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TheOtherDave comments on Nash Equilibria and Schelling Points - Less Wrong
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I'm not sure that regretting correct choices is a terrible downside, depending on how you think of regret and its effects.
If regret is just "feeling bad", then you should just not feel bad for no reason. So don't regret anything. Yeah.
If regret is "feeling bad as negative reinforcement", then regretting things that are mistakes in hindsight (as opposed to correct choices that turned out bad) teaches you not to make such mistakes. Regretting all choices that led to bad outcomes hopefully will also teach this, if you correctly identify mistakes in hindsight, but this is a noisier (and slower) strategy.
If regret is "feeling bad, which makes you reconsider your strategy", then you should regret everything that leads to a bad outcome, whether or not you think you made a mistake, because that is the only kind of strategy that can lead you to identify new kinds of mistakes you might be making.
If we don't actually have a common understanding of what "regret" refers to, it's probably best to stop using the term altogether.
If I'm always less likely to implement a given decision procedure D because implementing D in the past had a bad outcome, and always more likely to implement D because doing so had a good outcome (which is what I understand Quill_McGee to be endorsing, above), I run the risk of being less likely to implement a correct procedure as the result of a chance event.
There are more optimal approaches.
I endorse re-evaluating strategies in light of surprising outcomes.(It's not necessarily a bad thing to do in the absence of surprising outcomes, but there's usually something better to do with our time.) A bad outcome isn't necessarily surprising -- if I call "heads" and the coin lands tails, that's bad, but unsurprising. If it happens twice, that's bad and a little surprising. If it happens ten times, that's bad and very surprising.