\section{Methodology}


\subsection{Problem Formalization}

Mathematical formalization of agent-induced pricing distortions. Formal definition of potential loss mechanisms $\alpha D$

We consider a business across time during which we have an evolving vector $p_t \in \Re^N$ where $N$ is the number of products in our catalogue. our price vector is directly dependent on a demand function $q_t$ which we define as a linear method of a price elasticity matrix $B_t$. This is the same setup that Microsoft created in their research.

We gether interaction data from users interacting with a sample platform simulating a hotel/airline which generates interaction distributions $I_t = \{(p_t, q_t^\text{obs}, \pi_t)\}_{t=1}^T$


\subsection{Cost of Information Framework}

Mathematical demonstration and validation of the COI and citation backed evidence, and framework overview + show harm to user via other cost distortions. Maybe split into 3.2.1 (COI Theory) and 3.2.2 (Framework Design)

\subsection{System Architecture}
\begin{figure}[ht]
\centering
\begin{tikzpicture}[
  node distance=1.5cm and 2.5cm,
  box/.style={rectangle, draw, thick, minimum height=1cm, minimum width=3cm, align=center, fill=blue!10},
  kafka/.style={rectangle, draw=orange, thick, minimum height=1cm, minimum width=3cm, align=center, fill=orange!15},
  arrow/.style={thick,->,>=Stealth}
]

% Nodes
\node[box] (webapp) {Web Application \\ (Producer \& Consumer)};
\node[kafka, below=of webapp] (kafka) {Apache Kafka \\ Cluster};
\node[box, below=of kafka] (backend) {Backend Services / Microservices \\ (Producers and Consumers)};

% Connections
\draw[arrow] (webapp) to[out=210,in=150] node[above]{Publish} (kafka);
\draw[arrow] (kafka) to[out=50,in=330] node[below]{Consume} (webapp);
\draw[arrow] (backend) -- node[above]{Publish/Consume} (kafka);

% Optional: Kafka internal components
%\node[below=0.7cm of kafka, align=center] (topics) {Topics \\ Partitions};

% Optional background
\begin{scope}[on background layer]
  \node[draw, rounded corners, fill=orange!5, fit=(kafka), inner sep=0.3cm] {};
\end{scope}
\end{tikzpicture}
\caption{Technical Diagram}
\end{figure}

High level overview of how it works
\subsection{Experimental Design}
Study methodology and approach. Data acquisition strategy. Defined objectives and success criteria. Observable metrics and KPIs

\subsection{Dynamic Pricing Algorithm Analysis}
Deep dive into how the algorithm works, different kinds and justification for chosen appraoches + agent impact modeling and quantification.
\subsection{Reinforcement Learning Formulation}
How do we define the state space, action space and reward function breakdown and algorithm benchmarking.
POSSIBLY: Expand into full subsections: 3.6.1 (State-Action Space), 3.6.2 (Reward Design), 3.6.3 (Benchmarking)


\begin{algorithm}[t]
\DontPrintSemicolon
\KwIn{stepsize $\eta$, smoothing $\delta$, rank $d$}
\For{$t=1$ \KwTo $T$}{
  Sample $u_t$ on unit sphere; set $x_t^\prime=x_t+\delta u_t$\;
  Set $p_t \gets U x_t^\prime$ and observe $q_t, R_t(p_t)$\;
  $x_{t+1} \gets \Pi\_{\mathcal{X}}(x_t-\eta R_t(p_t) u_t)$\;
}
\caption{Online Pricing Optimization (template)}
\end{algorithm}