Gary Becker taught social interaction consumption models. He stressed two cases. In the "snob good" case, Jane's demand for a product declines if everyone else is consuming it. In the "bandwaggon case", Jane's demand for a product increases if everyone else is consuming it. In the case of female public transit use, I predict that the bandwaggon equilibrium will emerge due to a "strength in numbers". If young women anticipate that a large % of riders will also be female, they will feel safer and this will increase their likelihood of riding public transit. This is a type of community policing as other women will be looking out for their safety. The low price of public transit has a direct incentive effect encouraging women to ride it and it solves a co-ordination problem as women will know that other women will be riding with them.
This point has been studied in a more general setting by Pat Bayer and Chris Timmins.
In the traditional discrete choice demand model (such as choosing whether to commute by bus, car or by foot), people compare the exogenous attributes of each choice (so commuting by foot is cheap but slow while commuting by car is expensive but fast) and make their best choice. As the researcher observes each person's choice set and their actual choice, the researcher can use revealed preference logic to estimate the weights that the decision maker placed on the relative importance of each product attribute.
Bayer and Timmins extend this logic by arguing that not only does a product such as a Tesla (in the discrete car choice problem) have unique features but another feature of this product is the demographics of who consumes it.
So, suppose that Matt values the Tesla's raw acceleration power and he also likes the fact that the typical Tesla buyer is a "cool person". Note that this 2nd attribute is an emergent property of who buys the car. If Elon Musk could nudge more "cool people" to buy his car, then this would further increase the demand for the Tesla even if its physical attributes do not change.
The New Delhi policy makers have achieved this same "composition shift". While they haven't upgraded the public transit system, they have shifted its demographic ridership composition by encouraging more women to ride it.
This is actually a hard mathematical problem because the probability that a woman rides public transit is a function of whether the average woman rides public transit So, there are multiple equilibria here. The "free transit" helps to pick out a new equilibrium.
A good urban economics/development thesis could both study how much ridership increases and how much female quality of life and economic opportunity in New Delhi increases due to the "strength in numbers".
An interesting interplay between the demand elasticity parameters arises. The more price responsive are women to using public transit, then the larger will be the composition shift. The net increase in female benefits from this policy will be an increasing function of the price elasticity and the bandwaggon elasticity.
Now, throughout this piece I have ignored male behavior. Will men ride public transit more or less as more women are riding the system?