In this game, both players $A$ and $B$ follow specific rules:
1. A player cannot choose a number that has already been chosen by either player.
2. A player cannot choose a number consecutive to any number the player has already chosen.
3. The game is a draw if all numbers have been chosen; otherwise, the player who cannot choose a number anymore loses.

Player $A$ makes the first move, always taking numbers such that they cannot take consecutive numbers in subsequent moves.

To find the outcome of this game for any $n$, we need to analyze specific cases and determine the general pattern:

1. Base Cases:
- For $n = 1$, Player $A$ chooses the only number available, and since no numbers are left for Player $B$, the game ends with a draw.
- For $n = 2$, Player $A$ can choose either number $1$ or $2$. In either case, Player $B$ will have no available moves since the unchosen number is consecutive to $A$'s number, hence the game draws.
- For $n = 4$, Player $A$ can choose one number, then $B$ chooses another such that it creates two separate ranges. Player $A$ cannot continue a sequence, and if both sets of choices for $A$ result in step-wise valid but non-winning positions for $B$, resulting in a draw.
- For $n = 6$, similar to the $n = 4$, validating through moves results in non-winning sequences for both players if optimal play is performed, leading to a draw.

2. General Case Analysis:
- For $n > 6$, Player $A$'s strategy would lead to leaving isolated numbers for Player $B$. Using optimal strategies, Player $B$ can always respond by either splitting $A$'s group of non-consecutive numbers to maintain rounds until numbers are maximized or chosen, often resulting in Player $B$ winning.

3. Conclusion: By systematically analyzing numbers $n \leq 6$ and making extrapolated strategic observations, we find:
$$\text{For } n \in \{1, 2, 4, 6\} \text{, the game ends in a draw. Otherwise, } B \text{ wins.}$$

Hence, the outcome of the game can be represented as:
$$\boxed{\text{For } n \in \{1, 2, 4, 6\} \text{, the game ends in a draw, else } B \text{ wins}.}$$