Hi Vanessa, Thanks for your question! Sorry for taking a while to reply. The answer is yes if we allow for mixed policies (i.e., where an agent can correlate all of their decision rules for different decisions with a shared random bit), but no if we restrict agents to only be able to use behavioural policies (i.e., decision rules for each of an agent's decisions are independent because they can't access a shared random bit). This is analogous to the difference between mixed and behavioural strategies in extensive form games, where (in general) a subgame perfect equilibrium (SPE) is only guaranteed to exist in mixed strategies (and the game is finite etc by Nash' theorem).
Note that If all agents in the MAIM have perfect recall (where they remember their previous decisions and the information that they knew at previous decisions), then there is guaranteed to exist a SPE in behavioural policies). In fact, Koller and Milch showed that only a weaker criterion of "sufficient recall" is needed (https://www.semanticscholar.org/paper/Ignorable-Information-in-Multi-Agent-Scenarios-Milch-Koller/5ea036bad72176389cf23545a881636deadc4946).
In a forthcoming journal paper, we expand significantly on the the theoretical underpinnings and advantages of MAIMs and so we will provide more results there.
I know you've acknowledged Friston at the end, but I'm just commenting for other interested readers' benefit that this is very close to Karl Friston’s active inference framework, which posits that all agents minimise the discrepancies (or prediction errors) between their internal representations of the world and their incoming sensory information through both action and perception.