Subgame perfect Nash equilibrium & perfect Bayesian Nash equilibrium - Game theory

Question

For a week or so I have been struggling with the topics around the concept of subgame perfect Nash equilibrium (SPNE) and the perfect Bayesian Nash equilibrium (BNE). Namely:

1. Is it possible to apply backward induction (to obtain SPNEs) in dynamic games of complete but imperfect information? According to $\textit{Game Theory for Applied Economists}$ by $\textit{Robert Gibbons}$ it is possible. Or atleast, that what I understand from the text on page 128-129. But, then the SPNE would be implausible, since the player at the nonsingleton information set has no knowledge at which decision node he is. How is it then possible to apply backward induction to games of imperfect information?
2. On page 129 of the book of $\textit{Gibbons}$ it is mentioned that there is a second method to obtain an optimal solution/strategy profile by specifying a probability to each node in the nonsingleton information set. And (this is what confuses me) this yields a perfect Bayesian equilibrium. But wait a minute... the perfect Bayesian equilibrium occurs in games of incomplete information, so how would we solve games of imperfect information by regarding games of incomplete information? I thought that we could solve games of incomplete information with the help of solution concepts applicable in games of imperfect information, but the above tells the reverse (right?).

Hopefully one can help me out! Have a nice day.

Answer 1

As I read Gibbons, p. 128-129, He is saying you could try backward induction, but it wouldn't work for the reason you mention. One approach he hints at, going back to a non-singleton information set, is like Perfect Bayesian Equilibrium, which has one non-singleton information set at the very beginning (according to Osborne and Rubinstein, at least). Now a sequential equilibrium, which adds beliefs to the definition of equilibrium, has the one-shot deviation property, and may therefore be amenable to backward induction (again, see Osborne and Rubinstein, chapter 12.2). See also http://www.ssc.wisc.edu/~whs/teaching/711/lngt.pdf Hope this is a start.