Multi-agent influence diagrams (MAIDs) are probabilistic graphical models which represent strategic interactions between agents. MAIDs are equivalent to extensive form games (EFGs) but have a more compact and informative structure. However, MAIDs cannot, in general, represent settings of incomplete information -- wherein agents have different beliefs about the game being played, and different beliefs about each-other's beliefs. In this paper, we introduce incomplete information MAIDs (II-MAIDs). We define both infinite and finite-depth II-MAIDs and prove an equivalence relation to EFGs with incomplete information and no common prior over types. We prove that II-MAIDs inherit classical equilibria concepts via this equivalence, but note that these solution concepts are often unrealistic in the setting with no common prior because they violate common knowledge of rationality. We define a more realistic solution concept based on recursive best-response. Throughout, we describe an example with a hypothetical AI agent undergoing evaluation to illustrate the applicability of II-MAIDs.
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