The Aaronson Oracle

by Adi Gunawan 14 October 2026 19 min read 58x views

The story

Two machines in a corridor at Bell Labs

This test is seventy years old, and it was built out of relays before it was always written as software.

c. 1951
Bell Telephone Laboratories
Murray Hill, New Jersey

Some period in the first 1950s, at Bell Laboratories, an engineer named David Hagelbarger built a device to perform matching pennies against his colleagues. Two players all choose heads or tails in secret; one wins if the choices match, the another if they differ. Played against person really unpredictable, it is a coin flip and nothing more. Hagelbarger called his device SEER, for SEquence Extrapolating Robot.²

SEER did not example you in any profound way. It tracked three facts about the final two rounds: whether it had won or misplaced the perform before last, whether it had played the identical or differently, and whether it had won or misplaced the final play. Three yes-or-no facts provision eight possible situations, and for all of the eight the device remembered whether playing the identical had been working. It counted that on a small reversible oppose that stopped at affirmative and minus three, which Hagelbarger described alongside a expression value keeping: the stops "in consequence create the device ignore ancient history."²

PlateSEER's entire memory. Three yes-or-no facts concerning the final two rounds provision eight situations, and all situation keeps one small oppose of whether playing the identical has been working. The stops at affirmative and minus three are what Hagelbarger meant by making the device ignore ancient history.

Claude Shannon, who worked downward the hall, took note and built his own. His memorandum of 18 March 1953 opens by giving Hagelbarger the credit plainly, and afterward makes a difference that tells you exactly what benevolent of device this is:

That is the entire idea in one sentence. Game theory tells you the accurate way to perform matching pennies: flip a mental coin, be genuinely random, and no enemy can attack you complete time. Poe's detective does item alternatively entirely. He studies his enemy and guesses what sort of individual would choose what. Shannon built the detective, since the detective wins. Not against an ideal player. Against an genuine one.

Shannon's device looked for patterns and, in his words, "assumes that the participant volition prosecute the patterns the next period the same circumstance arises." When it had not seen a form reiterate at smallest twice, it moved at random, and the randomness came from a commutator spinning concerning ten times a second, sampled at the instant you pressed the button. Its unpredictability was harvested from the jitter in your own timing.¹

A wooden cupboard alongside a apparent top showing valves and wiring. A dish on the forefront says NIMWIT, and two columns of lamps below it are labelled PLAYER WINS and MACHINE WINS. — **Plate**Not the mind-reading machine, which is not photographed anyplace we could use, but one of Shannon's other game-playing boxes from the identical years: Nimwit, c. 1953, which plays Nim. It is current since of its forefront panel. Two columns of lamps, *player wins* against *machine wins*, is the scoreboard this entire leaf inherited. MIT Museum, Cambridge MA. Photograph by Daderot, released under CC0 1.0. Source.

He knew exactly how to attack it

What makes Shannon's memo notable is that he did not oversell it. He worked out how beatable the device was and published the method:

The device is not unbeatable. It is beatable three times out of four, on paper. You merely cannot run the method in your head during playing. That gap, between what is imaginable in regulation and what a individual can really do in the moment, is the complete topic of this page, and Shannon named it in 1953.

An aluminium maze of movable partitions on a flat tray, with a small dreary mechanical mouse position in one of the corridors and a award resting in another. — **Plate**Theseus, 1952. The mouse finds its way through the maze, and remembers the route. Shannon's habit of construction the idea fairly than lone penning it downward is the logic a memo about outguessing group came alongside a operating device attached. MIT Museum, Cambridge MA. Photograph by Daderot, released under CC0 1.0. Source.

The two machines played all other

Inevitably, person wondered which device was better. Hagelbarger recorded what happened:

PlateThe two machines were wired to a third that refereed between them. The scores shown are from the 2020 software replication, averaged complete games of a hundred plays; Hagelbarger's own study of the first equivalent put it at concerning 55 to 45.³

Shannon's simpler machine, built alongside concerning fractional as many relays, beat the additional detailed one. In 2020 a collection at Sibiu reimplemented the two in application and played them against all another ten times. Averaged over games of a hundred plays, Shannon's won 55.8 to 44.2. A sixty-year-old part of lab folklore, checked and confirmed.³

Player 3507
Machine 5010
the lifetime mark on the exhibit

Shannon's device motionless exists. William Poundstone went to see it at the MIT Museum's retention facility and did not perform it, since "that would have been nearly impious," he wrote, "for it recorded a final score: Player 3507. Machine 5010." Across all individual who always sat downward alongside it, the device won concerning 59% of the time. That is a life count on an display fairly than a controlled experiment, and it is the only machine-against-humans figure that survives at all.⁴

Fifty years later, in a address hall

The type you fair played comes from Scott Aaronson, who wrote it for a category he taught at Berkeley and describes it in Quantum Computing Since Democritus:

Hold on to two things there. The archetypal is that Aaronson could not attack his own program during knowing its origin code, the identical admission Shannon made concerning his relays. The second is the student, whom we will arrive rear to at the end, since how he did it is the item fairly than the exception.

Where our type departs

Ours is not a copy of either machine. Shannon's tracked whether it was winning and whether you had changed your choice, giving it eight situations and no recollection of your genuine sequence. Ours ignores winning and losing entirely and merely counts what you pressed following all recent run of presses. Aaronson's used runs of five, and so does ours. His forgot nothing; Hagelbarger's deliberately forgot; ours additionally never forgets, so a habit you autumn first keeps counting against you for the remainder of the session.

PlateThree machines, seventy years apart, and what all one really keeps. The figure of 63 is all run of five presses or fewer: 32 of dimension five, 16 of four, and so on downward to the empty run.

How it works, in English

What fair happened to you

There is no example of behavioral discipline in this program, and nothing about liberated will. It keeps a notebook.

Every period you media a key, it writes downward what you had fair pressed beforehand, and what you pressed next. Not once, but at six different lengths at the identical time. Suppose your final five presses were F, D, D, F, F, and afterward you media D. The notebook gains six entries: after FDDFF you pressed D; following DDFF you pressed D; following DFF you pressed D; following FF you pressed D; following F you pressed D; and, ignoring environment entirely, you pressed D.

PlateOne press, six entries. The notebook records the identical event at all environment dimension at once, from the complete run of five downward to no environment at all. Nothing alternatively is stored: no timing, no former of who was winning, no example of you.

To conjecture your next press, it says the notebook backwards. It takes your final five presses and asks: have I seen this exact run before, and if so, what did you do next? If it has seen that run at smallest twice and you did not do the two things likewise often, it answers alongside whichever you did more. If the run is new, or it has lone seen it once, or you split evenly, afterward it shortens the question. What concerning the final four presses? The final three? Two? One? And if all of that fails, it flips a coin, and that coin flip counts against its mark akin any another guess.

That shortening is the lone clever item in the program, and it is value understanding, since it is anywhere most naive versions go wrong. A lengthy run is a particular match: it tells you a lot whenever it applies. But first in a meeting you have not produced many five-press runs, so most of them have been seen exactly once, and a sole study is not evidence. It is a coincidence. A abbreviated run is vaguer but far better attested. The program prefers the longest run it has seen often adequate to trust, and slides downward to shorter, better-supported questions whenever the particular ones are thin.

That notebook is the example you have been playing against. The bands at the top of the leaf are the six questions the device asks, stacked: your final five presses at the top, downward to no environment at all — every media you have always made — at the bottom. Every kernel in a collection is one period you produced that run and afterward pressed something. Grains sit left of the centre row if you followed the run alongside D and correct of it if you followed it alongside F. So how broad a collection is says how much the machine knows, and how lopsided it is says what it knows.

The conjecture falls downward the centre. A run it has never seen is an empty collection and holds nothing. A run seen exactly formerly is a sole grain, which is a coincidence fairly than evidence. A run you followed alongside F exactly as frequently as alongside D is absolutely symmetrical, and the expert downward its center is open. The conjecture drops through all one of those and comes to remainder on the archetypal collection that is neither empty, nor thin, nor equal — then leans the way that collection leans. That autumn is backoff. It is the only clever item in the program, and it happens in forefront of you on every sole press.

Notice that the bands get wider as they get shallower, and that this is not a example decision. The top collection asks concerning one exact run of five and has nearly nothing in it; the base collection counts all media you have always made. Evidence thins as the inquiry gets additional specific, always, and that is the complete logic the device have to be consenting to provision up on its finest inquiry and ask a worse one.

Notice what this means. The device is not clever and it is not study you. It is asking, complete and over, one question: last time things looked akin this, what did you do? Everything alternatively on this leaf is a consequence of the fact that you keep answering.

Figure 2Your own notebook, from the meeting you fair played. Each row is a run of presses; the bar shows how frequently you followed it alongside F fairly than D. Rows the device trusted are marked. With no meeting stored, this shows a example instead, clearly labelled.

How it works, actually

Counting, and knowing whenever to stop

The appropriate name for the notebook is an n-gram frequency model. An n-gram is fair a run of n symbols. Here the symbols are your two keys, so there are 2⁵ = 32 imaginable runs of five, 16 of four, and so on downward to the bare run, which is the figure of every media you have always made.

Write c for a run of presses, called the context, and N(c, F) for the figure of times you pressed F immediately following that run. The prediction regulation is: obtain the longest environment c matching your latest presses specified that

N(c, F) + N(c, D) ≥ 2 and N(c, F) ≠ N(c, D),

and answer alongside whichever of F or D is larger. If no context qualifies, answer at random. Dropping to a shorter environment whenever a longer one is unavailable or unreliable is called backoff, and it is norm custom in statistical tongue modelling, anywhere the identical issue appears: particular evidence is better, but particular evidence is rare.

Backoff
prefer the longest run you have seen frequently adequate to trust

The threshold of two is doing genuine work. Without it, a run seen a sole time, a 1–0 count, would outrank a two-press environment seen forty times as 30–10, purely since it was longer. That is the classic sparse-data failure, and it makes the example noisier than counting alone would be.

Aaronson described his first in nearly these terms. Asked what the Berkeley program did, he replied:

Note the hedge, "there power have been." He is suggesting backoff, not recalling it. Our threshold and our backoff are our own choices, taken on his suggestion, and not a reconstruction of the program he ran in that address hall.

Figure 3One genuine prediction from your session, held motionless — and the entire notebook this time, not fair the six runs the device was asking about. Each circle is a run the device has recorded and its size is how frequently it happened; the lit way is your final five presses, the traversed circles are the runs it gave up on, and the ringed one is anywhere the conjecture came to rest.

Read the machine

The complete program

Below is the predictor, in full. Not a summary of it, not a cleaned up type for the article, but the document this leaf loaded and ran while you were playing. It is a few dozen lines long, and its shortness is the lesson: this is all it takes.

Loading oracle.js…

Figure 4The predictor, loaded from oracle.js, the identical document this leaf is running.

Two functions. document writes one media into every environment dimension at once; foretell walks from the longest environment to the shortest and returns the archetypal one it trusts. There is no training phase, no parameters fitted in advance, no example of you beyond the count it has built during this one session. It arrived knowing nothing.

What it says concerning us

Why you lost

A program this small should not be capable to do this. It wins because of item concerning you, and that item has been measured for decades.

The project you were given, producing a sequence alongside no form in it, is called random sequence generation, and it is among the additional studied failures in experimental psychology. A 2023 assessment states the finding without hedging: "A average finding of RSG studies is that group are bad randomizers."⁷

The most dependable way group neglect is the one Aaronson named in passing whenever he stated players "have too many alternations." The same assessment calls it the alternation bias: "the avoidance of repetitions, additionally called negative recency effect or alternation bias, anywhere an excess of alternations and suppression of repeating choices can be observed."⁷ Asked to be random, group toggle too frequently and reiterate too seldom. A fair coin produces a run of five identical flips additional frequently than feels acceptable, and so group who are imitating a coin reduce those runs away.

It goes deeper than switching too much. Analysing sequences people produced whenever asked to be random, one study concluded that they "avoid figure repetitions and systematically deviate from mathematical randomness," and that the results "emphasize the idea of a complex hidden Markov regulation scheme that underlies humanly generated random number sequences."⁸ That expression is worth pausing on. A hidden regulation scheme alongside recollection is additional or small exactly what an n-gram example is built to find. The oracle is not a general mind-reader; it is a equipment shaped to fit a particular defect, and the defect is real.

The bias additionally runs in the another direction. People not lone produce too many alternations, they believe too many alternations look random: shown binary sequences to judge, participants charge the ones with the most switching as the most likely, alongside the highest location about a 0.6 to 0.8 charge of alternation against a true value of 0.5.⁹ Sequences alongside additional structure, HHHHH fairly than HTTHT, are judged less probable.¹⁰ We create wrongly and we class wrongly, in the identical direction.

PlateThe form of the mistake, drawn from the finding fairly than from data: asked which binary sequences appearance most random, group highest location about 0.6 to 0.8 alternation, during a fair coin alternates fractional the time.⁹ The curve is an example of that reported peak, not a storyline of published measurements.

The education itself hurt you

Here is the part that ought to annoy you, and it is why the two experiments on this leaf are not equivalent. That 2023 assessment also reports: "if randomness is not overtly requested but fairly an implicit requirement, as in rivalrous games specified as matching pennies, randomness seems to be higher."⁷

Shannon's players were playing a equivalent and trying to win. You were told to be random. The writings says the second framing is the worse one, and that consciously attempting randomness makes you more predictable than merely trying to attack an opponent. The education at the top of this leaf was, in a small way, operating against you.

Figure 5How the device learned you: its running accuracy throughout your session, against the 50% a coin would manage.

The rematch

Now that you know

You have peruse the source. You cognize it counts runs of up to five, that it needs to have seen a run twice before trusting it, that it never forgets, and that it falls rear to a coin flip whenever it has nothing.

Play it again.

Most group do concerning as seriously the second time, and the two people finest placed to cognize stated so first. Shannon, having published the exact method for beating his device three-to-one, noted that it was "extremely difficult to transport out this program mentally." Aaronson, having written his, stated flatly: "I couldn't equal attack my own program, knowing exactly how it worked."

Knowing the algorithm gives you nothing, since the algorithm was never the difficult part. The difficult part is generating the sequence, and you are the one generating it, alongside the identical machinery that produced the biases in the archetypal place. Understanding a bias from the exterior does not toggle it off from the inside.

The dependable methods
all amount to the identical move: obtain the decision out of your own head

If you do attack it, the engaging inquiry is how. The dependable answers all have one item in common: they obtain the decision out of your head. Poundstone, playing a contemporary implementation, established he did improved whenever he stopped observing the score. "I established I did better whenever I tried to disregard the feedback, and improved yet whenever I made certain I couldn't see the bars"⁴, and better again whenever he deliberately repeated choices his instincts wanted to alternate. Others peruse digits off item external, or run a regulation in their caput that is cheap to compute but difficult to detect.

Which brings rear Aaronson's student, the one the program predicted exactly 50% of the period and who stated he "just used his liberated will." We have no document of what he really did. But any it was, it was a method, a way of choosing that his own habits could not reach. That is not a smaller item than liberated will. It is fair a additional helpful description of it: not the deficiency of a cause, but the capability to hand the decision to item another than your reflexes.

The device below is the identical file, alongside the identical rules, sealing its conjecture before all press.

This test needs JavaScript.

References

Sources

Every actual assertion complete traces to one of these. Where an original was rearward a paywall, the quotation is taken from a origin that reproduces it, and that is stated plainly.

Claude E. Shannon, "A Mind-Reading (?) Machine," Bell Laboratories Memorandum, 18 March 1953; reprinted in Claude Elwood Shannon: Collected Papers, IEEE Press, 1993, pp. 688–690. Full content (PDF).
D. W. Hagelbarger, "SEER, A SEquence Extrapolating Robot," IRE Transactions on Electronic Computers, EC-5(1), 1956, pp. 1–7. We could not get the original; the two quotations here are reproduced from citation 3, which marks its quotations explicitly.
Macarie Breazu, Daniel Volovici, Daniel I. Morariu, Radu G. Crețulescu, "On Hagelbarger's and Shannon's matching pennies playing machines," International Journal of Advanced Statistics and IT&C for Economics and Life Sciences, 2020, pp. 56–66. PDF.
William Poundstone, "How I Beat the Mind-Reading Machine," 30 July 2015. Article.
Scott Aaronson, Quantum Computing Since Democritus, Cambridge University Press, 2013, Chapter 18 ("Free Will"). The passage additionally appears in the openly accessible address notes the publish grew from: Lecture 18.
Scott Aaronson, individual communication to Nick Merrill, reproduced in the aaronson-oracle README. Not a published source. Repository. Aaronson links that implementation himself, and uses the name "the Aaronson Oracle," in his answer to Roger Penrose.
Maja Guseva, Carsten Bogler, Carsten Allefeld, John-Dylan Haynes, "Instruction effects on randomness in sequence generation," Frontiers in Psychology, 14:1113654, 2023. Open access.
Marc-Andre Schulz, Barbara Schmalbach, Peter Brugger, Karsten Witt, "Analysing humanly generated random figure sequences: A pattern-based approach," PLOS ONE 7(7): e41531, 2012. Open access. Its header prediction figures involvement nine-option numeral sequences, not two-key sequences, and are not quoted here.
Giorgio Gronchi, Marco Raglianti, Stefano Noventa, Alessandro Lazzeri, Andrea Guazzini, "Modeling the overalternating bias alongside an asymmetric entropy measure," Frontiers in Psychology, 7:1027, 2016. Open access. These figures depict how group judge sequences, not how they create them.
Stian Reimers, Chris Donkin, Mike E. Le Pelley, "Perceptions of randomness in binary sequences: Normative, heuristic, or both?" Cognition, 172, 2018, pp. 11–25. DOI · accepted type (PDF).