Fixed-camper script equilibria in a finite collision-scoring game

Abstract

Chair Monarchy is a finite simultaneous-move scoring game: N players choose among N descending-value chairs for N regulation rounds, and a chair pays only when exactly one player chooses it. The highest chair A is tempting, visible, and blockable. This paper studies the exact 2P and 3P finite game, the exact 3P and 4P script normal forms against a fixed A-camper, a bounded 5P scheduled-pressure grammar, and several sensitivity probes. The load-bearing exact results mostly concern the fixed-camper script game, where P1 is committed to A and the challengers choose regulation scripts. The main conclusion is not that anti-camp play is mass rebellion. It is sparse pressure: one challenger blocks A in a small number of selected rounds while lower lanes stay clean enough for someone else, or sometimes the blocker themself, to harvest the win.

Evidence Ledger

1. Model

Players choose chairs simultaneously. Chair values are A=N, B=N-1, and so on down to 1. A player scores the chair value in a round only if no other player chose that chair. Otherwise every player on that chair scores 0 for the round.

This is a congestion/anti-coordination game with one conspicuously high-value resource. Chair A is both prize and pressure point: it is the best private target when uncontested, but also the cheapest public way to deny a camper.

Regulation lasts N rounds. If regulation ends with a unique leader, the game ends. If first place is tied, the game enters overtime for up to N more one-round cycles. After each overtime round, a newly unique leader wins immediately. If all N overtime rounds remain tied for first, the championship is awarded by tied-leader representation: count which players appeared in the overtime tied-first sets, and split the championship among the players with maximal representation.

Player utility is lexicographic: championship share first, then final margin over the best opponent, then raw score. This is why a player may prefer a blocking collision that preserves a draw over a lower chair that scores points but concedes the crown.

Formally, if terminal scores are x and terminal championship shares are s, player i ranks outcomes by the ordered tuple U_i = (s_i, x_i - max_{j != i} x_j, x_i) under lexicographic order.

Camp-A script game. For a fixed round count K, the Camp-A script normal form fixes P1 to play A in every regulation round and lets each challenger choose a full length-K regulation script before play. Unless explicitly stated otherwise, these script games are regulation-only terminal games: after the scripted rounds, tied leaders split championship share immediately using regulation scores. The full adaptive solver uses the overtime rule above; the script normal forms are a separate fixed-camper slice designed to isolate rebellion mechanics.

2. Exact 2P and 3P Full-Game Facts

The one-round stage facts already show how utility changes the game:

In 2P, the exact pure-SPE correspondence from the initial state has one terminal outcome: scores (0, 0), shares (1/2, 1/2), path AA -> AA -> AA -> AA. The chair A move is both an attack and a block, so equal-start 2P collapses into mutual A-collision and a representation split.

The intuition is tiny but important. If one player ever takes B while the other takes A, A scores enough to control the crown; if both take A, neither scores but neither concedes. With the representation overtime rule, the equilibrium logic keeps returning to mutual A because a sure shared crown outranks a lower-chair point that hands the opponent sole first place.

In 3P, the full finite pure-SPE solver returns 219 win/margin equilibrium records and 105 raw-regulation equilibrium records. The full adaptive correspondence is large, so the cleaner anti-camp statements below use script normal forms against a fixed camper. Freezing P1 to Camp A is not a claim that P1 must play that way in the full game; it is a controlled slice that asks when challengers can dethrone a committed throne-camper.

3. Exact Camp-A Script Theorems: 3P

Fix P1 to AAA. P2 and P3 each choose one length-3 script in {A,B,C}^3, giving 729 script pairs. Every pair is scored, and every unilateral script deviation is checked.

Theorem (3P Two-Block Lemma). In the 3P, 3-round script normal form against P1=AAA, every pure Nash equilibrium uses exactly two challenger A-blocks, and every such equilibrium excludes P1 from sole victory. The equilibrium set has 12 profiles: 6 P2 wins and 6 P3 wins.

This is now backed by an independent reduced best-response proof: the only mutual best-response cycle is between an ABB-type script and an ABC-type script with staggered A-blocks. The full table is in outputs/chair_monarchy_proofs.md.

The canonical witness is P2=ABB, P3=BAC, with path AAB -> ABA -> ABC and final scores (3, 4, 3). The lesson is already visible: the anti-camp pattern is not constant blocking. It is scheduled sacrifice plus lower-lane order.

4. Exact Camp-A Script Theorems: 4P

Fix P1 to AAAA. Each challenger has 4^4 = 256 scripts, so the normal form contains 256^3 = 16,777,216 challenger profiles. The exhaustive pass finds 4755 pure Nash equilibria.

P1 is excluded in 3060 NE (0.643533). Exactly two A-blocks occur in 4392 NE (0.923659). Every NE has zero multi-block rounds (4755 of 4755).

The P1-excluded equilibria split by block stratum as follows: 2700 are two-block equilibria (2232 sole challenger wins and 468 challenger splits), and 360 are three-block challenger splits. No four-block equilibrium excludes P1.

4P refinement. The strict 3P statement does not generalize as "all NE have two blocks": 4P also has 360 three-block split NE and 3 four-block split NE. The sharper exact statement is that every NE with a sole challenger winner is in the two-block stratum.

The zero multi-block invariant has a direct proof: if two challengers block A in the same round, one can move to an empty lower chair while leaving A blocked. For the sole-winner claim, b=0 and b=1 are ruled out by score bounds, while the b=3 and b=4 strata are ruled out by a reduced finite best-response table. See outputs/chair_monarchy_proofs.md for the proof notes.

5. 4P Species Taxonomy

The first taxonomy pass classifies all 4,755 4P Camp-A script NE by combinatorial path shape. The Birth/Viability index is metadata only; it does not influence classification.

Operationally, the main species are:

The new second-pass result is that the 432 strict-classifier other sole anti-camp rows are not a junk class. They form a clean candidate species: self_picket_farmer.

shared_harvest is intentionally not a pure two-block species: 468 rows are two-block challenger splits and 360 rows are three-block challenger splits. This is why its average A-block count is above two.

The self_picket_farmer cluster covers 432 rows. In every one, the winner blocks exactly once, farms B in the other three rounds, both A-block rounds are clean picket rounds, both unblocked rounds are ordered lower-lane rounds, and there are no lower-lane collisions.

Canonical 4P witnesses:

Picket Conveyor is the outsourced-labor form: the winner never blocks. Self-Picket Farmer is the self-financed form: the winner pays one A-block cost and then wins from B. Jammed Conveyor shows the failure mode: two A-blocks with lower-lane collision, and in this reduced game it never excludes P1. Stone Court is the frozen lock case, rare here but taxonomically useful.

6. Adaptive 4P Continuation Slices

The full adaptive 4P game from the initial state is not solved. We do have exact restricted continuation slices from selected public prefix states, with P1 still fixed to A and challengers adaptive.

These slices do not prove full adaptive 4P SPE. They do show that the script species are not merely decorative: after several scheduled prefixes, the remaining adaptive continuation still frequently excludes the camper. In other words, the species are compatible with some adaptive continuations; this does not imply that initial-state 4P SPE must follow these species.

7. Bounded 5P Scheduled Pressure

The full 5P script normal form against Camp A has 3125^4 challenger profiles and is not enumerated. The bounded pass studies a scheduled-pressure candidate grammar: each round chooses either no A-blocker or one challenger A-blocker, and all non-blockers are placed on distinct lower chairs by one of eight deterministic assignment policies (top, bottom, rotations, and player-favoring variants).

Important scope note: the grammar restricts which candidates are generated, but not which unilateral deviations are tested. Each surviving candidate is checked against all 3125 length-5 scripts for each challenger, so these are genuine full-script best-response witnesses discovered inside a restricted candidate family. The full normal form is still unsolved because candidates outside the grammar were not globally enumerated.

The grammar generates 21,834 candidates. 162 survive full unilateral deviation checks against all challenger scripts. Of those, 160 exclude P1 and 162 use exactly two A-blocks.

Representative Two-Beat Cartel witness:

These are exact full-deviation witnesses found inside the scheduled-pressure candidate family. They should be read as a bounded witness family, not as a complete 5P theorem.

8. Sensitivity and Robustness

8.1 Chair values

The 3P two-block result depends on the value schedule. Rebuilding the 3P Camp-A script normal form across value schedules gives:

Reading: sparse pressure is calibrated, not magic. Standard (3,2,1) sits in a useful zone. Very steep A values push equilibria toward three blocks; near-flat schedules can return the throne to P1 or to splits. The steep case is mechanically intuitive: if uncontested A pays too much, the camper's passive ceiling rises and challengers need more A-pressure just to bring that ceiling back into range.

8.2 Round count

The cleanest surprise is the 3P K=4 hole: exact enumeration finds no anti-camp NE there, even though K=2, K=3, and K=5 do have anti-camp equilibria. In 4P, by contrast, the zero-multi-block property holds for every NE at every K tested.

The reduced count-vector proof sketch suggests a parity-flavored explanation: with four rounds and values (3,2,1), the mutual best-response count cycles that survive either leave P1 tied at 6 or leave P1 sole at 6. The exact script enumeration confirms the gap, but a short conceptual proof remains open.

8.3 Trembling-hand noise

Noise probes are sampled, not exact. Each row below uses 4,000 Monte Carlo trials with P1 fixed to Camp A and challengers following a named script except for random-chair mistakes with probability epsilon.

Sparse pressure is efficient in exact play but fragile under mistakes. The canonical 4P Picket Conveyor gives P1 average share 0 at eps=0, but already leaks to 0.24075 at eps=0.05. The eps=0.5 endpoint is essentially random play and is less theoretically meaningful; the small-noise headline is that sparse equilibria require near-exact coordination. Redundant blocking is less efficient as a win mechanism but more robust as denial.

9. Proof Notes

The accompanying proof appendix is generated at outputs/chair_monarchy_proofs.md. Its current proof targets are:

• 3P Two-Block Lemma: finite best-response proof over reduced script types.
• 3P K=4 anti-camp gap: exact enumeration plus reduced count-vector proof sketch.
• No multi-block rounds against fixed Camp A: direct local deviation proof.
• 4P sole anti-camp winners require two A-blocks: score bounds plus reduced finite tables for b=3 and b=4.

The important honesty flag is the last item: it is already exact inside a small reduced table, but the next paper-quality improvement is to replace that reduced table with explicit deviation certificates for each high-block case.

10. Sparse Pressure Principle

Sparse Pressure Principle. In the exact 3P/4P Camp-A script games and the bounded 5P scheduled-pressure grammar, successful anti-camp play is concentrated in a small number of single-blocker A attacks paired with collision-free lower-lane scoring. Over-blocking tends to produce mutual-hostage splits rather than a successor monarch.

This is a principle, not a general theorem over all N. The evidence is strong enough to guide search: find the minimum A-pressure needed to lower the camper's ceiling, then preserve lower-lane order. The 4P taxonomy sharpens the principle: anti-camp success comes in at least two clean forms, outsourced (picket_conveyor) and self-financed (self_picket_farmer).

Taxonomically, picket_conveyor is sparse pressure with outsourced blocking labor, self_picket_farmer is sparse pressure paid partly by the winner, and the bounded 5P two_beat_cartel witnesses are sparse pressure extended through two scheduled A-beats in a larger lower court.

11. Limits and Open Problems

• Full adaptive 4P from the initial state remains unsolved.
• Full 5P script normal form remains unsolved.
• Mixed equilibria are not analyzed.
• The 5P results are full-deviation witnesses from a bounded candidate family, not a global enumeration.
• The noise results are Monte Carlo probes, not equilibrium statements.
• A closed-form generic-N theorem is still missing.

Natural next questions:

• Can the reduced b=3 and b=4 4P tables be replaced by explicit deviation certificates?
• Can self_picket_farmer be given a theorem-level operational definition across N?
• What is the smallest adaptive 4P state space that preserves the 4P species split?
• Does the 3P K=4 anti-camp gap have a simple parity proof?
• Which 5P grammar restrictions can be relaxed without losing tractability?

Reproducibility

Run the full current pass in this order:

python3 -m unittest -v
python3 analyze_chair_monarchy.py
python3 simulate_strategies.py --games 200
python3 rabbit_holes.py
python3 analyze_four_player.py
python3 classify_four_player_species.py
python3 analyze_four_player_species_deep_dive.py
python3 analyze_five_player.py
python3 phase2_dives.py
python3 synthesize_chair_monarchy_paper.py

The main generated paper is this file. Companion reports include:

• outputs/chair_monarchy_report.md
• outputs/camp_a_4p_report.md
• outputs/camp_a_5p_report.md
• outputs/chair_monarchy_proofs.md
• outputs/campa4p_sole_anticamp_deep_dive_report.md
• outputs/phase2_dives_report.md

Companion proofs

Chair Monarchy Proof Notes

This appendix records the current proof state behind the generated paper. The goal is not to hide computation behind rhetoric: each theorem is tagged by proof type, and the finite-table steps are stated as finite lemmas rather than as magic.

Unless otherwise stated, script-game proofs use the regulation-only Camp-A script normal form: P1 is fixed to A in every scripted round, challengers precommit to regulation scripts, and tied regulation leaders split championship share immediately. The full adaptive game uses the separate overtime rule from the main paper.

A. 3P Two-Block Lemma

Statement. In the 3-player, 3-round script normal form against P1=AAA, every pure Nash equilibrium uses exactly two challenger A-blocks, and every pure Nash equilibrium excludes P1 from sole victory.

Proof type. Independent finite best-response proof over reduced script types. The exhaustive CSV is outputs/two_block_lemma_3p.csv, but the argument below does not rely on trusting the full 729-profile table.

With P1 fixed to A, a single round has the following payoff table for the two challengers. Entries are (P1, row challenger, column challenger).

row/col	A	B	C
A	(0,0,0)	(0,0,2)	(0,0,1)
B	(0,2,0)	(3,0,0)	(3,2,1)
C	(0,1,0)	(3,1,2)	(3,0,0)

Since regulation payoffs add across rounds and P1's script is fixed, a challenger's pure best response can be computed type-by-type up to permuting round labels. The score triple is (P1, responder, opponent) for a representative placement.

Opponent type	Responder best response type(s)	Representative score
AAA	BBB	(0,6,0)
AAB	BBC	(3,5,2)
AAC	BBB	(3,6,1)
ABB	BAC or BCA	(3,3,4)
ABC	BAB	(3,4,3)
ACC	BAB or BBA	(3,4,2)
BBB	ACC, CAC, or CCA	(6,2,6)
BBC	ACB or CAB	(6,3,5)
BCC	ABB	(6,4,4)
CCC	AAB, ABA, or BAA	(3,2,3)

Mutual best response can occur only between an ABB-type script and an ABC-type script, with their A-blocks in different rounds. Counting these gives: choose the winner, choose the winner's A-block round, then choose the other challenger's A-block round among the remaining two. The B/C placement is then forced: in the winner's A-block round, the other challenger must take C, and in the remaining unblocked round they must take B, otherwise a lower-lane collision creates a profitable deviation. That is 2 * 3 * 2 = 12 equilibria. Their scores are permutations of (3,4,3), so P1 is never the sole winner. Each such profile contains exactly two challenger A-plays. This proves the lemma.

B. 3P Four-Round Anti-Camp Gap

Statement. In the 3-player, 4-round script normal form against P1=AAAA with chair values (3,2,1), no pure Nash equilibrium excludes P1 from first place.

Proof type. Exact enumeration plus reduced count-vector proof sketch over opponent count vectors (A,B,C). The exact enumeration gives 48 pure NE, with zero P1-excluded equilibria.

Opponent counts	Responder best-response counts	Representative score
(0,0,4)	(2,2,0)	(6,4,4)
(0,1,3)	(2,2,0)	(6,4,5)
(0,2,2)	(2,2,0)	(6,4,6)
(0,3,1)	(2,1,1)	(6,3,7)
(0,4,0)	(2,1,1)	(6,1,6)
(1,0,3)	(2,2,0)	(3,4,3)
(1,1,2)	(1,3,0)	(6,6,4)
(1,2,1)	(1,2,1)	(6,5,5)
(1,3,0)	(1,1,2)	(6,4,6)
(2,0,2)	(1,3,0)	(3,6,2)
(2,1,1)	(1,3,0)	(3,6,3)
(2,2,0)	(1,2,1)	(3,5,4)
(3,0,1)	(0,4,0)	(3,8,1)
(3,1,0)	(0,3,1)	(3,7,2)
(4,0,0)	(0,4,0)	(0,8,0)

The only mutual count cycles are (1,3,0) <-> (1,1,2) and (1,2,1) <-> (1,2,1). At the count-vector level, the first gives P1 a share of first place; the second gives P1 sole first place. The exact script-level enumeration then checks all realizations of those cycles and finds that every surviving pure NE keeps P1 at least tied for first. Therefore no pure equilibrium excludes P1. This explains the K=4 hole in the round-count sensitivity table.

C. No Multi-Block Rounds Against a Fixed Camper

Statement. In any fixed Camp-A script normal form with N players, positive lower-chair values, and lexicographic win/margin/score utility, no pure Nash equilibrium can contain a round in which two or more challengers play A.

Proof type. Direct local deviation.

Suppose a candidate pure NE has a round t with m >= 2 challengers playing A. Pick one of those challengers. If that player changes only round t from A to an empty lower chair, then at least one challenger remains on A with P1, so P1 is still blocked in that round. An empty lower chair exists because there are N-1 lower chairs and only N-1-m challengers using lower chairs in that round. The deviator now scores a positive lower-chair value in round t; no other player's score changes. Championship share weakly improves, and the margin component strictly improves because own score rises while the best opponent score is fixed. This profitable deviation contradicts Nash equilibrium.

This proves the observed multi_block_rounds == 0 invariant directly; it is not merely a computational accident.

D. 4P Sole Anti-Camp Winners Require Two A-Blocks

Statement. In the 4-player, 4-round script normal form against P1=AAAA, every pure NE with a sole challenger winner has exactly two challenger A-blocks.

Proof type. Score bounds plus reduced finite best-response tables. This is theorem-level inside the reduced tables, but the remaining paper-polishing target is to replace those tables with compact deviation certificates.

Let b be the number of blocked A-rounds. By the no-multi-block theorem, b is also the total number of challenger A-plays. If b=0, P1 scores 16, while any challenger can score at most 12. If b=1, P1 scores 12; the blocker can score at most 9, and a non-blocking challenger can score at most 12. Thus b <= 1 cannot produce a sole challenger winner.

The high-block cases reduce to the following finite best-response facts.

b	A-block distribution	Pure-NE terminal form	Winner form
3	(3,0,0)	(4,x,5,5), x in {2,3} up to permutation	two challengers split
3	(2,1,0)	(4,5,6,6) up to permutation	two challengers split
3	(1,1,1)	(4,5,6,6) up to permutation	two challengers split
4	(4,0,0)	(0,0,0,0)	full draw
4	(3,1,0)	no mutual best response	none
4	(2,2,0)	no mutual best response	none
4	(2,1,1)	no mutual best response	none

Hence b=3 can exclude P1, but only through challenger splits; b=4 only survives as complete lock. Since b <= 1 cannot dethrone P1 and b >= 3 cannot produce a sole challenger winner, every sole challenger winner must lie in the b=2 stratum. The exact 4P enumeration then identifies the two major two-block families: outsourced picket_conveyor and self-financed self_picket_farmer.

E. Current Proof Targets

1. Replace the 4P b=3 and b=4 reduced tables with explicit deviation certificates for each A-block distribution.
2. Turn self_picket_farmer from a discovered finite cluster into a short operational theorem.
3. Find the smallest adaptive 4P state space where the script taxonomy survives subgame-perfect reasoning.
4. Decide whether the 3P K=4 gap has a conceptual parity explanation or is best left as a finite best-response fact.