A Nonatomic Game Involving Ambiguity

We consider a nonatomic game whose players have ambiguities over external factors. On top of a player $i$'s own action $a$ and the joint distribution $\delta$ of other players' identities and actions, also influencing the player's return $r$ is a state of the world $\omega$. Knowledge about the latter would be reflected by a signal $\sigma$ that an individual player can base her decision on. When all other players adopt strategies that amount to action distributions based on their signals, a player $i$ would act in accordance with her own preference on the vector made up of the distributions on returns anticipated to come her way under all states of the world $\omega$ allowed by her own signal $\sigma$. These preferences across all $(i,\sigma)$-players would constitute the game's ambiguity profile. Much like the finite version studied by Yang \cite{Y18}, there can be two equilibrium notions, namely, the action- and distribution-based ones that depend on whether a player controls individual actions or merely their distributions. Besides the existence of equilibria of the two varieties, we also study their continuities with respect to return functions and ambiguity profiles. Relationships between the two notions are also characterized for various special cases, with the most special one being the expected-utility Bayesian game. Here, the two would merge into the conventional notion. Moreover, nonatomic equilibria are shown to serve as approximate equilibria for their finite counterparts