Suppose I have a game in extensive form as follows. Player 1 moves first and chooses R, M or L. If he chooses R, the game ends. Otherwise, the game reaches a non-trivial information set of player 2. At this information set, player 2 chooses either action l or r and the game ends.
The payoffs (in normal form) are as follows:
[Ll Lr Ml Mr Rl Rr] =
=[4,1 0,0 3,0 0,1 2,2 2,2]
The NE/SPNE are: (L,l), and (R,(q,1-q)) with q\in[0;1/2], the latter of which includes the pure strategy NE (R,r).
Clearly l is sequentially rational as long as the belief that player to plays L is larger than 0.5. Also, such beliefs are consistent because if l is played, L is best response which implies belief equals 1 by consistency. Hence, (L,l) with belief equal to 1 is PBE.
For (R,r), the belief must be weakly smaller than 0.5 by seq rationality (and consistency of beliefs has no bite as action r is not played in equilibrium), and thus for it to be PBE.
Now, for the remainder partially randomised equilibria, I am not sure. I am reasoning that in order for player 2 to find it sequentially rational to mix between l and r, it must be that both l and r are sequentially rational (so that player 2 is indifferent), which implies that the belief is 1/2. Furthermore, the beliefs are consistent since in the partially randomised equilibria player 1 plays R and the information set of player 2 is off the equilibrium path (so any beliefs are consistent). Is this correct?
Thank you in advance.
There are slightly different definitions of consistency for beliefs. Your notion of PBE assumes (what some people call) weak consistency: off the equilibrium path, any belief is weakly consistent. Under weak consistency, your argument is correct.