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I want to ask about the intuition/where-to-look/what-to-try if I want to prove that an action value function optimal for a problem is also optimal for another reformulation of that smae problem. For example, a common approach to solve DEC-POMDP is to recast the problem into a common-knowledge-MDP. In this case (and in any similar case) What is the approach to follow in order to prove that the optimal action-value function optimal for the single agent reformulation is also optimal for the multi-agent formulation?

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The reduction of a Dec-POMDP to a centralized MDP that you mention is discussed at length in the technical report "Dec-POMDPs as Non-Observable MDPs" by Oliehoek and Amato.

Further background that can help understand the method can be found in "Sufficient plan-time statistics for decentralized POMDPs" in IJCAI'13 also by Oliehoek. A related approach dealing with common/individual knowledge and information gathering is "Multi-agent active perception with prediction rewards" in NeurIPS'20 by Lauri and Oliehoek. (Disclaimer: I am one of the authors).

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