My first idea is this.
In plain english, the probability is taken over the pseudorandomness of the good pair of direct incentives and all choices of purchase and sale sequences. The rest of this post just makes this idea more precise.
For sufficiently large N and K, denote by S_N^K the set of all rational numbers smaller than N with precision 1/K. Precisely,
S_N^K = \{\frac{m}{K} \in \mathbb{Q} \mid m \in \mathbb{Z}, |m| < NK\}
We can model activity on the market by an ordered list of elements in S_{N}^K. That is, we can model activity on the market by an element of (S_{N}^K)^n, for sufficiently large n.
For instance,
(-5000000, 100, -5, 1000000, 0, 0, \ldots, 0).
means, starting at peg, someone sold 5M FEI, then someone bought 100 FEI, then someone sold -5 FEI, then someone bought 1M FEI. There’s a sufficiently large n precisely because reweights are always in the finite horizon. I think it’s fair to call each of these vectors in (S_N^K)^n a path.
If we think about a constant product AMM, there exists a function p_{cp}: (S_N^K)^n \to \mathbb{R} that takes any path to the marginal price at the end of the path, assuming the market started at peg. The fact that a constant product AMM is path independent is precisely the statement that there exists a function p_{cp}^*: \mathbb{R} \to \mathbb{R} such that, for all x \in (S_N^K)^n,
p_{cp}(x) = p_{cp}(x_1, x_2, \ldots, x_n) = p_{cp}^*(\sum_{i=1}^n x_i)
Now, if we allow direct incentives to be pseudorandom with seed k \in \{0, 1\}^m, for any directly incentivized constant product AMM, there exist two keyed functions s_{di}^k:(S_N^K)^n \to \mathbb{R} and b_{di}^k:(S_N^K)^n \to \mathbb{R}. The first takes any path to the marginal selling price at the end of the path, assuming the market starts at peg. The second takes any path to the marginal buying price at the end of the path, assuming the market starts at peg.
This allows us to state more precisely what the probabilistic guarantee is. Let \Omega = (S_N^K)^n \times \{0, 1\}^m, let \mathcal{F} be the set of all subsets of \Omega, and let P be the counting function. Then (\Omega, \mathcal{F}, P) is a probability space with random variables p_{cp}, s_{di}^k and b_{di}^k. The probabilistic guarantee I have in mind is
P\left( s_{di}^k < p_{cp} - \delta_s, b_{di}^k < p_{cp} - \delta_b \right) \geq \frac{1}{2} + \epsilon
for some \epsilon > 0 and adequate choices of \delta_s, \delta_b > 0. Perhaps we can weaken this condition further to
P\left( s_{di}^k < p_{cp}, b_{di}^k < p_{cp} \right) \geq \frac{1}{2} + \epsilon
And, thinking about it a little more, it occurs to me that what we really want is a lower bound on the conditional probabilities we get when we fix x \in (S_N^K)^n. Otherwise, there are going to be paths where the condition doesn’t hold often enough. And that’s obviously undesirable. So
\forall x \in (S_N^K)^n, P\left( s_{di}^k < p_{cp}, b_{di}^k < p_{cp} \mid x \right) \geq \frac{1}{2} + \epsilon