## General informations

Current history : 0↓0↑1↓0↑1↓0↓0↑1↑0↑1↓1↓1↑

Next markets :

0 : https://manifold.markets/dionisos/binary-induction-game-v1-if-the-nex-38725abedcf2

1 : https://manifold.markets/dionisos/binary-induction-game-v1-if-the-nex-9a0dc1e418e4

### Direct help to discover the pattern

Given I misjudged the difficulty of this game, here is some help

If the last two tokens are XY (where X is a 0/1 token), then on the counterfactual where the players has choosen another 0/1 token for X, the last two token would have been not(X)not(Y).

So if the current history is : 0↓0↑1↓0↑ another correct history could have been : 0↓0↑1↓1↓

The next token (↑ or ↓) in the pattern is determined by only 3 informations :

The lenght of the history

The last token in the history (0 or 1)

The 0/1 token before that.

So, if the history is currently "0↓0↑1↓0↑1", then the next token is determined by :

Lenght = 9

Last token = 1

Previous 0/1 token = 0

## Story

Manifolds Markets was doing some market stuffs like a well-behaved prediction market, but suddenly there is a truck that appears out of nowhere at high speed.

Less than two second latter Manifold feel some deep discontinuity inside itself, and the sense that connect it to the meta-physical nature of its universe lose focus, blurs, and regain focus on a completely different set of meta rules :

## Game rules

The game takes place in several steps, at each step, the "current history" of the universe grows (exemple : "0↓0↓1↑" → "0↓0↓1↑1↓"), the goal is to find the patterns that this universe follows.

At the beginning of each step, there will be two markets, one for "0", and one for "1", about whether they will be up "↑" or down "↓".

At the end of each step, the market nearer to 50% of these two, will be selected, and the other one will resolve N/A.

The selected market will resolve ↑ or ↓, following an unknown [to the players] deterministic rule (this rule/pattern can take into account the entire history, not just the last outcome).

Then the history of the universe will be updated, and the next step begin.

(Did you except heroic fantasy and magics ? Manifold is a market, not a person)

Parallel to that, if the correct rule/pattern is at more than 90% in this free response markets, for the duration of a step, then this answer is selected, and the game end.

A step end when :

There is more than 2 days since the last step.

There is more than 10 participants in both the "0" and "1" markets. (I can decide to not stricly follow this rule)

I see it, and create the markets for the next step.

Two conditions should be satisfied for the game to end :

There is one option at 90% in this free response markets.

The option describe the correct pattern.

When the game end, this market resolve to the option describing the correct pattern.

Will you help Manifold to better understand its new world ?

The md5sum of the rule is : 6a28abf346c601a5ab24bbafec70acf4

I will not bet on any of these markets.

A possible rule/pattern could be : If there is an odd number of 1 in the current history, it end with ↑, and ↓ otherwise, which could result in something like : 0↓1↑1↓1↑0↑0↑…

## Related questions

What is special about 3 and 7 in the range 1-7? Other than they are the only n-th prime numbers from the range, where n is odd.

If someone gives me an answer that (combined with my reasoning so far*) gives me the actual pattern, I will share with you 20% of my profits** from this game after it ends.

** *which can be trash

** if I have any

@roma Are you sure you need to rule out 1? I don't know what you're thinking, but there is an undefined parameter in the first arrow's function, and 1, 3, and 7 are all 2^n-1 (ie. all 1s in binary)

Here is what GitHub copilot thinks (the "= ↓" at the end is produced by AI):

Edit:

If the last two tokens are XY (where X is a 0/1 token), then on the counterfactual where the players has choosen another 0/1 token for X, the last two token would have been not(X)not(Y).

So if the current history is : 0↓0↑1↓0↑ another correct history could have been : 0↓0↑1↓1↓

With this in mind, Copilot can't be right. ↓ in both cases would be wrong

@roma At first I didn’t understand you wrote everything except the last token, I thought the AI guessed the pattern for each line (with some not displayed prompt) I was like "wtf !" 😂

@dionisos Yeah, black text is prompt, and grey is the output. Would be funny if AI cracked it, but we couldn't get the pattern from it. But it's obviously wrong.

So the number is determined by traders, and the arrow is determined by the pattern?

And arrow has nothing to do with the number that follows it, right? Because at first I thought that:

0 + ↑ = 1

0 + ↓ = 0

1 + ↑ = 1

1 + ↓ = 0

@roma Yeah, I'd like a clarification on whether the new advice means the next 0/1 token is nondeterministic. I have a theory, but that would take away half the significance of it being right last time

@roma

> So the number is determined by traders, and the arrow is determined by the pattern?

Yes, this is correct.

> And arrow has nothing to do with the number that follows it, right?

Yes, this is correct, an arrow only depends on the history before it.

Made a sheet with the data if anyone wants to test formulas: https://docs.google.com/spreadsheets/d/16-TuUYV93pHHB5x0KewApoHQJm5qlGrDcl3Y8c9kUX0/edit?usp=sharing

@KongoLandwalker will the game continue infinitely if every time some option is above 90% it is the wrong rule?

IS it "option is selected = the market resolves to the option"?

There is two conditions for the game to end :

- There is one option at 90% in this free response markets.

- The option describe the correct pattern.

will the game continue infinitely if every time some option is above 90% it is the wrong rule?

Yes.

Except if nobody is interested anymore into trying to play the game, and some just want to know what the pattern was.

What does being selected men in terms of resolution?

So we control digits of the sequence and the rule controls arrows?

I want to test whether with 0 it will be arrow up next, do i bet on market 1 to lead it away from 50%?

Can the rule read outside the string of digits and arrows: read the news, date, day of the week, final percentages of the markets, amount of bettors? Is the string (history) the only arguement for the rule?

What does being selected mean in terms of resolution?

If 0 is "selected" by the players, then I look at the rule to see if the next token will be ↑ or ↓, if it is ↑ it resolves YES, otherwise it resolves NO.

(the other market for 1 resolve N/A)

So we control digits of the sequence and the rule controls arrows ?

Yes

I want to test whether with 0 it will be arrow up next, do i bet on market 1 to lead it away from 50%?

Yes, the idea is that what get selected is the token for which the players are the most uncertain about how it will resolve. It avoids a situation where the players know perfectly the rule for 0, but not for 1, and always choose 0 (making a profit because they bet correctly), without gaining more information about the rest of the rule.

Can the rule read outside the string of digits and arrows: read the news, date, day of the week, final percentages of the markets, amount of bettors? Is the string (history) the only arguement for the rule?

The history is indeed the only argument to the rule (so, no to all other question here)

I am starting to fear the pattern I chose is too hard.

I am considering if it would be a good idea to give some advice between markets. (I will definitely not do if anyone is against it)

@Stralor Yes, sorry this part was probably unclear. Only the history matter for inferring the next states.

(the markets are only useful to choose between 0 and 1)

ie : The parameter of the program will be (and only be, for the next state) either "0↓0↑1↓0↑1↓0" or "0↓0↑1↓0↑1↓1" (0 or 1 being chosen by the players)

And it will output either ↑ or ↓