  We propose a game-theoretic framework that incorporates both incomplete
information and general ambiguity attitudes on factors external to all players.
Our starting point is players' preferences on payoff-distribution vectors,
essentially mappings from states of the world to distributions of payoffs to be
received by players. There are two ways in which equilibria for this preference
game can be defined. When the preferences possess ever more features, we can
gradually add ever more structures to the game. These include real-valued
utility-like functions over payoff-distribution vectors, sets of probabilistic
priors over states of the world, and eventually the traditional
expected-utility framework involving one single prior. We establish equilibrium
existence results, show the upper hemi-continuity of equilibrium sets over
changing ambiguity attitudes, and uncover relations between the two versions of
equilibria. Some attention is paid to the enterprising game, in which players
exhibit ambiguity seeking attitudes while betting optimistically on the
favorable resolution of ambiguities. The two solution concepts are unified at
this game's pure equilibria, whose existence is guaranteed when strategic
complementarities are present. The current framework can be applied to settings
like auctions involving ambiguity on competitors' assessments of item worths.
