Thank you @JamesMartin . For the moment, I would start to focus on (i). I just added a question in the "Edit section" about the problem itself: I am now wondering whether the problem becomes more interesting if each player can always distinguish between an attempt to occupy the same location chosen by another player during the same round, and a location which was already occupied in the previous rounds.

@JamesMartin OK, I see: the strategy of each player may be dependent on the round number, but must be the same.

@JamesMartin how can I formalize the concept that no player knows the strategies of the others?

I see your point @JamesMartin. Your doubt is definitely legitimate. Could you please suggest a modification of the problem which makes it non-trivial? Should all the players play the same strategy?

Thank you, you are right about the misleading sentence on the (possible) determinism of the strategy, because the strategy must be randomized. Now I edit that part of the question. However, I do not understand your statement "[...] if you allow any strategies then of course player $i$ choosing location $i$ on the first round is good" because, for instance, player $p_1$ does not know her/him index ($1$ in this case). Hence, it is not possible to that each player $p_i$ chooses $\ell_i$ on the first round with probability $1$. Anyway, the strategy may differ from player to player.