Anyone riding a bike on city streets around here routinely encounters motorists inappropriately stopping and offering to yield the right of way at intersections.
It seems safe to assume that the majority are just being nice. But it does present a problem.
If you accept the offer to go ahead, you are only reinforcing this counterproductive notion of treating cyclists like pedestrians. (When, in fact, they should be treated like people in cars.)
But if you want to decline the offer, that poses a dilemma. How do you indicate to the driver that you wish him or her to proceed ahead while you stay put? Hand gestures, of course. But what kind?
Nobody likes drivers at intersections who signal to other motorists in an impatient, annoyed way. You know, as if the other driver's presence on the planet is a huge inconvenience for them. So obviously, cyclists don't want to be like them.
But if your gestures are indecisive or open to interpretation, that can cause its own problems.
Here's what I do. I smile in a weary way and wave at the driver, as if to say "I see you and realize you are offering me the chance to go in front of you."
Then I make a sweeping "No, thank you -- after you" motion with my right arm.
Often it works. Sometimes it doesn't.
Remember when I said the majority of drivers are just being nice when they offer to yield? Well, there's a reason I didn't suggest that all drivers are.
Some motorists regard all cyclists as a pain the ass. These drivers want to send the message that bike riders are so erratic and such a nuisance that they have to be coddled and given special treatment even though they have no business being on the road.
There's a temptation to say that there is another hand gesture more appropriate for communicating with them. But the cyclist would have to be a mindreader to know when someone is trying to be nice and when someone is engaging in performance forbearance.