There is an average amount (or more appropriately, percentage of the maximum) employees spend on travel expenses. This amount is almost certainly extremely close to the maximum allowed (or 100%), and certainly not the minimum possible. Employees should be compensated for spending less than this average amount, for instance, 50% of the savings. As an example, if I spend $60/night at a hotel instead of the average $90/night travelers spend at that location, I'll receive $15/night compensation for using the cheaper hotel. The same goes for rental cars.
If adopted, this will inevitably encourage people to save the government money, as the traveler will be compensated for their effort/sacrifice. The new process could be implemented with negligible effort since there is no additional paperwork (e.g. receipts) that needs to be filed, and only a simple calculation would be added to the Defense Travel System, GovTrip, or whichever system is used to process travel vouchers.
Considering the minimal cost involved in implementing my suggestion (i.e. minor changes to regulations and voucher system coding), the only way to NOT save money is if the average reimbursement amount in an area is equal to the lowest possible lodging or rental car rate there. There are no savings in this case, but neither is there any harm or burden since the status quo is preserved.
Granted, there are "prudence requirements" in travel regulations such as the JTR, but the regulations themselves concede there is no way to enforce them. Also, certain government programs exist to facilitate and encourage adherence to the prudence requirements (e.g. FedRooms), but there are still many cheaper options available. Regardless, the fact is that spending habits of travelers, and any laws related to them, are reflected in the average percentage of travelers’ allotted per diem they spend on lodging and rental cars, and compensation to the traveler would be based on that amount.