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In practice, how does TCAS decide which aircraft to climb and which to descend; or what factors does it take into account.

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The threads linked in the comments provide an answer, but only when read in conjunction with each other.

In isolation, for a level intruder, TCAS will produce a descent - which would tell the other aircraft (via Mode-S coordination) to climb.

In case of an encounter at same altitude with a non TCAS equipped aircraft, assuming that neither climb nor descent are inhibited, the TCAS will produce a descent advisory (part 2, sec. 2.2.4.1.2.1.5 - sense selection).

What RA will be generated if intruder traffic is at our level in TCAS II?

But what if the other aircraft is "thinking" the same thing? That's the heart of this question. The answer is, whichever TCAS makes a decision first, and then in case of a tie, the lower Mode S address "wins":

When a RA is selected, it is transmitted to the other aircraft. When the other aircraft receives that message, it will only use the opposite sense {climb/descent} for its own RA. In the rare case that both aircraft transmit their RA intent at the same time, the aircraft with the higher Mode S address will give in and reverse its RA sense if it is conflicting with the other.

How does TCAS work?

For further details, go back to the threads linked here; they have good sourcing of their answers. This answer is focused on exactly the question asked here: when everything is same-same, who goes up & who goes down? In short, the aircraft with the lower Mode S address will descend.

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