# ICM spectator calculation — 2026-09-06

Question: can an unchanged spectator stack lose ICM equity after two other players exchange chips?

These are synthetic valuations, not empirical results, poker strategies or ICM Trainer outputs. Three-player stacks are Hero/A/B = 20/60/20 in units of 1,000 chips. Prizes are 500/300/200 dollars. A and B transfer 20 units in either direction. The zero stack receives the already-earned third prize; the two survivors compete for the first two prizes. There is no simultaneous elimination.

## Reproduce

Download generate.py, verify.py, results.json, probability-sensitivity.csv, transfer-sensitivity.csv, outcomes.svg, hero.svg and MANIFEST.sha256 into one folder, together with this README. Python 3 standard library only. From that folder run:

```
python generate.py
python verify.py
```

The generator writes deterministic LF-terminated JSON/CSV beside itself. The verifier independently enumerates finishing orders and checks all published outputs. A fresh generator run should reproduce the distributed data bytes exactly. The manifest covers every file other than itself; SHA-256 is for artifact identity, not proof of mathematical correctness.

## Models and outputs

The generator uses the closed three-player ICM formula: first-place probability s_i/T; second-place probability sum over other j of (s_j/T)*(s_i/(T-s_j)); third is the remainder. Value is the probability-weighted prize sum. The verifier uses a separate permutation implementation. Both use exact rational arithmetic. JSON includes exact fractions and floating display equivalents; CSV carries exact rational values. Displayed article cents are independently rounded, so rounded rows may sum to $1,000.01 while exact values conserve $1,000.

Three endpoint states: 20/60/20, 20/80/0 and 20/40/40. Probability sensitivity: 101 p values from 0 to 1 in steps of 0.01; p means A wins. The exact spectator break-even probability 5/32 is solved algebraically and stored in JSON, not rounded to this grid. Transfer sensitivity: amounts 0 through 20 inclusive, equal 50/50 outcomes, three profiles (500/300/200;1000/0/0;300/300/300), 63 rows total. The equal-prize profile totals $900; conservation applies separately within each profile. Zero transfer, winner-take-all and equal prizes give no spectator change. Nonzero fair transfers with the original ladder give a positive average change.

No random sampling, iterations, seed, real hand records or proprietary ranges are involved. Amounts are artificial dollars and thousands of chips. The result does not value a call, fold or raise and excludes card distributions, action trees, dead-money/forced-bet adjustments, bounties, rake/fees, skill, future blinds, additional players and simultaneous eliminations. Before/after stack distributions must include all chips exactly once.

## Evidence and originality

Gilbert, The Independent Chip Model and Risk Aversion (2009), section3 and Theorem2: https://arxiv.org/pdf/0911.3100 — accessed2026-09-06. The paper supplies the ICM definition and the qualified expected-value theorem. This casebook's unequal-stack endpoint calculations, probability threshold and sensitivities are independently constructed examples, not a new theorem. The theorem requires a nontrivial chip-fair wager between exactly two participants, nonincreasing payouts, and some remaining player receiving less than the second-place prize. It does not promise a gain in every outcome. The 10% example is not chip-fair.

The two SVGs are original GTO Gecko diagrams created2026-09-06 from these calculations, with titles/descriptions. outcomes.svg is embedded beside a full HTML data table. hero.svg is the source of the article's WebP hero. No third-party imagery or user data is included. No product binary was tested for this work.