Is Dollarization a good decision for a small open economy with an economic cycle that weakly correlates with the anchor country? I have been reading the work of Alesina http://scholar.harvard.edu/files/alesina/files/dollarization.pdf, about the pros and cons of dollarization (i.e. adopting the currency of other country, say the dollar). It is not the case of the European Union by the way, here the small open economy completely gives up on having its own currency. 
Countries like Panama, El Salvador, and Ecuador have adopted such monetary policy and have done well during expansionary part of the cycle. Now, most commodities prices, of which these countries depends are down, and evidence will be available of the costs of dollarization. However, here I ask about theoretical reasons of why this extreme fixed exchange rate is better with a flexible scheme. The trade-off is between stable and credible monetary policies and flexible possibly manipulable ones in countries with weak institutions.
Like many questions in economics, optimal behavior depends on the set of alternatives. The alternatives of Panama, El Salvador, and Ecuador to dollarization are likely a heavily politicized monetary and exchange rate policy. And so for the hypothetical small open economy considered in the question, the key issue is "What's the institutional effectiveness and independence of the potential monetary authority?"

There are small, open economies with well run central banks (Australia and Switzerland) but also less well run ones (Turkey).  If Panama could have Australia's quality of central bank they'd probably prefer that to dollarization. That might explain why Chile and Costa Rica, with some of the best institutions in Central America choose not to peg their currency to the dollar. If Panama did manage its own monetary affairs without a peg they'd get likely get results more like Turkey. They likely prefer dollarization to the results they'd expect from independent monetary policy.

And indeed, when you peek into the past of many countries with pegged currencies, you often find an episode of poorly conducted monetary policy just before dollarization. Not Panama (never had its own currency since independence), nor El Salvador, but Ecuador introduced dollarization in response to a severe banking and financial crisis. Argentina also seemed to introduce dollarization in response to monetary crisis.  

There are of course other considerations related to optimal currency areas. In particular, if you do a lot of trade with the USA dollarization is less of a sacrifice with respect to the best possible independent currency and monetary policy then if you do most of your trade with a neighboring country that uses a different currency. If Mexico adopts a pegged currency at some point the dollar makes a lot more sense than the Euro when 70% of their trade is with the USA and trade is a big part of their economy. But neighboring countries often have similar trading partners, and are in a similar position with respect to other optimal currency area criteria (no labor mobility to the reference 
country, no fiscal transfer mechanism shared with the reference country, and probably similar business cycles) so institutions are going to play a important role in explaining cross-sectional differences in the decision to dollarize, and that is why I highlight that in this answer.