In a deck of $N$ cards, the cards are denoted by $1$ to $N$. These cards are dealt to $N$ people twice. A person $X$ wins a prize if there is no person $Y$ who got a card with a smaller number than $X$ both times. Determine the expected number of prize winners.
1. **Define the problem in terms of permutations:**
   - We have a deck of \( N \) cards, numbered from \( 1 \) to \( N \).
   - Each card is dealt to \( N \) people twice.
   - A person \( X \) wins a prize if there is no person \( Y \) who got a card with a smaller number than \( X \) both times.

2. **Label the people by the card number they received on the first deal:**
   - Let the card received by person \( n \) on the first deal be \( n \).
   - On the second deal, let the card received by person \( n \) be \( \sigma(n) \), where \( \sigma \) is a random permutation of the set \(\{1, 2, \ldots, N\}\).

3. **Determine the condition for winning a prize:**
   - Person \( n \) wins a prize if \( \sigma(n) \) is a left-to-right minimum of the permutation \( \sigma \).
   - A left-to-right minimum in a permutation \( \sigma \) is an element \( \sigma(n) \) such that \( \sigma(n) \) is the smallest among the elements \( \{\sigma(1), \sigma(2), \ldots, \sigma(n)\} \).

4. **Calculate the probability of being a left-to-right minimum:**
   - For a given \( n \), the probability that \( \sigma(n) \) is a left-to-right minimum is \( \frac{1}{n} \).
   - This is because \( \sigma(n) \) must be the smallest among \( n \) elements, and in a random permutation, each element is equally likely to be the smallest.

5. **Sum the probabilities to find the expected number of prize winners:**
   - The expected number of prize winners is the sum of the probabilities that each person wins a prize.
   - Therefore, the expected number of prize winners is given by the sum of the probabilities for each \( n \) from \( 1 \) to \( N \):
     \[
     \mathbb{E}[\text{number of prize winners}] = \sum_{n=1}^{N} \frac{1}{n}
     \]
   - This sum is known as the \( N \)-th harmonic number, denoted by \( H_N \).

6. **Express the final result:**
   - The \( N \)-th harmonic number \( H_N \) is defined as:
     \[
     H_N = \sum_{i=1}^{N} \frac{1}{i}
     \]

The final answer is \(\boxed{H_N = \sum_{i=1}^{N} \frac{1}{i}}\).