To outpromise means to make a promise that is better than someone else's promise.
Outpromising can show that someone is more confident or more convincing.
Examples:
I am outpromising my friend by promising to buy her a bigger present.
She outpromised me by saying she would take me to the best place in the world.
He is outpromising his brother by promising to take him to the moon.
We are outpromising each other with bigger and bigger promises.
The outpromising went on and on until they both promised to buy each other a whole store!
Stop outpromising me with your big promises!