There's a couple of points to address here.
When you form a Yaku (let's say a 1 point Yaku : Tane, 5 animals),
you've got the possibility to call Koi-Koi and keep playing the turn
instead of scoring your points. In this case, the turn's winner will
double his points.
If you declare koi-koi, then it is your opponent that will get double points if that he/she manages to complete a yaku before you. Declaring koi-koi by itself does not give the one who did it a doubling of points. Rather, it's a risk-reward tactic to create a better hand and score more points. After all, if you score 7 or more points they will double as well, but doing often requires declaring koi-koi, rather than settling for a lower score. This opens you up to possible retaliation if your opponent steals the koi-koi.
The remainder depends on the rule set being used.
Your opponent forms a 1 point Yaku, and also call Koi-Koi
In fact, the question could be different. To avoid this kind of situation, is it allowed to call Koi-Koi if my opponent has already did it, or am I forced to score my points?
Sometimes it's only allowed to declare koi-koi once per hand played. So someone calling koi-koi after you did might not be allowed. In the rules as specified by the booklet with the Nintendo hanafuda decks, it is indeed indicated koi-koi can only be declared once. Since koi-koi rules tend to vary depending on who's playing, it may however be perfectly fine to allow it. I myself prefer playing games where it's allowed but it tends to make for tense games, although sometimes it can be exploited when someone's got a strong hand and their opponent hardly has anything to counter with. Possible variations are
- Koi-koi only allowed once per hand
- Koi-koi allowed after opponent called it, but not twice in a row by the same player
- Koi-koi always allowed
The first would make for the tightest play but with some limitations in how much a game can turn around or how well you can retaliate. The second allows for some more dramatic comebacks. The third is the most lenient, but can also lead to exploiting a hand too much and monstrous scores. In the end players must decide on what they enjoy most. It seems hard to pin any "official" rules. Kind of like how there's multiple betting structures in poker variants (for example, you could play Texas-hold-em no limit or pot limit).
Then there's the matter of what happens if someone calls koi-koi, but the game finishes without any new yaku being made. If koi-koi can be called only once per game, the options are to have the player who called it keeping the points of the hand they had, or to award no points. The latter option makes koi-koi a bit more dangerous, because now you not only have to bet againt your opponent making a new yaku before you, but also that you can complete a new yaku before the end of the game. Also, it would be typical to lose the state of oya (dealer) if you called koi-koi but then failed to make a new yaku and get no points. Clubhouse Games on the Nintendo DS, featuring koi-koi, lets the player keep their points if they call koi-koi but the game ends without new yaku. It also allows koi-koi only once. The booklet with Nintendo hanafuda decks from the European edition seems to indicate that failing to make a new yaku after koi-koi ends the game on a draw, with no points scored and the oya going to the opponent. Again, I think it's a matter of choice for the players and a source of variation in the game. Not awarding points when the cards run out makes the game more exciting and introduces more risk-reward mechanics.
If koi-koi can be called multiple times per game, you either still have the choice of ending a on a draw without points awarded, or letting the last person to declare koi-koi get their points. If they did so after the other player had called koi-koi before, it would make sense to give them double points because they stole the koi-koi attempt.
I think the strictest set of rules would be:
- Only one koi-koi allowed per hand
- Ending by running out of cards awards no points (draw)
- Ending on a draw after calling koi-koi makes the opponent the oya