Actually, having a winner and a loser is the definition of a competitve interaction. Without those "classes", it is not a competition. By definition for it to be a competiton, there has to be something you are competing for, food, a mate, a job, a trophy, bragging rights, whatever.
If I am doing some jiyu waza training and I ask my partner to attack strongly and try to counter my techniques, that doesn't mean I am competing with him. If I am sucessful at keeping him off or controlling him, I haven't won anything and he hasn't lost anything. Competition might make it possible to formalize and structure these interactions in some good ways, but it would also change the nature of the interaction, and in my opinion, the nature of that art itself.
BTW. This is not a value judgement of you or your dojo. The training you describe honestly sounds like a lot of fun.
Then according to the the definition above, would "haves" and "have nots" be more appropriate competitive classifications? Erick has a good point - I think we want to differentiate dualism from competition. Competency is a singular evaulation of a qualitative ability. Competition is a hierarchy of competency.
For example: In high school, I participated in a spelling B. Did I win against the word if I spelled it correctly? Of course not, spelling the word qualified my spelling skill as competent; the number of words that I spelled correctly ranked my competency against other competent spellers in a hierarchy.
In aikido, I may engage in jiyuwaza. If I cannot control my partner, I nave not demonstrated comptency. If I can successfully control my partner I demonstrate a minimum level of competency. If I can control and pin my partner I demonstrate an advanced level of compentency. If I can control multiple partners, I demonstrate a high competency level and so on.
As a side note: Competition already exists in aikido - we call it the belt system. The ranking system was designed to allow strangers to evaluate the [relative] expected skill level of their partner. I don't remember fighting anyone for my belts, but I sure remember tests that pushed me to the limit of my skill. The belt I was given placed me within a hierarchy of aikido people. Unfortunately, poor instruction, microtesting environments, nepotism, and other factors have degraded the reliability of the skill level of the individual as represented by the belt.
I remember a seminar where I overheard two black belts training. Technique got physical and as nage was preparing to throw uke koshinage, uke said, "I don't take breakfalls." Nage said, "Are you injured? would you prefer ukemi?" uke said, "I don't like to fall." After a second of bewilderment nage looked at uke and said, "I understand. Thank you." and chose another partner. After class I approached the black belt (nage) and asked him what was the deal. He said (tongue and cheek), "Anyone would is so good they don't have to fall if they don't wanna is way outta my league so I chose a different partner." Brilliant. The truth of the situation was that the other black belt was senior and did not want to be thrown because he took poor ukemi.