CSPRNG opponent
AW.precommit.chanceControl supplies 4,096 outcomes from crypto.getRandomValues. Its exact 95% Clopper-Pearson interval should contain 50%. This control is allowed to fail.
not run yet
A theorem about your next moves
The promise that cannot be broken
You may choose every outcome after seeing the machine's prediction. It will still finish within a sealed number of mistakes of the best fixed expert on this page. Then you can make it wrong every single time and see what that promise was worth.
M - m* ≤ BFIXED BEFORE PLAY
The published anchor, reproduced first
The machine earns the right to make a promise.
Before any play is enabled, this browser recomputes the 1994 Weighted Majority constant and exhausts all 4,096 binary sequences of length 12 with the same implementation used below.
PUBLISHED FORMLittlestone and Warmuth, Theorem 2.1 and Corollary 2.1: M ≤ [m* ln(1/β) + ln N] / ln[2/(1+β)].
β = 1/2 CONSTANTcomputing
FINITE REPRODUCTIONcomputing all 4,096 sequences
TIGHTNESS WITNESScomputing
The game stays locked until this finishes.
Choose, compute, seal
The number is shown before your first move.
The two constant experts guarantee m* ≤ floor(T/2). Substituting that into the published theorem gives a pre-play regret allowance B. At T = 48 and N = 8, it is 41.054363. Large, but not larger than every logically possible regret.
Sealed regret allowance B
41.054363
mistakes above the best fixed expert
waiting for the published reproduction
B = theorem(floor(T/2)) - floor(T/2)
The full preimage appears after your first move.
The live game
I commit. You answer.
The prediction is public before you choose. LEFT wins weighted ties. Every expert that is wrong loses half its weight, whether the combined machine was right or wrong.
ROUND 1 OF 48
predictionLOCKED
Nothing you choose leaves this browser. There are no page fetches, analytics, or third-party requests.
Run the anchor and seal a game to unlock play.
MACHINE MISTAKES M0no rounds played
BEST FIXED EXPERT m*0chosen only in hindsight from the visible pool
REALISED REGRET M - m*0sealed B = 41.054363
LIVE THEOREM CEILING7.228M must not cross this comparator-dependent ceiling
Every comparator, exposed
| fixed rule | next vote | mistakes | weight |
|---|
Accuracy can collapse
Machine and best expert mistakes
machinebest fixed expert
No outcomes yet.
The theorem can remain intact
Realised regret and its allowance
realised regretsealed B
The comparator is fixed before the sequence begins.
The second half
The promise is not accuracy.
Press the red button. The machine announces a deterministic label, then the adversary chooses the other one. The machine misses every round precisely because it committed first.
The theorem survives because it asks a narrower question: did this machine lose much more often than the best one of eight fixed rules? The adversarial sequence makes those rules lose too. Small regret can coexist with no predictive power at all.
The comparator is fixed before the sequence begins. It cannot be a new rule invented after seeing your moves. A richer pool changes the question, but the logarithmic price of that pool changes the allowance too.
WHAT THE SEAL PROMISES
M - min fixed expert mistakes ≤ B
WHAT IT DOES NOT PROMISE
M is small
accuracy exceeds 50%
the expert pool contains a useful rule
the machine predicts you
WHY THE ADVERSARY EXISTS
A deterministic forecast revealed before the outcome can always be contradicted. A randomized forecaster recovers only by committing to a distribution, not a revealed sample.
M - min fixed expert mistakes ≤ B
WHAT IT DOES NOT PROMISE
M is small
accuracy exceeds 50%
the expert pool contains a useful rule
the machine predicts you
WHY THE ADVERSARY EXISTS
A deterministic forecast revealed before the outcome can always be contradicted. A randomized forecaster recovers only by committing to a distribution, not a revealed sample.
Two live controls
Chance should look like chance. Pattern should look like pattern.
Both controls create a fresh instance of the exact same predictor class used in the game. Neither resets it between rounds. Only the opponent changes.
Predictable opponent
The same machine receives LEFT 256 times. A useful update rule should move weight toward always LEFT. The reported accuracy includes an exact interval.
not run yet
Three results, kept separate
Deterministic mistakes are not randomized loss.
Deterministic Weighted Majority
“Let S be any sequence of instances and binary labels.”
M ≤ [m* ln(1/β) + ln N] / ln[2/(1+β)]Littlestone and Warmuth, Theorem 2.1 and Corollary 2.1
This is the theorem operated by the live game. At β = 1/2 its multiplier is 2.409420839653, giving the familiar 2.41(m* + lg N) form.
Hedge
“For any sequence of loss vectors l1, ..., lT”
LHedge(β) ≤ [L* ln(1/β) + ln N] / (1-β)Freund and Schapire, Theorem 2, equal initial weights
Hedge commits to a distribution and bounds mixture loss. It does not justify attaching its denominator or constants to the deterministic machine above.
Fixed-horizon tuned form
regret ≤ ln N / η + Tη / 8
η = √(8 ln N / T)
regret ≤ √(T ln N / 2)
η = √(8 ln N / T)
regret ≤ √(T ln N / 2)
Cesa-Bianchi and Lugosi, Theorem 2.2, for convex losses in [0,1]. The horizon T is known in advance. This is not the deterministic all-or-nothing mistake guarantee played here.
The lower wall
supT,N VT(N) / √[(T/2) ln N] ≥ 1
Cesa-Bianchi and Lugosi, Theorem 3.7, for binary outcomes and absolute loss. In their stated minimax sense, the leading square-root scale of Theorem 2.2 cannot be uniformly improved away.
The check
What could make this page wrong.
PUBLISHED ANCHORThe browser and offline verifier rederive the β = 1/2 constant from the paper's formula. The browser exhausts all 4,096 length-12 sequences with the live implementation.
FALSIFIERAny prefix with machine mistakes above [m* ln(1/β)+ln N]/ln[2/(1+β)] falsifies the implemented guarantee. The page checks this after every move and turns red if it happens.
NEGATIVE CONTROLThe offline verifier disables weight penalties. That broken machine violates the bound on ten RIGHT outcomes: M = 8, m* = 0, ceiling = 7.228263.
CHANCE CONTROLThe unmodified stateful predictor plays crypto.getRandomValues for 4,096 rounds. Accuracy is shown with an exact interval. The bound can hold at the same time.
TIGHTNESS SEARCHAmong the 4,096 anchor sequences, 111001010010 reaches 54.4% of its additive theorem allowance: regret 7 against allowance 12.865946. This is evidence of scale, not a claim of exact optimality.
FREE CHOICESLEFT wins ties. β is fixed at 1/2. T is selected from 16, 24, 48, or 96. N selects a visible prefix of the eight named experts.
UNCERTAINTYA random control can land outside a 95% interval about one run in twenty even when correct. One reader's play is one sequence, not a population sample.
LIMITThe proof is why the universal statement holds. A large finite sweep can catch an implementation fault, but it cannot replace the proof over every finite sequence.
Primary sources
The statements behind the seal.
N. Littlestone and M. K. Warmuth, The Weighted Majority Algorithm, 1994Information and Computation 108(2):212-261. Theorem 2.1 supplies the total-weight bound; Corollary 2.1 substitutes an individual expert's mistakes.
Y. Freund and R. E. Schapire, A Decision-Theoretic Generalization of On-Line Learning and an Application to Boosting, 1997Journal of Computer and System Sciences 55(1):119-139. Theorem 2 states the Hedge loss bound reproduced above.
N. Cesa-Bianchi and G. Lugosi, Prediction, Learning, and Games, 2006Theorem 2.2 gives the fixed-horizon convex-loss upper bound. Theorem 3.7 gives the matching minimax lower scale for absolute loss.