Background
A full-knowledge, two-player, alternating, finite, deterministic game is modeled by a tree T with labeled leaves and no infinite branches. The game starts at the root of the tree (level 0). Player A plays at even levels by choosing a node one level up the tree from the current node to move to, and Player B likewise plays at odd levels choosing a node one level up the tree from the current node to move to. The two players thus alternate, forming a path through the tree. The leaves are all labeled either "A" or "B", and Player A wins if the branch ends at a leaf labeled "A", while Player B wins if the game ends at a leaf labeled "B". At all times, both players know the complete state of the game (full-knowledge).
More Background
A "bounded-resource strategy" is a function from valid game states to valid game states (suitably represented) which can be computed by a finite state automaton. Player A, for example, might plug in the current state of the game into the automaton, and then the automaton will spit out the next move, and they repeat this throughout the whole game (like someone cheating in chess using StockFish). A "non-deterministic bounded-resource strategy" is one that uses a non-deterministic automaton rather than a deterministic automaton.
One strategy "dominates" another strategy if the player using the first strategy always wins when they use it against it against a player using the other strategy. For example, "always play rock", in rock-paper-scissors dominates "always play scissors".
A game exhibits "essential length k intransitivity of M-bounded strategies" if there is a collection of bounded-resource strategies $S_0, S_1, ..., S_{k-1}$ each with automaton size at most M such that the following conditions hold:
$S_{j}$ dominates $S_i$ whenever $j - i = 1, 2, ...,(k-1)/2$ (mod k). Here $k$ must be odd and at least 3.
There is no M-bounded strategy $S'$ dominating all the $S_i$.
Rock-paper-scissors exhibits essential length-3 intransitivity of M-bounded strategies for any $M \geq 1$: "always play rock" dominates "always play scissors" which dominates "always play paper", which dominates "always play rock". However, rock-paper-scissors is not a full-knowledge alternating game: you do not know what the other player played until the game is over (after you have already made your move).
It is easy to prove that for a full-knowledge, two-player, alternating, finite, deterministic game, there is always a winning unbounded strategy: a strategy which always wins against any other, which means for such games there cannot be essential length-k intransitivity of unbounded strategies.
Question
For which $M, k$ do there exist full-knowledge, two-player, alternating, finite, deterministic games with with essential length-k intransitivity of M-bounded strategies? Does this change if we allow non-deterministic M-bounded strategies?