How would you write that to a developer in a formal email? I'm looking for polite and still friendly way to inform a bidder that he was not the winner, to minimize his disappointment?
The specific words you are looking for may be "handed to". "handed over to" may be used in case another person has been made in-charge of an existing project that this person was in-charge of. In usage, I envision something like
I hope this mail finds you well. I am grateful for your time and patience in bidding for the project. However, you have been outbid by another developer for the same work, and therefore, the project has been handed to them. I once again take this opportunity to thank you for your time and effort.
EDIT: Another valid answer would be to use "awarded to". I have seen both usages fairly equally in articles. Thanks to @BoldBen for the suggestion.