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↑…