Wouldn't the extra money allow their message to get better "saturation"? If they can get their message out and cover more ground, be repeated more often, etc. Doesn't that give them an advantage?
That seems likely, but the research is clear that it's not what actually happens.
I haven't looked at the research myself, but among Economists this is an established fact.
To me, this is good news. The cynical view that anyone can buy an election by outspending their opponent is depressing, and I'm glad it isn't true.
BTW, it is true that the candidate who spends more usually wins. But that mostly just reflects that they're the most most popular candidate, so they get the most donations.