In this thesis I propose a framework for normal and extensive form games where players can use Knightian uncertainty strategically. In such Ellsberg games, ambiguity-averse players may render their actions objectively ambiguous by using devices such as Ellsberg urns, in addition to the standard mixed strategies. This simple change in the foundations leads to a number of interesting phenomena.

While Nash equilibria remain equilibria in the extended game, there arise new Ellsberg equilibria with distinct outcomes. This happens especially in games with an information structure in which a player has the possibility to threaten his opponents. I illustrate this with the example of a negotiation game with three players. This mediated peace negotiation does not have a Nash equilibrium with peace outcome, but does have a peace equilibrium when ambiguity is a possible strategy. That a game with more than two players can have interesting non-Nash Ellsberg equilibria is traced back to results on subjective equilibria.

Ellsberg equilibria are mathematically characterized by the Principle of Indifference in Distributions. In an Ellsberg equilibrium, players are indifferent between all mixed strategies contained in the Ellsberg equilibrium strategy. Furthermore, I observe that in two-player games players can immunize against strategic ambiguity by playing their maximin strategy (if a completely mixed Nash equilibrium exists).

I analyze Ellsberg equilibria in two-person games with common and conflicting interests. I provide a number of examples and general results how to determine the Ellsberg equilibria of these games. The equilibria of conflicting interest games (modified Matching Pennies) turn out to be consistent with experimental deviations from Nash equilibrium play.

Finally, I define extensive form Ellsberg games. Under the assumption of dynamically consistent (rectangular) Ellsberg strategies, I prove a result analog to Kuhn’s theorem: rectangular Ellsberg strategies and Ellsberg behavior strategies are equivalent.