Consider a deck with $n$ cards labeled $1, 2, \ldots, n$ arranged in some initial order. We need to determine under what circumstances Anna, who starts the game, has a winning strategy. The strategy depends on the number $k$ on the topmost card at each player's turn.

### Game Description:

1. At each turn, the player observes the topmost card, which has the number $k$. The allowed move is to rearrange the top $k$ cards in any order.
2. If the top card remains as $k$ after rearrangement the same player loses, and the game ends.

### Strategy Analysis:

Let's consider what gives Anna a winning strategy:

- Anna examines the number $k$ on the topmost card. She has the freedom to rearrange the $k$ topmost cards.
- If $k$ is the smallest among these $k$ cards, any rearrangement will necessarily keep $k$ as the topmost card. This results in Anna losing immediately because the topmost card after her rearrangement is still $k$.

However, if $k$ is not the smallest card among the top $k$ cards, Anna can always rearrange these cards such that a card smaller than $k$ becomes the topmost card, hence avoiding losing:

- This move changes the topmost card to a number smaller than $k$, passing control to Beatrice without losing right away. With each player's optimal play, Beatrice faces the same situation: if the number on the topmost card during her turn is the smallest among the cards she can rearrange, she will lose.

Thus, Anna has a winning strategy if and only if the number $k$ on the topmost card is not the smallest of the $k$ topmost cards:

$$\boxed{\text{Anna has a winning strategy if and only if } k \text{ is not the smallest of the } k \text{ topmost cards.}}$$
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