Two players in turn play a game. First Player has cards with numbers $2, 4, \ldots, 2000$ while Second Player has cards with numbers $1, 3, \ldots, 2001$. In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card with a larger number, scores 1 point. Then both cards are discarded. First Player starts. After $1000$ turns the game is over; First Player has used all his cards and Second Player used all but one. What are the maximal scores, that players could guarantee for themselves, no matter how the opponent would play?
To solve this problem, we need to analyze the strategies of both players and determine the maximum scores they can guarantee for themselves. Let's denote the First Player's cards as \(2, 4, \ldots, 2000\) and the Second Player's cards as \(1, 3, \ldots, 2001\). The game consists of 1000 turns, and the First Player starts.

1. **First Player's Strategy:**
   - On his turn, the First Player places the lowest card in his hand.
   - On the Second Player's turn, if the Second Player places a card lower than 2001, the First Player counters by playing the next card higher than the Second Player's card.
   - If the Second Player places the card with number 2001, the First Player should pick the lowest card in his hand.

2. **Second Player's Strategy:**
   - On his turn, if it is not the last turn, the Second Player plays the lowest card in his hand.
   - On the First Player's turn, the Second Player picks the next card higher than the First Player's card.
   - On his last turn, the Second Player plays the highest card he currently has (2001).

Let's analyze the outcomes of these strategies:

- **First Player's Turn:**
  - The First Player places the lowest card in his hand, which is 2, 4, 6, ..., 1998, 2000.
  - The Second Player will always have a card that is one less than the First Player's card (1, 3, 5, ..., 1997, 1999). Therefore, the First Player wins all these rounds.

- **Second Player's Turn:**
  - The Second Player places the lowest card in his hand, which is 1, 3, 5, ..., 1997, 1999.
  - The First Player will counter with the next higher card (2, 4, 6, ..., 1998, 2000). Therefore, the First Player wins all these rounds as well.

- **Last Turn:**
  - The Second Player places the highest card, which is 2001.
  - The First Player cannot counter with a higher card, so the Second Player wins this round.

Given the above strategies, the First Player wins all rounds except the last one. Since there are 1000 turns, the First Player wins 999 rounds, and the Second Player wins 1 round.

The final answer is \(\boxed{999}\) for the First Player and \(\boxed{1}\) for the Second Player.