I have asked this question in Mathematics StackExchange, but there is no response yet. I've just realized that here is the right forum for asking research level questions... :'(


In game theory, in the attempt to define sequential equilibrium, for every tuple $b$ of behavioral strategies for each player, the corresponding *belief system induced by $b$* is defined. A tuple $(b,μ)$ is called an *assessment* if $μ$ is the belief system induced by $b$.


Then, the criterion of an assessment being *sequentially rational* is defined.


Finally, *sequential equilibrium* is defined as an assessment $(b,μ)$ such that:


1. $(b,μ)$ is sequentially rational.
2. There exists a sequence of assessments $(b\_i,μ\_i)$ such that each $b\_i$ assigns nonzero probabilities to all decisions and the sequence converges to $(b,μ)$.


My question is: can there be a sequence of assessments $(b\_i,μ\_i)$ satisfying condition number 2 and converging to a sequential equilibrium but with each element being not sequentially rational?


Just in case the answer is YES, what if it is given that: *each* sequence of assessments satisfying condition number 2 -- and converging to an assessment $(b,μ)$ -- has $(b\_i,μ\_i)$ being not sequentially rational for all *large enough* $i$? Can then it be concluded that $(b,μ)$ is not sequentially rational?


I have a strong intuition that the answer to the second question must be YES, because if it isn't, then the definition of sequential equilibrium seems pointless.



On the limit of assessments that are not sequentially rational
Consider a situation in which some player has a strictly dominated strategy. Any strategy profile that assigns a strictly positive probability to all actions at all information sets will involve a player not playing something sequentially rational given any beliefs.


If we would not allow this, even the prisoner's dilemma would have no sequential equilibrium. 


