He thinks about himself first. And there's not too much reason for him to do otherwise, since he can't channel and is surrounded by people who can (Eggy, Nyn, Rand, Moiraine).
When they're all in danger, he'll step up, but it's easy for him to see this as a big mess all to do with the "bloody power." He knows how easy it is for heroes and other folks to get caught up in these sorts of battles, and he doesn't want to end up as fodder. Unless he sees someone good in danger, and then he's compelled to move.
Mat is always at his least selfish "in the moment." Out of the moment, it's about (a) what he wants and (b) the obligations he can't get out of.