Credible Threat in the Perfect Bayesian Equilibrium Consider a simple game between P1 and P2 shown in Figure 1.


We consider a Perfect Bayesian Equilibrium. P1 uses a cutoff strategy based on his $c_1$ that has the Cumulative Distribution Function of $F$. If his $c_1$ is lower than a threshold $c_1^*$, which depends on P2's strategy, P1 takes the strategy $x$.

Similarly, P2 uses a cutoff strategy based on his $c_2$ that has the Cumulative Distribution Function of $G$. If his $c_2$ is lower than a threshold $c_2^*$, which depends on P1's strategy, P2 takes the strategy $a$.

We denote the optimal cutoff strategies as $F(c_1^*)$ and $G(c_2^*)$.

Suppose the game has slightly changed as shown in Figure 2.


Now P1 has one more turn. Suppose that, given the initial $F(c_1^*)$ and $G(c_2^*)$, we set P1's gain from the strategy j as always being equal to the strategy k. We assume that P1 takes the strategy j if P1 is indifferent between j and k.

In this case, does P2 still continue to use the same cutoff strategy $G(c_2^*)$? For P2, the node from the path $k$ is never reached while P2 cannot verify it. As long as P2 uses the same cutoff strategy $G(c_2^*)$, P1 will take the strategy $j$. I am wondering if P2's threat is credible.

In the new game, P2's gain remains unchanged. However, P2 may try to increase his gain by deviating from the optimal cutoff strategy $G(c_2^*)$ if P1 never use the strategy k. But, P1 will be aware of it and may use the strategy k again. If so, P2 will again use the same optimal cutoff strategy $G(c_2^*)$. I got stuck here.

I appreciate any help.
I am going to assume that $F$ and $G$ are continuous distribution functions (I.e. players are playing properly mixed strategies). If this is not the case there is not enough information to answer your question. Given that assumption, player 1 and 2 must be indifferent among their actions in game 1 given the strategies of the other player. 

In the second game, you are assuming that the node given by $x,k$ is never reached (since $j$ and $k$ give the same payoff and when indifferent, player 1 always choose $j$). 

Under that behavioral assumption (which must be verified to be consistent with equilibrium), player 2 must believe she is at the node "$y$" with probability 1. Since Player 2's beliefs have changed with respect to game 1, the mixed strategy given by $G$ is, in general, no longer optimal (unless $a$ and $b$ give the same payoff regardless of the action of player 1). 

In short, player 2 will, in general, not use the same strategy in the first and the second game; so threatening to use $G$ is not credible. 

Btw all threats are credible in a PBE since strategies are best responses given some beliefs that are Bayesian.