Dominated Strategies in an Infinitely vs Finitely Repeated Game How does the concept of weak dominance work with infinite games? The abundance of concepts seems to muddy things. 

In particular, suppose two players play the following game an infinite number of times.

$$\begin{array}{l*{2}{c}r}
             &amp; A &amp; B  \\
\hline
A &amp; 1,1 &amp; 0,0 \\
B &amp; 0,0 &amp; 0,0
\end{array}$$

In the one shot game, it is clear that playing $A$ weakly dominates the action $B$. 

However, what if we consider a bizarre kind of grim trigger:


In the first period, a player plays $B$.
If the other played $A$ in the first period, play $B$ forever.
Otherwise, play $A$ forever.


For a patient enough player, it is weakly best to respond to this strategy with the same one. Any strategy that begins with $B$ and then $A$ afterwards is a best response (the off-path details can be different, so that a best response may not stipulate $B$ forever if the other chooses $A$ at the beginning). 

The pareto efficient equilibria involve always playing $A$ on the equilibrium path, and it's tempting to say that playing $A$ no matter what should be weakly dominant, but that doesn't seem to be the case. Has much been written on this kind of thing? What kind of ideas exist to rule out the kind of strategy described above? 

Is the strategy in question itself weakly dominated? I would guess that we could concoct an equally sinister trigger strategy $s'$ so that the strategy in question is not dominated by any other $s''$.
A usual refinement concept used to deal with weakly dominated strategies is the trembling hand perfect equilibrium. (I do not know others but this one works quite well.

The strategy in question is indeed weakly dominated by the following strategy


In the first period, player plays $B$.
In all other periods, player plays $A$.