You are actually dealing with two separate problems here: one is the bragger and the other is your receptive supervisor.
A bragger is programmed to brag. You and your co-workers should not think that you can turn her into a paragon of humility. Her need to wax eloquent about herself is coming from some needs that are well beyond anything that you need to be handling on the job. People who work with braggers often tend to adjust their own behavior, rather than trying to adjust that of the bragger. The most common adjustment is to place the bragging comments in the same category as the background noise from the air conditioner, elevator, or traffic. This keeps the bragger happy, and the rest of the staff sane.
Your supervisor appears to be a living tribute to the advertising adage that if you hear something often enough, you start to believe it. If he actually has the time to listen to her pronouncements and then take even more time to tell you how wonderful she is, there may be some serious questions about his managerial prowess.
Nonetheless, he has given you an important message in return: he is a buyer when it comes to employee announcements about themselves. If you honestly believe that your performance equals or surpasses that of your bragging associate, you should make sure that your supervisor hears your self-advertisements too.
In making your pitch, it is best to avoid any comments about the bragger. Rather, focus on your own measurable successes on the job. This is obviously not something you should do every day, but if your supervisor is going to be using such information to help determine raises and promotions, you should make sure that you are getting enough visibility.
There is no question that promoting yourself plays an important role in getting yourself promoted.
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