Intelligence, awareness, strength, patience, followed by flexibility. I might move flexibility up ahead of strength depending on if you mean flexibility physically speaking or if we are talking more about the ability to change and adapt instead.
I rank intelligence first because knowledge of ones art and those techniques that it includes and the ability to know when you should use or employ what. That is extremely important in using those skills and to often I see people making poor choices of this like picking up a pliers to hammer in a nail. Can you hammer a small or medium size nail into something with a pliers; certainly but the ideal tool is obviously a claw hammer for carpentry.
Awareness goes partly hand in hand with intelligence I think in being able to correctly see and read a situation and by doing that then being able to more easily determine the best skill, technique, or strategy to employ. Without this then you are just basically taking a shot in the dark and hoping to get lucky and what are the chances of success with such an approach as that? I am sure you will agree not good probably.
Flexibility I might put third if we are talking about the ability to adapt, change, or make adjustments. While your skills and technique and your thinking and actions should be so quick that your opponent has no chance to react that is not always the case. Your ability to see that an adjustment is in order and being able to make that change or adjustment on the fly is really important rather than blindly continuing your efforts, especially if they are not going to meet with any success or bare the results that you expect, want, and need.
I would have to put strength next or maybe even in front of flexibility if we are not talking about the ability to change or adapt. You have to be committed to your actions at some point and executing them. Half hearted attempts will only get you into trouble as well as make you weak and vulnerable to your opponent and his more stronger actions. Strength implies that to me more than the physical aspect of being strong per say.
In this forum we talk a lot about technique and the importance of it and that it will beat strength the majority of the time. However your technique, even if it is as good as or better than that of your opponent's will not suffice if you are not committed to it. Actions or potentially successful actions without commitment will just leave you in a situation where you might struggle or even flounder and here again make it more easily for your opponent to gain the upper hand and dominate you instead of you prevailing over or dominating him.
Patience I would put last. Its a needed quality in moderate doses I think but when we are talking about committed, direct action and intelligence and awareness being present then I would put it last. A trained persons intelligence and awareness will direct their actions as well as when also to maybe wait or hesitate or take the long view or approach to things rather than the short term view or approach.
Patience at the wrong time or in the wrong situation when direct action is required is really just hesitation and can have disastrous results for you. While I think martial arts stresses patience I think that term and aspect is often times misinterpreted and mis-taught. To many people think you should not employ or use your skills or training unless that other person has hurt you and left you no choice. Unfortunately you have to survive that initial attack or onslaught and to often people don't. That adversary's intended action and starting or beginning that is really enough and you should not have to wait till he actually hits you before you realize its time to defend yourself or to turn and run or take whatever action you are going to take.
Thats my spin on things and its a good question and one that I will look forward to reading some of the other answers on.
On a competitive level: Awareness of you opponents style of karate is important. As it is for self defense.
awareness
strength
flexibility
patience
intelligence - The reason this is last is that awareness & patience is just a sub category of intelligence.You need to be more specific about what TYPE of intelligence is of most importance.
Don't forget speed is also very important.