Small aircraft Flight Instructor reported an NMAC with another light aircraft in the vicinity of FFZ airport. Flight Instructor took evasive action to avoid a collision.

Narrative

Flying inbound to Falcon field at the conclusion of a training flight I was flying on the east side of Mesa gateway headed to pass west of the AJ Landfill VP when I saw on my EFB traffic advisory that there was an outbound aircraft; Aircraft Y; from Falcon field was approaching head-on with me at my altitude. I was unable to establish visual with the head-on aircraft because of the position of the setting sun but altered course to the right regardless to comply with right of way rules. Aircraft Y did not take action to avoid a collision by altering course to the right; but rather; altered course to the left; repositioning us on a converging course with my aircraft to their right. They continued on an altitude approximately the same as the altitude as my aircraft. I continued my altered heading but was still unable to see the outbound traffic because of the position of the sun; and approximately half a mile from the converging traffic according to my EFB traffic display and traffic warning I initiated a descent while continuing on my heading. I descended down to 2900 while continuing my scan searching for traffic to assure they weren't a collision hazard; and established visual contact with the other aircraft as I passed directly beneath them approximately 400 feet below. I chose to take action to avoid converging traffic; despite me being the aircraft with the right of way; because of previous personal experience with other traffic in similar situations where incorrect traffic avoidance procedures have been employed by traffic in converging and head-on traffic scenarios resulting in other near misses; which; though not as severe as this near miss; were also dangerous situations caused by incorrect procedures which have been plainly laid out in 14 CFR 91.113(e). Regardless of who had the right of way in the situation; I took what I felt was safe evasive action to avoid a fatal mid air collision. I feel the greatest contributing factor to this near miss is Aircraft Y's incorrect traffic avoidance procedures; and failure to follow proper right of way rules. I don't believe Aircraft Y was unaware of my aircraft's position; as I am familiar with the systems in the aircraft they fly which feature ADSB-in display with TCAS. I was also flying a distinctly colored; not white; aircraft which would have stood out in stark contrast with the blue sky and the surrounding terrain in the late afternoon sunlight.