On Earth ancient cultures developed marriage system of polygamy where one husband had multiple wives. What environmental or social differences would you need to create a world with an ancient culture that had one wife with multiple husbands?
The tech level will be on par with ancient Greece or Rome. The setting will take place in a parallel world similar to our own with subtle differences inhabited by humans.
That is what is called polyandry, and often (but not always, oddly enough) is paired to a matriarchal society. There are still some examples today of such society in Tibet where several distinct groups organized in that way. It's what you would call "classic polyandry" In this system the land is rather scarce so society evolved following the most logical path of having several men working one plot rather than trying to find one property each. Often in these societies one woman is married to 2 or more men that are all brothers (the men are related to each other, not to the woman). This allows the owned land to remain within one family undivided as these societies are not matrilineal (matrilineal = property passes from mother to daughter) Another reason that moved some societies towards polyandry was a matter of protecting the household while the husband was away. With multiple husbands there would have always been someone present (Eskimo, Inupiaq, Iglulik, Alutiq and Inuit were an example of this) As arrangement, polyandry was widely used: South-american tribes used it (Bari, Yanomamamo, Cubeo, Aymara, Panoan Matis, Cashinaua, Guaja, Zo'e, Suri, Kaingang, Ache and others), North-american Natives used it (Aleuts, Comanche, Tlingit, Shoshoni, Cherokee, Blackfoot, Pomo, Utes, Innu, Pavitoso and Pawnees are just some of the most notable examples cited in literature, according to researchers many more have been reported but overlooked in the past), Arctic populations used it (as mentioned) was common in Asia and Africa and it's not completely disappeared yet, researchers say that there are at least 80 different societes in the modern world that still practice it in one way or the other. In Sri Lanka for example this is recognized under Kandyan marriage law. Even in Europe, ancient Germans and Britons were some tribes that researchers mention as practicing some form of polyandry.
A paper by Katherine E. Starkweather and Raymond Hames "A Survey of Non-Classical Polyandry" not only shows how it was a practice more common that what was thought but also gives an explanation on why it was thought that way (from here):
So how is it that, in spite of all this evidence of polyandry accumulating steadily in the literature, anthropologists for so long passed along the "it's virtually non-existent" story? Starkweather and Hames suggest anthropology has been accidentally playing a scholarly version of the Telephone Game.
In 1957, George Murdock defined polyandry in a seminal text as "unions of one woman with two or more husbands where these [types of union] are culturally favored and involve residential as well as sexual cohabitation." Using such a strict definition, Murdock could accurately say polyandry was extremely rare; almost no cultures have polyandry as the dominant and most preferred form of family life.
Then subsequent scholars mis-repeated Murdock's remark; polyandry went from being understood as "rarely culturally favored" to "rarely permitted." Thus mating diversity that was known to exist became relatively invisible in the big story told by anthropology about human mating. (If you write off every exception to a supposed rule, you will never think to challenge the rule.)
In an email interview with me, Starkweather remarked, "I don't think that anyone, including Murdock, was operating from an explicitly sexist standpoint. However, I do think that the definitions of polyandry, and thus perceptions about its rarity, may have been due at least in part to the fact that an overwhelming percentage of anthropologists collecting data and shaping theory at the time were men." During Murdock's time, "there seemed to be a fairly pervasive belief that polyandry didn't make any sense from a male's perspective."
Polyandry does not mean that women are the head of the household, in many cases you find that is the eldest husband the one that takes decisions. Matriarchal societies (= headed by females) on the other hand do not imply that there is polyandry and matrilineal societies do not mean polyandry or matriarchy is included in the deal. Nothing prevents you to put all three together and have a society where the women took the role men had in our more recent, and well known, western society. Despite having found quite some people online calling it the end of humankind its not at all impossible that the same environmental, cultural and religious situations that brought so many groups to practice matriarchy, matrilineality and polyandry cannot be put all together in an alternate world.
Some references for the above are (there are many more, it would be a long list, its long as it is and i wont bother formatting it as this isn't a school paper): Johnson and Zhang ‘Matriarchy, polyandry, and fertility amongst the Mosuos in China’; Ellis ‘On Polyandry’; Levine and Silk ‘Sources of Instability in Polyandrous Marriages’; Smith ‘Is Tibetan Polyandry Adaptive? Methodological and metatheoretical Analyses’; Steward ‘Shoshoni Polyandry’; Starkweather ‘Exploration into Human Polyandry: An Evolutionary Examination of the Non-Classical Cases’; Starkweather and Hames ‘A Survey of Non-Classical Polyandry’; Berreman ‘Pahari Polyandry: A Comparison’; Cassidy and Lee, ‘The Study of Polyandry: A Critique and Synthesis’; Childs, ‘Polyandry and population growth in a historical Tibetan society’; McLennan 'Primitive Marriage: An Inquiry into the Origin of the Form of Capture in Marriage Ceremonies'
In The Moon is a Harsh Mistress, Robert Heinlein posited Lunar penal colonies analogous to what Britain did with Australia in real life. Due to the far higher incidence of incarceration among men, the population was heavily skewed toward males. That left the dominant family structures as some sort of wife-sharing arrangement: two or more husbands per wife. The "co-husbands" were often also partners in a family business as well. As time went on, these families raised boys and girls born in roughly equal numbers, reducing the imbalance, while the various nations of Earth continued to ship convicts, still far more male than female, helping to maintain the imbalance, albeit at a lower ratio.
If you can devise a mechanism by which females are in chronic short supply relative to an abundance of males, you can have your polyandry-dominated society. One possible way to do that is to have a neighboring polygynous culture that has extra males who can't find mates there, and would rather emigrate to the wife-sharing culture.
If you’re looking for environmental reasons in addition to cultural ones, then there’s no better place to look than sex-linked genetic defects. Colorblindness, for example. 1 in every 20 males is red-green colorblind, but only 1 in every 400 females is colorblind. This is because colorblindness comes from a recessive mutation in the X chromosome, which means that males, who only have one X chromosome, only need one mutated gene to be colorblind, but females have two X chromosomes, and need two mutations. And because this gene is recessive, it’s even rarer. A colorblind female can only be the result of a colorblind father, and either a colorblind, or carrier mother, someone who has only one mutated X chromosome, not both. There’s some more stuff about X-Linked recessive inheritance here.
Anyway, instead of colorblindness, let’s replace that with, say, resistance to some horrible, flesh-melting disease. This hypothetical disease is responsible for killing so many people, that if you’re not immune, there’s about a 10% chance of survival, tops. So, because this immunity is X-linked and recessive, this means that a pretty decent population of males are immune, but not a lot of females. This encourages the “many husbands, one wife” social structure for two reasons, A) many pairings of an immune female with immunized, carrier males increases the chance of more immunized females, and immunized children overall being born, and B) There literally aren’t enough females to go around anymore, so society had to shift.
Couple this with general societal inclination and matriarchal social structures, and there you go!
I think you're looking for matriarchy. And they have existed.
Edt: Since at least one of my downvotes is for the matriarchy part I'll explain a little more.
I'm using the definition my sociology professor used.
A matriarchy is a society in which females, especially mothers, have the central roles of political leadership, moral authority, and control of property,
This isn't a feminazi dictatorship. And the ones that have actually existed (very few) have tended to be polyamorous in their relationships. So while this isn't the only answer, it is certainly AN answer to the question.
If I remember my sociology correctly (been years!), in most matriarchies the women are the owners of land and parentage follows the line of women, which actually makes sense when you realize there is never a question if a child is a woman's or not.
Often in the Matriarchy, men and women are not tightly coupled. The women raise the children and the men have a looser relationship, making it very easy for a women to have multiple men, but likely sharing those men with other women as well.
The primary way for women to have multiple men requires that men do not 'possess' women. (of course this would also require monogamy to not be the expected norm)
I also forgot about Polyandry, which is where the woman has are more than 1 husband.
There is also polyamory which includes both polyandry and polygamy.
The Toda People practice Fraternal Polyandry
Also
In contemporary Hindu society, polyandrous marriages in agrarian societies in the Malwa region of Punjab seem to occur to avoid division of farming land.[14]
and
Some forms of polyandry appear to be associated with a perceived need to retain aristocratic titles or agricultural lands within kin groups, and/or because of the frequent absence, for long periods, of a man from the household. In Tibet the practice was particularly popular among the priestly Sakya class.
These are some of the factors allowing men to marry more than once:
They are considered heads of households in most societies around the world.
They are (primarily) responsible for financially supporting their families.
They are physically stronger than women. Not implying that marriage is a wrestling match for physical superiority, but women feel safer with physically strong men. It is a psychological factor.
Men handle outdoor and legal/social affairs of the family.
Children inherit father's names.
If you have a society where these roles are filled by women, then a matrilineal social structure can be formed. You may want to read a list of matrilineal societies and find the commonalities between them.
Polyandry and polyginy are two drastically different things. Polyginy emerge when a lot of men die constantly (who said war?) as a way to repopulate the tribe. It makes it so you can send all valid men to a certain death, have 10% of them survive and still have as much children in the next generation as you would have if everyone survived (something you can't do with polyandry + send women for obvious biological reasons)
polyginy is good when your population becomes too low because of high mortality. Polyandry might emerge during prolonged peace period (no male mortality) in a place with limited resources (so population growth is detrimental)
Obviously I'm only talking about "natural" polygamy and not about cultural polygamy however rites and customs have to come from somewhere and it can sometimes come from past environmental pressure and then persist even if those pressures no longer exist
Some food for thought...while it doesn't necessarily fit the tech level of ancient Greece or Rome, take a look at the current conditions in poor yet urbanized area of the United States. In these communities, a large percentage of the men are either incarcerated or are busily killing each other off. While some of AndreiROM's conditions are met, others are exactly opposite. But it's an interesting parallel.
I'm surprised no one has mentioned Pandora's Genes as a source of inspiration. In a post-apocalyptic Earth, science has loosed, among other things, a disease that largely effects women, killing them sometime between conceiving a child and shortly after giving birth. As a result, each woman generally takes 2 husbands (often but not always brothers, or at least good friends). Even if you've already got enough inspiration for your world building (I see you've already accepted an answer, but I just had to post), it's worth a read.
Another poster brought up that the idea of polygamy might have come about because one male can impregnate multiple females but a female can only be impregnated once at a time. If the setting provides it, it is possible that a female can be impregnated multiple times by multiple fathers (like cats).
Having the society be (at the very least) a matrilineal one would make it so that it doesn't really matter who the father is, the mother's name is the one who matters and there's no contention that the child is really hers. Inheritance might be slightly different. In a society where you can't take a paternal test, there is an uncertainty on if the child is actually the father's (aka, the concept of virginity and unbroken hymens). In a matrilineal one under this concept, you could expect that child rearing is mostly communal (at least in the family units proposed), because all males can reliably expect that at least one of the children is their's.
Of course, that also means that there would be more like a litter of children than a single birth.
EDIT: the matrilineal society brought up in other replies came about because of lack of arable land space. aka, having one large family takes up less space economically then three smaller ones.
That's a tough one, mostly because it's not within human nature to live within that particular arrangement. Psychologically and instinctively that wouldn't work.
A lot of work would have to go into convincing the men to accept the arrangement. Most likely the society in question would have to be matriarchal, and possibly worship a female goddess, such as Gaia, whose express wishes are that women rule the world, etc.
A few things that you would need to establish are:
I'll think of some more points, but I think that's a pretty solid foundation to start with.
