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 \(f:X\rightarrow Y\) 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 \(f:X\rightarrow[0,1]^X\)?
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 \(f:X\rightarrow[0,1]^X\) 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, 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 \(f_{UB}(A)=1f_A(UB)\) 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 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).
