Scoring with "Triumphal Arch" and only one Action card in the deck

Question

Yesterday my son and I played the "Advanced Introduction" of the Dominion Empires edition. Among other cards, it contains the "Triumphal Arch" whose description states:

When scoring, 3 VP per copy you have of the 2nd most common Action card among your cards (if it’s a tie, count either (*)).

The rule clarifications include the following text:

The tie can be for either first or second place: if you had 7 copies each of Villa and Village, you would score 21VP. If you had 7 copies of Villa, then 4 copies each of Village and Wild Hunt, you would score 12VP.

So far, all is clear. However, in our second game, my son focused so much on the Wedding, that in the end he only had exactly one single action card.

We decided to grant him 3 VP for this card, but weren't sure if this was the correct way to do it. Since he didn't have a "second most common" action card and since he didn't have two tying for first place, either, the rules didn't seem clear on how to handle this situation.

Is the "second most common" action card count zero, because there was no tie for first place and nothing for the second place? Or was our way of falling back to the single action card count correct?

It didn't really make much of a difference with just the single card, but in a game where a player only bougt one single type of action card, but had bought 8 of them, the difference between 0 and 24 would be quite significant.

(*) Interestingly enough, in our German version, the translation of the card is slightly different from the English text, it translates back as "if it's a tie, count both" which I initially interpreted in a way that would have counted 7 villas, 4 villages and 4 wild hunts to yield 24VP instead of 12VP.

Answer 1

Most importantly, have fun playing with your son!

For your question, in case there is no second most common Action card in a player's deck, that number would be 0. Or maybe it's non-existent, but you certainly don't count the most common one (that's kind of the strategic point of the card). So your son should have gotten 0 VP from Triumphal Arch in that game.

Another way to think about this conceceptually is that at the end of the game, you write down how many copies you have of each Action card. You then omit then names of the Action cards (so you only have the numbers left), then order those numbers, starting with the highest. Whatever number is in the second spot after the ordering is the number you multiply with 3 to get your VP from Triumphal Arc.

The German translation is wrong in this case. I was confused as well first (not having seen the English text before), thinking you could increase the VP by deliberately producing a tie (say 4x most common Action card, 4x second most common Action).