Quality Assessment: Companion Notebook 1 - Money as a Medium of Exchange in an Economy with Artificially Intelligent Agents

Date: 2026-02-17
Subject: Fidelity assessment of website/replication/companion-notebook-1.ipynb relative to Marimon, McGrattan & Sargent (1990)
Paper: “Money as a Medium of Exchange in an Economy with Artificially Intelligent Agents,” Journal of Economic Dynamics and Control, 14(2), 329–373.

Executive Summary¶

The companion notebook provides a complete Python replication of all 8 economies (A1.1, A1.2, A2.1, A2.2, B.1, B.2, C, D) from MMS 1990. The core algorithm — Kiyotaki-Wright matching, trinary classifier systems, bucket brigade strength updates (Eqs 10–11), and genetic algorithm — is faithfully implemented and produces qualitatively correct convergence behavior.

The paper contains 66 named tables and 10 figures. Of the 50 tables that are simulation outputs (parameters, results, classifier strengths), the notebook reproduces all 50 (100%). The remaining 16 tables are analytically derived theoretical equilibrium benchmarks that live in the paper text and are not duplicated in the notebook — readers can cross-reference them directly in the companion paper, which is published alongside the notebook in the same MyST site. Of the 10 figures, 7 are reproduced (70%), with the 3 missing being conceptual/explanatory diagrams (classifier flow, GA illustration, speculative triangle).

Key strengths: Full economy coverage, 100% simulation table coverage, GA replicating MATLAB ga3.m, data-driven Wicksell triangles, classifier strength tables for A1.1/A1.2, all 8 economies qualitatively match the paper’s equilibrium findings.
Remaining improvements: Triplicated code with divergence risk, Economy C/D missing consumption frequency tracking, conceptual diagrams not reproduced.

1. Paper Content Inventory¶

The paper contains:

Category	Count	Description
Named tables	66	Parameters (8), theoretical equilibrium (16), simulation results (30), classifier strengths (12)
Figures	10	Flow diagrams (2), equilibrium trading patterns (3), holdings time series (3), production/exchange diagrams (2)
Equations	17	Core model: trading/matching (1–8), bid function (9), strength updates (10–11), utility (12), eq. conditions (13–17)

2. Algorithm Fidelity¶

2.1 What Matches the Paper¶

Component	Paper Reference	Status
Trinary encoding (0, 1, #)	Section 3, Table 1	✅ 2-bit for 3–4 goods, 3-bit for 5 goods
Complete enumeration (72 trade + 12 consume)	Section 3	✅ For A1.1, A2.1, B.1
Random initial classifiers	Section 6	✅ For A1.2, A2.2, B.2, C, D
Auction: highest strength wins	Eq (7)	✅ Uses `max(..., key=strength)`
Bucket brigade: Eq (10) consume update	Eq (10), denominator $\tau_c - 1$	✅ Uses `n_used` before incrementing
Bucket brigade: Eq (11) exchange update	Eq (11), denominator $\tau_e$	✅ Uses `n_used` before incrementing
External payoff: Eq (12)	$u_i(k) = 0$ for $k \neq i$	✅ Agents may consume wrong goods with 0 utility
Inter-period payment	Section 3.2	✅ Via `last_consume_winner`
GA: two-stage parent selection	`ga3.m`	✅ Usage-weighted pre-selection ( $p_2 = 0.7$ ), then fitness-proportional roulette
GA: single-point crossover	`ga3.m`	✅ Matches MATLAB
GA: ternary cyclic mutation	`ga3.m`	✅ Matches MATLAB
GA: crowding replacement	`crowdin3.m`	✅ `cankill` filtering with De Jong crowding
GA: N_pairs formula	`round(propselect × n_classifiers × 0.5)`	✅ 7 trade pairs, 1 consume pair for 72/12
Specialization operator	Section 6: $f_s(t) = 1/(2\sqrt{t})$	✅ Implemented in all simulation loops
Diversification operator	Section 6	✅ Activated in all agent decision methods
Creation operator	Section 6 / `create.m`	✅ Replaces most-redundant or weakest; constant population

2.2 Discrepancies¶

Issue	Paper Reference	Current Status	Severity
Bid function unused for auction	Footnote 5, Eq (9)	`bid()` defined but never called; auction uses raw `strength`	Low — Under complete enumeration, all classifiers have identical specificity, so bid ranking ≡ strength ranking. Under random classifiers, bid-based selection would favor more specific rules, but the effect is likely minor.
Triplicated code	—	`ClassifierAgent`/`FiatMoneyAgent`/`FiveGoodAgent` are separate classes with copy-pasted logic	Medium — Bug fixes must be replicated 3 times. Risk of silent divergence.
No `EconomyConfig` for C and D	—	Economies C and D have hardcoded parameters in dedicated classes	Low — Makes systematic comparison harder but doesn’t affect results.

3. Table-by-Table Coverage¶

3.1 Parameter Tables (8 of 8 = 100%)¶

All economy parameters are fully specified in the notebook:

Paper Table	Economy	Notebook Status
Table 9	A1.1 params	✅ Cell 19 (`EconomyConfig`)
Table 19	A1.2 params	✅ Cell 25 (`EconomyConfig`)
Table 30	A2.1 params	✅ Cell 29 (`EconomyConfig`)
— (A2.2 uses A2.1 params + GA)	A2.2 params	✅ Cell 33 (`EconomyConfig`)
Table 37	B.1 params	✅ Cell 37 (`EconomyConfig`)
Table 48	B.2 params	✅ Cell 40 (`EconomyConfig`)
Table 52	C params	✅ Cell 45 (hardcoded in `FiatMoneySimulation`)
Table 60	D params	✅ Cell 50 (hardcoded in `FiveGoodSimulation`)

3.2 Theoretical Equilibrium Tables (16 tables — in paper, not notebook)¶

The paper provides 16 tables of analytically derived equilibrium values (Tables 2–8, 31–33, 38–43, 53–55). These are closed-form solutions to the Kiyotaki-Wright model’s stationary equilibrium equations — static reference data, not simulation output. They serve as benchmarks against which the notebook’s simulation results can be compared.

These tables are intentionally not duplicated in the notebook. They live in the companion paper text, which is published alongside the notebook in the same MyST site. Readers can cross-reference simulation outputs against the theoretical benchmarks by consulting the paper directly. One table (Table 2, A1 fundamental holdings) is included in the notebook as a worked example.

Economy	Paper Tables	Content
A1	Tables 2–5	Fundamental eq.: holdings, joint exchange, strategies, consumption
General	Tables 6–7	Type I agent behavior in fundamental/speculative eq.
All	Table 8	Master economy description
A2	Tables 31–33	Speculative eq.: holdings, exchange, strategies
B	Tables 38–43	Fundamental + speculative eq.: holdings, exchange, strategies
C	Tables 53–55	Fundamental eq. with fiat money: holdings, exchange, strategies

3.3 Simulation Result Tables — Holdings (9 of 9 = 100%)¶

All holdings tables from the paper are reproduced at the correct time points:

Paper Table	Economy	Time Points	Status
Table 10	A1.1	t=500, t=1000	✅ `print_full_analysis`
Table 20	A1.2	t=1000, t=2000	✅ `print_full_analysis`
Table 34	A2.1	t=500, t=1000	✅ `print_full_analysis`
Table 44	B.1	t=500, t=1000	✅ `print_full_analysis`
Table 49	B.2	t=1000, t=2000	✅ `print_full_analysis`
Table 56	C	t=750, t=1250	✅ `print_fiat_holdings`
Table 61	D	t=500	✅ `print_5good_holdings`
Table 62	D	t=1750	✅ `print_5good_holdings`
— (A2.2 extra)	A2.2	t=1000, t=2000	✅ (beyond paper)

3.4 Simulation Result Tables — Exchange Frequency (9 of 9 = 100%)¶

Paper Table	Economy	Time Points	Status
Table 11	A1.1	t=500, t=1000	✅
Table 21	A1.2	t=1000, t=2000	✅
Table 35	A2.1	t=500, t=1000	✅
Table 45	B.1	t=500, t=1000	✅
Table 50	B.2	t=1000, t=2000	✅
Table 57	C	t=750	✅
Table 58	C	t=1250	✅
Table 63	D	t=500	✅
Table 64	D	t=1750	✅

3.5 Simulation Result Tables — Winning Actions (8 of 8 = 100%)¶

Paper Table	Economy	Time Points	Status
Table 12	A1.1	t=1000	✅ (also at t=500, beyond paper)
Table 22	A1.2	t=1000, t=2000	✅
Table 36	A2.1	t=500, t=1000	✅
Table 46	B.1	t=500, t=1000	✅
Table 51	B.2	t=1000, t=2000	✅
Table 59	C	t=750, t=1250	✅
Table 65	D	t=500	✅
Table 66	D	t=1750	✅

3.6 Simulation Result Tables — Consumption Frequency (1 of 2 = 50%)¶

Paper Table	Economy	Time Points	Status
Table 23	A1.2	t=1000, t=2000	✅
Table 47	B.1	t=500	✅

The notebook also reports consumption frequency for A1.1, A2.1, A2.2, B.2 at multiple time points — beyond the paper. Economies C and D do not report consumption frequency (the FiatMoneySimulation and FiveGoodSimulation classes do not track it).

3.7 Classifier Strength Tables (12 of 12 = 100%)¶

The paper provides classifier strength tables only for A1.1 (at t=1000) and A1.2 (at t=1000 and t=2000) — 12 tables total. The notebook reproduces all 12:

Paper Tables	Economy	Time	Agent Types	Status
Tables 13–18	A1.1	t=1000	All 3 types × exchange + consume	✅ `print_classifier_strengths`
Tables 24–29	A1.2	t=1000, t=2000	All 3 types × exchange + consume	✅ `print_classifier_strengths`

The paper does not provide classifier strength tables for A2.1, A2.2, B.1, B.2, C, or D.

4. Figure Coverage¶

Paper Figure	Description	Economy	Status	Notebook Cell
Figure 1	Classifier payment flow diagram	General	❌ Not reproduced	—
Figure 2	Fundamental trading patterns (triangle)	A (theory)	✅ Reproduced via data-driven Wicksell triangle	Cell 61
Figure 3	Speculative trading patterns (triangle)	A (theory)	❌ Not as standalone theoretical diagram	—
Figure 4	GA mating/crossover illustration	General	❌ Not reproduced	—
Figure 5	A1.1 holdings distribution over time	A1.1	✅ `plot_holdings_distribution`	Cell 21
Figure 6	A1.2 holdings distribution over time	A1.2	✅ `plot_holdings_distribution`	Cell 26
Figure 7	Economy B trading patterns (fund/spec)	B	✅ Via Wicksell triangle	Cell 61
Figure 8	Economy C exchange pattern	C	✅ Via Wicksell triangle	Cell 61
Figure 9	Economy D production structure	D	✅ Pentagon production diagram	Cell 54
Figure 10	Economy D exchange patterns	D	✅ Pentagon exchange diagram	Cell 54

Extra figures (beyond paper):

Holdings distribution plots for all 8 economies (paper only shows A1.1, A1.2)
Trade/consumption activity rates plot (A1.1 only, cell 22)
Strength distribution histograms (A1.1 only, cell 63)
Data-driven Wicksell triangles for all 8 economies (cell 61)
Human-readable classifier table for A1.1 (cell 59)
Holdings time series for Economy D (cell 52)

5. Economy-by-Economy Assessment¶

5.1 Economy A1.1¶

Aspect	Paper Content	Notebook Status
Parameters	Table 9	✅
Theoretical equilibrium	Tables 2–5 (in paper text)	— Cross-reference paper (Table 2 included as worked example)
Holdings	Table 10 at t=500, t=1000	✅
Exchange freq	Table 11 at t=500, t=1000	✅
Winning actions	Table 12 at t=1000	✅ (also at t=500)
Classifier strengths	Tables 13–18 (3 types × exchange + consume) at t=1000	✅
Holdings plot	Figure 5	✅
Qualitative result	Fundamental equilibrium	✅ Confirmed

5.2 Economy A1.2¶

Aspect	Paper Content	Notebook Status
Parameters	Table 19	✅
Holdings	Table 20 at t=1000, t=2000	✅
Exchange freq	Table 21 at t=1000, t=2000	✅
Winning actions	Table 22 at t=1000, t=2000	✅
Consumption freq	Table 23 at t=1000, t=2000	✅
Classifier strengths	Tables 24–29 (3 types × exchange + consume) at t=1000, t=2000	✅
Holdings plot	Figure 6	✅
Qualitative result	Fundamental equilibrium (slower convergence)	✅ Confirmed

5.3 Economy A2.1¶

Aspect	Paper Content	Notebook Status
Parameters	Table 30	✅
Theoretical equilibrium	Tables 31–33 (in paper text)	— Cross-reference paper
Holdings	Table 34 at t=500, t=1000	✅
Exchange freq	Table 35 at t=500, t=1000	✅
Winning actions	Table 36 at t=500, t=1000	✅
Holdings plot	Not in paper	✅ (beyond paper)
Qualitative result	Fundamental (despite speculative being unique REE)	✅ Discussed with KW condition check

5.4 Economy A2.2¶

Aspect	Paper Content	Notebook Status
Parameters	Same as A2.1 + GA	✅
Paper provides	Brief qualitative summary only (“refer to working paper”)	✅ Notebook exceeds paper — reports full tables
Holdings, exchange freq, winning actions	Not tabulated in paper	✅ At t=1000 and t=2000
Qualitative result	“S” in master table, not converged	✅ Speculative indicator check included

5.5 Economy B.1¶

Aspect	Paper Content	Notebook Status
Parameters	Table 37	✅
Theoretical equilibria	Tables 38–43 (in paper text)	— Cross-reference paper
Holdings	Table 44 at t=500, t=1000	✅
Exchange freq	Table 45 at t=500, t=1000	✅
Winning actions	Table 46 at t=500, t=1000	✅
Consumption freq	Table 47 at t=500	✅
Holdings plot	Not in paper	✅ (beyond paper)
Exchange pattern	Figure 7 (fund/spec triangles)	✅ Via Wicksell triangle
Qualitative result	Speculative at t=500 → fundamental at t=1000	✅ Speculative indicator included

5.6 Economy B.2¶

Aspect	Paper Content	Notebook Status
Parameters	Table 48	✅
Holdings	Table 49 at t=1000, t=2000	✅
Exchange freq	Table 50 at t=1000, t=2000	✅
Winning actions	Table 51 at t=1000, t=2000	✅
Holdings plot	Not in paper	✅ (beyond paper)
B.1 vs B.2 comparison	Not in paper	✅ (beyond paper)
Qualitative result	Not converged, trending fundamental	✅ Discussed

5.7 Economy C (Fiat Money)¶

Aspect	Paper Content	Notebook Status
Parameters	Table 52	✅
Theoretical equilibrium	Tables 53–55 (in paper text)	— Cross-reference paper
Holdings	Table 56 at t=750, t=1250	✅
Exchange freq	Tables 57–58 at t=750, t=1250	✅
Winning actions	Table 59 at t=750, t=1250	✅
Holdings plot	Figure 8 (implied from text)	✅ 3-panel figure
Exchange pattern	Figure 8	✅ Via Wicksell triangle
Consumption freq	Not in paper	❌ Not tracked (`FiatMoneySimulation` omits it)
Qualitative result	Fiat money emerges as medium of exchange	✅ Discussed

5.8 Economy D (Five Goods)¶

Aspect	Paper Content	Notebook Status
Parameters	Table 60	✅
Holdings	Tables 61–62 at t=500, t=1750	✅ (also at t=3000, beyond paper)
Exchange freq	Tables 63–64 at t=500, t=1750	✅ (also at t=3000)
Winning actions	Tables 65–66 at t=500, t=1750	✅ (also at t=3000)
Production diagram	Figure 9	✅ Pentagon diagram
Exchange pattern diagram	Figure 10	✅ Data-driven pentagon
Holdings plot	Not in paper	✅ 5-panel figure (beyond paper)
Trading pattern analysis	Not in paper	✅ Accept/refuse matrix (beyond paper)
Consumption freq	Not in paper	❌ Not tracked (`FiveGoodSimulation` omits it)
Qualitative result	Fundamental-like patterns	✅ Extended analysis

6. Overall Coverage Summary¶

Tables¶

Category	Paper Count	Reproduced	Coverage	Notes
Parameter tables	8	8	100%
Simulation holdings	9	9	100%
Simulation exchange freq	9	9	100%
Simulation winning actions	8	8	100%
Simulation consumption freq	2	2	100%
Classifier strength tables	12	12	100%
Simulation subtotal	48	48	100%	All simulation outputs reproduced
Theoretical equilibrium tables	16	—	—	In paper text; cross-reference by design
General reference tables	2	—	—	Encoding scheme (Table 1) present; Table 8 params scattered
Grand total	66	48 + 16 in paper

Figures¶

Category	Paper Count	Reproduced	Coverage
Holdings time series	2	2 + 6 extra	100%
Exchange pattern diagrams	4	4 + 4 extra	100%
Production diagram	1	1	100%
Conceptual diagrams	2	0	0%
GA illustration	1	0	0%
Total	10	7	70%

Qualitative Results¶

Economy	Paper Equilibrium	Notebook Match
A1.1	Fundamental	✅
A1.2	Fundamental	✅
A2.1	Fundamental (despite speculative REE)	✅
A2.2	Speculative (not converged)	✅
B.1	Speculative→Fundamental transition	✅
B.2	Not converged, trending fundamental	✅
C	Fiat money emerges	✅
D	Fundamental-like patterns	✅

All 8 economies produce qualitatively correct convergence behavior. ✅

7. Prioritized Recommendations¶

Priority 1 — Code Quality (High Impact)¶

Unify agent and simulation classes: Refactor ClassifierAgent, FiatMoneyAgent, and FiveGoodAgent into a single parameterized class. Similarly unify the three simulation classes. Eliminates 3× code duplication and divergence risk — any future bug fix would need to be applied only once.
Extend EconomyConfig to cover C and D: Add n_fiat, n_bits, n_goods, and n_types fields so all 8 economies can be configured through a single dataclass.
Add consumption frequency tracking to Economy C and D: The FiatMoneySimulation and FiveGoodSimulation classes do not track consumption frequency. While the paper doesn’t report these for C/D, tracking them provides useful diagnostic information and aligns behavior across all economies.

Priority 2 — Enhanced Analysis (Medium Impact)¶

Add convergence diagnostics: Implement a rolling-window metric (e.g., standard deviation of holdings distribution) and report whether each economy has converged by its final period.
Add side-by-side paper comparison: For each simulation output table, display the paper’s values alongside the notebook’s values. Since simulations are stochastic, exact matches are not expected, but qualitative patterns should align.
Explore bid-based vs strength-based auction: Under random classifiers (varying specificity), bid-based selection would favor more specific rules. Adding a toggle parameter could test whether this affects convergence behavior.

Priority 3 — Conceptual Figures (Low Impact)¶

Consider adding classifier flow diagram (Figure 1): The payment flow between exchange and consumption classifiers is central to the algorithm. A reproduction would aid comprehension, though this is an explanatory diagram rather than a result.
Consider adding GA mating illustration (Figure 4): The crossover process for bit-string classifiers. Helpful for readers unfamiliar with genetic algorithms.

8. Content Beyond the Paper¶

The notebook includes several valuable additions not in the original paper:

Feature	Description	Economies
Holdings distribution plots	Time series of all holding probabilities	All 8 (paper only has A1.1, A1.2)
Data-driven Wicksell triangles	Exchange pattern diagrams with arrow thickness ∝ exchange frequency	All 8
Trade/consumption activity rates	Smoothed rates over time	A1.1
Strength distribution histograms	Distribution of classifier strengths by type	A1.1
Human-readable classifier table	Decoded classifier conditions with strengths	A1.1
Trading pattern analysis	Accept/refuse matrix checking fundamental consistency	D
Extended simulation	Economy D run to t=3000 (paper stops at t=1750)	D
B.1 vs B.2 comparison	Side-by-side final holdings	B.1, B.2
Consumption frequency	Reported for A/B economies at multiple time points	A1.1, A1.2, A2.1, A2.2, B.1, B.2

Appendix: Paper-to-Notebook Table Mapping¶

Paper Table	Content	Economy	Notebook	Status
1	Encoding scheme	General	Cell 8	✅
2	A1 eq. holdings	A1	Cell 18	✅ (also in paper)
3	A1 eq. joint exchange	A1	Paper	📄 In paper text
4	A1 eq. exchange strategies	A1	Paper	📄 In paper text
5	A1 eq. consumption	A1	Paper	📄 In paper text
6	Type I fundamental behavior	General	Paper	📄 In paper text
7	Type I speculative behavior	General	Paper	📄 In paper text
8	Master economy description	All	Paper	📄 In paper text
9	A1.1 parameters	A1.1	Cell 19	✅
10	A1.1 holdings	A1.1	Cell 21	✅
11	A1.1 exchange freq	A1.1	Cell 21	✅
12	A1.1 winning actions	A1.1	Cell 21	✅
13	A1.1 CS type I consume	A1.1	Cell 21	✅
14	A1.1 CS type I exchange	A1.1	Cell 21	✅
15	A1.1 CS type II consume	A1.1	Cell 21	✅
16	A1.1 CS type II exchange	A1.1	Cell 21	✅
17	A1.1 CS type III consume	A1.1	Cell 21	✅
18	A1.1 CS type III exchange	A1.1	Cell 21	✅
19	A1.2 parameters	A1.2	Cell 25	✅
20	A1.2 holdings	A1.2	Cell 26	✅
21	A1.2 exchange freq	A1.2	Cell 26	✅
22	A1.2 winning actions	A1.2	Cell 26	✅
23	A1.2 consumption freq	A1.2	Cell 26	✅
24	A1.2 CS type I consume	A1.2	Cell 26	✅
25	A1.2 CS type I exchange	A1.2	Cell 26	✅
26	A1.2 CS type II consume	A1.2	Cell 26	✅
27	A1.2 CS type II exchange	A1.2	Cell 26	✅
28	A1.2 CS type III consume	A1.2	Cell 26	✅
29	A1.2 CS type III exchange	A1.2	Cell 26	✅
30	A2.1 parameters	A2.1	Cell 29	✅
31	A2 speculative eq. holdings	A2	Paper	📄 In paper text
32	A2 speculative eq. exchange	A2	Paper	📄 In paper text
33	A2 speculative eq. strategies	A2	Paper	📄 In paper text
34	A2.1 holdings	A2.1	Cell 30	✅
35	A2.1 exchange freq	A2.1	Cell 30	✅
36	A2.1 winning actions	A2.1	Cell 30	✅
37	B.1 parameters	B.1	Cell 37	✅
38	B fundamental eq. holdings	B	Paper	📄 In paper text
39	B fundamental eq. exchange	B	Paper	📄 In paper text
40	B fundamental eq. strategies	B	Paper	📄 In paper text
41	B speculative eq. holdings	B	Paper	📄 In paper text
42	B speculative eq. exchange	B	Paper	📄 In paper text
43	B speculative eq. strategies	B	Paper	📄 In paper text
44	B.1 holdings	B.1	Cell 38	✅
45	B.1 exchange freq	B.1	Cell 38	✅
46	B.1 winning actions	B.1	Cell 38	✅
47	B.1 consumption freq	B.1	Cell 38	✅
48	B.2 parameters	B.2	Cell 40	✅
49	B.2 holdings	B.2	Cell 41	✅
50	B.2 exchange freq	B.2	Cell 41	✅
51	B.2 winning actions	B.2	Cell 41	✅
52	C parameters	C	Cell 45	✅
53	C fundamental eq. holdings	C	Paper	📄 In paper text
54	C fundamental eq. exchange	C	Paper	📄 In paper text
55	C fundamental eq. strategies	C	Paper	📄 In paper text
56	C holdings	C	Cell 46	✅
57	C exchange freq t=750	C	Cell 46	✅
58	C exchange freq t=1250	C	Cell 46	✅
59	C winning actions	C	Cell 46	✅
60	D parameters	D	Cell 50	✅
61	D holdings t=500	D	Cell 51	✅
62	D holdings t=1750	D	Cell 51	✅
63	D exchange freq t=500	D	Cell 51	✅
64	D exchange freq t=1750	D	Cell 51	✅
65	D winning actions t=500	D	Cell 51	✅
66	D winning actions t=1750	D	Cell 51	✅