My current candidate definitions, with some significant issues in the footnotes:
A fair environment is a probabilistic function F(x1,...,xN)=[X1,...,XN] from an array of actions to an array of payoffs.
An agentA is a random variable
A(F,A1,...,Ai−1,Ai=A,Ai+1,...,AN)
which takes in a fair environment F[1] and a list of agents (including itself), and outputs a mixed strategy over its available actions in F. [2]
A fair agent is one whose mixed strategy is a function of subjective probabilities[3] that it assigns to [the actions of some finite collection of agents in fair environments, where any agents not appearing in the original problem must themselves be fair].
Formally, if A is a fair agent in with a subjective probability estimator P, A’s mixed strategy in a fair environment F,
A(F,A1,...,Ai−1,Ai=A,Ai+1,...,AN)
should depend only on a finite collection of A’s subjective probabilities about outcomes
{P(Fk(A1,...,AN,B1,...BM))=[X1,...,XN+M]}Kk=1
for a set of fair environments F1,...,FK and an additional set of fair[4] agents[5]B1,...,BM if needed (note that not all agents need to appear in all environments).
A fair problem is a fair environment with one designated player, where all other agents are fair agents.
I might need to require every F to have a default action dF, so that I don’t need to worry about axiom-of-choice issues when defining an agent over the space of all fair environments.
I specified a probabilistic environment and mixed strategies because I think there should be a unique fixed point for agents, such that this is well-defined for any fair environment F. (By analogy to reflective oracles.) But I might be wrong, or I might need further restrictions on F.
Grossly underspecified. What kinds of properties are required for subjective probabilities here? You can obviously cheat by writing BlueEyedBot into your probability estimator.
This is an infinite recursion, of course. It works if we require each Bm to have a strictly lower complexity in some sense than A (e.g. the rank of an agent is the largest number K of environments it can reason about when making any decision, and each Bm needs to be lower-rank than A), but I worry that’s too strong of a restriction and would exclude some well-definable and interesting agents.
My current candidate definitions, with some significant issues in the footnotes:
A fair environment is a probabilistic function F(x1,...,xN)=[X1,...,XN] from an array of actions to an array of payoffs.
An agent A is a random variable
A(F,A1,...,Ai−1,Ai=A,Ai+1,...,AN)
which takes in a fair environment F[1] and a list of agents (including itself), and outputs a mixed strategy over its available actions in F. [2]
A fair agent is one whose mixed strategy is a function of subjective probabilities[3] that it assigns to [the actions of some finite collection of agents in fair environments, where any agents not appearing in the original problem must themselves be fair].
Formally, if A is a fair agent in with a subjective probability estimator P, A’s mixed strategy in a fair environment F,
A(F,A1,...,Ai−1,Ai=A,Ai+1,...,AN)
should depend only on a finite collection of A’s subjective probabilities about outcomes
{P(Fk(A1,...,AN,B1,...BM))=[X1,...,XN+M]}Kk=1
for a set of fair environments F1,...,FK and an additional set of fair[4] agents[5] B1,...,BM if needed (note that not all agents need to appear in all environments).
A fair problem is a fair environment with one designated player, where all other agents are fair agents.
I might need to require every F to have a default action dF, so that I don’t need to worry about axiom-of-choice issues when defining an agent over the space of all fair environments.
I specified a probabilistic environment and mixed strategies because I think there should be a unique fixed point for agents, such that this is well-defined for any fair environment F. (By analogy to reflective oracles.) But I might be wrong, or I might need further restrictions on F.
Grossly underspecified. What kinds of properties are required for subjective probabilities here? You can obviously cheat by writing BlueEyedBot into your probability estimator.
This is an infinite recursion, of course. It works if we require each Bm to have a strictly lower complexity in some sense than A (e.g. the rank of an agent is the largest number K of environments it can reason about when making any decision, and each Bm needs to be lower-rank than A), but I worry that’s too strong of a restriction and would exclude some well-definable and interesting agents.
Does the fairness requirement on the Bm suffice to avert the MetaBlueEyedBot problem in general? I’m unsure.