The player who initiates the game can win up to three crowns in the first round. In a random game, it is possible for the other player to have already won some crowns. This might happen if the player had already started the game before you two were selected to play together. In that case, you will be able to get more than three crowns in your first round.
When you are matched with another player immediately after starting a random match, the other player is always one turn ahead of you. That is the reason your opponent might have already won some crowns before you.