1. John got more votes than Jones.
2. Garcia's 'Change' slogan got 26% of the votes.
3. Marie's 'Action' slogan got more votes than 'Progress.'
4. Mrs. Smith got 14% of the votes.
5. The 'Honesty' slogan scored fewer votes than Lucy.

Clue 1 lets you X out three squares: John is not Jones, John is not fourth place, and Jones is not first place. Clue 2 tells us that Garcia's slogan is Change. But the 26% also tells us exactly where Garcia placed. Clue 4 tells us that Smith is a woman, and the 14% tells us where she must have placed.

Clue 2 says that Garcia (Change) got 26% of the vote. This couldn't be first place, because the winner got a majority (more than 50%). But it must have been second place, with third and fourth place splitting less than 24% of the vote. Our grid now shows us that Brown took first place. That leads us to Jones taking fourth place.

Clue 3 matches Marie with the slogan 'Action' and eliminates the possibility of 'Progress' as first place. Clue 4 puts Smith in third place (14% is less than 26% but more than the leftover single-digit percentage). Clue 5 eliminates 'Honesty' as first place and sets the winner as 'Action.' Marie Brown had this winning slogan. Our grid looks like:

Clue 4 tells us that Smith is a woman (Mrs.), and Lucy (not John) is a woman's name. 'Honesty' must be fourth place (fewer votes than Lucy), so 'Progress' is Lucy Smith's third place slogan. John Garcia used 'Change' as his second place slogan. Carlos Jones took fourth place with 'Honesty.' The winner was Marie Brown with 'Action' as her slogan.