Men with dominant, aggressive faces (high fWHR) are preferred for short term relationships
Facial Width-Height Ratio (fWHR) is a proportionate measure of a man's facial width to his height, measured laterally from the edges of the zygomatic processes and vertically from the mid-brow to the top of the upper lips. Two examples of how this measure works are posted under 'Figures' below.

Valentine et al. (2014) performed a study utilizing a speed dating format which consisted of young male and female subjects (n=159;78 young men and 81 young women) who weren't compensated in any way for their participation, and thus were participating in the study because they were "(looking for the) chance to find a real-life partner, suggesting the choices were primarily motivated by actual mating interests."

The males in the study were rated by independent raters on the following metrics:

Dominance (unisex raters)
How aggressive would this person be if provoked? (unisex raters)
Facial adiposity (unisex raters)
Attractiveness (female raters only)
Inter-rater reliability was high for all four dimensions (as measured by Cronbachs Alpha). Thus it was demonstrated that raters could come to a clear consensus on how dominant, aggressive, overweight, and attractive each man appeared.

Analysis of outcomes showed that men's fWHR explained fully 34% of the variance in women's interest in men for short-term relationships. fWHR was a very powerful predictor of being chosen for future dates by the women, even when higher fWHR faces were not deemed more attractive by women. High fWHR men were perceived as more dominant and thus more desirable to women for short-term sexual involvement.

Quotes:

The direct and indirect effects of fWHR combined explain 34% of the variance in women's interest in short-term relationships
Men with wide faces are not more physically attractive to women, but are preferred for short-term relationships and future dates.
Thus, results support our model that a higher male fWHR leads to perceptions of greater dominance, which in turn makes men attractive for short-term relationships
The results are also consistent with the non-mutually exclusive explanation that women may be favoring dominant-looking men to gain protection in a short term context at the expense of having long-term investment.
References:

Valentine KA, Norman PLI, Penke L, Perret DI. 2014. Judging a Man by the Width of his Face: The Role of Facial Ratios and Dominance in Mate Choice at Speed-Dating Events. Psychological Science. 25(3): 806-811





Women who have experienced domestic violence find men with higher fWHRs more attractive
Liberz et al. (2018) conducted a study to examine victimized (pertaining both sexual molestation and domestic violence) women's perception of the facial and behavioral cues of potential male aggressive behavior.

These women were presented with a series of male faces that varied in fWHR, and were questioned as to their level of physical attraction to these males and their perceptions of the males potential tendencies towards behavioral aggression.

They were further asked to imagine these men in close physical proximity to themselves, and were asked to press an arrow key up (closer) or down (away) to assess their preferences for physical closeness to these males.

Concurrently, the actual levels of behavioral aggression of several of the men in the photographs were assessed by the researchers utilizing a simulated economic game, in which participants could 'punish' opponents by stealing points from them, to no advantage in 'winning' the game.

It was found that the women who had suffered sexual molestation during childhood perceived the men (in general) as more sexually attractive, a finding stronger in those who had been victim to IPV (intimate partner violence). The researchers also found that while the victims of IPV viewed men with higher fWHRs as more potentially aggressive, they perceived them as significantly more physically attractive then the control group of women. The victimized women perceived men with higher levels of behavioral aggression as measured by the economic game as more attractive.

Quotes:

Moreover, (women) who had experienced IPV rated men with higher fWHRs and men with higher values of actual aggression to be more attractive as compared to (women) without histories of IPV.
A reduced appraisal of threat signals and an attraction to wider-faced and more aggressive men might increase the risk for revictimization.
It is possible that those men with masculine facial features signal more protective behavior and security, attributes that revictimized women might desire. These "psychological barriers" seem to be important predictors of coping responses and should find more consideration in prevention programs.
Reference:

https://www.sciencedirect.com/science/article/abs/pii/S0165178117303815





High fWHR is associated with greater lifetime reproductive success
Loehr & O'Hara (2013) examined data gathered from Finnish national archives pertaining to Finnish conscripts who fought in the Winter War with the Soviet Union that lasted from 30 November 1939 - 13 March 1940. The researchers examined photographs of the soldiers to calculate their fWHR (Facial-width to height ratio), which has been stated in previous studies to be associated with greater survival rates from traumatic injury, mating success, and aggression.

They found that men with wider faces had higher lifetime reproductive success (controlling for their chance of surviving the war). However, fWHR was not found to be significantly correlated with the chance of surviving the conflict. The researchers stated that this was possibly due to the use of modern weaponry in the war, as opposed to previous studies finding that fWHR increased rates of combat survival only for close-range (melee) encounters.

It was also found that the Finnish soldiers with thinner faces attained higher rank in the Finnish military, seemingly contradicting previous findings in the United States that fWHR was associated with higher ranks in the United States military. The researchers stated that this could be due to men with higher fWHRs being perceived as less trustworthy, and trustworthiness could have been more culturally valued and thus rewarded by the World War II era Finnish military hierarchy then in the contemporary American military.

Quotes:

On 30 November 1939, the Soviet Red Army invaded Finland, starting the 3.5 month long Winter War. We used Finnish archives to collate data on survival and number of offspring for soldiers who fought in this war to explore the correlation of fWHR with rank, fitness and survival.
Soldiers with wider faces had more children after controlling for wartime survival, (analysis with full data; a soldier with a face 1 s.d. wider has 1.88 times as many children.
Previous research has found that wider-faced males are less likely to die violent deaths, but only when close physical contact is involved (e.g.death by knife wounds or strangling), and not when technology is used (death by gunshot or poisoning.)
References:

Loehr J, O'Hara RB. 2013. Facial morphology predicts male fitness and rank but not survival in Second World War Finnish soldiers. Biology Letters. 9: 20130049.





Facial shape predicts perceived leadership ability and election outcomes
Quotes:

Split-second judgments of competence from facial images are positively correlated with real-life electoral success. Judgments of competence from briefly presented (i.e., 1/10 s) face images have predicted outcomes in elections for United States (US) congress, governor, and president. Quick leadership judgments from faces have also been found to predict voting decisions in the United Kingdom, Canada, Australia, Ireland, Italy, and Japan. Children's judgments of leadership can predict electoral success as well, and closely match leadership judgments made by adults.
Recent research has demonstrated that facial width-to-height ratio (bizygomatic face width, with length of the face defined as the distance from the upper eyelid to the top of the upper lip, see Fig. S1.) predicts leadership success in businesses with low levels of management complexity and predicts achievement drive in U.S. presidents. Recent studies have found that business leaders in the United Kingdom have higher width-to-height ratios than age and sex-matched counterparts. Facial width-to-height ratio correlates with perception of dominance and aggressive and untrustworthy behaviour, traits that likely impact leadership success.
Baby-faced individuals appear less competent which could influence leadership perception.
Leadership selection is also influenced by perceived facial masculinity (sexual dimorphism in face shape). For example, masculine face structure is preferred in leaders' faces in times of intergroup conflict, while more feminine faces are preferred during periods where within-group relationship maintenance is emphasized.
References:

https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0080957