A Game Theoretical Model of Radiological Terrorism Defense A Game Theoretical Model of Radiological Terrorism Defense

Radiological dispersal devices (RDD) pose a threat to the United States. Healthcare facilities housing high-risk radioactive materials and devices are potentially easy targets for unauthorized access and are vulnerable to malevolent acts of theft or sabotage. The three most attractive candidates for use in RDD considered in this study are: 60 Co (radiosurgery devices), 137 Cs (blood irradiators) and 192 Ir (brachytherapy high dose radiation device). The threat posed by RDDs has led to evaluating the security risk of radioactive materials and defending against attacks. The concepts of risk analysis used in conjunction with game theory lay the foundations of quantitative security risk management. This paper develops a two player non-cooperative one-shot simultaneous defender-attacker game. The defender (healthcare facility) chooses to defend one of the three high-risk radioactive material targets and the attacker (terrorists or adversaries) chooses to attack one of the three high-risk radioactive material targets. A risk-informed approach is used to model players’ payoffs or expected utilities for each choice of strategies. A game-theoretic model (RDD game) captures the strategic interaction between competing players who act rationally to maximize their expected utility. The evaluation of the RDD game results in a von Neuman max-min strategy solution being preferable to a mixed strategy Nash equilibrium solution. The von Neumann max-min strategy solution of the defender defending cobalt and the attacker attacking cesium is found to be the most prescriptive result, thus favoring the current efforts of phasing out cesium blood irradiators and replacing them with alternative technologies. The RDD game not only gives the defender strategic options to budget scarce security resources but also helps healthcare facilities make optimal choices under severe uncertainty about the terrorist threat.


Introduction
The global economy has several critical infrastructure sectors with political and national security importance that are potentially vulnerable to deliberate attacks by terrorists and other motivated adversaries.Considering the strategic nature of the attacker, protecting such structures against intentional attacks is fundamentally different from a random accident or acts of nature.Healthcare facilities and university campuses are examples of infrastructures that face an increase of perceived security threats stemming from radiological terrorism.Healthcare facilities around the world that routinely use radioactive materials to diagnose and treat illnesses are well-trafficked and purposely open to the public, making them highly susceptible to a terrorist attack.Radiological terrorism, including the use of a radiological dispersal device (RDD) or radiological exposure device (RED), are among the most likely weapon scenarios because of their relatively simple technology and widespread use of radioactive material.Therefore, it is imperative for the healthcare sector to use the principles of graded and riskinformed approach toward building the defenses for source security.
The classical risk assessment approach takes the perspective of a single entity (industry, individual, defender) in identifying the threats that could negatively impact its ability to conduct business.The elements of game theory, on the contrary, assumes a rational opponent and evaluates the incentives and actions of both the entities (defender and attacker) affecting each other, with a goal of maximizing their own individual outcome.Rajbhandari & Snekkenes [1] and Cox, Jr [2] provide an articulate and a detailed comparison of how game theory fits into and aids the risk assessment process to effectively manage threats from adversaries.This paper demonstrates the mapping between the two approaches by adapting the specific steps of risk assessment outlined in the precursor parts of this work [3] and applying it to the game theoretic model workflow.The main contribution of this paper is to show how game theoretic analysis could be an effective way to both defend against an attacker whose choice of target is unknown and selectively deploy security resources based on the current evaluation of threat.To enable the readers to have a better understanding of both methods, we structured the remainder of the paper is as follows.In Sect.II, we present the quantitative model of the Potential Facility Risk Index (PFRI) and a summary of its mathematical framework.In Sect.III, we summarize the key game theory concepts, notations, assumptions, and mathematical formalism.Sect.IV provides a more detailed application of game characteristics and mapping between the two approaches.Conclusion and discussion of our findings are given in Sect.V.

II. The Potential Facility Risk Index (PFRI) -Background
The iterative process of risk-informed approach [4] forms the basis for the quantitative model of the Potential Facility Risk Index (PFRI).The PFRI can be defined as a mathematical framework that uses the triplet definition of risk by identifying the threat, evaluating the vulnerabilities, and calculating the resulting consequences, given the occurrence of the attack [3].The PFRI, unique to the facility, can be used by radiological facilities to conduct self-assessments and gain a better understanding of the threat they face.Rane & Harris [3] formally introduces the novel PFRI framework by presenting and applying each element of risk to a hypothetical medical facility.Figure 1 presents the complete PFRI framework.The threat component of the PFRI model identifies threat as: (1) threat 'from'adversaries who may attempt a malicious act, and (2) threat 'to'radioactive or other nuclear material assets that the adversary might seek to harm.A multi-attribute utility function is used to solve for the asset preference and intentions of the threat adversaries.The material utility function [], mathematically presented as a product of [] and [], integrates the attributes of the relative attractiveness of radiological material based on the International Atomic Energy Agency (IAEA) categorization system and the physical from (metallic, powder, etc.) of the radionuclide Eq (1) to (3).

𝑈[𝑚𝑎𝑡] = 𝑈 𝑖 [𝑎𝑡𝑡𝑟𝑎𝑐𝑡𝑖𝑣𝑒𝑛𝑒𝑠𝑠] × 𝑈 𝑖 [𝑓𝑜𝑟𝑚]
(1) (3) where  is the index of the radionuclide (weapon chosen as a dispersal device);  is activity of the radionuclide in TBq;  is the danger value of the radionuclide in TBq;   is the mass of the radionuclide in kg; and   is the physical form index of the radionuclide (metallic = 1, powdered salt = 2) The adversary utility, U[adv], assesses the adversarial mindset of the attacker as a measure of symbolism or intent, X_SY, casualties or life loss, X_LL, from the attack, and the degree of economic damage, X_ED, from the attack.Swing weights, quantified on a scale of 0 to 1, are used to rank the attributes, X_k, based on the analyst's perspective of the adversaries' value tradeoffs.Once a complete set of fundamental objectives or motivations is identified, the multi-attribute utility function is linearly additive To further the profiling of the threat event, observers develop a set of plausible attack scenarios, evaluating each asset separately.To realistically represent the malicious intent of theft or sabotage, observers make assumptions on the following parameters: the physical protection system, adversary capabilities, probability of detection, number of entry and exit points, and adversary task times.
Coefficients denoted by   and   for before and after the RDD event, respectively, are obtained by dividing each economic variable entry (  )by its corresponding column total (∑   )Where, , is the index of economic variables and t is the index of the states of the economy (i.e., before and after the RDD attack).
The net consequence loss (  ) is calculated by taking the average of   and   .
A detailed description of the above parameters is provided in our previous work Rane & Harris [3].A numeric score is allocated to each of the risk triplet of threat, vulnerability (discussion of which is omitted due to its inapplicability in this paper), and consequence to devise one composite number of the PFRI metric, unique to the facility Eq (10).The PFRI risk chart, quantified on a scale of 1-10, with a score of 1 meaning "very low risk" and a score of 10 meaning "very high risk", can be used to communicate risk effectively and succinctly to the public.
The background of the PFRI is prominently featured in the discussion of this paper is because it provides probabilities of a successful attack and consequences for pairs of attacker-defender strategies.Obtaining this information is often seen as the heart of the practical problem that defenders need solved [2].The PFRI framework, including threat profiling, pathway analysis, PRA of how events may unfold during and following an attack, and consequence modeling of the results, is essential for developing a game-theoretic model.

III. Introduction and Basics of Game Theoretical Model
Game theory is an abstract mathematical theory for analyzing interactions among multiple decision makers, also known as players.Game-theoretic models are well suited to examine the possibility of achieving an optimum stable solution between the adversary and the defender.The decision makers may be nations, people, robots, or even corporations [7].The preferences of each player are specified by utility functions, as described in Section II, that quantify the amount of benefit resulting to each player from possible outcomes of the game; this benefit is referred to as the payoff.A player's strategy in a game is a complete plan of action for whatever situation might arise.The strategy fully determines the player's behavior.Each player has two or more strategies or specific choices.Strategy profiles, which are the possible combinations of strategies that can be used by the players, give different payoffs to each player [7].In this context of radiological source security, players are: (1) the defense forces of the healthcare facility on one side and (2) the terrorist or the attacker on the other side.This paper examines the strategic interaction between the two.
The work presented uses elements of non-cooperative game theory.Cooperative and non-cooperative theories are the two leading frameworks for analyzing games.Non-cooperative games are those in which the sets of possible actions of individual players give an outcome.Cooperative games are those in which the sets of possible joint actions of groups of players give an outcome.The players in a noncooperative game compete against each other, and each player is selfishly interested only in their own payoff.In some noncooperative games the players have perfect information about the game (such as chess), while in other cases, the players may have incomplete or asymmetrical information (such as many card games).
Equilibrium states are possible for one-shot games (games played only once), finitely repeated games, or infinitely repeated games.Nash equilibrium, named after Nobel laureate John Forbes Nash, is the most used solution concept in game theory.This notion captures a steady state play of a strategic game in which each player holds the correct expectation about the other player's behavior and acts rationally [7,8].If each player has chosen a strategy and neither player can increase their payoff by choosing an action different from his current one, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.
In this paper, a simultaneous one-shot non-cooperative game is applied to a healthcare facility (defender or player 1) housing radiation emitting devices and radioactive sources.The healthcare facility is defending its assets against a terrorist RDD attack (attacker or player 2).

A. Notations and the Mathematical Formalism
For this study, we define the following sets and functions: where Player 1 is the healthcare facility, or the "defender" and Player 2 is the terrorist or the "attacker".
The study limits the asset (radioactive material) list to the highest value targets (i.e., high likelihood of success and high impact) available, rather than all the potential targets in the medical facility.Of the hundreds of radioactive materials available, the three generally found in healthcare facilities are considered the most attractive candidates for use in RDD: 60 Co (radiosurgery devices), 137 Cs (blood irradiators) and 192 Ir (brachytherapy HDR device).The sources threatened with attack are the set  ∈  = {, , } with  being the atomic symbol for cobalt,  being the atomic symbol for cesium, and  being the atomic symbol for iridium.
Let   be the strategy space comprising each of the possible strategies   ∈   , where the  ℎ source is targeted by the  ℎ player.The strategy space of player 1 is  1 = { ,  ,  }.
The strategy space of player 2 is  2 = { ,  ,  }.The pure strategy profile is a vector of the form  = [ 1 ,  2 ] that gives a particular combination of pure strategies that the players can choose.The Cartesian product  1 ⨯  2 is the set of all possible pure strategy profiles in the game 1 .
A mixed strategy   is a randomization over pure strategies.Let   denote the space of player 's mixed strategy probabilities,   (  ), where   is the probability assigned to the player  for defending or attacking the  ℎ source such that for each player i,   (  ) ∈ [0,1] and ∑   (  )  = 1.
= [  ( , ),   ( , ),   ( , )],   ∈ Θ  , are the mixed strategy row vectors available to player , where  is the index of possible mixed strategy vectors available to the  ℎ player. 1 ×  2 is the set of all possible mixed strategy profiles.
It is convenient to denote -i as the index of "all other players" than player i.For each player , we define a von Neumann-Morgenstern utility (payoff) function   : S 1 × S 2 → ℝ (a function whose domain is the set of pure strategy profiles and whose range is the set of real numbers) so that for each pure strategy   ∈ S  that the players could choose,   ( ,  − ) is the player 's payoff in the game .Von Neumann-Morgenstern utility functions are a result of the Von-Neumann-Morgenstern utility theorem stating that because of certain widely accepted axioms of rationality, a decision maker considering random outcomes with a known probability distribution will act to maximize the expected value of a function weighing some measure of the benefits of each outcome by the probability of that outcome.
We extend the definition of a payoff function to mixed strategies by using the concept of expected value.
We define the pure strategy payoff matrix   : where T denotes transposition.

B. Assumptions
Assumption 1 (Rationality & Intelligence): Both players in this game are rational and intelligent.Rationality entails a player making all decisions with a view to maximizing their expected utility.
Intelligence entails that a player knows the rules of the game and can accurately compute payoffs from all combinations of players' actions that can occur in the game.
Assumption 2 (Common Knowledge & Complete Information): Each player in this game knows their own set of strategies and utility function and the set of strategies and utility function of the other player.It is common knowledge to both players that each player in the game knows the set of strategies and utility function of the other player.It is common knowledge that each player in the game is rational, intelligent, and aware of their own set of strategies and utility function.Common knowledge results in circularity of knowledge that can be stated as, "Player 1 knows that the game is being played, player 2 knows that player 1 knows that the game is being played, player 1 knows that player 2 knows that player 1 knows the game is being played, and so on..."

C. Definitions
Definition 1 (Mixed and Pure Strategies): A strategy is a complete and contingent plan determined by a player in advance of starting the game [7].
In the simultaneous one-shot game considered here, a pure strategy,   ∈   , results in only one of the  ℎ player's possible strategies being played with a probability of 1 and all other possible strategies being played with a probability of zero.Each mixed strategy,   , is a vector of probabilities   (  ) of the  ℎ player playing each of their pure strategies, so every pure strategy is represented by a unique The extended form of RDD game, shown in Figure 2, displays the decision nodes and payoffs for each player in the form of a game tree diagram.The branches of the diagram represent a possible strategy that could be chosen at the corresponding node, and branches terminating on an oval shape are unknown to the other player.This game assumes complete information, which is distinct from perfect information [8].Perfect information entails that any player can always observe the actions of the other throughout the game, meaning that in a simultaneous game of perfect information, the players would select their strategies simultaneously and with instantaneous knowledge of the decision made by the other player.The RDD game is simultaneous but has imperfect information, meaning that players select their strategies simultaneously but without being instantaneously informed of the outcome of the other player's decision.

Figure 2. The RDD game tree with decision nodes and payoffs
The pair of utility functions behind each branch indicate the gains and losses of both the defender and the attacker.The assets Co, Cs, and Ir indicate the available decision options at each decision node.Both players, who are assumed to be intelligent and rational agents, will act on the decision option that maximizes their individual payoffs.
The utility functions for the defender and the attacker are derived from the quantitative PFRI model summarized in Section II [3].The attacker's and defender's expected utilities are functions of the attacker's success probability of theft (  ).The attacker's success probability of theft is assigned a value of zero for any pure strategy profile ( 1,  2 ) where  1 =  2 .. The defender's utility function,  1 , gives the defender's disutility resulting from loss of life and economic loss consequences: The attacker's disutility from a failed attack outcome is assumed to be −0.1 across all radionuclides and attack scenarios for the purpose of the RDD game.We define  2 , the attacker's utility function: The normal form of a two-player game presents the payoffs from each strategy profile in the form of a matrix of ordered pairs giving the payoffs to each player from each pure strategy profile.In principle, the payoff numbers entered in the cells of the RDD matrix (  ) are (von-Neumann Morgenstern) expected utilities, computed using Eq (11) and Eq (12).
The normal form of the RDD game is the matrix URDD of ordered pairs of elements from the payoff matrices U1 and U2: The payoffs of the normal form of the RDD game given below in Table 1 resulted from evaluating the utility functions  1 and  2 , developed for the hypothetical facility using the PFRI methodology.The "matching pennies" game is a classic example in game theory without any pure strategy Nash equilibria.The "matching pennies" game, as shown in Table 2, is played between two players -Even and Odd.Each player has a penny and must secretly turn the penny to heads or tails.The players then reveal their choices simultaneously.If the pennies match (both heads or both tails), then Even keeps both pennies, so wins one from Odd (+1 for Even, −1 for Odd).If the pennies do not match (one heads and one tails) Odd keeps both pennies, so receives one from Even (−1 for Even, +1 for Odd) [8].Like the "matching pennies" game, the RDD game lacks any pure strategy Nash equilibria.The RDD game has no dominated strategies, so there is no dominated strategy solution or solution resulting from the iterated elimination of dominated strategies (IEDS).

Table 2. A simple example game of "Matching Pennies"
Matching Pennies Odd Heads Tails Even Heads 1, -1 -1, 1 Tails -1, 1 1, -1 Although there is not an established solution concept providing a pure strategy solution for the RDD game, applying a variation of the max-min solution concept results in a pure strategy solution that could be of interest to the defender.The max-min criterion states that it is rational for a conservative player to choose the strategy that maximizes their minimum possible payoff in the "worst-case" outcome resulting from the possible strategies of their opponent.The literature on max-min strategies describes them as "safety strategies" or "security strategies" because they enable the player to be certain that they have maximized the lower bound of possible outcomes of an otherwise highly uncertain game.
Since it is common for the health physics profession to take a conservative approach to radiation safety, it seems appropriate for the relatively conservative max-min strategy to be adopted by the facility defender rather than any of the more risk-loving strategies that are available.It is not self-evident that the attacker would also use the max-min strategy.The more aggressive max-max strategy, in which the strategy allowing the maximum possible payoff is chosen, could be a better fit to the attacker psychology.If the defender commits to the max-min strategy, choosing to prevent a worst possible payoff of -0.36 by playing  1 , the attacker's use of the max-max strategy resulting in the play of  2 would benefit the defender, giving the defender their best-case payoff of 0.
In the RDD game, the attacker is indifferent among their available pure strategies on the max-min criterion because their worst-case payoff is -0.1 for each pure strategy.Under the complete information assumption, the attacker would know that the defender is conservative.Thus, it would be rational for the attacker to infer that a conservative defender would play  1 to satisfy the max-min criterion if the game is limited to pure strategies.If the attacker infers that the defender would play a pure strategy of  1 , the attacker's best response would be to play  2 , resulting in a pure strategy equilibrium solution of ( 1 ,  2 ) under a variation of the max-min equilibrium solution concept.Any unilateral deviation by the attacker from ( 1 ,  2 ) would result in a worse payoff for the attacker and a better payoff for the defender.
According to Nash [12], every simultaneous one-shot game has at least one mixed strategy Nash equilibrium solution.For any strategy profile that is a Nash equilibrium, neither player could obtain a greater payoff by unilaterally deviating from the strategy profile.Hence player i would be indifferent between playing any of their pure strategies against the Nash equilibrium mixed strategy of their opponent,  The results of the mixed Nash equilibrium solution are shown in Table 3.  .3shows that the attacker maximizes the expected damage to the defender by attacking Ir with probability 0.56.The defender minimizes its expected loss by defending Co with probability 0.49.For these choices, the expected return for defending Co is -0.09 and the expected return for attacking Ir is 0.40.The results of the mixed strategy Nash equilibrium show that if either player deviates from its strategy, then the terrorist adversary can do no better, and his or her opponent (the healthcare facility) can do no worse, than the equilibrium-strategy payoffs.

V. Discussion and Conclusion
The RDD game uses recursive functions to model the adaptive response of terrorist adversary to the defensive countermeasures of healthcare facilities.It's assumptions of rationality, common knowledge, and the availability of mixed strategies may, however, not be realistic in RDD scenarios [13].The mixed strategy Nash equilibrium solution of the RDD game has the payoff  1 ( 1 * ,  2 * ) = −0.09,whereas the pure strategy solution under a variation of von Neumann's max-min solution concept has the payoff  1 ( 1 ,  2 ) = −0.15.The mixed Nash solution is preferable to the pure von Neumann max-min solution if two necessary conditions for its existence obtain: (1) mixed strategies are feasible for both players; and (2) both players correctly believe that their opponent is committed to the mixed strategy Nash equilibrium profile.Conditions (2) is not provided by the definition of common knowledge.
Condition ( 1) is unlikely to be satisfied for the terrorist attacker or the healthcare defender.There is some evidence that terrorists randomize their strategies, e.g., Timothy McVey claimed that he randomly turned to a phone book page to target the Alfred P. Murrah building in Oklahoma City [14].However, it is unlikely that many terrorist adversaries would be sufficiently familiar with game theory to compute Nash equilibrium mixed strategies [15].Condition (1) appears unlikely for a real healthcare facility to satisfy due to the difficulty of randomizing defenses, which are typically static and continuously operating at full capacity.Mixed strategies have been implemented for the Department of Homeland Security to randomize patrols or surveillance of vital large-scale infrastructure, e.g., the assistant for randomizing monitoring over routes (ARMOR) deployed at the Los Angeles International Airport [16].Although the deployment at healthcare facilities of automated surveillance systems or enhanced security patrols could be randomized, it would be difficult to persuade decision makers to invest in these costly security upgrades only for the purpose of deploying them randomly in support of a mixed strategy.
We have shown that condition (1) is unlikely to be satisfied in a realistic RDD game.If condition (1) is not satisfied, condition (2) cannot be satisfied because both players need to correctly believe that their opponent is committed to a mixed strategy, and such a belief cannot be correct if mixed strategies are infeasible.If the necessary conditions for a Nash equilibrium are unlikely to exist in a real instance of the RDD game, the Nash equilibrium solution is not robust for determining the optimal defense policy of the healthcare facility.
The max-min solution concept is highly robust under conditions of severe uncertainty because it gives the certain result that the lower bound on the uncertain payoffs is maximized.The necessary assumptions for the max-min equilibrium solution to exist in a real RDD game are rationality and common knowledge.Real world players do not possess the perfect rationality and common knowledge of an idealized gametheoretic model, but human behavior in real conflicts between terrorists and security forces is a reasonable approximation of these assumptions [17].Thus, we find that the von Neumann max-min solution of (s1Co, s2Cs) is the most prescriptive result of the RDD game from the standpoint of healthcare sector security policy.
If the max-min based prediction about attacker's behavior is wrong, then the defender can only do better ( 1 ( 1 ,  2 ) = 0   1 ( 1 ,  2 ) = −0.084),but not worse, than if the prediction is right.Thus, based on the max-min equilibrium solution of the RDD game, the healthcare facility could either direct its scarce resources towards defending Co and accepting the payoff of −0.15 or it could replace cesium blood irradiators by alternative technologies, resulting in a payoff of -0.084.Consideration of technological alternatives to radionuclide radiation sources has been recommended by national and international organizations like the IAEA, the Nuclear Regulatory Commission (NRC), the National Nuclear Security Administration (NNSA), the Health Physics Society (HPS), and others [15].Implementing a policy of replacing the cesium source with an X-ray technology in the hypothetical healthcare facility scenario would both support a more effective outcome of the game and provide an additional incentive to the current cooperative risk mitigation efforts.
As shown in Table .4, the max-min solution to the updated RDD game after the replacement of cesium blood irradiator with X-ray technology gives a strategy profile ( 1,  2 ) with the payoffs  1 ( 1,  2 ) = −0.084and  2 ( 1,  2 ) = 0.44.This solution is favorable to the defender because the defender's payoff is the second best possible (their best outcome would be u1=0) and the attacker's payoff is the second worst possible (their worst outcome would be u2 = -0.1).The defender's strategy has influenced the attacker to target iridium, which has significantly lower consequences for society than an RDD attack targeting cobalt or cesium.The RDD game results shows that game-theoretic reasoning can augment risk indexes such as the PFRI by providing decision makers with the capability to optimize their defenses against the predicted behavior of terrorist adversaries.The RDD game gives the defender strategic options that can be interpreted as possible allocations of a defense upgrade available for only one of the three sources at a time.This simplified idealization captures the trade-offs inherent in budgeting scarce security resources.A realistic policy prescription following from the RDD game would be to replace the cesium source with an alternate technology and divide the available security resources equitably between the two remaining sources, cobalt and iridium.
Risk metrics developed from probabilistic risk analysis, such as the PFRI, do not capture the strategic interaction between adversaries that is shown in game theoretical models.However, probabilistic risk analysis provides content for a game theoretic matrix that cannot be provided by game theory alone.The RDD game presented in this paper fills its payoff matrix with the success probability of theft, Ps, and utility functions developed in the prior publication about the PFRI.The,   , and utility functions used to compute payoffs for the RDD game are adopted from functions given for the hypothetical medical facility analyzed in the prior publication [3], and additional success probabilities of theft for  and  obtained using the PFRI methodology are presented in this paper.Although great care must be taken to gather reliable information for PFRI studies of particular healthcare facilities, the core idea of mapping the PFRI methodology to a game theoretical model produces sensible insights for allocating defensive resources [18].As future work, a risk informed cost-benefit analysis drawing on input from the PFRI and the RDD game can ensure that each healthcare facility uses its security budget optimally to reduce the RDD threat.

Table 1 . The RDD game with pure strategy defender-attacker payoffs RDD game -St. Benedict Healthcare
− .It follows that for the RDD game there exists a Nash equilibrium mixed strategy profile (  Û  (  * )  = Û  (  * )  = ̂ (  * ) where , ̂,  ̂ are unit row vectors.The following system of equations is solved to determine the mixed strategy probabilities that are the components of the vectors   * and   * : *