Actually the pre-merge companies in NYC didn’t use shared stations, except in the joint IRT/BMT section (#7 east of and including Queenboro Plaza). That is why inter-division transfers are typically by tunnel.
Other than that, it is interesting to see which of the two competing systems attract more customers. However, inter-division transfers may complicate the matter. In the pre-TA era, customers wishing to make inter-division transfers have to pay seperate fares for each division. It discouraged customers from making such transfer, benefiting the competing companies (by lowering each other’s revenues). If a unique station (or all the square stations) is served by only one division, you may want to ride solely in that division, given the choice. However, there are cases where inter-division transfer is benefical, e.g. only stations belonging to one division are near you, or the other division provides a short cut. Making rules about and programming these factors may be hard. If we open up free transfers, and let customers decide their itneraries as they currently do, then the competition comes down to how many stations each division serves that the other does not, and for customers making inter-division transfers, there will be no gain for either division.
The implementation could be like this: at the start of each week, the two players choose their upgrades (Week 2 onwards) and take turns to plan their systems, one by one. To avoid advantages by order, the turns continue until one of the players ends his/her turns without making any changes, thus allowing one to react to the other’s actions. Then the system is locked in and operates for a week. Standard ending rules apply.