The power chieftains hold in these positions is usually:
* Organising and directing the hunters, and dividing their kill.
* The power to declare war, and to divide the spoils.
* Powers relating to treaties and trade with other tribes.
* Adjudicating disputes between families.
The chief and his retinue are often the only recognisable political offices in such societies, in the sense that:
* There is an open and explicit process for selecting the office holder.
* Everyone openly acknowledges their authority.
* They make open proclamations with regards to their rules and determinations.
* They are held accountable for their decisions and their outcomes.
They constitute the public, universal aspect of power — formal authority. This is not the only kind of power possible.
As an earlier poster notes, in most East Asian societies from time immemorial women held the purse strings. Many societies are organised in such a way, that women have great power in aggregate but men hold the official, political positions. Susan Carol Rogers’ “The Myth of Male Dominance: Female Forms of Power” explores this dynamic in French peasant society. Patriarchy and women being in practice more powerful can co-exist.
There are virtually (actually?) no matriarchies in history or on the earth today, in the sense that women hold the majority of positions where power is open, accountable and governed by law (or custom). In most historical societies, women held none of these positions. That doesn’t mean they just rolled over and played dead while men ran riot with the power.
Perhaps as modernity got underway, the official power of men became much more significant. It wasn’t always the case that getting a new road in your town required approval at the state/province level. Informal, local power receded from view and it was only reasonable that women sought the formal, political kind in modernising societies. This change also makes it hard for us to recognise the importance of informal power in simpler societies.