You must have been there: cycling, slightly uphill, then you sense someone's behind you. Then a tranquil ride back home turns into a war of wills until:
a) You regain your distance and leave the challenger well behind
b) You react too little too late, the challenger rides past you without breaking sweat
c) You struggle for a bit, but in the end yield to the inevitable, honour preserved having fought the good fight
I'm usually into (b) or (c). Of course, it can get more complicated. More than once I've been 'chased' by newbies who, for instance, try to 'under' take, passing between the kerb and myself - they must feel 'safer' that way. Very irritating. I have in those cases tried very hard to keep ahead without getting too close to the kerb myself, until a widening on the road would enable me to let them past. Pillocks!. Usually, these newbies' speed is down to straining hard on their top gear, 'walking' on the pedals for far longer than it is sensible if you want your knees to remain serviceable when you are 50.
And there are the cheaters. Yes: those you leave behind because clearly you are the stronger cyclist. Then you stop at the traffic light... and they don't.
The funny thing is, I always say to myself 'I'm not the competitive sort, I'm just commuting' but most of the time the desire not to be out-done kicks in.
Ocassionally, I'm overtaken by the friendly sort - they tend to be the serious performance, tour-de-Manchester cyclists, who glide effortlessly past you, often smile at you, perhaps even share some comment or witticism. Bless them.