In modern economics, what are the least restrictive conditions shown to theoretically achieve Pareto Efficiency? In researching this question, I've come across plenty of sources that regurgitate intro econ textbooks that say markets equilibrate to Pareto efficiency in conditions of perfect competition (where specific and minimal definitions are also hard to find). I'm sure they're correct that this happens in perfect competition, but I've also read that various economists have simplified and minimized the conditions necessary for a market to theoretically reach pareto efficiency, but I can't find any details on those.

So what are the least restrictive conditions necessary for all markets that fit those conditions to theoretically progress toward a Pareto efficient equilibrium?
Well, when considering the minimal conditions necessary for an allocation to be Pareto optimal we must go to the primitives. First, we need the fact that all agents have rational preferences. What this means is that preferences are complete and transitive. Another thing we need is for preferences to be strictly convex.

Let $\succcurlyeq$ be the preference relation "weakly preferred to":

$\textbf{(A) Transitivity:}$ Given allocations $x,y,z \in X$, if $x\succcurlyeq y$ and $y\succcurlyeq z$ then $x\succcurlyeq z$

$\textbf{(B) Completeness:}$ Given allocations $x,y\in X$, either $x\succcurlyeq y$ or $y\succcurlyeq x$ or both $(x\sim y)$ 

$\textbf{(C) Strict Convexity}:\forall\, x,y \in X$ and $\forall \lambda\in[0,1]$ we have:
$$\lambda f(x)+(1-\lambda) f(y)&gt;f(\lambda x+(1-\lambda)y)$$

Also, let $X$ be the set of goods. We need $X$ to be finite.

These four axioms are sufficient for the existence of a competitive equilibrium.

Now, we must build a bare-bones theoretical market. What we need is the following:

$\textbf{(1) Complete Market:}$ Everyone has complete information and there are no transaction costs.

$\textbf{(2) Price Taking:}$ There is one price which everyone pays (and by the above statement, everyone knows this price).

$\textbf{(3) Locally Non-Satiated Preferences:}$ Basically, giving an agent more of any good will always increase their utility.

If we have $(A)-(C)$ and we have a finite number of goods, we will have a competitive equilibrium, and if we have $(1)-(2)$, we have a competitive market. Using these two and $(3)$, we can use the First Fundamental Theorem of Welfare Economics to say that our equilibrium is Pareto optimal.