  Classical game theory treats players as special---a description of a game
contains a full, explicit enumeration of all players---even though in the real
world, "players" are no more fundamentally special than rocks or clouds. It
isn't trivial to find a decision-theoretic foundation for game theory in which
an agent's coplayers are a non-distinguished part of the agent's environment.
Attempts to model both players and the environment as Turing machines, for
example, fail for standard diagonalization reasons.
  In this paper, we introduce a "reflective" type of oracle, which is able to
answer questions about the outputs of oracle machines with access to the same
oracle. These oracles avoid diagonalization by answering some queries randomly.
We show that machines with access to a reflective oracle can be used to define
rational agents using causal decision theory. These agents model their
environment as a probabilistic oracle machine, which may contain other agents
as a non-distinguished part.
  We show that if such agents interact, they will play a Nash equilibrium, with
the randomization in mixed strategies coming from the randomization in the
oracle's answers. This can be seen as providing a foundation for classical game
theory in which players aren't special.
