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Augmented Probability Simulation Methods for Sequential Games

Tahir Ekin, Roi Naveiro, Alberto Torres-Barrán, David Ríos-Insua

TL;DR

This paper addresses solving sequential defend-attack games under both complete and incomplete information by leveraging augmented probability simulation (APS). It introduces a scalable two-stage framework that combines Monte Carlo (MC) and APS approaches, with attacker and defender BRs via modes of augmented distributions; for incomplete information, Adversarial Risk Analysis (ARA) is employed to model uncertainty about attacker utilities and probabilities. Key contributions include (i) a MC-based baseline for complete-information games, (ii) a novel nested MH-based APS method that avoids discretization of continuous decisions, (iii) a detailed sensitivity/robustness analysis framework, and (iv) a cybersecurity application illustrating practical benefits in high-dimensional settings. The results demonstrate that APS can outperform MC in high-precision or continuous-space scenarios and provide richer sensitivity information, offering scalable tools for security and adversarial decision-making. The framework thus offers principled, computationally efficient means to derive and compare equilibrium decisions and resilience under model uncertainty.

Abstract

We present a robust framework with computational algorithms to support decision makers in sequential games. Our framework includes methods to solve games with complete information, assess the robustness of such solutions and, finally, approximate adversarial risk analysis solutions when lacking complete information. Existing simulation based approaches can be inefficient when dealing with large sets of feasible decisions; the game of interest may not even be solvable to the desired precision for continuous decisions. Hence, we provide a novel alternative solution method based on the use of augmented probability simulation. While the proposed framework conceptually applies to multi-stage sequential games, the discussion focuses on two-stage sequential defend-attack problems.

Augmented Probability Simulation Methods for Sequential Games

TL;DR

This paper addresses solving sequential defend-attack games under both complete and incomplete information by leveraging augmented probability simulation (APS). It introduces a scalable two-stage framework that combines Monte Carlo (MC) and APS approaches, with attacker and defender BRs via modes of augmented distributions; for incomplete information, Adversarial Risk Analysis (ARA) is employed to model uncertainty about attacker utilities and probabilities. Key contributions include (i) a MC-based baseline for complete-information games, (ii) a novel nested MH-based APS method that avoids discretization of continuous decisions, (iii) a detailed sensitivity/robustness analysis framework, and (iv) a cybersecurity application illustrating practical benefits in high-dimensional settings. The results demonstrate that APS can outperform MC in high-precision or continuous-space scenarios and provide richer sensitivity information, offering scalable tools for security and adversarial decision-making. The framework thus offers principled, computationally efficient means to derive and compare equilibrium decisions and resilience under model uncertainty.

Abstract

We present a robust framework with computational algorithms to support decision makers in sequential games. Our framework includes methods to solve games with complete information, assess the robustness of such solutions and, finally, approximate adversarial risk analysis solutions when lacking complete information. Existing simulation based approaches can be inefficient when dealing with large sets of feasible decisions; the game of interest may not even be solvable to the desired precision for continuous decisions. Hence, we provide a novel alternative solution method based on the use of augmented probability simulation. While the proposed framework conceptually applies to multi-stage sequential games, the discussion focuses on two-stage sequential defend-attack problems.

Paper Structure

This paper contains 22 sections, 2 theorems, 11 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Sequential games with complete information
  3. Monte Carlo simulation for games
  4. Augmented probability simulation for games
  5. Example 1
  6. Defender non strategic judgments.
  7. Attacker judgments.
  8. Complete information case.
  9. Sensitivity analysis
  10. Example 1. (cont.)
  11. Sequential games with incomplete information: ARA
  12. Monte Carlo based approach for ARA
  13. Augmented probability simulation for ARA
  14. Example 1 (Cont.)
  15. Sensitivity analysis of the ARA solution
  16. ...and 7 more sections

Key Result

Proposition 1

Assume that the Attacker's and Defender's utility functions, decision sets, probability distributions and proposal generating distributions satisfy conditions a), b), c) and d). Then, if the best response sets $a^*(d)$ are finite for each $d$, Algorithm alg:MHdefenderAPS2 defines a Markov chain with

Figures (11)

  • Figure 1: Basic two player sequential defend-attack game. White nodes affect solely the Defender; grey nodes affect only the Attacker; striped nodes affect both agents.
  • Figure 2: Attacker problem solutions for each defense
  • Figure 3: Solutions of Defender problem
  • Figure 4: Sensitivity analysis of the solution of the game with complete information
  • Figure 5: Influence diagrams for Defender and Attacker problems.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2