The TCAS system has in its data base the service ceiling of the airplane and its registration number. If neither airplane is near its service ceiling, the system uses the registration number to decide which airplane will receive a climb indication and which one will receive a descend indication. Since the system between airplanes "talk" to each other through the transponders, each "knows" what the other's command is. It appears the system work as it should and the crew followed ATC instructions instead of the TCAS alert. ATC does not receive this information and I don't know what it would take to receive it. One of the problems, in a heavy air traffic zone, is that the screens are so cluttered with data, it makes reading the screens difficult at times.