Grand Rapids Area Boardgamers
During a round: If you are the first to have at least 10 icons at the top of your played uramaki cards, you immediately score 8 points. Move your pawn on the score track and then immediately discard your uramaki cards. The next player to collect 10 uramaki icons scores 5 points, and the third player scores 2. Again, these points are scored immediately and the cards are discarded. If multiple players reach 10 or more icons on the same turn, ties are broken by the highest total number of uramaki icons on the tied players’ cards. If players are still tied, they both receive the full points for the place they are contesting. If this happens, the next place is skipped in scoring.
End of the round: The player with the most uramaki icons still in front of them scores points for the next unclaimed place (8 points if no one has scored uramaki in the round, 5 if only first place has scored, 2 if both first and second place has scored, 0 if all three places scored). If multiple players tie for the most, they each receive the full amount. It’s possible that one player may be awarded multiple placings for uramaki in a single round.
So at the end of the game, only one uramaki award is given (barring ties). It depends, then, on what you mean by "the end of your game" in your question. If only one uramaki award was granted during the game, and the game ends before second was claimed, then second place is awarded to whoever has the most remaining. Third is not given.