Translation:With the debate, the parties fight for votes.
I think it's because there's a part that is implied:
om stemmen (te krijgen).
At least, that's my interpretation!
Any natives who could confirm?
There is a slide difference in meaning. Difficult to explaine. Vechten voor stemmen implies that you fight for those votes. Vechten om stemmen implies you fight to get the votes. (Don't know if it is clearer now).
That's what I was trying to say.
Can I correct a few details?
In Dutch, are commas not used as often as in English to make sentences clearer?
I didn't understand this sentence until I saw the translation, because I thought 'debat' was the subject of the verb 'vechten'.
If there were a comma after 'debat' (as there would be in English) it would have been clearer to me.
In my opinion it's perfectly clear that with the debate cannot be the subject of this sentence since it's an Adjunct (category: prepositional phrase) (in this particular case I'd call it Adjunct of Instrument).
For further information, please start by reading this: https://en.m.wikipedia.org/wiki/Adjunct_(grammar)
That being said, what you said about a comma being necessary after a multi-word adjunct that is placed at the beginning of the sentence is true (I was going to say that it's not always necessary, but I decided to check: http://www.grammar-monster.com/glossary/adjuncts.htm)
Hope this helps!