  One of the reasons why stochastic dynamic games with an underlying dynamic
system are challenging is since strategic players have access to enormous
amount of information which leads to the use of extremely complex strategies at
equilibrium. One approach to resolve this challenge is to simplify players'
strategies by identifying appropriate compression of information maps so that
the players can make decisions solely based on the compressed version of
information, called the information state. For finite dynamic games with
asymmetric information, inspired by the notion of information state for
single-agent control problems, we propose two notions of information states,
namely mutually sufficient information (MSI) and unilaterally sufficient
information (USI). Both these information states are obtained with information
compression maps independent of the strategy profile. We show that Bayes-Nash
Equilibria (BNE) and Sequential Equilibria (SE) exist when all players use
MSI-based strategies. We prove that when all players employ USI-based
strategies the resulting sets of BNE and SE payoff profiles are the same as the
sets of BNE and SE payoff profiles resulting when all players use full
information-based strategies. We prove that when all players use USI-based
strategies the resulting set of weak Perfect Bayesian Equilibrium (wPBE) payoff
profiles can be a proper subset of all wPBE payoff profiles. We identify MSI
and USI in specific models of dynamic games in the literature. We end by
presenting an open problem: Do there exist strategy-dependent information
compression maps that guarantee the existence of at least one equilibrium or
maintain all equilibria that exist under perfect recall? We show, by a
counterexample, that a well-known strategy-dependent information compression
map used in the literature does not possess any of the properties of MSI or
USI.
