Reflective Oracles: A Foundation for Classical Game Theory
This work addresses a foundational issue in game theory for researchers and theorists, offering a novel approach to modeling players as part of the environment, though it is incremental in building on existing decision-theoretic frameworks.
The paper tackles the problem of providing a decision-theoretic foundation for classical game theory where players are not treated as special entities, by introducing reflective oracles that avoid diagonalization issues through randomization. The result shows that agents using these oracles interact to play a Nash equilibrium, with randomization in mixed strategies derived from the oracle's answers.
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.