The reason he is running is the same reason as Christine Todd Whitman wrote "It's My Party Too." He sees the likes of Carson, Cruz, Santorum, Huckabee and others in the race and thinks someone should stick up for the 'Rockefeller Republicans' who want to make the party more like conservative parties in the UK and Canada (more inclusive and secular). I think David Frum's recent essay
What Republicans Can Learn From British Conservatives articulates what Pataki (and to some extent others like Kasich) hope for the future of the GOP.