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:
- $(b,μ)$ is sequentially rational.
- 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 (but nevertheless converges to an assessment that is 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 must be YES, because if it isn't, then the definition of sequential equilibrium seems pointless.