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refactor: unify notation and improve clarity in COI equations

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Daniel Alves Rösel
2025-12-29 14:23:08 +00:00
parent 516e8ecb1f
commit 182e32ac7f
1 changed files with 5 additions and 5 deletions
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10
paper/src/chapters/03-methodology.tex
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@@ -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]} &= \mathbb{E}_{p\sim\pi(\tau)}[p] - \min_{\tau^\prime\in\boldsymbol{\tau}}{\mathbb{E}_{p\sim\pi(\tau^\prime)}[p]}
\end{align} \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: 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 % Unify notation of underline p and p_min which now means same things
\begin{align} \begin{align}
\int_{\underline{p}}^{\rho} (\rho - p) \, dF(p) &= c \\ \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 \\ c &> 0 \\
\therefore p^* = \rho \wedge \rho &> p_\text{min} \therefore p^* = \rho \wedge \rho &> p_\text{min}
\end{align} \end{align}
@@ -73,7 +73,7 @@ We then prove that:
\begin{theorem} \begin{theorem}
\begin{align} \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_{(1)} &= \min (p_1, p_2, \ldots, p_n) \\
P(p_{(1)} > p) &= [1-F(p)]^n \\ P(p_{(1)} > p) &= [1-F(p)]^n \\
\underline{F}(p) &= P(p_{(1)} \leq p) \\ \underline{F}(p) &= P(p_{(1)} \leq p) \\
@@ -81,9 +81,9 @@ P(p_{(1)} > p) &= [1-F(p)]^n \\
&= 1 - [1 - F(p)] \\ &= 1 - [1 - F(p)] \\
\text{survival functions...} \\ \text{survival functions...} \\
\mathbb{E}[\underline{F}(p)] &= \underline{p} + \int_{\underline{p}}^{\overline{p}} [1 - F(p)]^n \, dp \\ \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 \\ \cdots \\
\int_{\underline{p}}^{\overline{p}} 0 dp &= 0 \\ \int_{\underline{p}}^{\overline{p}} 0 \, dp &= 0 \\
\end{align} \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​. % 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): %Applying the Lebesgue Dominated Convergence Theorem (since the integrand is bounded by 1 on a finite interval):