gnat has identified two options that have been shown to work on Stack Exchange.
The first is the "dupe hammer" where someone with proven knowledge of an area can close a question as a duplicate with a single vote - they can also vote to reopen with a single vote too in case they change their mind. As you point out this only works with a fairly large community and/or one with a large number of questions.
The second is informing the asker that they might have asked a duplicate once one or more people vote/flag to close the question as a duplicate. This allows askers to cast a deciding vote should they agree, and it also (hopefully) teaches them to search in future.
One other suggestions that has been raised (but not implemented) on Stack Exchange several times is to remove the reputation (or "seconds" in your case) from answers posted to questions that are subsequently closed as duplicates. The idea behind this is to discourage people from answering. The main drawback with this that I see is that it's a stick rather than a carrot.
Even with the loss of rep/seconds people may still find it easier to answer than close as a duplicate as it can be hard and/or time consuming to go hunting for duplicates. If your system can automatically identify possible duplicates you could present this list to people trying to answer in the same way that Stack Exchange presents possible duplicates when you ask a question. This removes a big barrier to closing - the effort needed to find the duplicate.