ending a game on a tie?

I've never heard it that way. That usually means it's British English.

I don't know whether it is "right" or not, but everyone is going to understand what you mean and no one is going to get upset that you used a slightly different word. I wouldn't worry much about this one.

That makes sense. Thank you.

It's not British English.

"The game ended in a tie" is common usage here. I've never seen "on" used.

(In fact, we usually say "the game ended in a draw".)

Thanks for your confirmation. Funnily enough, the textbook from which the expression comes is compiled by British experts. But I believe it is pretty old and not accurate enough.