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08/21/2016 03:06 PM by name withheld; beliefs in a partially randomised perfect bayesian equilibrium

Dear all,

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 inform[ation set of player 2. At this information set, player 2 chooses either action l or r.

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,l) is PBE 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>0.5.

For (R,r), the belief must be smaller than 0.5 for it to be PBE.

Now, for the remainder partially randomised equilibria 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.
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