It's an essential aspect of decision making for an agent to figure out where it might be. Thought experiments try to declare the current situation, but they don't necessarily need to be able to convincingly succeed. Algorithmic induction, such as updating from Solomonoff prior, is the basic way an agent figures out which situations it should care about, and declaring that we are working with a particular thought experiment doesn't affect the prior. In line with updatelessness, an agent should be ready for observations in general (according to which of them it cares about more), rather than particular "fair" observations, so distinguishing observations that describe "fair" thought experiments doesn't seem right either.

My current candidate definitions, with some significant issues in the footnotes:

A fair environment is a probabilistic function $F(x_1,...,x_N)=[X_1,...,X_N]$ from an array of actions to an array of payoffs. 

An agent $A$ is a random variable 

$A(F,A_1,...,A_{i-1},A_i=A,A_{i+1},...,A_N)$ 

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 $F$,

$A(F,A_1,...,A_{i-1},A_i=A,A_{i+1},...,A_N)$

should depend only on a finite collection of $A$'s subjective probabilities about outcomes 

$\left\{P\right(F_k(A_1,...,A_N,B_1,...B_M))=[X_1,...,X_{N+M}]{\}}_{k=1}^K$ 

for a set of fair environments $F_1,...,F_K$ and an additional set of fair^[4] agents^[5] $B_1,...,B_M$ 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.

1. ^{^}

   I might need to require every $F$ to have a default action $d_F$, so that I don't need to worry about axiom-of-choice issues when defining an agent over the space of all fair environments.

2. ^{^}

   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 $F$.

3. ^{^}

   Grossly underspecified. What kinds of properties are required for subjective probabilities here? You can obviously cheat by writing BlueEyedBot into your probability estimator.

4. ^{^}

   This is an infinite recursion, of course. It works if we require each $B_m$ 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 $B_m$ 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.

5. ^{^}

   Does the fairness requirement on the $B_m$ suffice to avert the MetaBlueEyedBot problem in general? I'm unsure.

See Jessica's comment. Yeah it's primitive recursive assuming that your deductive process is primitive recursive. (Also assuming that your traders are primitive recursive; e.g. if they are polytime as in the paper.) There's probably some other parameters not necessarily set in the implementation described in the paper, e.g. the enumerator of trader-machines, but you can make those primrec.

If some function g is computable in O(f(n)) time for primitive recursive f then g is primitive recursive, by simulating a Turing machine. I am pretty sure a logical inductor would satisfy; while it's super exponential time, it's not so fast-growing it's not primitive recursive (like with the Ackerman function).