Identification with BBL In the last few years, the estimator proposed by Bajari, Benkard, and Levin ('07) for dynamic games has been gaining popularity. It is relatively straight forward and is one of the only viable options for estimating dynamic games with both continuous state space and continuous decision variables. I have heard from a few people, though, concerns about what it is really identifying (possibly not the structural parameters it's supposed to be identifying).

My question is three-fold. 1) what are the specific concerns about identification with BBL, 2) when do they (not) matter, and 3) is there a way of getting around the identification issues without having to, say, approximate the state/actions as discrete.
After searching for a while, this is the best answer I can so far come up with.

1) A formalized argument for why identification could break down under BBL is from Srisuma ('13). He gives two specific examples in the online appendix  where identification is lost because of using additive rather than multiplicative perturbations to construct off-equilibrium value functions (as was suggested in the original BBL paper). This is indicative of a broader issue with BBL that there might be off-equilibrium parameters that satisfy the BBL minimum-distance estimator.

2) The two examples given in the appendix are both quite basic and standard (single-agent and Cournot). This suggests that the phenomenon could be a problem in many/most applications.

3) Be creative with policy perturbations. Although Srisuma doesn't show the benefit of multiplicative over additive perturbations in general, the couple examples given show that multiplicative perturbations could improve the estimator. Formalizing optimal perturbations seems a good place for further research.