The open block: the same four tables, nothing at the centre
4,000,000 complete games at five, four, three and two players — no Prism, no neutral anchor, no guaranteed connection.
Studies 5–8 replay the campaign's second half with the wild centre tile removed. The first placement commits to a real colour, openings carry information, and a hand that cannot connect must pass — a move that does not exist in practice when the Prism anchors the table. Each study is paired deal-for-deal with its prism twin: identical shuffles, identical goals, one rule changed. (The head-to-head between the blocks is the synthesis report, next.)
Within the block, the prism studies' pattern repeats — a persona ladder that table size cannot bend, and a set of metrics that slide monotonically as real minds leave the table — plus two things the prism block never showed: a first-mover edge that is simply gone, and 1,727,760 forced passes.
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 ladder is rigid — not one crossing. Across all four sizes the standings never change: The Honest One, The Hunter, The Hoarder, The Spender, The Opportunist, The Liar, in that order, at every table. The Honest One wins 1.10× / 1.10× / 1.11× / 1.09× its fair share at 5 / 4 / 3 / 2 players; The Liar never climbs above 0.94×. Even the prism block's one rank swap — the Hoarder yielding to the Spender in the duel — does not happen here: the Hoarder slips from 1.06× to 1.02× but keeps its rung (Spender: 1.01×).
FindingThe first-mover edge is gone. Moving first is worth 0.99× / 0.99× / 0.99× / 0.99× fair share across the four sizes — within 0.7% of exactly fair everywhere, the flattest line in the lab. Without a wild centre, the opening placement buys no guaranteed connection; it only commits information first. What the prism block measured as a first-mover premium was never really about moving first.
FairnessSeats and goal colours stay flat everywhere: no seat deviates more than 2.6% of fair share in any study, no colour more than 60.2%. Remove the anchor and the game stays structurally fair.
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
Games with a forced pass
Average board area
Turn order at every size
Win rate by real-player turn order — k bars for k players. With the Prism these bars tilt toward the opener at every size; here they sit level with the fair line. The flattest bars in the lab.
Turn order
5 players
Turn order
4 players
Turn order
3 players
Turn order
2 players
Quirks at the middle of the table
Non-monotonePasses per game peak at 3 players — 0.40 / 0.45 / 0.48 / 0.40 across 5 / 4 / 3 / 2 players — while the share of games touched by a pass climbs monotonically (5.1% / 6.0% / 6.9% / 7.8%). The duel resolves the tension: its games are the likeliest to contain a lock-out, but at ~26 moves they are also the shortest — a stuck hand tends to stay stuck for several rounds, and short games leave fewer turns in which to stay locked out.
Non-monotoneShared victories peak at 3 players again — 1.78% / 2.05% / 2.10% / 1.81% — the same three-player quirk the prism block showed, at roughly three times the rate. Whatever makes the three-player table the friendliest to simultaneous finishes, it is a property of the table, not of the anchor.
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 | 32.3 | 30.8 | 28.6 | 25.8 |
| Median moves | 29 | 27 | 25 | 22 |
| Chain endings | 97.5% | 96.6% | 95.4% | 93.8% |
| Exhaustion endings | 2.5% | 3.4% | 4.6% | 6.2% |
| Shared victories | 1.78% | 2.05% | 2.10% | 1.81% |
| Wins ended by an action | 16.4% | 17.6% | 19.5% | 22.1% |
| Forced passes per game | 0.40 | 0.45 | 0.48 | 0.40 |
| Games with a forced pass | 5.1% | 6.0% | 6.9% | 7.8% |
| Stealth wins (goal unproven) | 26.4% | 29.5% | 33.3% | 38.2% |
| Avg opponents who knew the winner’s goal | 1.53 | 1.43 | 1.28 | 1.11 |
| Goal changes per game | 9.56 | 8.89 | 7.94 | 6.82 |
| Wins with an acquired goal | 61.8% | 59.8% | 57.2% | 54.4% |
| First mover vs fair share | 0.99× | 0.99× | 0.99× | 0.99× |
| Avg board area (cells) | 32.1 | 30.9 | 29.2 | 27.1 |
| Avg tiles left in deck at end | 14.0 | 15.2 | 17.0 | 19.5 |