  The ways in which natural selection can allow the proliferation of
cooperative behavior have long been seen as a central problem in evolutionary
biology. Most of the literature has focused on interactions between pairs of
individuals and on linear public goods games. This emphasis led to the
conclusion that even modest levels of migration would pose a serious problem to
the spread of altruism in group structured populations. Here we challenge this
conclusion, by analyzing evolution in a framework which allows for complex
group interactions and random migration among groups. We conclude that
contingent forms of strong altruism can spread when rare under realistic group
sizes and levels of migration. Our analysis combines group-centric and
gene-centric perspectives, allows for arbitrary strength of selection, and
leads to extensions of Hamilton's rule for the spread of altruistic alleles,
applicable under broad conditions.
