{ "formatVersion": 1, "study": { "id": "decision-method-sensitivity-20260916", "title": "Arithmetic or geometric? 1,771 weighting runs on one fictional decision", "studyDate": "2026-09-16", "stepPercent": 5, "criteria": [ { "id": "conversation", "name": "Conversation" }, { "id": "preparation", "name": "Low preparation" }, { "id": "weather", "name": "Weather resilience" }, { "id": "novelty", "name": "Novelty" } ], "options": [ { "id": "discussion", "name": "Indoor discussion", "scores": { "conversation": 5, "preparation": 4, "weather": 5, "novelty": 2 } }, { "id": "walk", "name": "Neighborhood photo walk", "scores": { "conversation": 4, "preparation": 5, "weather": 1, "novelty": 5 } }, { "id": "workshop", "name": "Postcard workshop", "scores": { "conversation": 2, "preparation": 3, "weather": 5, "novelty": 4 } } ], "status": "Deterministic sensitivity analysis using fictional inputs. No people, products, venues, or observed outcomes were evaluated.", "question": "How often do an arithmetic weighted mean and a geometric weighted mean select different leaders when the same four weights vary across their full five-point grid?" }, "methods": { "arithmetic": { "formula": "sum(score × weight / 100)", "interpretation": "Compensatory: a strong score can offset a weak score." }, "geometric": { "formula": "exp(sum((weight / 100) × ln(score)))", "interpretation": "Multiplicative: a low score exerts a stronger penalty. All scores in this study are greater than zero." }, "winnerRule": "The option with the largest method score is the winner. Equal method scores within 1e-10 are recorded as a tie." }, "grid": { "constraint": "Four integer percentage weights sum to 100.", "stepPercent": 5, "includesZero": true, "includesOneHundred": true }, "summary": { "runCount": 1771, "winnerCounts": { "arithmetic": { "discussion": 817, "walk": 703, "workshop": 204, "tie": 47 }, "geometric": { "discussion": 869, "walk": 587, "workshop": 309, "tie": 6 } }, "agreement": { "sameSingleWinner": 1576, "differentSingleWinner": 144, "atLeastOneTie": 51 } }, "limitations": [ "The option scores and decision context are fictional, so the winner counts do not estimate behavior or outcomes in the world.", "The grid gives each five-point weight combination one run. That is a coverage choice, not a probability distribution over real preferences.", "Multiplying one-to-five preference scores assumes ratios that the illustrative scales do not establish. The geometric result is a method stress test, not a recommendation to use that model.", "The run holds scores fixed. Different evidence, criteria, thresholds, correlations, or uncertainty models can change every result." ], "version": { "number": "1.0", "published": "2026-09-16", "corrections": [] }, "license": { "name": "CC0 1.0 Universal", "url": "https://creativecommons.org/publicdomain/zero/1.0/", "appliesTo": "Original generated run data in runs.json and runs.csv.", "excludes": "Linked third-party source material, names and marks, site design, article prose, and chart presentation." }, "citation": { "author": "Awesome Patel", "title": "Arithmetic or geometric? 1,771 weighting runs on one fictional decision", "publisher": "Steady Ideas", "year": 2026, "url": "https://steadyideas.com/guides/arithmetic-geometric-weight-sensitivity" }, "runs": [ { "run": 1, "weights": { "conversation": 0, "preparation": 0, "weather": 0, "novelty": 100 }, "arithmetic": { "scores": { "discussion": 2, "walk": 5, "workshop": 4 }, "winner": "walk" }, "geometric": { "scores": { "discussion": 2, "walk": 5, "workshop": 4 }, "winner": "walk" }, "sameWinner": true }, { "run": 2, "weights": { "conversation": 0, "preparation": 0, "weather": 5, "novelty": 95 }, "arithmetic": { "scores": { "discussion": 2.15, "walk": 4.8, "workshop": 4.05 }, "winner": "walk" }, "geometric": { "scores": { "discussion": 2.09376, "walk": 4.613404, "workshop": 4.044879 }, "winner": "walk" }, "sameWinner": true }, { "run": 3, "weights": { "conversation": 0, "preparation": 0, "weather": 10, "novelty": 90 }, "arithmetic": { "scores": { "discussion": 2.3, "walk": 4.6, "workshop": 4.1 }, "winner": "walk" }, "geometric": { "scores": { "discussion": 2.191916, "walk": 4.2567, "workshop": 4.090261 }, "winner": "walk" }, "sameWinner": true }, { "run": 4, "weights": { "conversation": 0, "preparation": 0, "weather": 15, "novelty": 85 }, "arithmetic": { "scores": { "discussion": 2.45, "walk": 4.4, "workshop": 4.15 }, "winner": "walk" }, "geometric": { "scores": { "discussion": 2.294674, "walk": 3.927575, "workshop": 4.136152 }, "winner": "workshop" }, "sameWinner": false }, { "run": 5, "weights": { "conversation": 0, "preparation": 0, "weather": 20, "novelty": 80 }, "arithmetic": { "scores": { "discussion": 2.6, "walk": 4.2, "workshop": 4.2 }, "winner": "walk+workshop" }, "geometric": { "scores": { "discussion": 2.402249, "walk": 3.623898, "workshop": 4.182558 }, "winner": "workshop" }, "sameWinner": false }, { "run": 6, "weights": { "conversation": 0, "preparation": 0, "weather": 25, "novelty": 75 }, "arithmetic": { "scores": { "discussion": 2.75, "walk": 4, "workshop": 4.25 }, "winner": "workshop" }, "geometric": { "scores": { "discussion": 2.514867, "walk": 3.343702, "workshop": 4.229485 }, "winner": "workshop" }, "sameWinner": true }, { "run": 7, "weights": { "conversation": 0, "preparation": 0, "weather": 30, "novelty": 70 }, "arithmetic": { "scores": { "discussion": 2.9, "walk": 3.8, "workshop": 4.3 }, "winner": "workshop" }, "geometric": { "scores": { "discussion": 2.632764, "walk": 3.085169, "workshop": 4.276938 }, "winner": "workshop" }, "sameWinner": true }, { "run": 8, "weights": { "conversation": 0, "preparation": 0, "weather": 35, "novelty": 65 }, "arithmetic": { "scores": { "discussion": 3.05, "walk": 3.6, "workshop": 4.35 }, "winner": "workshop" }, "geometric": { "scores": { "discussion": 2.756189, "walk": 2.846627, "workshop": 4.324924 }, "winner": "workshop" }, "sameWinner": true }, { "run": 9, "weights": { "conversation": 0, "preparation": 0, "weather": 40, "novelty": 60 }, "arithmetic": { "scores": { "discussion": 3.2, "walk": 3.4, "workshop": 4.4 }, "winner": "workshop" }, "geometric": { "scores": { "discussion": 2.8854, "walk": 2.626528, "workshop": 4.373448 }, "winner": "workshop" }, "sameWinner": true }, { "run": 10, "weights": { "conversation": 0, "preparation": 0, "weather": 45, "novelty": 55 }, "arithmetic": { "scores": { "discussion": 3.35, "walk": 3.2, "workshop": 4.45 }, "winner": "workshop" }, "geometric": { "scores": { "discussion": 3.020668, "walk": 2.423447, 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