The prism block: one game, four table sizes
4,000,000 complete games at five, four, three and two players — every table anchored by the Prism.
Studies 1–4 played one million games each at a shrinking table. Every empty chair becomes an imaginary player: dealt a goal and a hand like anyone else, but never acting and never winning — goals that sit in the rotation as pure information. So these four corpora ask one question four ways: what does removing real minds from the same game do to it?
Two kinds of answer showed up. Some numbers refuse to move — they are properties of the rules, not the table. Everything else slides monotonically with player count, in one direction, across all four studies.
The four studies
What table size cannot touch
Normalize every persona's win rate by its table's fair share (1/5, 1/4, 1/3, 1/2) and the four studies collapse onto one ladder. Order fixed by the five-player standings — a crossing would be visible, not resorted away.
Personas
Persona win rate as a multiple of fair share, by table size
FindingThe hierarchy is structural. The Honest One wins 1.14× / 1.15× / 1.16× / 1.15× its fair share at 5 / 4 / 3 / 2 players — never below 1.14×. The Liar never climbs above 0.84×. Straightforward play beats sustained bluffing by roughly the same margin whether the bluff has four audiences or one — consistent with the pre-registered expectation that bot-vs-bot deception pays nobody, because there are no human minds to deceive.
The one crossingThe Hoarder’s patience stops paying in the duel. At five and four players the Hoarder edges the Spender (1.03× vs 1.01× at five); by three players they have swapped, and at two the Hoarder slips to 0.99× — the only persona whose relative standing moves with table size. Hoarding tempo is a bet that the table stays busy long enough to cash it.
FairnessSeats and goal colours stay flat everywhere: no seat deviates more than 1.6% of fair share in any study, no colour more than 60.2%. The structural-fairness claim of the single-study reports survives the size sweep intact.
What shrinks with the table
Every metric below moves in one direction as real players leave — no reversals, across three million games of separation. Left to right in each tile: five players down to the duel.
Mean moves per game
Exhaustion endings
Wins ended by an action
Stealth wins
Goal changes per game
Wins with an acquired goal
First mover vs fair share
Average board area
Turn order at every size
Win rate by real-player turn order — k bars for k players, imaginary seats hold goals, never turns. The edge is real at every size, and largest where the most opponents wait behind you.
Turn order
5 players
Turn order
4 players
Turn order
3 players
Turn order
2 players
The three-player quirk
Non-monotoneOne number breaks the pattern: shared victories peak at three players — 0.51% / 0.59% / 0.66% / 0.60% across 5 / 4 / 3 / 2 players, with the maximum at 3. Every shared victory in the block is an exhaustion tie, and the two-imaginary-seat table is where equal-length chains most often survive to the last tile together.
The full table
Every compared metric, all four studies. Per-study detail, methodology and chart provenance live in the individual reports.
Appendix
| Metric | 5 players | 4 players | 3 players | 2 players |
|---|---|---|---|---|
| Games | 1,000,000 | 1,000,000 | 1,000,000 | 1,000,000 |
| Mean moves per game | 22.3 | 21.4 | 20.3 | 18.7 |
| Median moves | 19 | 18 | 17 | 16 |
| Chain endings | 99.3% | 99.0% | 98.6% | 98.1% |
| Exhaustion endings | 0.7% | 1.0% | 1.4% | 1.9% |
| Shared victories | 0.51% | 0.59% | 0.66% | 0.60% |
| Wins ended by an action | 23.1% | 24.5% | 26.6% | 28.7% |
| Actions, share of all moves | 27.2% | 26.8% | 26.2% | 25.4% |
| Stealth wins (goal unproven) | 46.1% | 48.9% | 51.7% | 55.3% |
| Avg opponents who knew the winner’s goal | 0.91 | 0.85 | 0.79 | 0.70 |
| Goal changes per game | 5.54 | 5.23 | 4.83 | 4.33 |
| Wins with an acquired goal | 46.7% | 45.5% | 44.4% | 43.2% |
| First mover vs fair share | 1.05× | 1.03× | 1.02× | 1.02× |
| Avg board area (cells) | 24.8 | 24.1 | 23.1 | 21.7 |
| Avg tiles left in deck at end | 23.7 | 24.7 | 26.0 | 27.8 |