Could someone give me a list of real use cases of Paxos. That is real problems that require consensus as part of a bigger problem.
Is the following a use case of Paxos?<
In the case you describe, you're right, Paxos isn't really necessary: A single central authority can generate a permutation for the deck and distribute it to everyone at the beginning of the hand. In fact, for a poker game in general, where there's a strict turn order and a single active player as in poker, I can't see a sensible situation in which you might need to use Paxos, except perhaps to elect the central authority that shuffles decks.
A better example might be a game with simultaneous moves, such as Jeopardy. Paxos in this situation would allow all the servers to decide together what sequence a series of closely timed events (such as buzzer presses) occurred in, such that all the servers come to the same conclusion.