Now that anyone can set a bounty on other peoples question, it might be nice to know who actually award it.
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Hover over the bounty award, it should show you who awarded the bounty.
I think the best location for names of bounty assignment would be as part of revision history. The only other reasonable place is in the tooltip. However, while separate bounties on one post are recorded separately, the tooltip will only show the combined total. See the research by Jon and myself here, and a related feature request for improving the tooltip.
Adding names (which can be long, and numerous) will make the tooltip potentially unwieldy. Comparatively, attaching it to a revision history event, in which each individual bounty can be isolated, will look cleaner. Also, the name of whoever gave a bounty isn't something that's immediately necessary knowledge, so it doesn't hurt it as much to require a click to determine. There's no other reasonable location to place this information, so revision history seems best.