The official rules are:
When two aircraft are converging, the one "on the right" has the right of way. "On the right" is from the point of view of an observer in the narrower angle betyween the two flight paths. This is why airplanes have red lighs on the left wing and green on the right; if you see a red nav light the other craft has the right of way. Of course, it's smart for *both* craft to simply maneuver to avoid each other.
In the event of a head-on convergence, where both angles are equal, both aircraft are supposed to "break right": turn to their own right. This has the additional advantage of increasing the visual size of your aircraft to the other pilot: an airplane pointed at you is hard to see, but when banked into a turn the wings become much more visible.
These rules are actually based on the "rules of the road" for ships at sea. (Yes, I know there' s no road at sea, but that's what they're called.) Other rules borrowed from terrestrial navigation relate to the priority of various types of craft pover each other, such as sailplanes over powered craft, and airships over heavier-than-air craft. An aircraft in the process of landing has priority over all others.
The reason pilots will dive when in trouble is that they can almost always dive faster than they can climb. Birds know this too. Unfortunately "down" is the same direction for both airplanes, even if one of them is inverted. The pilot usually still knows which way down is.
That's one reason collision avoidance is based on lateral turns. You can reduce the closure rate and accomplish separation faster by *turning* than by any other means....especially if you can avoid turning in the same direction.
This all asusmes you *see* the other aircraft...not always possible. Sometimes one or both aircraft are in cloud, or view is obstructed by the aircraft itself...a high-wing and a low-wing plane in the oattern at the same time with the low wing above is a classic scenario. We do our best to avoid this with careful use of the radio and also Air Traffic Control when available.
In the case in point, there was only one ATC guy working and the conflict alert computer was being repaired.
A new collision avoidance system under development called ADS -B involves having aircraft continually broadcast their high-precision position, direction and speed digitally, allowing in-cockpit display of local traffic and on-board conflict alerting without involving anybody on the ground at all.
http://www.ads-b.com/