Goliath find the rule of this bot :-)

This is Goliath , Goliath is a basic bot, it has one rule and one exception to the rule. The only way Goliath can break its rule is if all legal move go against it.
Here Goliath is black with 2500 elo and Stockfish is white with 2000 elo
Appart from this, Goliath will behave like any other chess engine and just choose the best move.
If you know the rule beating Goliath become possible even with a huge elo difference, if you don't know the rule you are still playing a restricted bot (as all the moves have to follow the rule) and you can get lucky by going in a sequence that will destroy the bot.
Stockfish not understanding rule just play chess, in this condition if Goliath is at 2500 elo (the strength of Camille), the win rate over 100 games will be 0.56 vs Stockfish at 2000, 0.42 vs stockfish at 2200 and 0.24 vs stockfish at 2400. So Goliath 2500 elo is funcationally at 2100 elo, versus a human knowing the rule more 1400 elo.

Camille can control bots at 5 different level, current engines mostly have 1 and some can also do a second category in a limited fashion.

Once you have found the rule and exception any suggestion for making this bot more stable?

Same bot, following the same simple rule and one exception. Here white with 1700 elo losing vs Stockfish as black with 1000 elo.
With 700 elo difference the win rate should be 98%, but over 100 game it is 88%.
Even if stockfish cannot exploit the rule limitation directly, its limits the move Goliath can do, hence the drop in win rate.

I'm not sure what you are asking? Is it a puzzle for us to recognize the rule?

If I understand correctly the rule should lead to suboptimal moves, in the first game there is not even a single blunder by black so there is not much to go on...

The bot cannot blunder it is at 2500 elo, just it choose inside the move complying with the rule :-). And the restriction is something you are supposed to do anyway in MOST move.
Another game where Goliath at 2500 elo with the white, lose to a 1000 elo stockfish. It does not happen often but way more often that it should. Stockfish just sometime start a death spiral and Goliath is lost. Hence the name Goliath because you can make it fall way more easily that the elo rating suggest.

The point of a puzzle is that participants need to have enough information to solve it.

For example, based on the games you showed it seems the computer is not allowed to play a move that would make a 3 fold repetition.

Probably that's not the rule, but every other guess that someone can make based on the provided info is probably also not the rule ;-)

The rules and its exception are very basic in fact :-) .

To identify it, I would say the best way is the end game. Because that's when it can become VERY restrictive, that's when some move who would obviously been the best are not played.

Camille could do a bot that don't do 3 fold repetition, or that does not do 3 fold repetition until one rook goes on e4, or 2 knight are on the board , or after move 20 , or only for the bishop and the king.

The number of bots possible are in the trillions, here we just use one rule that is applied indiscriminately to all the pieces. Once you know it, it become very hard not to see it.

My earlier guess was actually sort of serious as there is a chance to draw via threefold repetition but instead 43. Bf1?? plays for a loss

Ha ha yes Ra1 was a better move, but it was also an impossible move, because against the rule and not an exception :-) . That's why Goliath lose often in the end game, too many good move are not allowed.

Same with Ne5 on the first game, that's was a better move than the one played by goliath but it was against the rule

So nobody found :-)

Goliath can only move on protected square, except when taking a piece.
This bot is quite good on the opening and middle game but if there is too much exchange taking place his mobility become limited and the endgame difficult.

Hence the name Goliath , look strong but if you know its limitation winning is doable even with a big elo difference

https://www.chess.com/member/botgoliath

The two last games are a good example , on the game it win against the 2000 elo bot, the knight was crucial to allow movement of the other piece in the end game.
On the game its lost today against a 2400 elo bot, the knight was useless in the end game because it had only one square where it could move and it would have been taken by the pawn.

https://www.chess.com/game/computer/1310948603

https://www.chess.com/game/computer/1316853237