# The Ubiquitous Converse Lawvere Problem

_(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](https://agentfoundations.org/item?id=1356), and give a motivation for this stronger property. This came out of conversations with Marcello, Sam, and Tsvi.

**Definition:** A continuous function $f:X\rightarrow Y$ is called ubiquitous if for every continuous function $g:X\rightarrow Y$, 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 $f:X\rightarrow[0,1]^X$?

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I will refer to the [original](https://agentfoundations.org/item?id=1356) 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 $f:X\rightarrow[0,1]^X$ 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 $U_A(B):=2f_B(A)-f_A(B)$, 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 $A^\prime\in X$ such that $f_{A^\prime}(B)=f_B(A)$. i.e. $A^\prime$ 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 $A^\prime$, so $f_{A^\prime}(B)=f_B(A)$.

Thus, for any fixed agent $A$, we have some other agent $A^\prime$ that responds to any $B$ the way $B$ responds to $A$. However, it would be nice if $A^\prime=A$, 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 $FB\in X$, such that $f_{FB}(A)=f_A(FB)$ for all agents $A\in X$.

**Proof:** Given an agent $B\in X$, there exists an policy $g_B\in [0,1]^X$ such that $g_B(A)=f_A(B)$ for all $A$, since $A\mapsto f_A(B)$ is continuous. Further, the function $B\mapsto g_B$ is continuous, since the function $A,B\mapsto f_A(B)$ and the definition of the exponential topology. Since $f$ is ubiquitous, there must be some $FB\in X$ such that $f_{FB}=g_{FB}$. But then, for all $A$, we have $f_{FB}(A)=g_{FB}(A)=f_A(FB)$.

Note that we may not need the full power of ubiquitous here, but it is the simplest property I see that gets the result.

Note that this FairBot is fair in a stronger sense than the FairBot of [modal combat](https://arxiv.org/pdf/1401.5577.pdf), in that it always has the same output as its opponent. This may make you suspicious, since the  you can also construct an UnfairBot, $UB$ such that $f_{UB}(A)=1-f_A(UB)$ for all $A$. This would have caused a problem in the modal combat framework, since you can put a FairBot and an UnfairBot together to form a paradox. However, we do not have this problem, since we deal with probabilities, and simply have $f_{UB}(FB)=f_{FB}(UB)=1/2$. Note that the exact phenomenon that allows this to possibly work is the fixed point property of the interval $[0,1]$ which is the only reason that we cannot use diagonalization to show that no CLS exists.

Finally, note that we already have a combat framework that has a true FairBot: the [reflective oracle](https://arxiv.org/pdf/1508.04145.pdf) framework. In fact, the reflective oracle framework may have all the benefits we would hope to get out of a UCLS. (other than the benefit of simplicity of not having to deal with computability and hemicontinuity).

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_This post was originally published on Oct 20th 2017, and has been brought forwarded as part of the AI Alignment Forum launch sequences._

_Tomorrow's AIAF sequences post will be 'Iterated Amplification and Distillation' by Ajeya Cotra, in the sequence on iterated amplification._