Polygamy is beneficial to
- the "desirable" men (i.e., men with power/status/wealth who can attract multiple mates) and
- "generic" women (i.e., women who cannot hope to monopolize a "desirable" man, but who can benefit from "upgrading" from being the one and only wife of a good-for-nothing bum to being the 7th wife of a rich man; this is an "upgrade" because a small share of a big fortune can be more than almost nothing).
At the very least the option (for men: taking an additional wife; for women: "upgrading") improves their negotiating power in marriage.
Note that polygamy leaves many "generic" men without a mate (and without even a hope of getting one) which puts them in a desperate position (leading to very risky behavior).
A promise of enforcing monogamy gives hope to the mass of "generic" males and thus is an extremely profitable move for any revolutionary in a polygamous society.
Observe that both Jesus and Muhammad used this move: Christianity outlawed polygamy completely, while Islam limited the number of wives to 4 and gave the promise of conquered women to the mujaheddin (jihad participants).
What really matters is not the legality of polygamy, but its availability. If a "desirable" man can have an "official" mistress and no one bats an eye, then the society can be considered to be de facto polygamous.
I would expect more correlation between the spread of polygamy (measured by, e.g., Gini index of number of women / men or the entropy of the distribution) and the violent crime rate (which is mostly perpetrated by males), because, as I already mentioned, men without a hope for a mate tend to engage in extremely risky behavior in the off-chance of radically changing their status.
The crime rate, however, can be moderated by an oppressive government or given an outlet in military adventures, so the formula would be something like
glm(crime ~ polygamy + oppression + aggression)
glm stands for, as DVK guessed, generalized linear model.
In other words, if we build a regression model
crime ~ A + B * polygamy + C * oppression + D * aggression
and I conjecture
A > 0
B > 0
C < 0
D < 0
This, of course, should not imply that that there are no other possible predictors, e.g., the economic inequality as expressed as, e.g., Gini index, is probably positively correlated with crime, while general wealth negatively.