The etymologically correct term is polygyny since polygamy is the case with multiple spouses for both genders (the term for one woman with many husband is polyandry).
I don’t know how to draw the game tree in blogspot so verbal description should do.
It is a two stage game where on the first stage the husband decides whether to do polygyny and tell his wife. I will only discuss the case of husband with poligyny inclination since otherwise nothing happened and the couple stay monogamous. On the second stage the wife decide whether to accept her husband intention to do polygyny or not.
For husband the preference in descending order is polygyny, monogamiy (with first wife) and divorce first wife to marry the incoming woman (before receiving complains, this condition not apply to all men/husbands). I will discuss the wife's preference a bit later.
Since the husband strictly prefer polygyny to divorcing the first wife to marry the incoming woman then the first wife could threat for divorce if she don’t want poligyny. The husband cancels his intention and they stay monogamous.
Why this is not always the case? The threat is not credible if the husband know the first wife’s preference is (in descending order) monogamous, polygyny & divorce. Meaning the husband know the wife in the end will choose to accept the second marriage than get divorce and be a widow.
The financial struggle of non-working women to get new job after long unemployed is an uphill struggle. There is also problem of society looking down on widows and the difficulty to get new husband after divorce.
First wife everywhere! You can, paradoxically, prevent your husband (if he has the inclination) from conducting polygyny by credibly demonstrating your willingness to divorce him.
So get a good job with decent salary, keep the social network (in case you need better job/ husband) and staybeautiful. You will keep your husband of yourself as long as you want him (caveat, if the assumptions used above are applicable in your case).












