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It may be obvious to you, but for the non-experts here:
What, exactly, is RARS?
How do booking engines use RARS in conjunction with the data from various sources contained in your screenshots?
RARS is the current "Rail Availability & Reservation Service". It manages seat bookings and availability (quotas for Advance tickets, as well as the ability to book flexible tickets on "mandatory reservation" services) for rail services in Great Britain.
It's a commercial off-the-shelf system that Rail Delivery Group purchased and have had some work done on top of to make it more suitable for GB use.
Inventory and pricing managers at train operators are responsible for using RARS for yield management for their services. Some train operators use software to partially or fully automate some of this.
Booking engines must consult RARS when selling tickets that are quota-controlled (Advances) or when selling tickets in conjunction with services that are in the timetable as "mandatory reservations". Only when there is availability, can most retailers sell these types of tickets. A practice that began during the pandemic, many train operators now lie about their trains actually being run with compulsory reservations in the timetable data.
Apart from the underlying class of travel, is there ever a legitimate need for an operator to make the availability of a reserved seat contingent on specific ticket types?
In my view, no. In the view of operators, "it allows innovation!" (e.g. Seatfrog, Standard Premium - which is of course a bodge in the fares data, these tickets are considered Standard Class)
Are there any ways other than ticket type, in which an operator can control the way booking engines determine availability of a reserved seat or counted place?
My understanding is that operators can use any of the following pieces of information to discriminate during availability and booking requests: from station, to station, route code, passenger discount (i.e. which railcard they have), retailer, sales channel (TVM, ticket office, web application).