Reluctant Hero

This would be where the character classified as an anti-hero often refuses to help people even when an opportunity presents itself. They continue to indulge in immoral and selfish behavior and are not a reliable agent of good. They aren't particularly evil--they are just very self-centered. This character was discussed in The Hero with a Thousand Faces by Joseph Campbell

Examples would be Han Solo of the Star Wars Saga and Gru in the Despicable Me series.