Compared to giving only one of your partner happiness, you need twice the time to make your partners happy.
- Is this の used as a nominalizer?
- Can の be replaced with こと?
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That の is a nominalizer in the sense that it plays the grammatical function of turning the verb phrase 一人の相手を幸せにする into a noun phrase, just like the の in, say, 本を読むのが好きです. However, the resulting noun phrase seems to have a more specific meaning than 一人の相手を幸せにすること, which refers to the act of making one person happy in general and abstract terms.
Your sentence has no subject or topic, but we can assume it to be a person because of 使える. It’s you (or someone else) who must be able to spend twice as much time, not the act of making other people happy. For this reason, the following sentence would sound unnatural (to me, at least).
This would compare one act against another, but the sentence would end up with a mismatching predicate.
The original sentence must be something like the following in its complete form.
This basically compares two amounts of time or efforts. I would translate 一人の相手を幸せにするの as something along the lines of “the time you need to spend to make one person happy” and this translation is not possible with こと.
In comparison, the following seem both acceptable as the predicate is matching.
So is this person talking about how to do 二股? Please always explain the context when you present a sentence based on an unusual context. Also, note that 一人の相手 here is "one girl", not "one of your partners" (see this). Translating 相手 as "partner" is often misleading (see this).