I can think of some ways data science would be trialed in an election but I'm not sure how many of them usable. I am sure some of them are employed already. Measuring outcome on a population is easy. Tailoring it and getting to a stage where you can do uplift modeling and STILL measure outcomes. that is where the problem is.
Here's an approach Trump could take:
Run a poll. 20 pounds if you give your name, date of birth, which candidate you are going going to vote for.
You can use the personally identifiable information to back it up into some credit bureau's data files.
Now slam a region with marketing activity -- maybe talk about how Obamacare sucks in one region, talk about how guns are awesome in another region.
Do a poll again, try to get results from same people. Now you know what kind of people are likely to change their opinion, given the marketing message is a) about Obamacare b) guns being awesome.
Now, for each person, you know associated characteristics(single, likely to visit gambling sites, blah blah) from the credit bureau files and some other third party data. Some uplift modeling here to understand what kind of segment is more likely to change their opinion if you're saying Obamacare sucks.
Now, instead of telling everybody Obamacare sucks, purchase data from your credit buerau or whomever and based on the segments you've defined, just run different messages. You have their e-mail IDs -- put it out on Facebook. You have their phone numbers, put it out on Twitter.
Your marketing investment is likely to give you better outcomes than a blanketed one-size fits all targeting. The problem is.. how do you know it has actually influenced election results? You'll never really know as you can't A/B test an electoral outcome.
Like I said, this is just about being a bit creative. It's not advanced math.
I do think the forces that influence elections are a lot more complicated than this.