Can gender-balanced social norms mitigate the gender differences in competitiveness that are observed in traditional patriarchic as well as in modern societies? We experimentally assess men's and women's preferences to compete in a traditional society where women and men have similar rights and entitlements alongside a patriarchic and a matrilineal society which have previously been studied. We find that, unlike in the patriarchic society, there is no significant gender difference in the inclination to compete in the gender-balanced society. We also find that women's decisions in our experiment are optimal more often than men's in the gender-balanced society - opposite to the pattern encountered in the patriarchic society. Our results highlight the importance of culture and socialization for gender differences in competitiveness and suggest that the large gender-differences in competitiveness documented for modern societies are a long-term consequence of a patriarchic heritage.