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* initial environemnt definitions * high level defintion * formlating the reward simply * improved implementation * tailored docker compose image for secondary tenaordboard * preliminary desriptions and babble * details on formulation and defintion of agent and its loop * typos one * more grammar issues * fluidity improvements and refactors * more decluttering and dnoising * finalizing introduction review * some methodology * somehow this disappeared * bit more of this and that * methodology of how we do architectuer and online DP * fix: compilation * expanding on the taxonomy and economic references * authoer notes * acks + google GCP * making space w new format nada lit review * stronger lit review and more sources * forgot about tables and graphs * dedupe citations * adding cloudflare * fixing env vars * updating docs with url * upating embed * fixing the url * paper badge * formaliztaion of rewards and adding definitions * noisy formulations * connecting some more dots here * adding significant weight in prices * fixing error * fixing typos and consistency * extra math formulations and refferenceot DRO * fixing diagram of loops * github mindmap * fixing erro and thiknig about big picture * enhancing the website * goals methodology and gitignore * some more references and theory links * talking about some wtp * feature: added wordcounter * forcing latex builds and fixining the bib # * refactor: update Cost of Information equations and notation for clarity * some more math and refactors * refactor: unify notation and improve clarity in COI equations * refactor: generalize master function for demand estimation and pricing strategies * we dont like math but we have to do it :( * refactor: enhance Cost of Information framework with additional context and illustration * refactor: enhance literature review and methodology sections with economic theory insights and system architecture details * alining format to fit the rubric * refactoring bibliography * fix: align * mdp additionally * trying different title * adding balance figure * agentic givergence, finally * fix: figure fonts adjusted to match
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\section{Methodology}
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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.
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\subsection{Problem Formalization}
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Mathematical formalization of agent-induced pricing distortions. Formal definition of potential loss mechanisms $\alpha D$
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We define a commercial environment where the platform interacts with a stream of sessions. Let $\mathcal{S}$ denote the set of all sessions. Each session $s \in \mathcal{S}$ is generated by an actor belonging to a latent class $Y_s \in \{H, A\}$, where $H$ denotes Human and $A$ denotes Agent.
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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.
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Each session produces a trajectory of observable events $\tau_s = (e_{s,1}, \ldots, e_{s,L_s})$. An event $e_{s,k}$ is a tuple defined as:
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\begin{equation}
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e_{s,k} = (a_{s,k}, i_{s,k}, t_{s,k})
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\end{equation}
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where:
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\begin{itemize}
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\item $a_{s,k} \in \mathcal{A}$ is the action taken (e.g., \texttt{view\_item}, \texttt{add\_to\_cart}).
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\item $i_{s,k} \in \{1, \ldots, N\}$ is the target item index.
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\item $t_{s,k} \in \mathbb{R}_+$ is the continuous timestamp.
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\end{itemize}
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We gether interaction data from users interacting with a sample platform simulating a hotel/airline which generates interaction distributions $I_t = \{(p_t, q_t^\text{obs}, \pi_t)\}_{t=1}^T$
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The platform does not directly observe the true underlying demand function $d(p)$. Instead, it observes a behavioral proxy $\hat{q}_t$, which is a composite signal derived from the mixture of actor types. We define the demand proxy for product $i$ at epoch $t$ as a weighted aggregation of events:
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\begin{equation}
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\hat{q}_{t,i} = \sum_{s \in \mathcal{S}_t} \sum_{k=1}^{L_s} \omega(a_{s,k}) \cdot \mathbb{1}[i_{s,k} = i]
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\end{equation}
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where $\omega: \mathcal{A} \to \mathbb{R}_+$ assigns weights to actions based on their signal strength regarding willingness to pay.
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\subsubsection{Actor Types and Demand Curves}
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We formalize the heterogeneity of actors by introducing a type space $\Theta$. An actor of class $Y_s$ is further parameterized by a type $\theta \sim \mathcal{D}_{Y}$. This type determines the actor's demand response function $d(p; \theta)$, sampled from a distribution of possible demand curves. The total observed demand is a stochastic process governed by the mixture:
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\begin{equation}
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Q(p) = (1-\alpha) \cdot \mathbb{E}_{\theta \sim \mathcal{D}_H}[d(p; \theta)] + \alpha \cdot \mathbb{E}_{\theta \sim \mathcal{D}_A}[d(p; \theta)] + \epsilon_t
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\end{equation}
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where $\alpha \in [0, 1]$ represents the contamination parameter (proportion of agents) and $\epsilon_t$ is non-stationary market noise.
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\subsection{Cost of Information Framework}
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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)
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\subsection{Cost of Information (COI) Framework}
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The \textit{Cost of Information} (COI) represents the markup a pricing policy $\pi$ attempts to extract from the market by leveraging demand signals. We define COI as the expected premium over the minimum viable price $\underline{p}$ (or marginal cost). This also speaks to the financial urgency as a consequence of information asymmetry between the platform and the actors.
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\begin{definition}[Cost of Information]
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Let $\pi(\tau)$ be a pricing policy mapping interaction histories to prices. The COI is defined as:
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\begin{align}
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\text{COI} &= \mathbb{E}[P] - \underline{p} \\
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&= \int_{\underline{p}}^{\bar{p}} (1 - F_\pi(p)) \, dp
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\end{align}
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where $F_\pi(p)$ is the cumulative distribution function of prices generated by $\pi$ under standard operating conditions.
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\end{definition}
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\subsection{System Architecture}
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\begin{figure}[ht]
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\centering
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\begin{tikzpicture}[
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node distance=1.5cm and 2.5cm,
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box/.style={rectangle, draw, thick, minimum height=1cm, minimum width=3cm, align=center, fill=blue!10},
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kafka/.style={rectangle, draw=orange, thick, minimum height=1cm, minimum width=3cm, align=center, fill=orange!15},
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arrow/.style={thick,->,>=Stealth}
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]
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\centering
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\begin{tikzpicture}[scale=1.2]
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% Define the Gaussian function: centered at 2
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\def\bellcurve(#1){1.5 * exp(-0.5*((#1-2)/0.6)^2)}
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% Nodes
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\node[box] (webapp) {Web Application \\ (Producer \& Consumer)};
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\node[kafka, below=of webapp] (kafka) {Apache Kafka \\ Cluster};
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\node[box, below=of kafka] (backend) {Backend Services / Microservices \\ (Producers and Consumers)};
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% Draw the main axis
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\draw[->, thick] (0, 0) -- (4.5, 0) node[right] {$p$};
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\draw[->, thick] (0, 0) -- (0, 2) node[above] {Density};
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% Connections
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\draw[arrow] (webapp) to[out=210,in=150] node[above]{Publish} (kafka);
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\draw[arrow] (kafka) to[out=50,in=330] node[below]{Consume} (webapp);
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\draw[arrow] (backend) -- node[above]{Publish/Consume} (kafka);
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\draw[thick, smooth, samples=100] plot[domain=0:4] (\x, {\bellcurve(\x)});
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\node at (3.2, 1.2) {$f_\pi(p)$};
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% Optional: Kafka internal components
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%\node[below=0.7cm of kafka, align=center] (topics) {Topics \\ Partitions};
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% Define p_min and E[p]
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\def\pmin{0.8}
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\def\mean{2}
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% Optional background
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\begin{scope}[on background layer]
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\node[draw, rounded corners, fill=orange!5, fit=(kafka), inner sep=0.3cm] {};
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\end{scope}
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\end{tikzpicture}
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\caption{Technical Diagram}
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% Vertical lines
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\draw[dashed] (\pmin, 0) -- (\pmin, 2.0);
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\draw[dashed] (\mean, 0) -- (\mean, 2.0);
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% Labels on axis
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\node[below] at (\pmin, 0) {$\underline{p}$};
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\node[below] at (\mean, 0) {$\mathbb{E}[p]$};
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\draw[<->, thick, red] (\pmin, 2.0) -- (\mean, 2.0) node[midway, above] {COI};
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\end{tikzpicture}
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\caption{Illustration of the Cost of Information (COI). The COI is defined as the difference between the expected price $\mathbb{E}[p]$ realized by the policy and the minimum viable price $\underline{p}$.}
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\label{fig:coi_illustration}
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\end{figure}
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High level overview of how it works
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We now formally demonstrate that standard dynamic pricing mechanisms are not incentive-compatible with high-frequency agentic traffic. As the number of independent competitive agents $N$ querying the system grows, the platform's ability to sustain a COI vanishes.
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\begin{theorem}[COI Erosion in the Limit]
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Let $N$ be the number of independent, utility-maximizing agents querying the platform. Let $p_{(1)}$ be the first order statistic (minimum) of the prices offered to these agents. As $N \to \infty$, the Cost of Information converges to 0.
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\end{theorem}
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\begin{proof}
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Let $p_1, \ldots, p_N$ be independent and identically distributed (i.i.d.) price samples drawn from the policy's distribution $F(p)$ with support $[\underline{p}, \bar{p}]$. The realizable price for an optimal searching agent is the first order statistic $p_{(1)} = \min(p_1, \ldots, p_N)$.
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The survival function (or reliability function) of the minimum price is given by:
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\begin{equation}
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S_{p_{(1)}}(t) = P(p_{(1)} > t) = [1 - F(t)]^N
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\end{equation}
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To determine the expected value $\mathbb{E}[p_{(1)}]$, we recall the property that for any continuous random variable $X$ with support $[A, B]$, the expectation can be expressed as the lower bound plus the integral of the survival function:
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\begin{equation}
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\mathbb{E}[X] = A + \int_{A}^{B} P(X > t) \, dt
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\end{equation}
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Applying this to our pricing statistic where the lower bound is $\underline{p}$:
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\begin{align}
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\mathbb{E}[p_{(1)}] &= \underline{p} + \int_{\underline{p}}^{\bar{p}} P(p_{(1)} > t) \, dt \\
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&= \underline{p} + \int_{\underline{p}}^{\bar{p}} [1 - F(t)]^N \, dt
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\end{align}
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Since $F(t)$ is a valid CDF, for any $t > \underline{p}$, we have strict inequality $F(t) > 0$, implying $0 \le 1 - F(t) < 1$. By the properties of limits, as $N \to \infty$, the term $[1 - F(t)]^N$ converges to 0 pointwise for all $t > \underline{p}$.
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Applying the Lebesgue Dominated Convergence Theorem (noting that the integrand is bounded by 1 on the finite interval $[\underline{p}, \bar{p}]$):
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\begin{equation}
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\lim_{N \to \infty} \int_{\underline{p}}^{\bar{p}} [1 - F(t)]^N \, dt = \int_{\underline{p}}^{\bar{p}} 0 \, dt = 0
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\end{equation}
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Substituting this back into the expression for COI:
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\begin{align}
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\lim_{N \to \infty} \text{COI} &= \lim_{N \to \infty} (\mathbb{E}[p_{(1)}] - \underline{p}) \\
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&= \lim_{N \to \infty} \left( (\underline{p} + 0) - \underline{p} \right) \\
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&= 0
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\end{align}
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\end{proof}
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This result proves that standard pricing policies $\pi$ fail to extract surplus in the presence of large-scale agentic search, necessitating a robust counter-mechanism.
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% The DRO objective creates a lower bound on COI extraction, effectively guaranteeing a minimum margin even in the presence of adversarial agents. we need to prove this and demonstrate that in a theorem.
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%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)
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\subsection{System Architecture: Hybrid Kappa-Lambda Architecture}
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In order for our research to have grounding in interactions we built a robust e-commerce web-platform. We initially conducted a survey of the leading platforms of airlines and hotel booking sites to identify the specific interface patterns that effectively manage complex travel data. Our analysis revealed a clear industry standard: while both sectors rely on tabbed service selection and left-sidebar filtering to streamline navigation, they diverge in result presentation: airlines utilize visual date-price bars and multi-step wizards to optimize for logistical transparency, whereas hotel platforms leverage image-led cards and scarcity triggers to drive emotional engagement and urgency. Our web framework defines a highly agnostic boilerplate which can be seeded with any data-modality with an easy-to-tailor pattern, which we leverage to define a \texttt{hotel} and \texttt{airline} mode. Both modes are then individually deployed via an environment level argument which adjusts the proxy routing with a custom middleware inside next.js to render only the desired mode. The purpose of this was to create a baseline adaptable to any use-case or desired commercial application.
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The architecture of this platform begins with the deployed web-apps posting interaction data to our backend which processes them and stores each ingested interaction into a kafka cluster. This serves as our data reservoir tracking and associating each interaction with its session and importantly with which experiment it belongs to. Not only do we track the behavioral interactions, but our pricing provider micro-service, once called by the frontend reports the observed/queried price-product into kafka. This kafka cluster is subscribed to by our pipeline which is configured on a schedule in Airflow, with the possibility of manual trigger. The final stage of the pricing pipeline, submits computed dynamic pricing results into a redis database for quick updates which is then read by the pricing provider and displayed on the webapp. This is a very generic end-to-end mechanism which is applicable to a variety of different e-commerce tasks. We intentionally put emphasis on the development of this infrastructure to establish a reproducible framework for interaction and to minimize any noise.
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\subsubsection{DevOps Principles}
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\subsubsection{Online Dynamic Pricing}
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The dynamic pricing done is handled by a pipeline which computes a demand estimate on a per-product basis of a specific window of the data, defined by the period $T$ which by default is 5 minutes. This dynamic pricing pipeline computes a demand estimate vector $\hat{q} \in \mathbb{R}^N$ by a weighted sum of interactions for each product, it additionally computes a price elasticity vector $\hat{\epsilon}$ in the same dimensions as our demand. The final features matrix is of the size $N \times 2$ which we translate to a new price vector $\hat{p} \in \mathbb{R}^N$. The transformation that governs this dynamic pricing is a very simple surge-based pricing (a special case of our later defined policy $\pi$):
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\begin{equation}
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\hat{p}_i = \begin{cases}
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p_{0,i} \cdot \lambda_{\text{surge}} & \text{if } \hat{q}_i \geq \theta_{\text{high}} \\
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p_{0,i} \cdot \lambda_{\text{disc}} & \text{if } \hat{q}_i \leq \theta_{\text{low}} \\
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p_{0,i} & \text{otherwise}
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\end{cases}
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\quad \forall i \in \{1, \ldots, N\}
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\end{equation}
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where $p_0 \in \mathbb{R}^N$ is the base price vector (which is seeded into our database distinctly for each mode of the commerce platform), $\theta_{\text{high}}, \theta_{\text{low}} \in \mathbb{R}$ are demand thresholds defining surge and discount regions, and $\lambda_{\text{surge}}, \lambda_{\text{disc}} \in \mathbb{R}^+$ are multiplicative factors with typical values $\lambda_{\text{surge}} = 1.2$ and $\lambda_{\text{disc}} = 0.9$. This piecewise function enables rapid price adjustment in response to observed demand without requiring complex elasticity estimation or historical calibration, allowing us to expose actors within our experiments to a system with a dynamic component of pricing.
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We will for our offilne experimental intents generalize a master function for encompasing distinct demand estimation and pricing strategies.
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\begin{align}
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V(\cdot) = \max_{p_t} \min_{Q \in \mathcal{U}(\hat{d})}{\mathbb{E}_{d\sim Q} [p_t \times d(p_t, x_t ; \theta) + \psi V_{t+1}(\cdot)]}
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\end{align}
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We follow differnet substitutouns which will server as hyperparameters later on.
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\subsection{Experimental Design}
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Study methodology and approach. Data acquisition strategy. Defined objectives and success criteria. Observable metrics and KPIs
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\subsection{Dynamic Pricing Algorithm Analysis}
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Deep dive into how the algorithm works, different kinds and justification for chosen appraoches + agent impact modeling and quantification.
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\subsection{Reinforcement Learning Formulation}
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How do we define the state space, action space and reward function breakdown and algorithm benchmarking.
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POSSIBLY: Expand into full subsections: 3.6.1 (State-Action Space), 3.6.2 (Reward Design), 3.6.3 (Benchmarking)
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The experimentation begins with the design of goals, with careful consideration to assure a uniform spanning across different variables within each product-architecture of either the hotel or airline platforms. Our crafted collection of goals (jobs to be done) is then tracked in a postgress database with one table to track goals and another table to track different experiment runs, and their associated goals in a experiment-goal one-to-one relationship.
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The purpose of this effort to gather data on interactions, is the first half of our research. With this collected data on behavioral characteristics, enhanced by our feature augmentation, we can create distribution separation into two bins $y \in \{A,H\}$ with a certain probability $p$ dependent on the session-specific features. To address the second loop of our system, we use this gained capability of discrimination to enhance the learner design involved in our surrogate dynamic pricing task which simulates an independent dynamic pricing scenario under which we can train a more controlled policy with the ability to account for true demand signals under conditions of contamination from non-human actors.
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\begin{algorithm}[t]
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\DontPrintSemicolon
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\KwIn{stepsize $\eta$, smoothing $\delta$, rank $d$}
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\For{$t=1$ \KwTo $T$}{
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Sample $u_t$ on unit sphere; set $x_t^\prime=x_t+\delta u_t$\;
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Set $p_t \gets U x_t^\prime$ and observe $q_t, R_t(p_t)$\;
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$x_{t+1} \gets \Pi\_{\mathcal{X}}(x_t-\eta R_t(p_t) u_t)$\;
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}
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\caption{Online Pricing Optimization (template)}
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\end{algorithm}
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Our approach can be well summarized by a three-stage division, first we intend to observe and \textit{vectorize} the behavioral interaction data from our experiments, we then develop the separability which helps us deepen the semantic understanding of the behavioral patterns. Finally we use our newly gained learner to leverage a defensive mechanism within the simulation stage of a controlled dynamic pricing loop.
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\begin{figure}[ht]
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\resizebox{\columnwidth}{!}{%
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\input{chapters/loop_figure.tex}
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}
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\caption{Overview of the Dynamic Pricing Tasks.}
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\end{figure}
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Study methodology and approach. Data acquisition strategy. Defined objectives and success criteria. Observable metrics and KPIs.
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\subsection{Generative Contamination and Separability}
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To develop a robust pricing agent, we require a simulation environment capable of generating realistic, contaminated interaction data. We achieve this by learning from our Phantom platform data using a two-stage approach.
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\subsubsection{GOFAI-Based Separability}
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We employ Good Old-Fashioned AI (GOFAI) heuristics to generate initial weak labels for separability. We define a set of rule-based predicates $\phi_j: \tau \to \{0, 1\}$ to partition the dataset $\mathcal{D}$ into high-confidence sets $\mathcal{D}_H$ and $\mathcal{D}_A$. We construct distinct MDPs per each behavioral profile of humans and agents and from those we establish $D_{KL}$. From initial findings we compute a KL divergence of $\approx 2.0236$ across transition probabilities between states which can be seen in \ref{fig:human_mdp_viz} and \ref{fig:agent_mdp_viz}.
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\begin{figure}[ht]
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\centering
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\includegraphics[width=0.8\textwidth]{chapters/mdp_human.pdf}
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\caption{Markov Decision Process visualization illustrating the behavioral transition dynamics for human actions.}
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\label{fig:human_mdp_viz}
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\end{figure}
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\begin{figure}[ht]
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\centering
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\includegraphics[width=0.8\textwidth]{chapters/mdp_agent.pdf}
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\caption{Markov Decision Process visualization illustrating the behavioral transition dynamics for \textbf{agent} behavior profiles. The state space and transition probabilities are learned from observed session trajectories to enable generative contamination.}
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\label{fig:agent_mdp_viz}
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\end{figure}
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\subsubsection{Transition Probability Estimation}
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For both subsets, we model the session dynamics as a Markov Decision Process (MDP) and estimate the transition kernel $\mathcal{T}$. The probability of transitioning to state $s'$ given state $s$ is estimated via maximum likelihood:
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\begin{equation}
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\hat{P}(s' \mid s) = \frac{N(s, s')}{\sum_{k \in \mathcal{S}} N(s, k)}
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\end{equation}
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where $N(s, s')$ is the count of observed transitions. This allows us to construct a \textit{Contamination Generator} $\mathcal{G}(\alpha)$. Given a clean trajectory dataset, $\mathcal{G}$ injects synthetic agent trajectories sampled from the learned transition matrix $\hat{P}_A$ until the effective mixing ratio reaches $\alpha$.
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\subsection{Distributionally Robust Reinforcement Learning (DR-RL)}
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We formulate the pricing problem as a Stackelberg Game where the Platform (Leader) sets prices $p_t$ and the Aggregate Demand (Follower) responds. However, the exact mixing parameter $\alpha$ and the demand distribution shift are non-stationary and unknown in online settings. Relying on a simple error term $\epsilon$ is insufficient. Instead, we adopt a Distributionally Robust Optimization (DRO) objective.
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\subsubsection{Ambiguity Set Construction}
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We define an ambiguity set $\mathcal{U}_p(\hat{P}_N)$ centered around our empirical reference distribution $\hat{P}_N$ (derived from the generator $\mathcal{G}$). We utilize the Wasserstein distance metric to define the set of plausible demand distributions the agent might face:
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\begin{equation}
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\mathcal{U}_\epsilon(\hat{P}_N) = \left\{ Q \in \mathcal{P}(\Xi) : W_p(Q, \hat{P}_N) \le \epsilon \right\}
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\end{equation}
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This set captures all distributions that are statistically close to our observed training data but allows for adversarial shifts (e.g., sudden bot spikes).
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\subsubsection{The Min-Max Objective}
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The robust policy $\pi^*$ is obtained by solving the maximin problem:
|
||||
\begin{equation}
|
||||
\pi^* = \arg \max_{\pi} \min_{Q \in \mathcal{U}_\epsilon} \mathbb{E}_{d \sim Q} \left[ R(p, d) - \lambda \cdot \text{COI}(p) \right]
|
||||
\end{equation}
|
||||
where $R(p, d)$ is the revenue function and $\lambda$ weighs the penalty for information leakage (COI).
|
||||
|
||||
\subsubsection{Actor Implementation}
|
||||
In our simulation, the "Follower" is implemented as a set of Actors. Each Actor is initialized with a type $\theta$ which samples a specific demand curve $d(p; \theta)$ from the latent distribution. This formalization ensures that our DR-RL agent does not overfit to a single deterministic demand function but learns a policy robust to the distributional uncertainty defined by $\mathcal{U}_\epsilon$.
|
||||
|
||||
|
||||
As part of our reward engineering we think about the UX factor ($UX \in [0,1]$) whic his our proxy for user experience degradation, this is computed as a mixture of contribution from the separability model metric of $\frac{1}{\text{Specificity}}$.
|
||||
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\resizebox{0.5\columnwidth}{!}{%
|
||||
\input{chapters/balance_figure.tex}
|
||||
}
|
||||
\caption{Introducing the UX index allows us to better distinguish the kind of impact different methods have and allows us to compare them on this Pareto-like scale.}
|
||||
\end{figure}
|
||||
|
||||
We also need to think about a policy like taxation to the agents Strategy-Proof Mechanism Design, specifically the Vickrey-Clarke-Groves (VCG) payment rule. We link and prove that this would create an incentive for the dominant strategy to become truth-telling.
|
||||
|
||||
\section{Heuristics as part of neuro-inspired steering systems}
|
||||
|
||||
Steve Burns, superior culliculus (face heuristics) we create this sort of part of the 'brain' + amortized inference.
|
||||
|
||||
We could say that a DQN for example is the learnin subsystem and then within our reward mechanism or some other computational method we introduce a steering subsystem which acts as the proposed ``pricing heuristic'' against the given non human transaction data.
|
||||
|
||||
\section{Market construction}
|
||||
|
||||
Reference in New Issue
Block a user