chore: rename to distinguishability

paper/src/chapters/03-methodology.tex

@@ -3,7 +3,7 @@
 % Extra notes and clarifications: we observed some humans and get their transition probabilities between event types
 % We modify behavioral profiles of transition matrices with price elasticity matrices generated by sample valuations of a distributing.
 
-This section details the theoretical and practical framework developed to address dynamic pricing under the influence of non-human actors. We begin by formalizing the problem environment and the nature of the actors. We then derive the \textit{Cost of Information} (COI) theorem, proving the erosion of pricing power in the limit of agent saturation. Following this, we outline our generative contamination strategy using GOFAI-driven separability and transition probability learning. Finally, we formulate the robust control problem as a Stackelberg game solved via Distributionally Robust Reinforcement Learning (DR-RL) with constructed ambiguity sets.
+This section details the theoretical and practical framework developed to address dynamic pricing under the influence of non-human actors. We begin by formalizing the problem environment and the nature of the actors. We then derive the \textit{Cost of Information} (COI) theorem, proving the erosion of pricing power in the limit of agent saturation. Following this, we outline our generative contamination strategy using GOFAI-driven distinguishability and transition probability learning. Finally, we formulate the robust control problem as a Stackelberg game solved via Distributionally Robust Reinforcement Learning (DR-RL) with constructed ambiguity sets.
 
 \subsection{Problem Formalization}
 
@@ -141,7 +141,7 @@ The architecture of this platform begins with the deployed web-apps posting inte
 
 \paragraph{Public Web Artifact} We transition the Kappa like architecture of the data collection to a Lambda architecture for actual learning in a surrogate environment. This allows us to move faster on data which is provided and helps us create a feedback loop for production deployment. To support further research in this intersection of fields we release P4P \footnote{\url{https://github.com/velocitatem/p4p}} as a public repository providing the interaction layer of the PHANTOM framework. This provides a configurable storefront which can be tailored to any commercial setting with a standardized session-level event tracking. We document the API adapters or what the framework expects in terms of schemas for pricing providers and log ingestion servicse. The repository is intended for controlled experimentation and method replication rather than production commerce deployment.
 
-\paragraph{Public Dataset Artifact} For reproducibility of the behavioral analysis and separability experiments, we also release the interaction dataset used in this thesis as \textit{WhoClickedIt}. The dataset is hosted on Hugging Face \footnote{\url{https://huggingface.co/datasets/velocitatem/whoclickedit}} and is distributed as one flattened event sheet (\texttt{whoclicked.csv}) with explicit labels (\texttt{actor\_type}, \texttt{is\_agent}, and \texttt{record\_type}). The associated dataset card specifies the schema, collection process, and known limitations; a full copy is included in Appendix~\ref{app:whoclicked_card}.
+\paragraph{Public Dataset} For reproducibility of the behavioral analysis and distinguishability experiments, we also release the interaction dataset used in this thesis as \textit{WhoClickedIt}. The dataset is hosted on Hugging Face \footnote{\url{https://huggingface.co/datasets/velocitatem/whoclickedit}} and is distributed as one flattened event sheet (\texttt{whoclicked.csv}) with explicit labels (\texttt{actor\_type}, \texttt{is\_agent}, and \texttt{record\_type}). The associated dataset card specifies the schema, collection process, and known limitations; a full copy is included in Appendix~\ref{app:whoclicked_card}.
 
 
 \subsubsection{DevOps Principles}
@@ -189,9 +189,9 @@ The human data collection involved 13 participants, all of whom provided explici
 
 To evaluate quality and realism of the setup, we store both structured event logs and full interaction transcripts. This lets us combine quantitative analysis with transcript-level qualitative findings. The result is an isolated system where we can control the interaction process while preserving realistic behavior.
 
-Operationally, goals and experiment runs are tracked in PostgreSQL (goal table, run table, and assignment mapping). This data-acquisition phase is the first half of the methodology and is intentionally a disconnected component that feeds the later contributions. The second half uses collected behavioral traces to separate classes $\theta \in \{A,H\}$ with session-conditioned probability estimates, then injects those estimates into the pricing learner.
+Operationally, goals and experiment runs are tracked in PostgreSQL (goal table, run table, and assignment mapping). This data-acquisition phase is the first half of the methodology and is intentionally a disconnected component that feeds the later contributions. The second half uses collected behavioral traces to distinguish classes $\theta \in \{A,H\}$ with session-conditioned probability estimates, then injects those estimates into the pricing learner.
 
-Our process follows three stages: (1) observe and \textit{vectorize} behavioral interactions, (2) learn separability to characterize human versus agent patterns, and (3) use the learned signal to train a defensive policy in a controlled dynamic-pricing simulator.
+Our process follows three stages: (1) observe and \textit{vectorize} behavioral interactions, (2) learn distinguishability to characterize human versus agent patterns, and (3) use the learned signal to train a defensive policy in a controlled dynamic-pricing simulator.
 
 \begin{figure}[ht]
   \resizebox{\columnwidth}{!}{%
@@ -298,15 +298,15 @@ In addition to behavioral events, the platform logs price observations to a sepa
 
 
 
-\subsection{Generative Contamination and Separability}
+\subsection{Generative Contamination and Distinguishability}
 
 To train a robust pricing learner, we need a simulator that can generate realistic interaction data under controlled contamination. We build this from Phantom data using a two-stage approach.
 
 
-\subsubsection{Ground-Truth Separability}
-Because sessions are collected under controlled experimental conditions where each actor is assigned a known type at the start of the trial, labels $\theta_s \in \{H, A\}$ are available as ground truth rather than as the output of a heuristic classifier. We therefore estimate separate transition kernels directly from each labeled partition $\mathcal{D}_H$ and $\mathcal{D}_A$, treating the resulting $\hat{\mathcal{T}}_H$ and $\hat{\mathcal{T}}_A$ as the ground-truth behavioral profiles for each class. We then ask a direct methodological question: are the kernels separable enough to justify downstream pricing control that depends on that separability?
+\subsubsection{Ground-Truth Distinguishability}
+Because sessions are collected under controlled experimental conditions where each actor is assigned a known type at the start of the trial, labels $\theta_s \in \{H, A\}$ are available as ground truth rather than as the output of a heuristic classifier. We therefore estimate separate transition kernels directly from each labeled partition $\mathcal{D}_H$ and $\mathcal{D}_A$, treating the resulting $\hat{\mathcal{T}}_H$ and $\hat{\mathcal{T}}_A$ as the ground-truth behavioral profiles for each class. We then ask a direct methodological question: are the kernels distinguishable enough to justify downstream pricing control that depends on that distinguishability?
 
-To answer this, we compute per-session KL divergence scores against both class-level centroids. For each session $s$ in either partition, we fit a session-level event transition kernel $\hat{\mathcal{T}}_s$ from that session's trajectory alone, then compute its average KL divergence to the human centroid ($\Delta_{H,s}$) and to the agent centroid ($\Delta_{A,s}$). The per-session separability score is the gap $\Delta_{H,s} - \Delta_{A,s}$: a negative value indicates proximity to human behavior, a positive value indicates proximity to agent behavior.
+To answer this, we compute per-session KL divergence scores against both class-level centroids. For each session $s$ in either partition, we fit a session-level event transition kernel $\hat{\mathcal{T}}_s$ from that session's trajectory alone, then compute its average KL divergence to the human centroid ($\Delta_{H,s}$) and to the agent centroid ($\Delta_{A,s}$). The per-session distinguishability score is the gap $\Delta_{H,s} - \Delta_{A,s}$: a negative value indicates proximity to human behavior, a positive value indicates proximity to agent behavior.
 
 The normality assumption cannot be made for KL divergence distributions, which are right-skewed and bounded below by zero, so we do not use a Student's $t$-test. Instead we apply a Mann-Whitney $U$ test \parencite{mann_test_1947} on the per-session gap scores between the two groups. The Mann-Whitney test is a rank-based nonparametric test that compares the stochastic ordering of two independent samples without distributional assumptions, making it appropriate for small samples drawn from skewed populations. We report $U$, the exact two-sided $p$-value, and group-level descriptive statistics for the gap scores.
 
@@ -470,7 +470,7 @@ We also consider taxation-like overlays for agent traffic under strategy-proof m
 
 \subsubsection{Pricing Mechanism Summary}
 
-We now present the complete pricing mechanism that integrates the behavioral separability, contamination estimation, and robust optimization components developed in the preceding sections. Algorithm~\ref{alg:phantom_loop_clean} formalizes the defensive pricing loop as a Stackelberg game where the platform (leader) sets prices and the aggregate demand (follower) responds through observed session trajectories.
+We now present the complete pricing mechanism that integrates the behavioral distinguishability, contamination estimation, and robust optimization components developed in the preceding sections. Algorithm~\ref{alg:phantom_loop_clean} formalizes the defensive pricing loop as a Stackelberg game where the platform (leader) sets prices and the aggregate demand (follower) responds through observed session trajectories.
 
 \begin{algorithm}[t]
 \caption{PHANTOM defensive pricing loop}