(This post was originally published on Oct 20th 2017, and is 1 of 4 posts brought forwarded today as part of the AI Alignment Forum launch sequence on fixed points.)

In this post, I give a stronger version of the open question presented here, and give a motivation for this stronger property. This came out of conversations with Marcello, Sam, and Tsvi.

Definition: A continuous function $f : X \to Y$ is called ubiquitous if for every continuous function $g : X \to Y$ , there exists a point $x \in X$ such that $f (x) = g (x)$ .

Open Problem: Does there exist a topological space $X$ with a ubiquitous function $f : X \to [0, 1]^{X}$ ?

I will refer to the original problem as the Converse Lawvere Problem, and the new version as the the Ubiquitous Converse Lawvere Problem. I will refer to a space satisfying the conditions of (Ubiquitous) Converse Lawvere Problem, a (Ubiquitous) Converse Lawvere Space, abbreviated (U)CLS. Note that a UCLS is also a CLS, since a ubiquitous is always surjective, since $g$ can be any constant function.

Motivation: True FairBot

Let $X$ be a Converse Lawvere Space. Note that since such an $X$ might not exist, the following claims might be vacuous. Let $f : X \to [0, 1]^{X}$ be a continuous surjection.

We will view $X$ as a space of possible agents in an open source prisoner's dilemma game. Given two agents $A, B \in X$ , we will interpret $f_{A} (B)$ as the probability with which A cooperates when playing against $B$ . We will define $U_{A} (B) := 2 f_{B} (A) - f_{A} (B)$ , and interpret this as the utility of agent $A$ when playing in the prisoner's dilemma with $B$ .

Since $f$ is surjective, every continuous policy is implemented by some agent. In particular, this means gives:

Claim: For any agent $A \in X$ , there exists another agent $A^{'} \in X$ such that $f_{A^{'}} (B) = f_{B} (A)$ . i.e. $A^{'}$ responds to $B$ the way that $B$ responds to $A$ .

Proof: The function $B \mapsto f_{B} (A)$ is a continuous function, since $B \mapsto f_{B}$ is continuous, and evaluation is continuous. Thus, there is a policy $B \mapsto f_{B} (A)$ in $[0, 1]^{X}$ . Since $f$ is surjective, this policy must be the image under $f$ of some agent $A^{'}$ , so $f_{A^{'}} (B) = f_{B} (A)$ .

Thus, for any fixed agent $A$ , we have some other agent $A^{'}$ that responds to any $B$ the way $B$ responds to $A$ . However, it would be nice if $A^{'} = A$ , to create a FairBot that responds to any opponent the way that that opponent responds to it. Unfortunately, to construct such a FairBot, we need the extra assumption that $f$ is ubiquitous.

Claim: If $f$ is ubiquitous, then exists a true fair bot in $X$ : an agent $F B \in X$ , such

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