We recently released two new papers on reflective oracles and agents.

The first is “Reflective oracles: A foundation for classical game theory,” by Benja Fallenstein, Jessica Taylor, and Paul Christiano.

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 co-players 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.

The second paper develops these ideas in the context of Solomonoff induction and Marcus Hutter’s AIXI. It is “Reflective variants of Solomonoff induction and AIXI,” by Benja Fallenstein, Nate Soares, and Jessica Taylor.

Solomonoff induction and AIXI model their environment as an arbitrary Turing machine, but are themselves uncomputable. This fails to capture an essential property of real-world agents, which cannot be more powerful than the environment they are embedded in; for example, AIXI cannot accurately model game-theoretic scenarios in which its opponent is another instance of AIXI.

In this paper, we define

reflectivevariants of Solomonoff induction and AIXI, which are able to reason about environments containing other, equally powerful reasoners. To do so, we replace Turing machines by probabilistic oracle machines (stochastic Turing machines with access to an oracle). We then usereflective oracles, which answer questions of the form, “is the probability that oracle machineMoutputs 1 greater thanp, when run on this same oracle?” Diagonalization can be avoided by allowing the oracle to answer randomly if this probability is equal top; given this provision, reflective oracles can be shown to exist. We show that reflective Solomonoff induction and AIXI can themselves be implemented as oracle machines with access to a reflective oracle, making it possible for them to model environments that contain reasoners as powerful as themselves.