---
id: CB-WP-0025
kind: product
title: "Could we have won: a path out of a lost game, and how hard the game actually is"
status: active
state_hub_workstream_id: "a866982f-94e7-432b-b0d5-2fefb781a574"
---

# Purpose

```
structural tier  L   (creates a new capability — a search over game state,
                      and a measurement the engine does not currently take;
                      both produce numbers ground-game will tune against)
chaos            d8 = 6  → no override
declared tier    L
```

Declaration 8 of chaos window 2. Tier L: separate survey, **adversarial
review**, ADR, then spec, then code.

## Two remarks, and why they are one pass

> *"I had a game where we lost and in this case I would have liked to know
> if and how we could have won… the best path is not computable I guess so
> a path to win is fine."*

> *"Do we have difficulty estimations? If so we should show them. It will
> help tuning the game. I felt it was too easy but then we lost, so who
> knows."*

They are the same machine asked two questions. *Was this game winnable?*
is a search from a recorded state. *How hard is this game?* is that search
run over many deals and counted. Building the second without the first
gives a win-rate with no witness; building the first without the second
gives one anecdote per game.

**"I felt it was too easy but then we lost, so who knows" is the finding.**
The maintainer cannot calibrate the game from play, and that is precisely
the gap `ground-game` is currently blocked in: **GROUND-WP-0005**
*(Difficulty tiers (threshold ratio) and optional Pressure dial)* is active
with **both its tasks in `wait`**. Tiers cannot be set without a measured
baseline, and clay-borg is the thing that can measure. This pass is what
unblocks it — which is the design-instrument aspect CB-WP-0022 is in the
middle of stating, arriving with a concrete demand.

## What already exists, so the survey does not re-find it

- **The state is replayable.** `cb-game-runtime` records sessions as
  scenarios; `replay.rs` and `make replay-test` already re-run them.
  A search does not need new persistence.
- **The move space is enumerable.** `legal_commands` exists and, since
  CB-WP-0023, is narrow enough to be worth trusting — SOLVE is offered
  only where it can act, so the branching factor is real rather than
  inflated by inert moves.
- **Bots exist.** `games/ground/src/bot.rs` has `GreedyPolicy` and
  `RandomPolicy`, wired through `bot_policy` (`table.rs:208`). A win rate
  over N seeds is reachable with what is already there — the question is
  whether that number *means* anything, which is the survey's problem.
- **The threshold is public.** `OutcomeView.total` / `.threshold` /
  `.group_success`. Difficulty has a denominator already.

## What makes this hard, and must not be waved through

**The game is not perfect-information and the search must respect that.**
A path computed with the deck known is a path the players could never have
found. `view.rs` hides the deck, other seats' hands, and face-down
selections *by rule* (GR-S02/S04, GR-R02/R04). A retrospective solver
running on `GroundState` sees all of it. So the ADR must decide, in
words, **which of these three the tool answers**:

- *was this deal winnable by an omniscient player* — cheap, honest,
  and answers a question nobody asked;
- *was it winnable from what the seats could see* — the question actually
  asked, and the expensive one;
- *did a reasonable line exist* — a bounded search from the losing seat's
  information, which may be the only affordable honest answer.

Getting this wrong produces a feature that tells the maintainer he could
have won by playing a card he had no way to know was there. **That is
worse than not shipping it.**

**And a difficulty number is a claim about a distribution.** One win rate
over one bot policy over N seeds is not "the difficulty"; it is that
policy's win rate. Whatever the spec adopts must name its policy, its N,
and its seed range, or `ground-game` will tune tiers against a number
whose meaning drifts the next time a bot improves.

## Task: survey

```task
id: CB-WP-0025-T01
status: done
priority: high
state_hub_task_id: "556cfd24-5992-4cc0-be90-0b60e989b5bb"
```

`research/CB-RES-0008-*.md`, with `tier: L` and the chaos roll recorded
(`loop-lint` checks both).

Per §Step 1 the survey is done when it can name a **benchmark-to-beat**
per dimension — a number or a reproducible comparison, not an impression.

- **Retrospective solvers in games with hidden information.** The prior art
  is real and should be named: determinized search (perfect-information
  Monte Carlo) and its known failure — *strategy fusion*, where a
  determinizing solver claims lines that require knowing which world it is
  in. That failure is exactly the trap in §What makes this hard. Bridge
  and Skat post-mortem tools are the closest analogues; poker solvers are
  the well-studied case and the wrong shape.
- **"A path to win" as a product, not a proof.** The maintainer already
  conceded optimality (*"the best path is not computable I guess"*). So
  the target is a **witness**: one concrete line of play that reaches
  `group_success`, or a defensible *no line found within bound B*. Name
  what a witness must carry to be checkable.
- **Difficulty as a measured quantity in co-operative games.** Pandemic and
  its relatives set difficulty by a dial with a published win rate. The
  benchmark-to-beat is: can we produce a win rate whose confidence
  interval is tight enough to distinguish two threshold settings?
- **Cost.** Search over an event-sourced aggregate with full `validate` on
  every branch has a per-node price. Measure it on our machine, on our
  scenarios — the runnable-baseline option applies here, since a search
  that cannot finish while the player is still looking at the page is a
  different feature.

**Done 2026-08-05.**
[CB-RES-0008](../research/CB-RES-0008-could-we-have-won.md), with a
runnable baseline (`games/ground/examples/difficulty-baseline.rs`).

> **Everything this record originally claimed was withdrawn by T02 the
> same day.** Kept as a pointer rather than rewritten, because a claim
> retracted silently is how three earlier wrong premises survived
> (ADR-0012 D5).

**Claimed:** a greedy bot wins 200/200 at five and six seats, so the game
is too easy there; the points-to-threshold ratio rises with seat count and
explains the curve; `legal_commands` costs 112–161 µs, so exhaustive
search is out; and all of this explains the maintainer's report.

**Withdrawn:** the win rate is `GreedyPolicy`'s, not the game's (C4 —
`FirstLegal` scores 0% on the same deals); the ratio is identical at 3p
and 4p, which differ by 12.5 points (C2); the cost figure was wrong by
30–50× and exhaustive search is affordable (C1, C6); and the maintainer's
losses were 3-player games on the pre-ruling deal (C5).

**What survives:** the harness exists and runs, the 6/9/12 arithmetic is
right against `Problems.csv`, and the branching widths (4.7–9.1) hold. See
[CB-RES-0008](../research/CB-RES-0008-could-we-have-won.md) for the
corrected text and
[the response](../history/260805-could-we-have-won-response.md) for the
full accounting.

**Prior art named the trap** — determinized search suffers *strategy
fusion* (Frank, Basin & Matsubara 1998) — **and T03 then established it
does not apply here.** After the game there is one world, so a line found
in it is executable in it. Fusion is an obstacle to a *playing* engine,
which this is not.

## Task: adversarial review

```task
id: CB-WP-0025-T02
status: done
priority: high
state_hub_task_id: "bdd9fc97-4706-4d8e-b286-91ff53339680"
```

Tier L requires it. Exactly one round: challenge, then response, trail in
`history/`, unpolished. Require an attempt at:

- **that the honest version is unaffordable** — that a search respecting
  the information rule is too expensive or too weak to find anything, so
  the shipped tool will quietly become the omniscient one with a
  reassuring label;
- **that a witness misleads more than it helps** — being shown a line that
  needed a card you could not know about teaches a wrong lesson about the
  game, and the tool would be better refusing to answer;
- **that the difficulty number is a bot benchmark wearing a difficulty
  costume**, and `ground-game` will tune the game against our bot rather
  than against play;
- **that this is CB-WP-0022's job** — the design instrument is being built
  right now, and a difficulty measurement is a finding-producing tool. The
  strongest counter is that the register records findings and this
  *produces* them, but the reviewer should press whether that is a
  distinction worth a separate capability.

**Done 2026-08-05.** Trail:
[challenge](../history/260805-could-we-have-won-challenge.md),
[response](../history/260805-could-we-have-won-response.md).
Separate agent, as in CB-WP-0022.

**Six of seven conceded. The survey's headline finding is WITHDRAWN.**

**C4 kills it and the reviewer ranked it fourth.** A `FirstLegal` policy —
`legal[0]`, no heuristic — scores **0% at five and six seats** where greedy
scores 100%, and **77.5% at two** where greedy scores 66%. Two
unsophisticated agents span the whole range, so *"too easy at 5–6 seats"*
was about `GreedyPolicy`, not GROUND.

**C1**: the node cost was wrong by 30–50× — the timer bracketed whole
games — and **the tell was in my own output**, falling as branching rose.
**C6**: exhaustive search is *not* out, which changes T03's premise.
**C3**: the finding failed the admissibility rule this project wrote nine
hours earlier. **C2**: 3p and 4p share deal, threshold and ratio and differ
by 12.5 points. **C5**: `lib.rs:2487` contradicts the "explains the
maintainer" claim — his losses were 3-player on the pre-ruling deal,
unwinnable at 6 against 7.

**T06 was pointed at GROUND-WP-0005, which is blocked on this number.**
Sending it would have been the fifth wrong premise to reach ground-game.
**Both tier-L reviews here have now caught a false headline that every
gate passed.**

## Task: decide

```task
id: CB-WP-0025-T03
status: done
priority: high
state_hub_task_id: "055630a1-56e0-4ad4-8796-35f444313371"
```

`decisions/ADR-0013-*.md`. (ADR-0012 is CB-WP-0022 T03's.) At minimum:

- **which question the solver answers**, from the three in §What makes
  this hard, and what it is called in the UI — the name must not overclaim;
- **the information boundary**: whether the search runs on `GroundState`
  or on a `GroundView`, and if on state, what stops it using what the view
  hides. Note that running on the view makes the rule structural rather
  than a promise, and that this is the cheapest guarantee available;
- **the bound**: depth, node budget, or wall clock, and what *no path
  found* means against it — a bounded search that says "unwinnable" is
  lying, and the wording must say "none found within B";
- **whether difficulty ships as one number or a small table**, and what it
  is a function of: policy, seat count, threshold, seed range;
- **where it lives** — a new crate, a mode of `cb-play`, or a tool under
  `tools/`. The tier was declared L on the assumption of a new capability;
  if the ADR concludes it is a mode of an existing one, say so, and the
  over-declaration is a chaos-window data point worth recording.

**Done 2026-08-05.**
[ADR-0013](../decisions/ADR-0013-could-we-have-won.md), seven decisions.
**Two of them are not what T03 was written expecting**, because the review
moved the ground under both.

- **D1 — strategy fusion does not apply, and that is why this is
  affordable.** Fusion is a defect of *aggregating over determinizations
  to choose a move*. **After the game there is one world**: the deal is
  known, so a search over it produces a line executable in the only world
  there is. The survey treated fusion as this pass's central obstacle; it
  is an obstacle to a *playing* engine, which we are not building.
- **D4 — difficulty is the WINNABLE FRACTION, not any bot's win rate.**
  C4 killed the bot rate: two trivial policies span 0–100% on the same
  deals, and improving the bot would make the game "easier" without a rule
  changing. The solver supplies a policy-independent measure — *over N
  deals, in what proportion does a winning line exist* — which is a
  property of the deal distribution and the threshold, and is what
  GROUND-WP-0005 actually needs. **The bot rate never was.**

**D2** searches `GroundState` — the survey's view-only boundary is not
implementable, since a view cannot fold events — and moves the guarantee to
a checkable per-move `visible`/`hidden` marking from `project()`. **D3**
bounded exhaustive over the last K rounds, with *"no winning line found in
the last K rounds"* normative, never *"unwinnable"*. **D5** makes the
harness an instrument before any figure is quoted (C3). **D6** no new
crate and no port — **the L declaration was an over-declaration, recorded
as one**. **D7** the node cost is disputed 5×; T04 benchmarks it and
neither existing figure may be cited.

## Task: specify

```task
id: CB-WP-0025-T04
status: done
priority: high
state_hub_task_id: "7c3ca25d-0001-44ef-a07d-0ab2d51b09a5"
```

`specs/` — extend `MetricsAndScenarios.md` or add a capability spec as the
ADR directs, with metrics, because a spec without them is prose.

Candidates, to be argued not adopted:

- **witness checkability** — every path the tool emits replays through the
  existing scenario runner and ends in `group_success`. Target 100%, and it
  is a hard gate, not a metric: a path that does not replay is a bug that
  says the opposite of the truth;
- **search cost** — nodes and wall clock at the chosen bound, on the
  recorded games we have;
- **difficulty resolution** — the smallest threshold difference the
  measurement can distinguish, with its N. This is the number
  `ground-game` needs, and stating it as *"we can tell 5 from 7 but not 7
  from 8"* is more useful than a win rate with no error bar.

Per `ground-game`'s ruling (GROUND-WP-0004 T02), **any arithmetic finding
this produces ships a runnable reproduction and a row-level table** — never
a summed figure. A difficulty number is arithmetic, and it is exactly the
kind that has already gone wrong twice.

**Done 2026-08-05.**
[specs/RetrospectiveAnalysis.md](../specs/RetrospectiveAnalysis.md) v1.0,
and `games/ground/benches/search.rs` for D7's disputed number.

**The benchmark's own first fixture was defective — the same defect class,
one layer down.** A fixed stop at step 20 put 2p and 4p where seat 0 had
**no legal commands**, so it timed an empty `Vec` (~120 ns) and silently
skipped `validate_fold`. It now advances to a real branch **and asserts
it**.

Measured at real decision points (table in
[RetrospectiveAnalysis §3.1](../specs/RetrospectiveAnalysis.md)):
`legal_commands` 4.06–4.76 µs, `clone` 378–639 ns, `validate+fold`
**0.5–3.8 µs**.

**`clone` is in there because a search must copy state per branch**, and
`iter_batched` excludes setup from timing — leaving the budget resting on
an unmeasured span, which is precisely C1's mistake. **Per-child cost is
not uniform**: `validate+fold` ranges 0.5–3.8 µs by command, so budgets
use the upper end (~5 µs/child).

**That settles D3's affordability with real numbers**: joint branching
over the last two rounds is ~5×10² / 1.6×10⁵ / 5.7×10⁵ at 2/3/4 seats →
negligible / **0.8 s** / **2.9 s**. `K = 2` holds at two to four seats and
**does not at five or six**, where the tool must reduce `K` and *say so*
rather than silently search less.

**§4.1 is a normative prohibition**, not a preference: a single-policy win
rate may not be reported as a difficulty. The spec carries the measured
reason — greedy 100% vs first-legal 0% on identical deals.

## Task: build the witness

```task
id: CB-WP-0025-T05
status: todo
priority: high
state_hub_task_id: "5f1cbda3-7427-4f23-8c67-b2e216b987a8"
```

The search, the bound, and the replayable path. Wire it to the ending page
so a lost game can be asked the question — the page CB-WP-0024 T01 is
already reworking, so land that first or expect a conflict.

**A game that was won is not asked the question.** The feature exists for
a loss.

**Controls:**
- every emitted witness replays to `group_success` through the existing
  runner — asserted, not spot-checked;
- a deal constructed to be unwinnable returns *none found*, and the test
  says which construction makes it so;
- the information boundary is mutation-provable: relax it, and a test
  naming *that* boundary goes red. If it cannot be mutated, it was a
  comment rather than a rule.

## Task: measure the difficulty, and hand it to ground-game

```task
id: CB-WP-0025-T06
status: todo
priority: high
state_hub_task_id: "723feae8-1ed4-4dd4-9330-8ec84625e5d9"
```

Run the measurement, ship it as a `make` target beside the other
instruments, and show the result in the game — the maintainer asked for it
to be visible, and a number in a file will not calibrate anything.

Then send it to `ground-game` **against GROUND-WP-0005**, which is active
with both tasks waiting on exactly this. Per CB-WP-0022 T06, it lands as a
file in their repo under their workplan, not only an inbox entry — *the
message that sat unread for four days is the baseline to beat*.

**Controls:**
- the number regenerates from a single command, and `facts.toml` carries
  it if anything else quotes it (§Single source of fact — `make
  facts-check`);
- the report carries the row-level table the ruling requires;
- **the seed range and policy are in the number's name**, not in a
  footnote.

## Task: evidence

```task
id: CB-WP-0025-T07
status: todo
priority: high
state_hub_task_id: "2616e06c-5d88-4680-be18-4452515dff21"
```

`evidence/CB-EV-0023-*.md`.

- **Was the game winnable**, for the maintainer's actual lost game. That is
  the acceptance test with a face on it.
- **What the honest search cost against the omniscient one**, since the
  review will have pressed hardest there.
- **Whether the difficulty measurement moved ground-game**, or sat.
- **What tier L cost against what it caught** — third full-weight L pass in
  the project, and the second in this chaos window.
- **Quote CB-WP-0024's cost by re-running the instrument.**