Kuhn's Theorem for Games of the Extensive Form with Unawareness

We extend Kuhn's Theorem to games of the extensive form with unawareness. This extension is not obvious: First, games of the extensive form with non-trivial unawareness involve a forest of partially ordered game trees rather than just one game tree. An information set at a node in one tree may consist of nodes in a less expressive tree. Consequently, perfect recall takes a more complicated form as players may also become aware of new actions during the play. Second, strategies can only be partially an object of ex ante choice in games with unawareness. Finally, nodes that a player may expect to reach with a strategy profile may not be the nodes that actually occur with this strategy profile, requiring us to define appropriate notions of equivalence of strategies. We show if a game of the extensive form with unawareness has perfect recall, then for each mixed strategy there is an equivalent behavior strategy but the converse does not hold under unawareness.