2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ I'm a bit fuzzy on this, but I think each $(b_i,\mu_i)$ can be let us say $\epsilon_i$-sequentially rational, with $\epsilon_i \to 0$, but none of them exactly sequentially rational. By this I mean that some player has a deviation that improves expected utility by up to $\epsilon_i$. $\endgroup$ – usul May 28 '18 at 8:43
  • $\begingroup$ A counterexample has been given here: math.stackexchange.com/questions/2797641/… $\endgroup$ – Mauri Ericson Sombowadile May 29 '18 at 16:23
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.