Several bodies of theory develop the idea that the intelligence of highly social animals – most interestingly, humans – is significantly organized around the adaptive problems posed by their sociality. By this ‘political intelligence’ hypothesis, sociality selects for, among other attributes, capacities for ‘manipulating’ information others can gather about one's own future behaviour, and for ‘mindreading’ such manipulations by others. Yet we have little theory about how diverse parameters of the games that social animals play select for political intelligence. We begin to address that with an evolutionary simulation in which agents choose between playing Prisoner's Dilemma and Hawk–Dove games on the basis of the information they can retrieve about each other given four broad information processing capacities. We show that political intelligence – operationally, the aggregate of those four capacities – evolves to its highest levels when co-operative games are generally more attractive than conflictual ones, but when conflictual games are at least sometimes also attractive.