Applying Game Theory and Computer Simulation to Fault Tree Analysis

Fault tree analysis is a useful probability theory-based tool for evaluating a system’s risk and reliability. Typically, fault trees are populated with basic event failure probabilities from a variety of quantitative and qualitative sources. This article presents a new methodology that combines simulation with game theory to populate a fault tree with strictly quantitative probability estimates for basic events in the fault tree. This new method is applied to an example ship self-defense scenario, and the probability of effectiveness against a group of small attack boats is calculated. The resulting fault tree is used to model a war gaming situation in which the players must choose optimal strategies and weapons. This articles describes a means for generating a fault tree in which the top event probability is optimized with the assignment of basic events probabilities in accordance with game theory. quantitative in nature. Data on the availability of the system’s components, collected through a data collection program or by using a manufacturer’s specification, are the first and preferred source. If this source of data is not available or is not appropriate for the basic events, the event probabilities may be derived from the second source: modeling and simulation techniques. The third data source, input by subject-matter experts (SMEs), is qualitative. This article explores a quantitative methodology based on game theory as a replacement for the third source of data. Note that the inputs to this study’s methodology are the same modeling and simulation results used to inform the second data source. Hence, the goal is to produce a fault tree consisting of data derived from purely quantitative sources. FTA began in 1961 when H. A. Watson of Bell Telephone Laboratories began looking for ways to quantify the reliability and safety of the Minuteman missile INTRODUCTION Fault tree analysis (FTA) is an analytic methodology commonly used to assess risk and reliability. FTA is a failure-based approach that begins with an undesired event (top or top-level event) and, through a systematic backward-step process, identifies the basic causes or combination of basic events that lead to the top-level event. The fault tree is a logical illustration of the events and their relationships that provide the necessary and sufficient means for the undesired event to occur. It computes the probability that the undesired event will occur and provides insight on the importance of the basic events modeled within the tree. Fault trees facilitate investigative methods to increase system reliability, reduce opportunities for system failure, and identify the most important contributors to system effectiveness, ultimately in an attempt to minimize risk. The probability of each basic event is determined from one of three primary sources, two of which are Applying Game Theory and Computer Simulation to Fault Tree Analysis Johns Hopkins APL Technical Digest, Volume 33, Number 3 (2016), www.jhuapl.edu/techdigest 217 launch control systems as a part of a contract with the U.S. Air Force.1 Since its application to the Minuteman system, the use of FTA has become widespread for analyzing the safety and reliability of complex dynamic systems such as nuclear reactors, processing plants, and power systems.2–4 For systems in which failure probabilities follow well-known distributions, calculating basic event probabilities is straightforward. However, in more complex systems, such as dynamic fault trees, in which failures must occur in a specified sequence, Monte Carlo simulation methods have been applied.5,6 The idea of applying fault tree methodology to human-based security systems is a relatively new concept. The concept of game theory is often brought up during the discussion of war games; in fact, sometimes the terms are used so closely in context they appear to be synonymous. However, war gaming and game theory are distinct concepts. War gaming is the simulation of a conflict situation, whereas game theory is a mathematical theory that can aid in identifying an optimal strategy or course of action when certain conditions within the conflict are met. Hence, game theory is a tool that can aid the decision-making process in a war game. In 1957, Walter Deemer and Clayton Thomas suggested that the new concepts of game theory may be used in place of traditional analytical war gaming to solve generalized tactical problems of limited scope.7 By the late 1950s, both Air Force Col O. G. Haywood and U.S. Navy CAPT R. P. Beebe had written papers supporting the use of game theory as a decision-making aid in war games.8–10 The game theoretic study of the two-person game fit naturally with war gaming scenarios during the Cold War, where the bipolar world was split into the United States versus the Soviet Union. Since the dissolution of the Soviet Union, the generalization that the current global state of affairs can be modeled as a two-person zeroor nonzero-sum game became impossible.11 However, on the small-scale, single-engagement level, the two-person game assumption still holds and the results from game theory still serve as a valuable decision-making tool for selecting an optimal strategy. This article will explore how the game theoretic concept developed in post-World War II war gaming can be applied in conjunction with the reliability and risk assessment tool of fault trees for small-scale engagements of a limited scope. This article considers an engagement scenario in which a Blue (friendly) ship is forced to defend itself against an attack by small Red (adversarial) boats. In this scenario, each side has the option to make one of two tactical decisions. Blue has the choice to select one of two different weapon types, while Red must select between two different engagement tactics. The goal in applying game theory to this scenario is to answer the following question: if both sides follow the best possible strategy during an engagement and also assume the rival will do the same, what is the optimized and best possible Blue system effectiveness for the scenario? For this scenario, system effectiveness is defined as the probability that the Blue ship succeeds in neutralizing all Red boats in a force-on-force engagement before the Blue force is neutralized by Red. Because the tactical actions will greatly affect system performance, it is necessary to determine the optimal frequency for each side to use a particular tactic. The analytic process used in this study models each combination of tactics using Surface Warfare Simulation (SuWSim), a Johns Hopkins University Applied Physics Laboratory (APL)-developed surface warfare modeling tool. The outputs of the simulation are analyzed using game theory to determine the optimal strategies for each side. The identified optimal decision strategy is then used to populate the basic event probabilities within the fault tree that would have traditionally been gathered from qualitative sources. Finally, a probability of effectiveness (PE) is computed using the simulation-derived basic event probabilities for the set of basic events.