only as one form of commitment. So we must replace money by commitments—agreements to provide labor, goods, or services. The flow of these commitments is dynamic, multi-dimensional, and polycentric.
Inspired by how Dirac expanded quantum mechanics to encompass multi- dimensional wave functions, we introduce a matrix-based framework for decentralized resource coordination. This framework captures:
Commitment Formation: Agreements made within and across pools,
Resource Flows: Exchanges between agents,
Fulfillment Velocity: The rate at which commitments circulate.
The Matrix Framework: Expanding MV=PY
I’ll motivate below how the classical equation evolves into: CV=P⋅R
Where:
C: Commitments matrix, tracking contributions and withdrawals across agents,
V: Velocity vector, representing the circulation rate of commitments,
P: Price vector, assigning relative value to resource commitments,
R: Resources matrix, capturing the flow of actual goods and services.
This equation generalizes the scalar relationships of MV=PY into a structure that can represent decentralized, polycentric systems.
Just as MV=PY is often referred to as the Exchange Equation, this expanded, matrix-based version: CV = P⋅R can be called the Multi-Dimensional Exchange Equation. It highlights how Commitments and Velocity (left side) balance with Relative Value (Prices) and Resources (right side) in a network of decentralized, polycentric exchanges.
*N.B. Read the recent article on Price Vectors here.
Why the Velocity of Commitments is Interesting
The velocity of commitments measures how actively commitments circulate within a system. It helps us look at the fluid movements underneath scalar approximations like speed, as well as credit and debt. Mark Burgess Promise Theory links such velocity with building trust. It provides key insights:
System Efficiency: Higher velocity suggests frequent exchanges, indicating a dynamic and efficient system. Lower velocity might reflect stagnation or inefficiencies.
Equity of Participation: Velocity highlights imbalances in contribution and fulfillment. Equal velocities across agents or pools indicate fairness.
Inter-Pool Dynamics: In multi-pool systems like https://viz.sarafu.network, velocity reveals the flows or movement of commitments across pools, revealing interconnected resource flows.
Example 1: A Single ROLA Pool
Initial Setup
Three families (A, B, C) participate in a Rotating Labor Association (ROLA) pool:
Each family seeds 2 commitments into the pool. They make their offering (and conditions) of providing farm labor known to the others.
Over three rounds (This can be a 3 week period in practice), families exchange and fulfill commitments for farming labor support.
Round-by-Round Transactions
Initial Pool (seeding round):
Each family starts with 2 commitments in the pool:
The rows and columns of this matrix represent the commitments of resources each family has made to each other family. Those along the diagonal can be considered commitments made (not yet held by any particular family) - while the off diagonal elements are between families.
Round 1 (Family A Acts):
Family A adds 2 commitments to the pool, withdrawing 1 each from Families B and C.
Updated commitments matrix:
Here we can see that Family A now has a total of 4 outstanding commitments and is holding commitments from family B and C (shown as off diagonal elements).
Round 2 (Family B Acts):
Family B adds 2 commitments to the pool, withdrawing 1 each from Families A and C.
Updated commitments matrix:
Round 3 (Family C Acts):
Family C adds 2 commitments to the pool, withdrawing 1 each from Families A and B.
Updated commitments matrix:
Final State:
At the end of the cycle, all families return to their original commitments in the pool:
*Exercise for the reader to work out the transformation matrix equations for each of these steps.
Resource and Velocity Analysis (single pool)
Resources Matrix (R)
Suppose each family provides an equal amount of labor or resources to the others:
Price (or Relative Value) Vector (P)
If all labor or resources have the same relative value:
Velocity of Commitments (V)
Plugging into CV=P⋅R one finds:
Each family’s Commitment Velocity being 1 suggests a balanced and fair circulation of commitments. This means that each of their commitments circulated a single time into the pool (after the initial seeding).
*Exercise for the reader to solve for V showing steps.
Example 2: Two Interconnected ROLA Pools
Initial Setup
Pool 1:
Families 1, 2, and 3 each seed 2 commitments.
Pool 2:
Families 3, 5, and 6 each seed 2 commitments.
In this example there are two agreements (pools) each involving 3 families. Family 3 is a member of both agreements or pools - can can serve as a conduit connecting the two networks.
Cross-Pool Dynamics
Step 1: Family 1 Exchanges in Pool 1
Family 1 places 2 additional commitments into Pool 1, withdrawing 2 from Family 2.
Step 2: Family 1 Uses Family 3’s Commitment in Pool 2
In Pool 2, Family 1 uses the commitments effectively borrowed from Family 3 in Pool 1 to access 2 commitments from Families 5 and gives those commitments back to family 5 after they fulfill their commitments (say for roof repair).
Step 3: Family 5 Exchanges in Pool 2
Family 5 then contributes 2 commitments to Pool 2, withdrawing 2 from Families 3 - and Family 3 helps them with roofing.
Step 4: Family 3 Restores Balance
Since Family 3 is in both pools, it coordinates the final step to re-balance each pool, returning them to the initial matrices.
*Exercise for the reader to work out the matrix equations for each of these steps ensuring all pools come back to their initial state.
Multi-Pool Dynamics
Cross-Pool Leverage
Families (like Family 3) that participate in multiple ROLAs (or pools) serve as bridges, allowing commitments to flow between pools. This creates broader opportunities for mutual aid but also demands careful record-keeping.
Velocity as a Diagnostic Tool
By analyzing how often commitments are exchanged or redeemed, one can identify bottlenecks or underutilized resources.
Scalable Fairness
The matrix-based framework scales gracefully, ensuring balanced exchanges even when multiple pools are interconnected. Every promise made (and fulfilled) is tracked at the matrix level.
Moving at the Speed of Trust
Why This Matters
Expanding the scalar exchange equation MV=PY into a Multi-Dimensional Exchange Equation —CV=P⋅R - reveals the complex interplay of commitments in decentralized systems. Classical economics centers on currency as a single medium of exchange, but in polycentric resource-sharing networks, National Currencies can be seen as just another type of commitment and instead promises themselves form the backbone of exchange. By seeing the flow visible of commitments, velocity, and resources, these systems can preserve fairness and enable interoperability and resilience.
Takeaways
Velocity of commitments is an essential metric for assessing how actively promises move through a system, highlighting both efficiency and fairness.
Matrix-based flow tracking allows us to pinpoint imbalances and design methods to correct them, ensuring all participants benefit.
Polycentric Pools (like connected ROLAs) can overlap and be harnessed to extend mutual support across wider communities, with certain participants acting as bridging nodes.
This is simply a lens - a way of looking at what is happening in resource coordination networks based on trust. The network of overlapping pools can be considered an intangible commons and through this lens, we see that the speed of trust—embodied by the velocity of commitments—can be just as vital as the raw quantity of resources.
As we expand our understanding of human trust relationships - we can imagine that we exist within a vast matrix of these overlapping pools already (albeit centralized through national currencies).
As we start to move beyond scalar units of measure and toward polycentric networks, we’ll need to start seeing the currents we are all part of (as Art Brock likes to say “Current-See” rather than currency).
“We move at the speed of trust.”- Adrienne Maree Brown
Brilliant stuff
How are commitments of various sorts and qualities quantified; how is equivalence determined?