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Personalized Pricing Decisions Through Adversarial Risk Analysis

Daniel García Rasines, Roi Naveiro, David Ríos Insua, Simón Rodríguez Santana

TL;DR

This paper addresses pricing under competition and uncertainty by replacing strong common-knowledge assumptions with Adversarial Risk Analysis (ARA). It develops a generic pricing template built on a multi-agent influence diagram (MAID) that jointly models the supported pricer, the customer, and competitors using random utilities and distributions, enabling Monte Carlo-based forecasts of strategic decisions. The approach is illustrated in two domains: a retail pricing case with price-only decisions and a pension-fund pricing case with richer features (return, tenure, penalties, and covariates). Results show how relaxing common-knowledge assumptions yields different optimal prices and purchase probabilities, and the framework provides interpretable, probabilistic guidance that complements machine-learning in estimation and forecasting. The work includes open-source code and discusses extensions to dynamics, additional decision variables, and broader applications such as transfer pricing and multi-customer markets.

Abstract

Pricing decisions stand out as one of the most critical tasks a company faces, particularly in today's digital economy. As with other business decision-making problems, pricing unfolds in a highly competitive and uncertain environment. Traditional analyses in this area have heavily relied on game theory and its variants. However, an important drawback of these approaches is their reliance on common knowledge assumptions, which are hardly tenable in competitive business domains. This paper introduces an innovative personalized pricing framework designed to assist decision-makers in undertaking pricing decisions amidst competition, considering both buyer's and competitors' preferences. Our approach (i) establishes a coherent framework for modeling competition mitigating common knowledge assumptions; (ii) proposes a principled method to forecast competitors' pricing and customers' purchasing decisions, acknowledging major business uncertainties; and, (iii) encourages structured thinking about the competitors' problems, thus enriching the solution process. To illustrate these properties, in addition to a general pricing template, we outline two specifications - one from the retail domain and a more intricate one from the pension fund domain.

Personalized Pricing Decisions Through Adversarial Risk Analysis

TL;DR

This paper addresses pricing under competition and uncertainty by replacing strong common-knowledge assumptions with Adversarial Risk Analysis (ARA). It develops a generic pricing template built on a multi-agent influence diagram (MAID) that jointly models the supported pricer, the customer, and competitors using random utilities and distributions, enabling Monte Carlo-based forecasts of strategic decisions. The approach is illustrated in two domains: a retail pricing case with price-only decisions and a pension-fund pricing case with richer features (return, tenure, penalties, and covariates). Results show how relaxing common-knowledge assumptions yields different optimal prices and purchase probabilities, and the framework provides interpretable, probabilistic guidance that complements machine-learning in estimation and forecasting. The work includes open-source code and discusses extensions to dynamics, additional decision variables, and broader applications such as transfer pricing and multi-customer markets.

Abstract

Pricing decisions stand out as one of the most critical tasks a company faces, particularly in today's digital economy. As with other business decision-making problems, pricing unfolds in a highly competitive and uncertain environment. Traditional analyses in this area have heavily relied on game theory and its variants. However, an important drawback of these approaches is their reliance on common knowledge assumptions, which are hardly tenable in competitive business domains. This paper introduces an innovative personalized pricing framework designed to assist decision-makers in undertaking pricing decisions amidst competition, considering both buyer's and competitors' preferences. Our approach (i) establishes a coherent framework for modeling competition mitigating common knowledge assumptions; (ii) proposes a principled method to forecast competitors' pricing and customers' purchasing decisions, acknowledging major business uncertainties; and, (iii) encourages structured thinking about the competitors' problems, thus enriching the solution process. To illustrate these properties, in addition to a general pricing template, we outline two specifications - one from the retail domain and a more intricate one from the pension fund domain.
Paper Structure (22 sections, 3 theorems, 28 equations, 10 figures, 2 tables, 2 algorithms)

This paper contains 22 sections, 3 theorems, 28 equations, 10 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement and solution
  3. Supported pricer's problem
  4. Customer's problem
  5. Competitors' problems
  6. Comments
  7. Pricing in retailing
  8. Problem formulation
  9. Modelling and algorithmic details
  10. Case
  11. Case 1. Benchmark. Common knowledge with known competitor's price and deterministic customer's behavior
  12. Case 2. Known competitor's price and uncertain customer's behavior
  13. Case 3. Uncertain competitor's and customer's behavior
  14. Complex pricing in the pension fund market
  15. Problem formulation
  16. ...and 7 more sections

Key Result

Lemma 1

If the utility function $u_1$ is continuous in $p_1$ for fixed $s$ and $c$, ${\cal P}_1$ is compact, and there exists an integrable function $\xi(c, s)$ such that $\vert u_1(p_1, c, s)\vert \leq \xi(c, s)$, then $p_1^*$ exists.

Figures (10)

  • Figure 1: Multi-agent influence diagram for the global pricing problem.
  • Figure 2: The three partial problems for the pricing problem. Only two producers reflected.
  • Figure 3: Global pricing problem in retailing with two producers.
  • Figure 4: Results for first case; $p_2 = 30$, $\sigma = 0.01$. Price selected, $29.5$ (best viewed in color).
  • Figure 5: Results for second case. $p_2 = 30$; $\sigma^2_{1,2}$ sampled from inverse-gamma priors. Optimal price, $26$ (best viewed in color).
  • ...and 5 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Lemma 3