Deterrence and Risk Preferences in Sequential Attacker Defender Games with Continuous Efforts

Transcription

1 Risk Analysis DOI:./risa.768 Deterrence an Risk Preferences in Sequential Attacker Defener Games with Continuous Efforts Vineet M. Payappalli, Jun Zhuang,, an Victor Richmon R. Jose Most attacker efener games consier players as risk neutral, whereas in reality attackers an efeners may be risk seeking or risk averse. This article stuies the impact of players risk preferences on their equilibrium behavior an its effect on the notion of eterrence. In particular, we stuy the effects of risk preferences in a single-perio, sequential game where a efener has a continuous range of investment levels that coul be strategically chosen to potentially eter an attack. This article presents analytic results relate to the effect of attacker an efener risk preferences on the optimal efense effort level an their impact on the eterrence level. Numerical illustrations an some iscussion of the effect of risk preferences on eterrence an the utility of using such a moel are provie, as well as sensitivity analysis of continuous attack investment levels an uncertainty in the efener s beliefs about the attacker s risk preference. A key contribution of this article is the ientification of specific scenarios in which the efener using a moel that takes into account risk preferences woul be better off than a efener using a traitional risk-neutral moel. This stuy provies insights that coul be use by policy analysts an ecisionmakers involve in investment ecisions in security an safety. KEY WORDS: Attacker efener games; homelan security; risk aversion; risk-seeking behavior. INTRODUCTION The attacks on the Worl Trae Center in New York on September, became a pivotal moment in the way we stuy an unerstan risks in security an safety. With numerous agencies such as the Office of Domestic Prepareness an the Nuclear Incient Response Team being create after these attacks, the amount of resources that have been allocate to unerstan an to prepare for these types of risks have grown exponentially. The 6 U.S. Department of Inustrial an Systems Engineering, University at Buffalo, The State University of New York, Buffalo, NY, USA. McDonough School of Business, Georgetown University, Washington, DC, USA. Aress corresponence to Jun Zhuang, Department of Inustrial an Systems Engineering, University at Buffalo, The State University of New York, Buffalo, NY 6-5, USA; jzhuang@buffalo.eu. buget for the Department of Homelan Security (DHS) is about $65 billion in total. () Such huge investments in the numerous counterterrorism an security efforts being launche every year eman a more careful an rigorous approach to stuy an unerstan these risks. The funamental question this article investigates is how the notion of risk preferences affects players equilibrium strategies in a sequential attacker efener (AD) game an what it implies for the notion of eterrence. In its most general form, eterrence is simply the persuasion of one s opponent that the costs an/or risks of a given course of action he might take outweigh its benefits. () The aim of this article is to try to narrow the gap between existing mathematical moels of AD games in counterterrorism literature an the extensive literature from behavioral economics an psychology that ocuments the ifferent attitues of 7-/7/-$./ C 7 Society for Risk Analysis

2 Payappalli, Zhuang, an Jose iniviuals towar risk an uncertainty. This stuy coul be of use to researchers, homelan security practitioners, policymakers, policy analysts, an other government agencies in unerstaning how players in AD games evelop strategies when risk preferences, an important aspect of human behavior an ecision making, are introuce in analytical moels. This woul improve ecision making over the large group of existing moels, which implicitly assume risk neutrality. To our knowlege, there is no existing paper that specifically examines the effects of risk preferences on eterrence. The iea of eterrence springs from the avantage that a first-mover may have to significantly affect the actions an choices of a seconmover player (a.k.a., the first-mover avantage). As Hausken () mentions, sequential games of this form are useful in enabling analysts to come up with analysis an recommenations that are preemptive (cf. Zhuang an Hausken, () Hausken an Zhuang, (5) an Jose an Zhuang (6) ). It is also of interest to note that there may be instances in which it is to the avantage of the efener to not always reveal her/his strategy by opting to play in a simultaneous fashion (e.g., see Zhuang an Bier (7) ). We believe that this article will serve as a first builing block in this research irection. Specifically, we present analytical results relate to the effect of attacker s an efener s risk preferences on the efense effort an their impact on the optimal eterrence level. Numerical illustrations an some iscussion of the effect of risk preferences on eterrence an the utility of using such a moel are provie, as well as sensitivity analysis of continuous attack investment levels an uncertainty in the efener s beliefs about the attacker s risk preference. A key contribution of this article is the ientification of specific scenarios in which the efener using a moel that takes into account risk preferences woul be better off than a efener using traitional riskneutral moels. The rest of this article is organize as follows. Section provies a literature review, an Section. introuces the continuous efense, iscrete attack (CDDA) moel, which is followe by some analytical results in Section.. Section. presents numerical illustrations relate to the propositions an shows the equilibrium responses of the AD in several interesting scenarios. Also shown in Section. is the importance of the risk-preference moel by comparing the results with a conventional riskneutral moel, an the section provies scenarios in which risk-preference moels give better results than risk-neutral moels. Section explores the extension where the attacker also has a continuous action space. Section 5 analyzes how the equilibrium is affecte if the efener has incomplete information about the attacker s risk preference. Section 6 conclues an presents future research irections. Finally, the Appenix gives calculations an a plot on which a iscussion at the en of Section.. is base.. LITERATURE REVIEW In the risk analysis literature, numerous stuies try to better unerstan how we eal with risks associate with aaptive/strategic aversaries, where game theory has often been use, with roots ating back to the 95s. One class of moels that has grown in popularity an use are AD games. (8 ) As Cox Jr. () mentions, these tools are constantly relie upon when oing risk analysis because of their ability to reorient current aversarial risk analysis to make it useful through the evelopment of useful preictive moels of causal relationships an improving a efener s ecision-making capabilities. Beyon counterterrorism, AD games have also been applie to other general risk analysis contexts such as cybersecurity ( ) an war gaming. (5) Hausken an Levitin (6) provie a comprehensive review of AD moels from a systems perspective. Developments relate to the applications in counterterrorism an corporate competition have supplemente the traitional statistical risk analysis with a new approach calle aversarial risk analysis. (7) Hausken (8) applies game theory in probabilistic risk analysis, thereby introucing a behavioral approach in assessing the reliability of systems. In general, the literature has consiere aversaries as strategic (9 ) as well as nonstrategic. (,) Decision making uner uncertainty has been the object of investigation in various isciplines for ecaes. (5) Bernoulli s proposal (6) that people maximize expecte utility an not expecte value was the first step towar introucing risk preferences in ecision making. Research on ecision making uner uncertainty has progresse a long way, with the evelopment of the von Neumann Morgenstern utility theorem, (7) a better unerstaning of the willingness to pay for risky investment options, regret theory, (8,9) an prospect theory. () Weber an Johnson (5) provie a historical context of these evelopments in the risk-preference literature. Although most of these evelopments have foun

3 Deterrence an Risk Preferences in Sequential Attacker Defener Games applications primarily in economics an finance, in the broaer context, the existence of risk preferences is a universal phenomenon. We attempt to translate some of these evelopments into the realm of critical national security issues. We fin that in almost all AD games in the literature, players are moele as risk neutral; i.e., they make ecisions that maximize expecte payoffs or minimize expecte losses. () This has conventionally been one following the economic traition of assuming agents to be perfectly rational as well as for moeling convenience. () However, extensive empirical an theoretical evience has shown that risk neutrality may not be realistic or preferre in practice. (6,7,, ) For example, Stewart et al. (5) suggest that policymakers within the U.S. government an its agencies (incluing the DHS) are risk averse for lowprobability high-consequence events because of the catastrophic or ire nature of these hazars. This woul imply that the U.S. government an its agencies shoul be treate as risk averse in some AD counterterrorism games. Some stuies have foun that certain terrorist organizations are also risk averse. (6 8) We also acknowlege that aversaries coul be nonrational (9) or isplay boune rationality. () However, in this article, we focus on rational aversaries who eviate from the traitional norms of risk neutrality. We think that this is a natural starting point in unerstaning the effect of risk preferences in issues such as eterrence. In the AD games literature, some authors have recognize the importance of risk preferences. For example, Zhuang an Bier (9) mention that risk aversion an risk-seeking behavior may impact the outcomes when they apply game theory in stuying resource allocation for countering terrorism an natural isasters; however, they stuy risk preferences by incorporating risk parameters only in part of the utility function. Other papers mention the notion of utility functions but often en up using linear utility (i.e., they assume risk neutrality yet use the term utility interchangeably with payoffs) or o not fully moel utility an risk preferences consistent with the ecision an risk analysis literature (e.g., Bell et al. () an Liu et al. () ). In the context of sequential games, several papers iscuss the notion of risk aversion not necessarily of players, but of strategies. These fall into the broa category of robust game theory where an analyst may want to etermine an minimize worst-case scenarios. For example, Yin et al. () an Qian et al. () stuy the notion of risk-averse strategies in a sequential Stackelberg game (which is a game between a leaer an a follower competing on quantity (5) ), where each player optimizes over a class of possible utility functions.. CONTINUOUS DEFENSE DISCRETE ATTACK (CDDA) MODEL.. Moel We consier a two-player sequential game. In the first stage, the efener chooses a continuous level of efense investment [, ) that maximizes her/his expecte utility. After observing the efener s level of investment, an attacker in the secon stage chooses to either attack (enote by A or a = A) or not attack (enote by NA or a = NA). If the attacker chooses to attack, his/her success probability P epens on how much the efener investe in efense. This probability success function P :[, ) [, ] is strictly ecreasing in. To remove trivial cases, we assume that the function P is not equal to or for any >. For each player, we efine three parameters. First, the efener an the attacker each values the target (the resource that the government tries to efen from terrorist attacks) at v an v a, respectively. In aition, each player has a unit cost for efening (c ) an attacking (c a ). For the attacker, we assume that v a > c a so that the ecision whether to attack or not oes not become trivial. The utility functions of the efener an the attacker are, respectively, enote by u an u a. Fig. provies the sequence of steps an Table I summarizes the notation we use in the article. In this article, we focus on the subgame perfect Nash equilibrium strategy for both players an analyze the impact of incorporating risk preferences in an AD game, focusing on its impact on eterrence... Analytical Results We begin by examining the best response of the attacker using backwar inuction. Observing the We assume in our moel that players are able to quantify their valuation of targets typically in monetary terms. For example, Shan an Zhuang (6) use the valuation of 7 U.S. urban areas provie by Willis et al. (7) to illustrate their moel. An equilibrium foun by applying rollback to the extensive form game is referre to as subgame perfect equilibrium. (8)

4 Payappalli, Zhuang, an Jose Stage Stage Defener Con nuous efense effort () A acker A ack Discrete a ack effort Not a ack, () ( ) ( ) Successful a ack, ( ) Unsuccessful a ack, ( ) Fig.. Sequence of moves in the AD game (CDDA moel) with the players utilities. Table I. Notation for Decision Variables an Parameters Use in the Article Decision Variables a Attacker s ecision (a {A, NA} in CDDA moel) or attacker s effort (attack investment level, a in CDCA moel) Defener s effort (efense investment level) Functions P() an P(a, ) The probability of successful attack u a Attacker s utility function u Defener s utility function U a (a, ) Total expecte utility of the attacker U (a, ) Total expecte utility of the efener â() arg max U A (a, ) Attacker s best response Parameters (a, ) Equilibrium strategy λ Coeffecient of efener s efense effectiveness c a Attacker cost for attacking c Defener s unit cost of effort v a Attacker s valuation of the target v Defener s valuation of the target efense level, the attacker chooses a {A, NA} that maximizes his expecte utility U a (a, ). Proposition provies an important property about the attacker s best response. Proposition. The attacker s best response â() is of a threshol type in ; i.e., there exists a threshol for which the attacker will attack (â() = A) when < an not attack (â() = NA) when. Proof. The attacker will maximize his/her expecte utility, i.e., his optimization problem is: max U a (a, )= P()u a (v a c a )+( P())u a ( c a ). In this case, the attacker will choose a = A if an only if P()u a (v a c a ) + ( P())u a ( c a ) > u a () u a () u a ( c a ) P() > u a (v a c a ) u a ( c a ) ( < P u a () u a ( c a ) u a (v a c a ) u a ( c a ) ). () The secon inequality hols because v a > c a an u a is nonecreasing in its argument. The thir inequality follows from the assumption that P is strictly ecreasing in. (We note that if we instea assume that P is nonincreasing, we can easily replace P by the generalize inverse P [ ] an with some work show that the result still hols.) This threshol iea tells us that there exists a level [, ) such that the attacker will choose not to attack for any, because the probability of successful attack is too low to provie him/her sufficiently large expecte utility for attacking. This threshol is what we will refer to as the eterrence level, which is given by: when r = P (r) when < r < () when r where r = u a() u a ( c a ) u a (v a c a ) u a ( c a ). The conitions r > an r < are trivial an o not arise for a strictly

5 Deterrence an Risk Preferences in Sequential Attacker Defener Games 5 nonecreasing function u a. Also, when u a is a strictly increasing function, there will always be a positive finite as < r <. First, we note that this quantity is well efine since we assume that P is strictly ecreasing in its argument. Next, we provie an interpretation for this quantity by analyzing the simple case when u a (x) is linear (i.e., a risk-neutral attacker). For a strictly increasing u a, the eterrence level in Equation () simplifies to P ( c a v a ) in the risk-neutral case, which represents the point where the expecte gain P()v a equals the cost of attacking c a. In the more general setting of nonlinear utility, we stuy how the eterrence level changes when an attacker is viewe to be more risk seeking (less risk averse). Using Pratt s () efinition, we say a player with utility function û is more risk seeking (less risk averse) than a player with utility function u if there exists an increasing convex function g such that û(x) = g(u(x)) for all x. The next proposition provies insights on the eterrence level as we consier the more general setting of risk preferences. Proposition. The eterrence level is (i) at least as high for a more risk-seeking (less risk-averse) attacker, (ii) nonecreasing in v a, an (iii) nonincreasinginc a for a risk-seeking attacker. Proof. Since P is ecreasing in, P must also be ecreasing. Therefore, it is sufficient to stuy whether K(u a,v a, c a ):= u a() u a ( c a ) u a (v a c a ) u a ( c a is increasing ) or ecreasing. For ease of notation, let x = c a an y = v a c a. Since we assume that v a > c a >, we have x < < y. (i) Consier two utility functions u a an û a, where û(x) = g(u(x)) an g is an increasing convex function. By the convexity of g, we know that the marginal utility ifferential between y an x is greater than the ifferential between an x, i.e., g(u a (y)) g(u a (x)) u a (y) u a (x) > g(u a()) g(u a (x)) u a () u a (x) u a () u a (x) u a (y) u a (x) > g(u a()) g(u a (x)) g(u a (y)) g(u a (x)) K(u a,v a, c a ) > K(g(u a ),v a, c a ). Therefore, the eterrence level associate with u a is lower than the eterrence level associate with û a, since P is ecreasing. (ii) For v a >v a, we note that u a (v a c a) u a (v a c a ) for any utility function u a. Hence, K(u a,v a, c a) = u a() u a ( c a ) u a (v a c a) u a ( c a ) u a () u a ( c a ) u a (v a c a ) u a ( c a ) = K(u a,v a, c a ). (iii) To prove that is nonincreasing in c a for a risk-seeking attacker, we nee to show that K(u a,v a, c a ) is increasing for a convex u a.we note that K c a = u a ( c a) u a (v a c a ) u a ( c a ) [u a() u a ( c a )](u a ( c a) u a (v a c a )) [u a (v a c a ) u a ( c a )], where u a = u a c a. The first term on the right-han sie is positive because all utility functions are nonecreasing in their arguments. The secon term is also positive for a convex u a because u a ( c a) u a (v a c a ) < ; i.e., increasing convex utility functions become steeper. Therefore, for convex u a,we have K c a. Proposition shows how the eterrence level is affecte by changes in the moel parameters. In particular, we notice that for a more risk-seeking (less risk-averse) attacker, a efener has to invest more to completely eter an attack. As expecte, we also see that as the value of the target to the attacker increases, the eterrence level increases. Finally, we note that the impact of the attacker s costs c a is not irectly evient in this moel, since analytically the effect of c a on epens on the utility function. Depening on the concavity of the attacker s utility function u a, we may not necessarily see a monotonic change in the eterrence level. For a risk-seeking attacker, we can show that is nonincreasing in c a. However, for a risk-averse attacker, this is not guarantee because the proof of Proposition (iii) is not applicable for concave (riskaverse) utility functions. Next, we consier the efener, who has the benefit of moving first. Her/his efense level choice can be a strategic ecision that results in eterrence; however, this choice has to be balance with the relative value of the target an the cost associate with such efensive investments. Proposition escribes the equilibrium strategy of the efener. Proposition. Let U ( int ):= P( int )u ( v c int ) + ( P( int ))u ( c int ). The equilibrium strategy of the efener [, ] is an interior point (which we enote by int ) if an only if U ( int ) > u () an

6 6 Payappalli, Zhuang, an Jose Defener s expecte utility, U/ 5.5 (a) * = (b) * = int (c) * = efense investment, int efense investment,.5 5 efense investment, Defener s utility, U / Deterrence effort, Optimal efense effort, * Fig.. A set of scenarios showing the three possible types of optimal solutions for the CDDA moel. Baseline values: z = z a =,v = v a = 6, c = c a = 5, an λ =, where λ is the efense effectiveness coefficient in the exponential success function P() = e λ..5 U ( int ) > [u a() u a ( c a )]u ( v c ) u a (v a c a ) u a ( c a ) + [u a(v a c a ) u a ()]u ( c ). u a (v a c a ) u a ( c a ) Proof. The efener faces the following optimization problem to maximize her/his expecte utility. Plugging in the attacker s best response in Equation () to the efener s optimization problem, we have: max U (â(), ) } P()u ( v c ) if â() = A () = +( P())u ( c ) u ( c ) ifâ() = NA. The value of that maximizes the efener s utility will be the equilibrium efener strategy. If we let int :={ : U () = }, then we have when { U () > max(u ( ), U ( int )) interior solution, = int { when U ( int ) > max(u (), U ( )) eterrence solution, when U ( ) > max(u (), U ( int )). The conitions provie in the proposition follow by expaning U () an U ( ). Proposition explains that there are three types of solutions to the efener s problem. Fig. illustrates the three possible scenarios numerically. First, can be when the efener oes not invest any amount into efense (Fig. (a)). This case happens in the extreme case when v is not sufficiently high so that the efener eems the target to be worth protecting or alternatively the probability of successful attack is sufficiently high so that the cost of efense is too high to have any significant impact or savings. The other extreme case = happens when the cost of efense is relatively cheap such that the efener has sufficient resources to invest in efense (Fig. (c)). Even though the efener can reuce the probability of successful attack by investing >, she/he woul not choose to o so. The result suggests that the level of investment nees not excee, i.e., there is no nee to overinvest in efense. The case = int is perhaps the most interesting since this represents the mile groun where the efener neither goes all the way nor oes she/he o nothing at all. This interior solution happens when the efener s investment is sufficiently high to minimize the expecte isutility while taking into account that she/he nees not spen too much in efening the target. From Proposition an Equation (), we notice that ue to the Boolean nature of the attack, int is inepenent of the attacker s utility function u a, an the eterrence efense level is inepenent of the efener s utility function u. A close-form solution oes not exist for int an hence for because of the specific way utility functions are efine. More etails are provie at the en of Section.. an in the Appenix... Numerical Illustration... Sensitivity Analyses To provie some aitional insights to this moel, we provie a few numerical illustrations that allow us to see in etail how the equilibrium strategies an payoffs epen on the moel parameters.

7 Deterrence an Risk Preferences in Sequential Attacker Defener Games 7 P() λ =. λ =. λ =. λ =. 6 8 Fig.. Probability of successful attack P() plotte for ifferent values of efense investment an efense effectiveness λ. Clearly, a higher value of λ ecreases P() an vice-versa. For the purpose of numerical illustration, we assume a few functional forms in this section. In particular, following Bier et al., (9) we assume an exponential success function P() given by: P() = exp( λ), where λ> is the efense effectiveness coefficient. This function is strictly ecreasing in an is boune between (, ] for [, ) (Fig. ). For the players utility functions, we use power utility functions (5) of the form: u a (x) = (z a + x) β a u (x) = (z + x) β, where the risk-preference parameters for the AD, β a an β > are parameters that primarily affect the curvature of the utility function. Finally, the terms z a an z are large positive constants introuce so that z a + x an z + x are always positive an well efine especially when β a an β (, ). In aition, the use of z a an z gives flexibility in the functional form estimation process an coul potentially be interprete as initial wealth (or enowment) if it makes sense in the context being stuie. An interesting an useful feature of the power utility function is that it covers the three main categories of risk preferences that we want to investigate. In particular, we are able to cover risk-averse ( <β<), risk-neutral (β = ), an risk-seeking (β >) behaviors. For sensitivity analyses, we focus on the changes that happen to these three variables: (i) efener s eterrence effort, (ii) efener s optimal effort, an (iii) attacker s optimal effort a. In particular, we examine these variables as the following seven parameters vary: (i) efener s target valuation v, (ii) efener s cost coefficient c, (iii) efener s risk-preference parameter β, (iv) attacker s target valuation v a, (v) attacker s cost effectiveness c a,(vi) attacker s risk-preference parameter β a, an (vii) efense effectiveness λ. The changes in the equilibrium behavior of the players are capture in the one-way sensitivity plots in Fig. that vary only one parameter at a time while holing all other parameters equal to the baseline case. The plots in Fig. show several important variables incluing the two equilibrium solutions an a. The baseline values are highlighte by the soli vertical line, while the critical point when the attacker strategy changes is highlighte by the ashe line. In Figs. (a) an (e), we note that the equilibrium investment of the efener increases, an then stays constant as the target valuation of the efener an attacker, respectively, increases. For the efener, we see that she/he is prompte to invest more to protect the resource as its valuation increases. The case for the attacker (Fig. (e)) follows the result provie in Proposition (ii), which implies that more effort is require from the efener to eter an attacker as the target valuation increases. This result hols irrespective of the attacker s risk preference. In analyzing the impact of costs, Fig. (b) shows that the efener woul invest less when her/his cost increases. This reuce investment may prompt an attacker to attack an the overall expecte utility of the efener woul then ecrease. Hence, a efener may want to later focus her/his attention on mitigating this risk by trying to improve other aspects of efense (e.g., improving the effectiveness of efense λ) when costs are beyon her/his control. On the other han, when c a increases, Fig. (f) shows that will ecrease an approach zero at a point where the attacker is worse off attacking, irrespective of the efener s investment level. To some extent, if governments are able to affect the cost of an attack; e.g., making it more ifficult an costly to launch an attack (e.g., increase cost of materials an components for bombs an increase cost for successfully moving resources), the overall efense effort may be significantly reuce. Relate to the risk-preference parameters, Figs. (c) an (g) provie some interesting results. Here, we see that a more risk-seeking (less riskaverse) efener woul efen less although she/he is certain that attack woul happen. This is interesting because we woul expect that the certainty of the attack makes the efener invest more, but that oes not necessarily happen. The reason is that, the efener knows that the high risk-seeking behavior of

8 8 Payappalli, Zhuang, an Jose (a) Target valuation of efener, v (b) Cost coefficient of efener, c (c) Risk preference parameter of efener, β 5 () Defense effectiveness, λ efense effort A NA efense effort NA A efense effort NA A efense effort A NA efense effort 6 8 v (e) Target valuation of attacker, v a NA A 6 8 v a efense effort 6 8 c (f) Cost coefficient of attacker, c a A NA 6 8 c a β (g) Risk preference parameter of attacker, β a efense effort NA A β a A NA λ Legen et : Deterrence efense effort * : Optimal efense effort Baseline value Critical point Attack Not Attack Fig.. One-way sensitivity analysis of attacker s equilibrium response (a ), optimal efense effort ( ), an eterrence level ( ), with respect to the parameters use in the CDDA moel. Regions marke with A an NA refer to the regions for which it is optimal for the attacker to Attack an Not Attack, respectively. Baseline values: λ =, z a = z =, v a = v = 6, c a = c = 5, an β a = β =. the attacker makes it very costly for her/his to eter the attack. Hence, the efener gambles on the outcome of the attack rather than investing a lot of resources up front. Fig. (g) provies an illustration of Proposition (i), where increases when facing a more risk-seeking (less risk-averse) attacker. Finally, Fig. () focusing on the efense effectiveness parameter λ shows that ecreases in λ. This happens because an increase in λ ecreases the probability of a successful attack making it more unlikely for an attacker to get a high expecte utility from attacking. In terms of the equilibrium level, we see that initially increases, an then eventually ecreases. Although the very high success probability of attack at very low values of λ forces the efener not to invest, the increase chance of unsuccessful attacks when λ increases encourages the efener to invest more. However, at higher values of λ, the increase in expecte returns from efense is overcome by the significant increase in the cost of investing, leaing the efener to invest less. Most of the time, this parameter is beyon the control of both players. If this parameter can be ajuste as well (e.g., by technology investments; see Jose an Zhuang (6) ), then this coul also be use as a powerful eterrence tool. It is important to mention here that espite using very simple but reasonable utility functions, it is not possible to erive a close-form solution for int an. Hence, we analyze a few more baseline scenarios to see if the int follows the same tren as in Fig.. The plots are given in the Appenix. It follows from Fig. A in the Appenix that ue to the highly nonlinear form of the close-form expression of int β, it oes not follow a specific tren, an hence it cannot be inferre whether int always follows the same ecreasing tren as in Fig.. However, in real applications, nonavailability of a close-form solution of int oes not necessarily hiner the ecisionmaking process of the efener because it may be possible to reasonably estimate the efener s an the attacker s equilibrium responses numerically, or simply use these examples to unerstan which scenario or realm the problem belongs to.... Moel Comparison We stuy in this subsection the usefulness of the new moel with risk preferences propose in this article. We efine the utility of the moel as the ifference between the efener s expecte amages between the moel in Section. an a moel in

9 Deterrence an Risk Preferences in Sequential Attacker Defener Games 9 Defener s expecte amage (a) λ =.5 5 β a (a) λ =.5 5 β a (a) λ =.5 5 β a Defener s expecte amage using new moel Defener s expecte amage using risk neutral moel Fig. 5. One-way sensitivity analysis to stuy the impact of the risk-preference parameter β a on the ifference in expecte amage between a CDDA moel that takes into account risk preferences (new moel) an another that oes not (risk-neutral moel). Baseline values: z a = z =, v a = v = 6, c a = c = 5, an β =. which the efener incorrectly believes that the attacker is risk neutral an acts accoringly. Using the same baseline parameter values as in Section.., we perform one-way sensitivity analyses to stuy the impact of the risk-preference parameter β a on the utility of the moel, as shown in Fig. 5. We present the baseline case of a risk-neutral efener. The expecte amage of the efener is shown in terms of the expecte costs, which explains the negative values. For all the values of λ consiere here, the ifference between the expecte values of the two moels is zero when β a = ; i.e., the perceive risk-preference level of the attacker is correct. Fig. 5(a) shows that when the efener s efense effectiveness is low (e.g., λ =.5), the new moel s performance is comparable to that of a risk-neutral moel, for the range of β a consiere. The ifference between the expecte amages of the two moels is zero because when the efense effectiveness is too low, the certainty of attack an the success rate of attack are high. Hence, the efener is better off by investing int (an interior solution, which is inepenent of β a because of the Boolean nature of the attack). Fig. 5(b) shows the case when the efense effectiveness is slightly higher (e.g., λ =.5). When β a <, a efener using a risk-neutral moel for the attacker woul prepare more for an attack an hence incur higher expecte costs of efening than a efener using the new moel that correctly consiers risk preferences. The ifference in expecte amage is zero for β a > because in such cases the efener is better off by investing int, which is inepenent of β a. Fig. 5(c) shows that for higher values of λ (e.g., λ =.5), when β a <, the ifference is positive because the efener s investment reaches the maximum,. When β a >, the efener using the risk-neutral moel for the attacker woul efen less than the level require to eter the attack. Hence, attack is certain an the efener s expecte amage is higher. On the other han, the efener using the new moel woul be able to eter the attack, so her/his expecte amage is lower. We woul expect similar results when the efener is risk averse or risk seeking because expecte amage is inepenent of her/his risk preference. From Fig. 5, the new moel gives lower expecte amage if the true type of the attacker is risk seeking than when he/she is risk averse. Also, the new moel oes not give consierably less expecte amages for very low values of efense effectiveness λ. This happens because when the attack is more likely to be successful, the efener woul not try to eter the attack an choose the interior solution, which is inepenent of β a. Hence, except in the case when efense effectiveness is very low, the efener is expecte to incur losses if she/he uses a risk-neutral moel. This example illustrates by how much our moel that correctly consiers risk preferences coul outperform a risk-neutral moel. For example, the sensitivity analysis in Fig. 5 coul be use to etermine the threshol value of efense effectiveness λ above which the new moel performs better than the risk-neutral moel against risk-averse as well as

10 Payappalli, Zhuang, an Jose Stage Stage Defener Con nuous efense effort () A acker Con nuous a ack effort (a) (, ) (, ) Successful a ack, ( ) Unsuccessful a ack, ( ) Fig. 6. Sequence of moves in the AD game (CDCA moel) with the players utilities. â() 5 5 Fig. 7. Attacker s best response â() as a function of efener s investment, in the CDCA moel. Baseline values: z a = z =, v a = v = 6, c a = c = 5, an β a = β =. risk-seeking attackers. For the baseline values consiere, when λ =.5 (moerate efense effectiveness) the new moel outperforms the risk-neutral moel only against a risk-averse attacker, whereas when λ =.5 (high efense effectiveness) the new moel outperforms the risk-neutral moel against risk-averse an risk-seeking attackers. These results suggest that users of AD moels coul value risk preferences less in certain situations but shoul also be aware of the potential savings/losses that coul be incurre in situations where these moels yiel substantial ifferences.. CONTINUOUS DEFENSE CONTINUOUS ATTACK (CDCA) MODEL.. Moel The moel that we present here is an extension of the CDDA moel in Section. The main ifference from the CDDA moel is that the attacker is able to choose from a continuous level of attack effort a [, ) that maximizes his/her expecte utility. The moifie game tree is given in Fig. 6. Since it is reasonable to say that the efener s an the attacker s efforts etermine whether the attack is successful or not, we nee to take this into account when assessing the probability of successful attack. We use the contest success function of the a a+.(5) form P(a, ) = This function is increasing in a, ecreasing in, an boune between an, a,. By backwar inuction, we begin by examining the best response of the attacker. Observing the efense level, the attacker chooses a that maximizes his/her expecte utility U a (a, ). His/her optimization problem is: max U a (a, ) = P(a, ) u a (v a c a a) + ( P(a, ))u a ( c a a) = a a+ u a(v a c a a) + a+ u a( c a a). The attacker s best response function â() is obtaine using the necessary first-orer conition a U a(a, ) =. That is, ( a â() = arg max a + a + u a( c a a) a + u a(v a c a a) () ). Hence, the efener s equilibrium investment is = arg max U (â(), ) an the attacker s equilibrium investment is a = â( ). Fining a general analytical solution for â(), a, or is not possible, an the following sections iscuss how the situation coul be analyze further in such cases... Numerical Illustration an Sensitivity Analysis For the purpose of illustrating insightful scenarios, we use the same functional forms for utility as in Section.. Uner power utility u a (x) = (z a + x) β a,

11 Deterrence an Risk Preferences in Sequential Attacker Defener Games Investment (a) Target valuation of efener, v Investment (b) Cost coefficient of efener, c a * * Investment (c) Risk preference parameter of efener, β 6 6 v () Target valuation of attacker, v a 6 6 c (e) Cost coefficient of attacker, c a β (f) Risk preference parameter of attacker, β a 6 Investment Investment Investment 6 v a 6 c a β a Fig. 8. One-way sensitivity analysis of the equilibrium investments of attacker (a ) an efener ( ) with respect to the parameters use in the CDCA moel. Baseline values: z a = z =, v a = v = 6, c a = c = 5, an β a = β =. Equation () becomes: ( a â() = arg max a a + (z a + v a c a a) β a (5) + ) a + (z a c a a) β a. Close-form solutions o not exist for â(), a, an ue to the properties of the general power form utility function, as mentione in Section.. Hence, we stuy the behavior of â(), a, an to observe possible trens an erive insights. Fig. 7 shows that the best response of the attacker first increases an then ecreases in the efener s investment, an approaches an stays at zero for high values of. Fig. 8 illustrates the changes in the equilibrium responses (a an ) as the parameters v a,v, c a, c,β a, an β change one at a time, keeping all others at the baseline values. For ease of comparison, we use the same baseline values that are use in the sensitivity analysis of the CDDA moel in Section.. (Fig. ). Due to iminishing marginal expecte utility of attacking a efener s valuable target, a increases in v an then stabilizes (Fig. 8(a)). This behavior is comparable to a in Fig. (a). However, exponentially increases in v in Fig. 8(a) in contrast to the marginally ecreasing in Fig. (a) ue to the possibility of eterrence inuce by the Boolean nature of attack in the CDDA moel. It is optimal for the efener to increase her/his investment as the attacker s valuation of his/her target (v a ) increases (Fig. 8()), an this keeps the attack completely eterre until a particular value of v a (as observe in Fig. (e)). However, the efener is better off by ecreasing her/his efforts when v a increases any further. Interestingly, both the attacker an the efener invest more at very low values for the cost parameter an invest significantly less when costs are very high (Figs. 8(b) an (e)). In fact, the attacker is eterre completely when the cost of attack is significantly high (Fig. 8(e)). Another interesting observation is that in Fig. 8, a in Fig. 8(b) is very similar to in Fig. 8(e). This is surprising because one woul not expect so much symmetry in a sequential game.

12 Payappalli, Zhuang, an Jose In aition, the trens of are ifferent in Figs. 8(b) an 8(e), but still similar in Figs. (b) an (f). The attacker s response (a ) to changes in β an β a is similar (Figs. 8(c) an 8(f)). However, eterrence is absent in these cases, which contrasts with the iscrete case where a risk-neutral (riskaverse) attacker is eterre against a risk-averse (risk-neutral) efener (Figs. (c) an (g)). In summary, the analysis here shows various scenarios in which there coul be similarities/issimilarities between the results from the CDDA an the CDCA moels. Solely by extening the attack effort from iscrete to continuous, significant variation is observe in the results. Thus, unerstaning the attacker s ecision-making process coul be a critical step for the efener in rafting more effective efense strategies. Optimal preemptive efense strategies against an attacker with continuous efense capabilities coul be quite ifferent from those against an attacker with iscrete attack capabilities. 5. CONTINUOUS DEFENSE CONTINUOUS ATTACK - INCOMPLETE INFORMATION (CDCAII) MODEL One realistic challenge in moeling risk preferences is etermining what level or type of risk attitue to incorporate. In many terrorism an counterterrorism contexts, it is ifficult to estimate specific forms of utility or estimate risk-preference parameters. For example, the efener might be uncertain about the attacker s risk attitues an other parameters. In some contexts, it may be possible to have a rough estimate of players risk preferences through reveale preferences that can be measure by specific actions taken in the past (e.g., Phillips (6 8) ). Other approaches to estimation coul be interviewing subject-matter experts. We moel this commonly encountere setting where the efener is uncertain about the attacker s risk preference by extening our moel to an incomplete information moel in which the players have some beliefs instea of the precise knowlege about their opponent s type. As the secon-mover, the attacker has the avantage of observing the efener s actions. It is also very likely for the attacker to be much more informe about the efener than vice versa, because the efener (such as a government entity) is often manate by transparency laws to ivulge a significant amount of information (e.g., efense buget) to the public. 5.. Moel In this section, we use abbreviations RA, RN, an RS to represent risk-averse, risk-neutral, an risk-seeking behaviors, respectively. In the moel consiere here, the efener has certain beliefs about the risk preference of the attacker, which is that the attacker coul be RA with probability p,rn with probability q, an RS with probability p q. We exten the CDCA moel in Section to evelop the CDCAII moel as follows. The attacker s best response function is â() = arg max a U a (a, ). The efener s equilibrium investment is: = arg max (p U (â RA (), a) (6) + q U (â RN (), a) + ( p q) U (â RS (), a)), where â RA (), â RN (), an â RS () represent the best responses of a risk-averse, risk-neutral, an riskseeking attacker as perceive by the efener, respectively. The attacker s equilibrium investment is a = â( ). 5.. Numerical Illustration an Sensitivity Analysis Although the efener is uncertain about the attacker s risk preference, the attacker has complete knowlege of the efener s risk preference. Hence, the attacker s best response function in this case is thesameasinfig.7. Consiering a iscrete setting where the efener believes that the attacker is RA, RN, or RS with probabilities p, q, an p q, respectively, the efener s expecte utility is calculate as the probability-weighte sum of three utilities (calculate by consiering the attacker as RA, RN, an RS), each weighte by the respective probability ( p, q, or p q). Using the same functional forms of utility an baseline values as in Sections. an., we show in Figs. 9 how the equilibrium investment of an RA, RN, an RS efener ( ), respectively, an that of an attacker (a ) changes as the efener s beliefs change. In cases where the attacker or efener (or both) is (are) consiere not to be RN, the risk preference values are set at.5 an. for RA an RS behavior, respectively. Since a continuous range of risk preference values is not use, iscrete step values are obtaine for a an.inall cases, the efener s investments are ientical for all three types of attackers, which makes sense because

13 Deterrence an Risk Preferences in Sequential Attacker Defener Games Fig. 9. Sensitivity analysis of equilibrium investments of a risk-averse (RA) efener ( ) an an attacker (a )withrespect to the parameters p an q. Baseline values: v a = v = 6, c a = c = 5, an β =.5. Fig.. Sensitivity of equilibrium investment of a risk-neutral (RN) efener ( ) ananattacker(a ) with respect to the parameters p an q. Baseline values: v a = v = 6, c a = c = 5, an β =. the efener oes not know the true type of the attacker. First, an RA efener invests the most when she/he strongly believes that the attacker is RN, an the least when she/he strongly believes that the attacker is RS (Figs. 9() (f)). An RN efener invests the most when she/he strongly believes that the attacker is RA an weakly believes that the attacker is RN; an less when she/he believes that the attacker is RS (Figs. () (f)). This is comparable to Fig. 8(f) in which ecreases in β a. An RS efener (Figs. () (f)) invests the most when she/he weakly believes that the attacker is RS; except when her/his beliefs of the attacker being RA an RN are high an low, respectively (when she/he invests the least). When facing an RA efener, the RA an the RN attackers invest the most when the efener s belief about his/her risk preference is accurate (Figs. 9(a) an 9(b)). Against any type of efener,

14 Payappalli, Zhuang, an Jose Fig.. Sensitivity of equilibrium investment of a riskseeking (RS) efener ( ) an an attacker (a )withrespect to the parameters p an q. Baseline values: v a = v = 6, c a = c = 5, an β =. an RS attacker invests the most when the efener invests the most (Figs. 9(c) an 9(f), (c) an (f), an 9(c) an 9(f)). Against an RN efener (Fig. ), an RA attacker invests less in response to larger investments from the RN efener; however, an RS attacker oes the opposite. Also, a of an RN attacker is higher for a broaer range of p an q than a of an RA or an RS attacker, which can be compare to Fig. 8(f) in which a is higher when β a = than when β a. An RN attacker invests less when the efener has a wrong belief about the attacker s riskpreference type, espite the large ifference in the possible efense investments (. or.). The attacker s response against an RS efener (Fig. ) follows a pattern similar to that against an RN efener: An RA attacker invests less in response to larger investments from the RN efener; however, an RS attacker oes the opposite. The illustrate moels in Figs. 9 highlight the impact of incomplete information about a player s risk-preference types. These are useful in AD games because they can significantly affect the equilibrium responses of the players. The CDCAII moel presente here aresses the ifficult issue of estimating the level of risk preferences but still provies insights into how results woul change base on incorrect belief about the attacker s risk preference type. 6. CONCLUSION AND FUTURE RESEARCH DIRECTIONS In this article, we consier a sequential, singleperio, single-target AD game, where the efener preemptively invests in efense an the attacker chooses whether to attack or not. Here, the effect of players risk preferences on the equilibrium behavior of these players is analyze, focusing on the notion of eterrence, an these results are presente analytically, numerically, an graphically. Numerical illustrations an sensitivity analysis of continuous attack investment levels an uncertainty in the efener s beliefs about the attacker s risk preference are also provie. One key contribution of this article is the ientification of specific scenarios in which the efener using our moel woul be better off than a efener using a risk-neutral moel similar to those use in most of the literature. We fin that this incorporation of risk preferences is appealing an that this woul certainty strengthen the policymakers an risk analysts unerstaning of moels. This incorporates a funamentally recognize behavioral an economic principle that is often not consiere in mathematical moels such as AD games for the purpose of convenience. AD game moels that incorporate risk preferences provie robustness to a recommenation or an analysis when the recommenation remains the same when parameters are change. In cases where the solution an equilibrium behavior coul significantly vary, it may be useful to inform ecisionmakers an risk analysts of such possibilities. In terms of future research irections, this work opens new questions an areas to be explore. One interesting question is how risk preferences propagate in multiperio games (e.g., Cole an Kocherlakota (5) an Jose an Zhuang (6) ). This puts forwar the question of whether changes in

15 Deterrence an Risk Preferences in Sequential Attacker Defener Games 5 eterrence solutions that are ue to risk preferences can be sustaine in equilibrium for multiperio games. It woul be of interest to unerstan how solutions may evolve if we also allow intertemporal changes in risk preferences. Another interesting extension woul be to unerstan how risk preferences can also affect the solution in these games when players have multiple objectives. Keeney (5) an Keeney an von Winterfelt (5) mention that multiple objectives woul be a fertile area of research in risk analysis, an we believe that this area coul further be enriche by incorporating the notion of risk preferences. The next step to the moel woul be to fin ways to valiate the moel an its preictions. This coul perhaps be one through behavioral stuies or experiments. Thereafter, we expect that incorporating other behavioral theories (e.g., prospect theory or regret theory) into the moel coul provie aitional insights into other specific contexts an applications. ACKNOWLEDGMENTS This research was partially supporte by the U.S. Department of Homelan Security (DHS) through the National Center for Risk an Economic Analysis of Terrorism Events (CREATE) uner awar number -ST-6-RE. This research was also partially supporte by the U.S. National Science Founation uner awar numbers 899 an 9. However, any opinions, finings, an conclusions or recommenations in this ocument are those of the authors an o not necessarily reflect views of the DHS, CREATE, or NSF. We thank the eitors an anonymous referees for their helpful comments. The authors assume responsibility for any errors. (a) λ =.,z =,v =8,c =.5 (b) λ =.5,z =,v =8,c = (c) λ =.,z =,v =8,c = int β int β int β -.5 β () λ =.,z =,v =,c =.5 - β (e) λ =.5,z =,v =,c = - β (f) λ =.,z =,v =,c =. int β int β int β -.5 β (g) λ =.,z =,v =8,c = 5 - β (h) λ =.5,z =,v =8,c = 5 -. β (i) λ =.,z =,v =8,c = int β int β int β -5 β (j) λ =.,z =,v =,c =. -5 β (k) λ =.5,z =,v =,c = - β (l) λ =.,z =,v =,c =. int β int β int β -. β - β -. β Fig. A. Plot for slope of int with respect to β.