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Tops-only aliasing at the Exchange (worked example) ​

What this page is for ​

This is research documentation: a concrete, hand-checkable demonstration that a tops-only view of the mat is not a sufficient state at the Poisoned-Gift Exchange. Two distinct true positions present the same tops to A yet require opposite optimal Gifts; choosing the wrong one flips a provable +1 win to a −1 loss. This is a representation exhibit (what the agent is allowed to see), distinct from the deployment exhibits on the home page (frozen line vs re-solving). It makes the front-page claim — the board snapshot alone is not a sufficient state representation — concrete.

Verified (development tree only — public psdg has no private/ directory): every value below comes from the bare reference solver (solve_exchange, solve_from_position, apply_exchange, evaluate) — not from the facing-blind action model that first surfaced the conflict. The independent re-derivation lives at private/psdg/ml/verify_exhibit.py.

The opening ​

Both positions are reached from the YouTube-demo opening:

  • Board (tops, ascending): 2, 2, 3, 4, 5, 6
  • A Red Crystal (top, facing): (2, 6)
  • B Red Crystal (top, facing): (1, 2)

Each player drafts 3 dice (six dice total), then the Exchange (Poisoned Gift) resolves.

The two positions ​

Each die is shown as (top, facing). The two positions are indistinguishable to an agent that sees only tops: A's tops are (3, 4, 5) and B's are (2, 2, 6) in both.

A crucibleB crucibleA topsB tops
Position 1(3, 1), (4, 6), (5, 3)(2, 4), (2, 4), (6, 5)3, 4, 52, 2, 6
Position 2(3, 6), (4, 1), (5, 6)(2, 1), (2, 6), (6, 5)3, 4, 52, 2, 6

The facings differ — and facings are exactly what Phase 2 (the Tumble) scores. A tops-only agent has thrown that information away.

The conflict ​

Both positions are won for A (+1) under optimal play, confirmed by two independent solver routes (solve_exchange and solve_from_position). But the optimal Gift is opposite, and cross-applying the other position's optimal Gift loses the game outright:

Position 1 — A crucible (3,1),(4,6),(5,3); B crucible (2,4),(2,4),(6,5)

  • Gift top-3 (facing 1) secures +1 ← optimal
  • Gift top-4 (facing 5) secures −1 ← Position 2's optimal Gift; B best-responds (0,3); Phase 1 1–1, Phase 2 0–0 → loss
  • Swing: +1 → −1 (regret 2)

Position 2 — A crucible (3,6),(4,1),(5,6); B crucible (2,1),(2,6),(6,5)

  • Gift top-4 (facing 1) secures +1 ← optimal
  • Gift top-3 (facing 1) secures −1 ← Position 1's optimal Gift; B best-responds (0,1); Phase 1 1–1, Phase 2 1–2 → loss
  • Swing: +1 → −1 (regret 2)

A tops-only agent sees one observation, must commit to one Gift, and is therefore wrong on at least one of the two positions — and being wrong costs the whole game, not a margin.

Mechanism (one line): keep the dice whose facing is 6 — they turn to gold at the Tumble. Which dice those are depends on facings, which the tops-only view cannot see.

Draft paths (reconstructable from scratch) ​

So a skeptic can rebuild each position from the opening (A moves first; entries are (top, facing) Twists):

Position 1

  1. A: (5, 3)
  2. B: (6, 5)
  3. A: (4, 6)
  4. B: (2, 4)
  5. A: (3, 1)
  6. B: (2, 4)

Position 2

  1. A: (5, 6)
  2. B: (6, 5)
  3. A: (4, 1)
  4. B: (2, 6)
  5. A: (3, 6)
  6. B: (2, 1)

How far this generalizes ​

This is not a one-off curiosity of the demo board:

  • Enumerated (demo board): of 14 tops-only Exchange groups, 4 have no common optimal Gift — an irreducible regret floor of 0.021 under win/loss. (public ml/aliasing_exchange.py)
  • Cross-opening: the conflict appears in ≈79% of 120 random openings; the demo board is a typical-to-mild case. (public ml/run_cross_opening.py)
  • Objective-invariant: the conflict footprint is identical under win/loss, score-margin, and lexicographic (win-then-margin) objectives — margin-only conflicts = 0. The failure source is representation (dropped facings), not objective coarseness. (private/psdg/ml/run_lexicographic.py, development tree)

So the effect is rare per position but maximal where it occurs (a full win → loss), and it is general across openings rather than special to this board.

Trained-agent companion: a tabular learner actually pays this cost — a full-state learner generalises to optimal play while a tops-only learner loses across openings, at both Draft and Exchange. See Learning the wrong state.

What this is not ​

  • Not hidden information. After setup PSDG is perfect-information; the facings are on the table. The agent simply isn't fed them when its state is a tops-only summary — aliasing, not poker-style secrecy. See FAQ — partial observability.
  • Not the deployment headline. The frozen-line-vs-re-solving result (~8.5% vs ~5.7%) is a protocol claim; this is a representation claim. They are complementary, not the same argument.
  • Not "tops-only agents lose." Outside the conflicting subset a tops-only Gift is fine; the point is that a sufficient state must include facings at the Exchange.