`quality()` answers "is this matchup fair". Callers picking which
comparison to run next need "is this matchup informative", and the two
coincide only for two evenly matched competitors. Without a principled
alternative, downstream code was reaching for hand-rolled heuristics
like `quality * sigma_a^2 * sigma_b^2`, which double-counts uncertainty:
the two factors are not independent.
Adds `expected_information_gain`, the outcome-weighted divergence
between current beliefs and the beliefs each result would produce:
EIG = SUM P(outcome) * KL(posterior_after(outcome) || prior)
Available standalone over `Rating`s, and as
`History::expected_information_gain` using current skills and the
history's own beta, drift and p_draw — so the outcomes it weighs are the
ones that would actually be fitted.
This is the mutual information between the outcome and the skills, which
gives an analytic ceiling: gain cannot exceed the entropy of the thing
being observed, so at most `ln k` nats for k outcomes. That bound is the
sharpest test available, because an acquisition function is unusually
exposed to returning finite, plausible, monotone numbers while being
wrong — it would simply select slightly worse matchups forever. A
prototype of this returned 4.77 nats from a sign error while passing
every monotonicity check; `never_exceeds_the_entropy_of_the_outcome`
catches that class unconditionally.
Measured against the ceiling the values are meaningful rather than
vacuous: 0.382 nats for an even matchup between diffuse priors against
an 0.693 ceiling, falling to 0.013 for a lopsided one and 0.000 for a
hopeless one.
`disagrees_with_the_quality_times_variance_heuristic` pins down that
this is not a monotone transform of the heuristic it replaces — the two
rank a lopsided matchup and a confident even one in opposite orders — so
a later "simplification" cannot quietly revert to it.
Cost is one inference pass per possible outcome, documented on the
public API alongside the shortlist-then-score pattern, so callers do not
discover it in production.
Also folds the duplicated key-gathering in `predict_quality` and
`performances` into one validated `member_skills`.
Refs #39
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_011hcFjNDmHXZF8URGLku5zZ
logaritmiskandClaude Opus 5
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