Democracy

$N$ people gather around a circular table to play Democracy. Each game of Democracy consists of several rounds.

A round begins with an Election procedure: each player places one card with their name into a pile. The cards are thoroughly shuffled, and one card is drawn at random from the pile, while the others are discarded. The person whose name is on the drawn card becomes the President for that round. As President, this player can issue humorous decrees, make funny sounds, send participants into exile until the end of the round, and in general, do whatever they wish.

When the President's term ends, the round concludes and the next round begins. If during any round, the President is someone who has already served as President in any previous round of the current game, the Election is declared rigged and Democracy ends.

Naturally, every participant wants to become President; however, it is not always possible for everyone to do so in one game. Therefore, an important question arises: on average, how many games must be played before each participant has been President at least once?