Tower Controller reported an NMAC requiring evasive action to avoid a collision.

Narrative

Training in progress on LC1 position. Aircraft X was in the right closed traffic pattern along with 2 other aircraft. Aircraft Y called inbound 9 miles NW of the field and was instructed to enter right downwind for XXR. The pilot requested to fly overhead to enter a left downwind for XXL. The trainee instructed Aircraft Y to overfly midfield at or above two thousand feet; the pilot replied 'Roger;' acknowledging the instructions. Pattern altitude for closed traffic is one thousand four hundred feet. Aircraft Y; abeam midfield reported Aircraft X in sight as he flew southbound to cross overhead midfield. Aircraft X then reported that he had to perform an evasive turn to avoid a 'near midair incident.' At the time of that report; Aircraft X indicated 017 and Aircraft Y indicated 018 on the radar display. The aircraft both continued on and landed on their assigned runway. This incident occurred because both aircraft involved flew in a manner other than what was instructed and expected of them. Aircraft Y descended well below his altitude restriction and Aircraft X was flying well above pattern altitude without notifying the tower of his intentions to do so. Both aircraft placed themselves in a position to create a conflict with each other. Due to the lag in radar display; the proximity of the aircraft was both unrealized and unexpected until the pilot had reported it to us.Pilots must be familiar with standard procedures as well as follow any instructions given them by ATC; if both of those had taken place; there would have been no incident.