From 182e32ac7f1a53595bc40ec337622d4ad85a20f4 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Daniel=20Alves=20R=C3=B6sel?=
 <60182044+velocitatem@users.noreply.github.com>
Date: Mon, 29 Dec 2025 14:23:08 +0000
Subject: [PATCH] refactor: unify notation and improve clarity in COI equations

---
 paper/src/chapters/03-methodology.tex | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/paper/src/chapters/03-methodology.tex b/paper/src/chapters/03-methodology.tex
index c8b0f6d..3102ca5 100644
--- a/paper/src/chapters/03-methodology.tex
+++ b/paper/src/chapters/03-methodology.tex
@@ -56,13 +56,13 @@ COI &= \rho - p_\text{min} \\
 &= \mathbb{E}_{p\sim\pi(\tau)}[p] - \min_{\tau^\prime\in\boldsymbol{\tau}}{\mathbb{E}_{p\sim\pi(\tau^\prime)}[p]}
 \end{align}
 
-Where the $p_0$ vector is both the initial state of the system and the base price for each product. We also define a pricing method at any time $t$ as $t: p_t \in \mathbb{R}_+^N$, satisfying a discrete cap $\{p \in \mathbb{R}_+^N \vert \quad \underline{p} \leq p \leq \overline{p}\}$ which act as our business constraints, limiting prices to the range of $(\underline{p}, \overline{p})$. We treat $p_t$ as the price vector shown to the an actor both experimentally and in-simulation.
+Where the $p_0$ vector is both the initial state of the system and the base price for each product. We also define a pricing method at any time $t$ as $p_t \in \mathbb{R}_+^N$, satisfying a discrete cap $\{p \in \mathbb{R}_+^N \mid \underline{p} \leq p \leq \overline{p}\}$ which act as our business constraints, limiting prices to the range of $(\underline{p}, \overline{p})$. We treat $p_t$ as the price vector shown to an actor both experimentally and in-simulation.
 
 Per product we follow a cumulative distrubtion $F(p)$ which we can leverage to prove the existence of COI under certain conditions of agent contamination. We state that:
 % Unify notation of underline p and p_min which now means same things
 \begin{align}
 \int_{\underline{p}}^{\rho} (\rho - p) \, dF(p) &= c \\
-\int_{\underline{p}}^{\underline{p} + COI} F(p) \, dp &= c \\
+\int_{\underline{p}}^{\underline{p} + \text{COI}} F(p) \, dp &= c \\
 c &> 0 \\
 \therefore p^* = \rho \wedge \rho &> p_\text{min}
 \end{align}
@@ -73,7 +73,7 @@ We then prove that:
 
 \begin{theorem}
 \begin{align}
-\lim_{N \to \infty} COI &= 0 \\
+\lim_{N \to \infty} \text{COI} &= 0 \\
 p_{(1)} &= \min (p_1, p_2, \ldots, p_n) \\
 P(p_{(1)} > p) &= [1-F(p)]^n \\
 \underline{F}(p) &= P(p_{(1)} \leq p) \\
@@ -81,9 +81,9 @@ P(p_{(1)} > p) &= [1-F(p)]^n \\
 &= 1 - [1 - F(p)] \\
 \text{survival functions...} \\
 \mathbb{E}[\underline{F}(p)] &= \underline{p} + \int_{\underline{p}}^{\overline{p}} [1 - F(p)]^n \, dp \\
-COI: \mathbb{E}[\underline{F}(p)] - \underline{p} \\
+\text{COI}: \mathbb{E}[\underline{F}(p)] - \underline{p} \\
 \cdots \\ 
-\int_{\underline{p}}^{\overline{p}} 0 dp &= 0 \\
+\int_{\underline{p}}^{\overline{p}} 0 \, dp &= 0 \\
 \end{align}
 % Since F(p) is a CDF, for any p>pmin​, F(p)>0, implying 0≤1−F(p)<1. By the properties of limits, as n→∞, [1−F(p)]n→0 for all p>pmin​.
 %Applying the Lebesgue Dominated Convergence Theorem (since the integrand is bounded by 1 on a finite interval):