So Ben, looks like it's time to cross swords again :)
1. Real world performance varies over time...
I don't really understand this complaint. With A/B testing, you have to collect enough data before making a decision that the conversion rate fluctuations average out. If you're worried about monthly fluctuations (which is reasonable), and you're doing A/B testing, you need to collect data for a few months. This means months where you aren't optimising. With a bandit algorithm conversion rate fluctuations will cause changes in the bandit behaviour, but they will average out over time just like A/B testing. We can prove this, so I don't know what justification there is for the statement that it is "a big issue for this approach". Indeed I think this points out a weakness of A/B approaches. Firstly, it's arguable that tracking fluctuations in conversion rate is a good thing. They may be temporary, but why shouldn't the algorithm react to temporary changes? Secondly, unlike A/B testing you're not spending months collecting data -- you're actually optimising while you collect data. Finally, what site can afford to spend months not changing, while data is collected?
2. (This is really a special case of #1 - but a very, very important special case...
I see two issues here: 1) the fact the traffic has changed and 2) how long does it take the system to recognise this change?
Addressing 1: Getting technical for a moment, A/B tests assume the distribution generating the data is stationary. This situation violates this assumption, so one shouldn't be using A/B testing here. I'd like to know what is meant by a properly constructed A/B test in this case. Does it mean throwing out all the data before the significant improvement? If you say that a non-stationary distribution is fine, then you can develop a bandit algorithms for non-stationary situations (e.g. http://arxiv.org/abs/0805.3415). Myna handles non-stationary situations, so long as they change quite slowly.
Point 2 is really about convergence rates. There is some work here (http://imagine.enpc.fr/~audibert/Mes%20articles/ALT11.pdf) and the upshot is that convergence rate can be a problem in theory though you can design for it. Notably we haven't observed this issue in practice.
3. Code over time gets messy...
This is a non-issue in my experience. At some point you decide that an experiment is no longer worthwhile. Either you're redesigning things so it doesn't make sense anymore, or the experiment has been running long enough that you are confident of no more changes. When you hit this point you remove the code. It can be a year or a hour after starting the experiment. Either way, you know with a bandit algorithm you will have optimised what traffic you did receive.
4. Businesses are complex, and often have multiple measures they would like to balance...
The glib answer to this is to set your reward criteria correctly, but more below.
5. Many tests perform differently on existing users and new users...
Both of these are really about performing extra analysis on the data. Myna doesn't help here, and we could make it better in this regard by, for example, exporting data. Some of this extra analysis, such as cohort analysis, we can and will automate in the future. There will always be analyses that we can't or don't perform, so there will always be jobs for people who perform these analyses :-) On the other hand, bandit algorithms are used enough in practice (e.g. Google uses them for AdWords) that their utility has been validated. It's a trade-off between time and return. We want to automate the common cases so the analysts can concentrate on the interesting parts.