Defending against multiple different attackers

European Journal of Operational Research 211 (2011) 370–384

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Innovative Applications of O.R.

Defending against multiple different attackers Kjell Hausken a,⇑, Vicki M. Bier b a b

Faculty of Social Sciences, University of Stavanger, N-4036 Stavanger, Norway Department of Industrial and Systems Engineering, University of Wisconsin–Madison, 1513 University Avenue, Room 3270A, Madison, WI 53706, USA

a r t i c l e

i n f o

Article history: Received 20 July 2008 Accepted 19 December 2010 Available online 24 December 2010 Keywords: War Attack Defense Asset Conflict Contest success function

a b s t r a c t One defender defends, and multiple heterogeneous attackers attack, an asset. Three scenarios are considered: the agents move simultaneously; the defender moves first; or the attackers move first. We show how the agents’ unit costs of defense and attack, their asset evaluations, and the number of attackers influence their investments, profits, and withdrawal decisions. Withdrawal does not occur in one-period (simultaneous) games between two agents, at least with the commonly used ratio-form contest success function, but can occur in two-period games between two agents. The presence of one particularly strong attacker can cause other attackers to withdraw from the contest if the advantaged attacker appropriates so much of the defender’s asset that it is no longer sufficiently attractive to interest other attackers. In such cases, the defender focuses exclusively on the strong attacker. An advantaged defender may be able to deter attacks by moving first, but will continue to suffer from attacks if moving second. This suggests the importance of proactive rather than reactive defense. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The article considers one defender of an asset and two (possibly heterogeneous) attackers. This type of situation can arise in many contexts. For example, in computer security, one may need to defend both against highly sophisticated hackers and against relatively naive ‘‘script kiddies’’. Similarly, a nation may have to defend itself against both foreign and domestic attackers, or against attackers with different goals and motivations. Such attacker groups may differ in multiple respects. We model two such differences here-differences in the values they ascribe to the defended asset, and differences in their unit cost of attack (where for example sophisticated hackers may have much more effective means of attack available to them than script kiddies). Agents are at a disadvantage if their unit costs are too high. 1.1. Related work Several aspects of the model developed here have been studied previously. In particular, a great deal of research in the past decade has been concerned with conflicts between a single defender and a single attacker. Bier (2004) lays out the general rationale for a gametheoretic approach to such conflicts. She considers both games between an attacker and a defender, and games between multiple defenders; in addition, she explores how game theory can be integrated with reliability theory in cases where the asset to be protected is a system with multiple components, rather than a single standalone asset. For a survey of work that examines the strategic dynamics of governments vs. terrorists, see Sandler and Siqueira (2009). They survey advances in game-theoretic analyses of terrorism, such as proactive versus defensive countermeasures, the impact of domestic politics, the interaction between political and militant factions within terrorist groups, and defense against terrorism in the face of fixed budgets. Bier et al. (2005) use game theory to identify the optimal defense against intentional threats to series and parallel systems. Payoffs may be measured either in the success probability of an attack (if all components have the same valuation or attractiveness), or in expected gain (to the attacker) or loss (to the defender) if components are allowed to have different values or attractiveness. However, the attacker’s strategy is limited to the choice of which component(s) in the system to attack (in the case of a series system); the possibility of attacker deterrence is not considered. Bier et al. (2007) provide a framework for analyzing how the optimal defense could change in the face of uncertainty about the attacker’s asset valuations. In this framework, the defender’s valuations of the assets are common knowledge; the attacker’s valuations are known to the attacker, but not to the defender. The attacker chooses which asset to attack in order to ⇑ Corresponding author. E-mail addresses: [email protected] (K. Hausken), [email protected] (V.M. Bier). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.12.013

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maximize the expected benefit from an attack (as given by the attacker’s asset valuation, time the success probability of an attack); the defender allocates defensive resources to the assets to minimize the expected loss from an attack (taking into account not only the success probability of an attack on each asset, but also the defender’s probability that a particular asset will be the most attractive to the attacker). Azaiez and Bier (2007) consider more complex system structures (beyond merely series or parallel systems). In addition, they consider the possibility that a sufficiently strong defense could deter the attacker from attacking. Zhuang and Bier (2007) adapt the basic model developed in the above research to a situation in which the defender must protect against not only a single intentional attacker, but also the possibility of a natural disaster (which is assumed to be exogenous, and does not respond to the nature of the defenses). Thus, this can be viewed as a limiting case of a model with two attackers, in which one of the attackers is not intelligent or adaptive. Most of this past work concludes that the defender always has a first-mover advantage, in the sense that an optimal defense of the most valuable assets will not only reduce the likelihood of a successful attack on those assets, but (all else equal) deflect attacks to assets that are less valuable to the attacker, or else deter attacks altogether. Thus, the results generally recommend disclosure of defensive investments. However, disclosure cannot always be optimal in practice, since otherwise, there would be no role for secrecy and/or deception regarding defensive investments. For a paper that departs from the idea of disclosure; see Zhuang et al., 2010. Several researchers have also considered games among multiple defenders (with the risk of attack often being considered exogenous). For example, Kunreuther and Heal (2003) consider a model in which agents are subject both to direct attacks (which can be prevented by investment in security) and to indirect attacks (resulting from direct attacks on other agents, which can be prevented only by the actions of those other agents); motivating examples involving such risk of ‘‘contagion’’ might include computer security and vaccination. Zhuang et al. (2007) extend this work to the case where the attack is delayed rather than immediate, so that the defenders’ discount rates play a role in determining their optimal strategies; they find that the mere existence of myopic agents can sometimes make it undesirable for non-myopic agents to invest in security, even when it would otherwise be in their interests to do so. Likewise, Keohane and Zeckhauser (2004) consider strategic interactions between terrorism defenses by government and by private individuals. In particular, they consider both defenses that involve positive externalities for other individuals, and those that create negative externalities. In fact, taking into account such externalities, Trajtenberg (2006) even recommends that ‘‘the government should spend enough on fighting terrorism at its source, so as to nullify the incentives of private targets to invest in their own security’’. Finally, Aspnes et al. (2006) models ‘‘containment of the spread of viruses in a graph of n nodes’’. In this model, each node is considered to be a player, which has the option to install antivirus software at some known cost, or risk infection and a loss. In this context, the authors show that a centralized solution can give a better total cost than an equilibrium solution. Most of the above work does not concern itself directly with issues of attacker strength or advantage. In general, classical theory holds that superior agents are advantaged relative to inferior agents. For example, in biology, a size advantage or formidable posture usually deters adversaries. However, inferior agents also have advantages. Thus, Sun Tzu (320 Before Christ) observes the fierce strength of agents fighting in ‘‘death ground,’’ with their backs against the wall. In early game-theoretic work on the ‘‘paradox of power,’’ Hishleifer (1991, page 177) studied how advantaged versus disadvantaged agents fare in conflicts, and concluded that ‘‘poorer or smaller combatants often end up improving their position relative to richer or larger ones. This is the paradox of power . . . initially poorer contenders are rationally motivated to fight harder’’. Thus, those who are in a disadvantaged position may be able to improve their position through conflict. Of course, the paradox of power does not always hold; for example, it fails when the decisiveness of conflict is extremely large (Hirshleifer, 1991), or when there are multiple agents fighting individually, and one has some advantage in the conflict-say, an incumbency advantage (Mehlum and Moene, 2006). However, the paradox of power has been supported experimentally by Durham et al. (1998). Thus, differences in attacker levels of resources or sophistication appear to be important in understanding how a defender should optimally defend itself against multiple heterogeneous attackers. Finally, some recent work considers multiple attackers. For example, Mavronicolas et al. (2008) analyze an information network with threats called attackers. Each attacker uses a probability distribution to choose a node of the network to damage. The defender cleans attacks from some part of the network, which it chooses independently using its own probability distribution. However, the attackers are considered to be homogeneous. Contests with multiple attackers can credibly be analyzed as rent-seeking models, at least as long as adding a new agent causes the fixed-size rent to be allocated among all rent seekers (Hausken, 2005), rather than enlarging the size of the rent (as might be the case in contests with multiple defenders, for example). The rent-seeking model has commonly been used to analyze lobbying, R&D races, and elections. In this approach, multiple agents play a game in which each agent exerts an effort that affects its success in acquiring a rent, for which the various agents compete. Typically, simultaneous-move games are considered, where all agents choose strategies simultaneously and independently. See Krueger (1974), Posner (1975), Tullock (1980) for early contributions to the literature on rent seeking, and Congleton et al. (2008) for a review of its 40-year history. See also Nitzan (1994), Hausken (2005) for models in which the size of the rent is enlarged by the addition of a new agent. A rent-seeking model is appropriate when multiple agents compete for something of value, which is represented by the rent. In that case, a higher level of effort yields a larger fraction of the rent. Examples of rents for which agents compete are private goods, control of territory, monopoly privileges, budgets fought over by competing interest groups (political parties, localities, research teams, etc.), the best security specialists or risk analysts, government support for different industries, promotion and election opportunities, allocation of public goods such as sanitation, and employment and welfare opportunities. Rent seeking is distinguished from profit seeking, in which agents seek to extract value by engaging in mutually beneficial transactions. A further distinction is between rents that are obtained legally versus illegally (e.g., through illegal war or occupation, terrorist attack, fraud, embezzlement, or theft), although this paper does not distinguish between them, and in principle applies to both types of rent. By contrast, an example of a phenomenon that cannot be modeled as a rent, and hence cannot be fully and adequately addressed by the model developed below in this paper, is a public good – i.e., a good that is non-rival (so that consumption by one agent does not reduce consumption by others), and non-excludable (so that no one can be excluded from consumption of the good, creating the possibility of free riding). For example, Sandler (2005) discusses preemption of terrorism as a public good, since ‘‘Proactive measures (e.g., preemptive strikes) against terrorists create external benefits for all at-risk nations.’’ Similarly, if one posits that one goal of the insurgents in Iraq is to end the US occupation of that country, any benefits associated with the end of the occupation might well be shared by all insurgent groups, encouraging some groups to free ride on the insurgent efforts of others.

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1.2. Contribution and comparison In this paper, we depart from the emphasis in much previous work on interactions between multiple defenders, or between a single attacker and a single defender, to consider the strategic interactions between multiple attackers, and how a defender can optimally protect against such threats. Moreover, we allow for attackers to be heterogeneous with respect to both their valuations of the asset to be protected, and their strength or capabilities. Thus, for example, our model can be used to answer questions such as: Is it adequate for a defender to protect itself against the strongest attacker, or must it protect itself against the totality of attacker strength and capability? If it is infeasible for the defender to protect itself against the strongest possible attacker (e.g., because that attacker has an overwhelming advantage), is it still advantageous to protect against weaker attackers? These are questions on which the existing game-theoretic literature on security and defense is virtually silent, but can be of significant importance in the real world (since a defender may be faced with threats from sophisticated computer hackers as well as ‘‘script kiddies,’’ from nation states as well as small terrorist groups, etc.). Also, while most past literature has focused on sequential games in which the defender moves first (as might be appropriate, for example, in decisions about long-lasting and observable capital investments, such as installation of vehicle barriers), this paper also considers games in which the defender moves simultaneously with the attackers (by keeping its defenses secret), and games in which the attackers act first, leaving the defender to move second. Examples of cases in which the attackers move first may be when the attackers announce (credibly) that a new attack will occur at some point in the future, or when the attackers commit resources to such an attack (and the defender gains intelligence about those investments). In such cases, the defender can take the attackers’ decisions as given when choosing its defensive strategy. In general, who moves first is likely to depend on the specific types of threats and defenses being considered; therefore, our paper considers all three options. This allows us to elucidate under what condition the defender and/or the attackers can benefit from a first-mover advantage. Again, this can be an important consideration in practice. For example, Clausewitz (1832, section 6.1.2) argued for the ‘‘superiority of defense over attack,’’ which applies to classical warfare: ‘‘The defender enjoys optimum lines of communication and retreat, and can choose the place for battle.’’ Surprise is an attacker advantage, but leaving fortresses and depots behind through extended operations also leaves attackers exposed.1 In World War II, tanks and aviation technology gave some increased advantage to attackers. As guerilla warfare and urban warfare have become more common, attackers have gained further advantages. In the cyber context, attackers generally have an advantage. In particular, Anderson (2001) argues that ‘‘defending a modern information system could . . . be likened to defending a large, thinly-populated territory like the nineteenth century Wild West: the men in black hats can strike anywhere, while the men in white hats have to defend everywhere’’. The use of a rent-seeking model for the interactions among the defender and the attackers is another novel feature of our work, compared to most recent security research. This is both a contribution and a limitation of our work. In particular, in defending a single asset (such as a computer system, or the city of New Orleans), a rent-seeking model is plausible, since the value of the asset might be significantly reduced after an initial, partially successful attack (e.g., defacing an organization’s web page, or blowing up the levees that protect New Orleans from flooding). However, the adoption of a rent-seeking model limits the relevance of our model to situations in which there are multiple assets to protect, in which increasing the number of attackers could significantly increase the total damage that can be done (rather than just dividing up the value of a single asset among more agents). 1.3. Summary of results In the remainder of this paper, we show how the defense and attack efforts and the profits of a single defender and multiple attackers depend on the unit costs of defense and attack, and on the agents’ valuations of the asset being defended. Three scenarios are considered: first, a simultaneous game in which all agents move simultaneously; second, a two-period game in which the defender moves first; and third, a two-period game in which the attackers move first. Throughout, we find that agents disadvantaged by high unit costs of attack or defense (relative to the value of the asset in question) generally withdraw from the conflict-choosing not to attack (in the case of high-cost attackers), or not to defend the asset (in the case of a high-cost defender). In the simultaneous game, we find that at least two agents remain actively involved in the contest-either one attacker and the defender, or two attackers; as a special case, we show that if all attackers have equivalent characteristics, the defender withdraws when sufficiently disadvantaged, but the attackers always attack. However, for the sequential game, it is possible for only one agent to remain in the contest, and enjoy the entire value of the asset. We also show that the defender gives up its asset more easily (i.e., at a lower unit cost of defense) when the attackers move first than in the simultaneous game or when the defender moves first; thus, the defender has a first-mover advantage. Interestingly, the results show that the presence of one particularly strong attacker can cause other attackers to withdraw from the contest, if the advantaged attacker appropriates so much of the defender’s asset that it is no longer sufficiently attractive to interest other attackers. Thus, a nation or company under severe attack by one adversary may no longer be an attractive target to others. In such cases, the defender can focus exclusively on the contest with the strong attacker. As long as the other attackers remain sufficiently disadvantaged to refrain from attacking, the optimal allocation of defensive resources no longer depends on the characteristics of those attackers who withdraw. In contrast, when no attackers withdraw, the defender needs to defend itself against the sum of the attacks from all attackers. Moreover, we find that a sufficiently advantaged defender will be able to deter all attackers by moving first in a sequential game, while at least one attacker will always remain in the contest in a simultaneous game, or when the attackers move first. Thus, even a highly advantaged defender will continue to suffer from attacks if not playing as the first mover. This demonstrates the importance of proactive defense, especially for strong defenders. These results suggest the importance of proactive rather than reactive defense, since making defenses known to potential attackers can decrease the number of threats against which the defender must protect itself.

1 Examples of features improving defense are the use of trenches (combined with the machine gun) in World War I, the use of castles and fortresses (with cannon fire from higher elevations), and the use of checks and guards (in the broadest sense of those terms). The superiority of the defense over the attack appears to be even larger for production facilities and produced goods than for Clausewitz’s mobile army (Hausken, 2004).

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1.4. Overview Section 2 presents the model. Section 3 considers the simultaneous game, with general analysis, special cases, corner solutions, and numerical examples. Section 4 presents the sequential (two-period) game when the defender moves first. Section 5 presents the two-period game when the attackers move first. Section 6 compares the three games, and Section 7 concludes. 2. The model 2.1. Notation n r Ri t Ti f Fi c Ci g Gi u Ui

number of attackers, n P 2 defender’s valuation of the asset, r > 0 attacker i’s valuation of the asset, Ri > 0 defender’s defense effort, t P 0 attacker i’s attack effort, Ti P 0 defensive expenditure (as a function of defense effort), f P 0 attack expenditure for attacker i (as a function of attack effort), Fi P 0 unit cost of defense, c > 0 attacker i’s unit cost of attack, Ci > 0 fraction of asset retained by defender, 0 6 g 6 1 fraction of asset retained by attacker i, 0 6 Gi 6 1 defender’s profit, u P 0 attacker i’s profit, Ui P 0

2.2. The basic model Players: In this paper, we consider a game with one defender and n attackers, which gives a set of n + 1 agents. The agents compete for a valuable asset controlled by the defender. The defender values the asset at r. Attacker i values the asset at Ri.2 As is common in the literature, we assume that all agents are risk neutral. In reality, defenders are likely to be risk averse, while attackers may be risk seeking. Accounting for risk attitude complicates the analysis substantially, however, so is left to future research. In the meantime, the assumption of risk neutrality provides a building block for future work, and does not fundamentally change the nature of the arguments. Strategies: Each attacker has the option of launching an attack, with the goal of acquiring a portion of the defender’s asset (or, alternatively, capturing, destroying, or disabling the entire asset with some probability). The decision variable of attacker i is the level of effort Ti 2 [0, 1) to devote to an attack (which will equal 0 when the attacker chooses not to launch an attack). Achieving a level of attack effort Ti is assumed to require an attack expenditure Fi(Ti), where @Fi/@Ti > 0. Likewise, the defender chooses an effective level of defense t for its asset. This means that the defender chooses the strategy t within the strategy set [0, 1), where t is a continuous free choice variable, t 2 [0, 1). The defensive expenditure required to achieve t is f(t), where @f/ @t > 0. For simplicity, we limit ourselves here to linear functions, given by f = ct and Fi = CiT, where c is the unit cost of defense (1/c is the efficiency of defense) and Ci is attacker i’s unit cost of attack (1/Ci is the efficiency of attack). These functions could in principle be more general, however-for example, reflecting decreasing marginal returns to expenditure. (The expenditures c t and CiTi can be interpreted as expenses in capital and/or labor, where c, t, Ci, and Ti are non-negative.) Contest success functions: The contest between the defender and the attackers for the asset is assumed to take the common ratio form frequently used in the rent-seeking literature. See Tullock (1980) for an early formulation and Skaperdas (1996) for an axiomatic approach. In particular, we consider the following contest-success functions:

g¼

tþ

t Pn

i¼1 T i

;

Gi ¼

tþ

Ti Pn

i¼1 T i

;

i ¼ 1; . . . ; n;

ð1Þ

where g and the Gi are the fractions of the defender’s asset that the defender and attacker i retain (or the probabilities of successful defense or attack), relative to their evaluations of the asset. In other words, the defender controls an asset. The defender and n attackers engage in a contest over the asset. The defender defends with level of effort t, attacker 1 attacks with level of effort T1, attacker 2 attacks with level of effort T2, etc., up to attacker n, which attacks with effort Tn. After the contest, the defender controls a fraction g of the asset, attacker 1 controls a fraction G1, attacker 2 controls a fraction P G2, etc., up to attacker n, which controls a fraction Gn. The fractions sum to one; i.e., g þ ni¼1 Gi ¼ 1. As expected, the fraction g satisfies the first-order conditions @g/@t > 0 and @g/@Ti < 0, where ‘‘@’’ means differentiation of g with respect to the free-choice variables t and Ti, respectively. That is, g increases when the defender increases t, and decreases when attacker i increases Ti. The fraction g also satisfies the second-order conditions @ 2g/@ t2 < 0 and @ 2 g=@T 2i > 0, where ‘‘@ 2’’ means double differentiation. That is, the defender benefits concavely from its own defense investment, and suffers convexly from the attacks launched by each of the attackers. Similarly, the fraction Gi satisfies the first- and second-order conditions @ Gi/@t < 0, @ 2Gi/@t2 > 0, @ Gi/@Ti > 0, @ 2 Gi =@T 2i < 0, @Gi/@Tj < 0, and 2 A special case is when r = Ri (e.g., measured in money), which means that the defender and all attackers value the asset equally. More generally, the asset may be given a more subjective economic value, human value, symbolic value, or other kinds of value. Values may be sales value, purchasing value, reimbursement value, use value, application value, exchange value, insurance value, etc., and these valuations may differ among the defender and the attackers.

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@ 2Gi/@Tj 2 > 0 for i, j = 1, 2, . . . , n, i – j. That is, each attacker benefits concavely from its own attack effort, and suffers convexly from both the defender’s investment and the attack investment launched by the other attackers. Payoffs: The profits (payoffs or utility functions) u(t, T1, . . . , Tn) of the defender and Ui(t, T1, . . . , Tn) of attacker i are given by

u ¼ uðt; T 1 ; . . . ; T n Þ ¼

tþ

t Pn

i¼1 T i

r  ct;

U i ¼ Uðt; T 1 ; . . . ; T n Þ ¼

tþ

Ti Pn

i¼1 T i

Ri  C i T i ;

i ¼ 1; . . . ; n:

ð2Þ

In other words, each actor incurs a cost of defense or attack, and enjoys a fractional contest success defined in Eq. (1), which is multiplied by that actor’s valuation of the asset in question (r for the defender, Ri for attacker i). We hereafter suppress t, T1. . . , Tn in the expressions for u and Ui, and i = 1, . . . , n, to simplify the notation. Sequence of play: In the next three sections, we consider three versions of the basic game outlined above. First, we consider the simultaneous game where the n + 1 agents choose their strategies simultaneously and independently. Then, we consider a two-period game where the defender chooses its strategy first, and the n attackers choose their strategies second. Finally, we consider a two-period game where the n attackers choose their strategies first, and the defender chooses its strategy second. 3. Analysis of the simultaneous game 3.1. General analysis We solve for Nash equilibrium, defined such that no agent can benefit by unilaterally deviating from his equilibrium strategy. Each strategy in such an equilibrium is a best response to all other players’ strategies in that equilibrium. The defender’s free choice variable is the defense t. The attackers’ free choice variables are the attack efforts Ti. The first-order and second-order conditions for an interior solution are

  Pn @U i Ri t þ j¼1;j–i T j ¼  C i ¼ 0; P @T i ðt þ nj¼1 T j Þ2   Pn @ 2 U i 2Ri t þ j¼1;j–i T j ¼ < 0; P @T 2i ðt þ nj¼1 T j Þ3

P @u r ni¼1 T i ¼  c ¼ 0; P @t ðt þ ni¼1 T i Þ2 P @2u 2r ni¼1 T i ¼ < 0; Pn 2 @t ðt þ i¼1 T i Þ3

ð3Þ

which constitutes a Nash equilibrium. The second-order conditions are always satisfied. Solving for the interior solution gives

n t¼

P n

  ðn  1Þ cr ; 2 P

  P n cr þ ni¼1 CRii  n CRii Ti ¼ ;  2 P c þ ni¼1 CRii r

Ci i¼1 Ri



c r

þ

n Ci i¼1 Ri

n X i¼1

Ti ¼ 

c r

þ

n2 cr Pn

Ci i¼1 Ri

ð4Þ

2 :

Using these results for the levels of defense and attack effort at equilibrium enables us in the next proposition to specify the conditions for when an agent is deterred in the simultaneous game (i.e., when the agent withdraws from the game). c 1 Proposition 1. Conditions   for deterrence in the simultaneous game:PThe defender gives up its asset if r P n1 Pn Cj n Cj Ci Ci 1 c c , which also can be expressed as P þ 6 n  ; for i ¼ 1; 2 . . . n. j¼1;j–i Rj j¼1 Rj n1 r r Ri Ri

Pn

Ci i¼1 Ri .

Attacker i ceases attacking if

Proof. Follows from requiring t and Ti, respectively, to be non-negative in Eq. P(4), t P 0 and Ti P 0. For example, from Eq. (4), we will have n Ci c an interior solution t > 0 whenever the numerator in the expression for t, n i¼1 Ri  ðn  1Þ r , is greater than zero. Rearranging terms and P P n n C C 1 c 1 i i dividing through by n gives cr < n1 i¼1 Ri . When this is not the case,r P n1 i¼1 Ri , we will have a corner solution with t=0, and the defender will give up its asset. h This proposition states that the defender gives up its asset if its unit cost of defense is high, its evaluation of the asset is low (so the asset is not worth defending), the attackers’ unit costs of attack are low (so that they can easily launch attacks), or the attackers’ evaluations of the asset are high (so that they are highly motivated to attack). Conversely, an attacker ceases attacking if its unit cost of attack is high, its evaluation of the asset is low, or the defender and the other attackers have low units costs and high asset evaluations. Inserting (4) into (2), the profits of the n + 1 agents are given by

Pn u¼

Ci i¼1 Ri c r

þ

 ðn  1Þ cr Pn C i

0

!2

c

r Ui ¼ @

r;

i¼1 Ri

þ

Pn

c r

Cj j¼1 Rj

þ

Pn

 n CRii

Cj j¼1 Rj

12 A Ri :

ð5Þ

While the defense and attack efforts in (4) depend only on the ratios of the unit costs to the asset evaluations, the profits additionally depend on the asset evaluations themselves. 3.2. Special cases of the simultaneous game For simplicity, we begin by considering the case where there is only one attacker. We define C, R, T, and U as the unit cost, asset valuation, attack, and profit for the attacker, respectively. We set C/R = c/(k r), which means that the defender is disadvantaged with a ratio c/r that is k times as high as C/R for the attacker. That is, when k > 1, the defender is disadvantaged, when 0 < k < 1, the attacker is disadvantaged, and when k = 1, both agents are equally advantaged. Inserting n = 1 and C/R = c/(k r) in Eqs. (4) and (5) gives

t¼

2

k 2

ðk þ 1Þ c=r

;

T¼

k

2

ðk þ 1Þ c=r

;

u¼

r ðk þ 1Þ

2

; 2

U¼

k R ðk þ 1Þ

2

;

u r : ¼ U k2 R

If r = R, the attacker invests k times more than the defender, and earns a profit k2 times higher than the defender.

ð6Þ

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For n attackers, we let Ci/Ri = C/R = c/(k r). That is, we allow for the possibility that the Ri may differ from R provided that Ci differs from C so that the equality holds. Substituting into Eqs. (4) and (5) gives

t¼

nkðk  ðk  1ÞnÞ 2

ðn þ kÞ c=r

2

;

Ti ¼

k n 2

ðn þ kÞ c=r

u¼

;

 2 k  ðk  1Þn r; nþk

 Ui ¼

k nþk

2 Ri :

ð7Þ

The attackers always invest in this case. If the defender is facing at least two attackers, then the interior solution for t is negative when k > 1 (disadvantaged defender), and the defender will instead cease investing, give up the asset, and earn zero profit. Using these more specific results for the levels of defense and attack efforts at equilibrium in a game with n homogeneous attackers enables us in the next proposition to specify how disadvantaged the defender must be (in terms of c/r compared with C/R) in order to be deterred from defending in the simultaneous game. Moreover, we find that attackers are never deterred from attacking. Proposition 2. Conditions for deterrence in the simultaneous game with homogeneous attackers: Assume that all attackers have equivalent characteristics, Ci/Ri = C/R. (a) If the defender is disadvantaged with a ratio c/r that is k P n/(n  1) times as high as C/R for the n attackers, Ci/Ri = C/R = c/(k r) for some k > 1, then the defender ceases investing, gives up the asset, and earns zero profit. (b) The attackers always attack. Proof. Follows from requiring t 6 0 in Eq. (7), and noting that Ti > 0 in Eq. (7). For example, from Eq. (7), we will have t > 0 whenever n k [k  (k  1) n] > 0. Dividing through by n k and rearranging terms gives k < n/(n  1) (since we have assumed that n P 2). Conversely, when k P n/(n  1), then we will have a corner solution with t = 0 (i.e., the defender will cease investing, give up its asset, and earn zero profit). h The defender withdraws when k P n/(n  1), which is the area above the highest curve in Fig. 1. In this figure, the horizontal axis shows the number of attackers, n. The vertical axis shows the degree of advantage or disadvantage faced by the defender, given by k = (c/r)/(C/R), where k > 1 is a disadvantage, and 0 < k < 1 indicates an advantage for the defender. As the number of attackers grows, the extent of disadvantage required for the defender to withdraw decreases, as we might expect. 3.3. Corner solutions of the simultaneous game when two agents remain Returning now to the general case of n (possibly heterogeneous) attackers, we explore the circumstances under which one or more agents may wish to withdraw from the game (corresponding to a corner solution of zero investment). If the inequality   P C Ci 1 c P n1 þ nj¼1;j–i Rjj in Proposition 1 is not satisfied for some i, then attacker i withdraws from the game (due to the assumption that Ri r Ti must be non-negative), does not invest, and does not enjoy a fraction of the asset. When one attacker is removed from the game in this manner, the parameter n is reduced by one, and the resulting (revised) inequality for each of the remaining attackers is reevaluated. Assume that this process continues until there are only n = 2 attackers left, i and j. If Ci/Ri P c/r + Cj/Rj, then attacker i ceases attacking and chooses Ti = 0, which gives Ui = 0. This means that only attacker j remains to compete with the defender. Solving the first-order conditions for the defender and attacker j gives

t¼ u¼

C j =Rj

;

T i ¼ 0;

; 2

U i ¼ 0;

½c=r þ C j =Rj 2 rC 2j =R2j ½c=r þ C j =Rj 

Tj ¼

c=r ½c=r þ C j =Rj 2

Uj ¼

Rj c2 =r 2 ½c=r þ C j =Rj 

1 Conversely, assume that the inequality cr P n1

Pn

Ci i¼1 Ri

; ; 2

Ci c Cj P þ : r Rj Ri

ð8Þ

in Proposition 1 is not satisfied. Then the defender gives up its asset and chooses t = 0,

which gives u = 0, and the n attackers compete for the asset. Solving the first-order conditions for the n attackers gives

Fig. 1. Regions where agents withdraw in the three games. The horizontal axis shows the number of attackers, n. The vertical axis shows the ratio of the defender’s unit defense cost and asset valuation, c/r, divided by the ratio of the attackers’ unit attack cost and asset valuation, C/R.

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t ¼ 0;

P  n Cj Ci ðn  1Þ j¼1 Rj  ðn  1Þ Ri Ti ¼ ; P 2 0Pn

u ¼ 0;

Ui ¼ @

n X

n Cj j¼1 Rj

12  ðn  1Þ CRi i A Ri ; Pn Cj

Cj j¼1 Rj

j¼1 Rj

i¼1

n1 T i ¼ Pn C ; i

i¼1 Ri

ð9Þ n c 1 X Ci P : r n  1 i¼1 Ri

When n = 2, (9) becomes

C 2 =R2

t ¼ 0;

T1 ¼

u ¼ 0;

U1 ¼

½C 1 =R1 þ C 2 =R2 2 R1 C 22 =R22 ½C 1 =R1 þ C 2 =R2 

T2 ¼

;

2

;

C 1 =R1 ½C 1 =R1 þ C 2 =R2 2

U2 ¼

R2 C 21 =R21 ½C 1 =R1 þ C 2 =R2 

T1 þ T2 ¼

;

2

;

1 ; C 1 =R1 þ C 2 =R2

c C1 C2 P þ : r R1 R2

ð10Þ

In this case, the investment by attacker i is inverse U shaped in the extent Cj/Rj of disadvantage suffered by the other attacker, i, j = 1, 2, i – j. That is, it first increases in the other attacker’s disadvantage, and then begins to decrease once the second attacker becomes so disadvantaged as not to pose a substantial challenge. By contrast, the profit earned by attacker i increases concavely in the disadvantage of attacker j, approaching a horizontal asymptote equal to the asset value of the attacker i. The above corner solutions are summarized in the following proposition, illustrating that deterrence can never lead to more than n  1 agents withdrawing from the game (assuming n attackers and one defender in the simultaneous game), leaving at least two agents in the game. Proposition 3. Conditions for deterrence in the simultaneous game when two agents remain: (a) When one attacker and one defender remain in the contest, the attacker does not withdraw regardless of how large Ci/Ri is, and the defender also does not withdraw. (b) When c/r is high, so that the defender withdraws, and two attackers remain, they never withdraw. Proof. Proposition 3(a) follows from Eq. (8), which shows that setting Ti = 0 leads to Tj > 0 and t > 0. Proposition 3(b) follows from Eq. (9), where setting t = 0 implies that Ti > 0 and Tj > 0. h One limitation of the ratio form of the contest success function in (1) is that at least two agents always exert effort at equilibrium. When these agents are adversaries (defender and attacker), neither one-sided withdrawal (submission) nor two-sided withdrawal (peace) is possible in a simultaneous game. When the two remaining agents are attackers, we have one-sided withdrawal (submission) in the sense that the defender withdraws, but peace is not possible, since the two attackers still attack. The so-called difference form of the contest success function discussed by Hirshleifer (1989:107) allows both one-sided and two-sided withdrawal, but that model has the limitation that an interior asymmetric solution is not possible. The logic of the difference form is that the winning probability depends only on the difference between efforts. Amegashie (2006) introduces a tractable noise parameter into the ratio form, which makes one-sided, two-sided, and n-sided withdrawal all possible in a simultaneous game. 3.4. Numerical examples for the simultaneous game We now illustrate the behavior of this model graphically. Fig. 2 illustrates Proposition 1 with four regions in parameter space; without loss of generality, we restrict ourselves for simplicity to the special case in which C1/R1 = 1. When the sum of c/r and C2/R2 is less than one, the triangle in the lower left corner illustrates that attacker 1 has a competitive disadvantage, and withdraws from attacking. As the sum of c/r and C2/R2 increases above one, attacker 1 starts to attack, leading to competition between all three agents, in the middle region of the

Fig. 2. Simultaneous game: four regions in parameter space for competition and withdrawal when C1/R1 = 1. The horizontal axis shows the ratio of attacker 2’s unit attack cost and asset valuation. The vertical axis shows the ratio of the defender’s unit defense cost and asset valuation, c/r.

K. Hausken, V.M. Bier / European Journal of Operational Research 211 (2011) 370–384

377

Fig. 3. Simultaneous game: efforts and profits as functions of the ratio of attacker 2’s unit attack cost and asset valuation, C2/R2, when c/r = C1/R1 = r = Ri = 1 for i = 1, 2.

Fig. 4. Simultaneous game: efforts and profits as functions of the ratio of attacker 2’s unit attack cost and asset valuation, C2/R2, when c/r = 1.5 and C1/R1 = 0.5, r = Ri = 1 for i = 1, 2.

figure. As c/r increases, the defender experiences a competitive disadvantage, and eventually gives up defending the asset, as illustrated by the upper left triangle. Finally, as C2/R2 increases, attacker 2 experiences a competitive disadvantage, and eventually ceases attacking, as illustrated by the lower right triangle in Fig. 2. When one agent is sufficiently disadvantaged relative to the other two agents, that agent withdraws. There is no withdrawal in the upper right corner of Fig. 2, regardless how high c/r is and C2/R2 as long as these are comparably sized. Even though attacker 1 with C1/R1 = 1 is significantly advantaged as c/r and C2/R2 approach infinity, neither the defender nor attacker 2 is sufficiently advantaged relative to the other. Assume c/r = 1 and let C2/R2 increase from zero. Fig. 3 illustrates the variables when c/r = C1/R1 = r = Ri = 1 for i = 1, 2. The defender and attacker 1 are equally advantaged, and hence t = T1 and u = U1. When C2/R2 is low, the advantaged attacker 2 enjoys a high U2, generated through a high and inexpensive T2. As C2/R2 increases, t = T1 and u = U1 increase, while T2 and U2 decrease, reaching zero when C2/R2 = 2, after which attacker 2 withdraws. When attacker 2 is no longer in the game, inserting the parameter values into Eq. (8) confirms that the defender and attacker 1 invest equally much and earn equal profits at t = T1 = u = U1 = 0.25. Fig. 4 lets the defender be 50% more disadvantaged, and attacker 1 be twice as advantaged; that is, c/r = 1.5 and C1/R1 = 0.5, r = Ri = 1 for i = 1, 2. Attacker 2 now faces less opposition from the defender, and more opposition from attacker 1, who appropriates a substantial part of the asset. The defender’s c/r high is disadvantageous, so it withdraws when C2/R2 < 1. As C2/R2 increases above 1, the defender is less severely disadvantaged relative to attacker 2, so it joins in the contest. As C2/R2 > 2, attacker 2 (who is now heavily disadvantaged) withdraws. 4. Two-period game when defender moves first 4.1. General analysis It is sometimes realistic for the defender to choose its investment in the first period, while the attackers choose their effort in the second period. We solve for a sub-game perfect equilibrium defined as follows: A strategy profile is a subgame perfect equilibrium if it represents Nash equilibrium of every subgame of the original game. More informally, this means that if (1) the players play any smaller game that consists of only one part of the larger game, and (2) their behavior represents a Nash equilibrium of that smaller game, then their behavior is a subgame perfect equilibrium of the larger game. A common method for determining subgame perfect equilibria in the case of a finite game is backward induction, which means that the second period is solved first, followed by the solution of the first period. The first-order and second-order conditions of the n attackers for the second period are as given in Eq. (3), and imply the following results:

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    Pn Pn @U i Ri t þ j¼1;j–i T j @ 2 U i 2Ri t þ j¼1;j–i T j ¼  C i ¼ 0; ¼ < 0; P  3 P @T i @T 2i ðt þ nj¼1 T j Þ2 t þ nj¼1 T j ! !! n n X X Ci 1 : ) Ti ¼ t þ Tj tþ Tj Ri j¼1 j¼1

ð11Þ

When the defender moves first, the first-order and second-order conditions for the defender in the simultaneous game of Eq. (3) are no longer valid. To derive the new first-order and second-order conditions, we begin by summing up T1 + T2 +    + Tn for the n attackers, yielding n X

Ti ¼

tþ

i¼1

n X

!"

Ti

n tþ

i¼1

n X

!

Ti

i¼1

# n X Ci : Ri i¼1

ð12Þ

Solving for the sum T1 + T2 +    + Tn then yields n X i¼1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P 4t ni¼1 CRii þ ðn  1Þ2 þ n  1  2t ni¼1 CRii Ti ¼ : P 2 ni¼1 CRi

ð13Þ

i

Inserting (13) into the denominator in (2) and simplifying gives the defender’s first-period profit as follows:

P 2t ni¼1 CRi i r  ct: u ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 4t ni¼1 CRii þ ðn  1Þ2 þ n  1

ð14Þ

Differentiating u with respect to t to determine the defender’s first-order and second-order conditions for an interior solution gives

Pn

@u @t

Ci i¼1 Ri

r ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  c ¼ 0; Xn C i 4t i¼1 þ ðn  1Þ2 Ri P 2  2 n Ci  ðn  1Þ2 cr i¼1 Ri ) t¼ : c2 Xn C i 4 i¼1 R r i

2r

2

X

Ci i¼1 R i n

2

@ u ¼ 3=2 < 0 Xn C i @t 2 4t i¼1 þ ðn  1Þ2 Ri

ð15Þ

As before, the second-order condition is always satisfied. Inserting (15) into (13) gives n X i¼1

n X 1 ðn2  1Þ 1 Ci þ P Ti ¼  ; 2c=r 4 ni¼1 CRi 4ðc=rÞ2 i¼1 Ri

tþ

n X

Pn Ti ¼

Ci i¼1 Ri

i¼1

i

þ ðn  1Þ cr : P n Ci c

2r

ð16Þ

i¼1 Ri

Similarly, inserting (15) into (22) gives

0Pn

12 0Pn C 13 j c þ ðn  1Þ cr j¼1 Rj þ ðn  1Þ r C i A41  @ A5: Pn C j Pn C j c c 2R

Cj j¼1 Rj

Ti ¼ @

2r

i

j¼1 Rj

ð17Þ

j¼1 Rj

r

Finally, inserting into (2) gives

P n u¼

Ci i¼1 Ri

 ðn  1Þ cr Pn C i c

4r

i¼1 Ri

2

2

r;

0Pn C 132 j c j¼1 Rj þ ðn  1Þ r C i @ A5 Ri : U i ¼ 41  Pn C j c 2Ri r

ð18Þ

j¼1 Rj

Using these results for the levels of defense and attack effort at the sub-game perfect equilibrium enables us in the next proposition to specify the conditions for when an agent is deterred in games where the defender moves first. Proposition 4. Conditions for deterrence in the sequential game when the defender moves first: (a) The defender gives up its asset, and is deterred P  Pn C i n Cj C i Pn C j Ci 1 c from investing in defense, if cr P n1 i¼1 Ri . (b) Attacker i ceases attacking if r 6 2Ri j¼1 Rj = j¼1 Rj  ðn  1Þ 2Ri ; i ¼ 1; 2 . . . ; n. Proof. Follows from Eqs. (15) and (17), and the fact that t and Ti must be nonnegative; i.e., t P 0 and Ti P 0. For example, from Eq. (15), we P 2  2 Pn C i n Ci 1  ðn  1Þ2 cr > 0. Rearranging terms and taking square roots, we get cr < n1 will have t > 0 whenever the numerator i¼1 R i¼1 R . i

i

When that does not hold, we must have t = 0, since t P 0 by assumption. h The conditions under which the defender gives up its asset remain the same as in the simultaneous game in Proposition 1. This follows from comparing (15) and (4), where requiring that t be nonnegative has the same implication. The conditions under which attackers withdraw from the game change from Propositions 1–4; however, there is no simple or universal way to summarize these changes, since depending on the parameter values, attackers may be either more or less likely to withdraw when the defender moves first than in the simultaneous game. Therefore, in the next section, we illustrate this result by highlighting several special cases.

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379

4.2. Special cases when defender moves first As before, we begin with the case of a single attacker. Letting n = 1 and C/R = c/(k r) in Eqs. (15), (17), and (18) gives

t¼

1 ; 4kc=r

T¼

2k  1 ; 4kc=r

u¼

r ; 4k

U¼

 2 2k  1 R; 2k

u kr ¼ : U ð2k  1Þ2 R

ð19Þ

The attacker invests 2k  1 as much as the defender, but earns a profit that is only (2 k  1)/k as high. When k > 1, the attacker has a secondmover disadvantage compared with the simultaneous game (where in an analogous situation the attacker invests k times as much as the defender, and earns a profit that is k2 times as high). When k 6 1/2, the attacker does not attack, and earns zero profit (in contrast to the simultaneous game, where the attacker always attacks). For n attackers, we let Ci/Ri = C/R = c/(k r). In that case, Eqs. (15), (17), and (18) give 2

t¼

n2  k ðn  1Þ2 ; 4knc=r

Ti ¼

½nðk þ 1Þ  k½nðk  1Þ þ k 2

4kn c=r

;

u¼

ðk  nðk  1ÞÞ2 r; 4kn

 Ui ¼

2 nðk  1Þ þ k Ri : 2kn

ð20Þ

Using these more specific results for the levels of defense and attack effort at equilibrium for the case of n homogeneous attackers when the defender moves first enables us in the next proposition to compare the defender’s and attackers’ investments and profits, and how the agents’ profits depend on the number of attackers. Proposition 5. Equilibrium solutions for the sequential game with homogeneous attackers and defender when the defender moves first: When C/R = c/r (so k = 1) and r = Ri, the defender invests as much as all n attackers taken together (t/Ti = n), and earns as much profit as all n attackers taken together (u/Ui = n). The defender’s profit is inversely proportional to n, while each attacker’s profit is inversely proportional to n2. 2

2

2n1 ¼ 4nc=r ; a similar substitution gives Proof. Follows from Eq. (20) when k = 1. For example, substituting k = 1 into Eq. (20) gives t ¼ n ðn1Þ 4nc=r 2 2n1 T i ¼ 4n2 c=r. Adding up n terms of the form Ti converts the n in the denominator to an n, showing that the defender invests as much as all n

attackers taken together. h These results are in contrast to the results of the simultaneous game, as given in Eq. (7) when k = 1. In the simultaneous game, the defender investment and profit are only as much as those of any one of the (homogeneous) attackers (i.e., t = Ti and u/Ui = 1), and both types of agents experience profits that are inversely proportional to (n + 1)2. As in the results for the simultaneous game in Eq. (7), the defender in the two-period game of Eq. (20) ceases investing when disadvantaged by a value of k > 1, if facing at least two attackers (n P 2): Based on the equilibrium defense and attack efforts for n homogeneous attackers, the next proposition specifies how disadvantaged the defender and the attackers must be (in terms of c/r compared with C/R) in order to be deterred, in games where the defender moves first. Proposition 6. Conditions for deterrence in the sequential game with homogeneous attackers when the defender moves first: Assume that all attackers have equivalent characteristics, Ci/Ri = C/R. (a) If the defender is disadvantaged with a ratio c/r that is k P n/(n  1) times as high as C/R for the n attackers, C/R = c/(k r) for some k > 1, then the defender ceases investing, gives up the asset, and earns zero profit. (b) The attackers will all withdraw whenever k 6 n/(n + 1). Proof. Follows from Eq. (20), and the fact that t and the Ti are non-negative; i.e., t P 0 and Ti P 0. For example, substituting in k = n/(n  1) into Eq. (20) gives t = 0. For k > n/(n  1), the numerator of the expression for t in Eq. (20) will be negative, but the denominator will remain positive. Since t P 0 by assumption, this means that we cannot have an interior solution for k > n/(n  1), so we must instead have t = 0. h The defender withdraws when k P n/(n  1), as in the simultaneous game (above the highest curve in Fig. 1). The attackers withdraw when k 6 n/(n + 1), which is below the lowest curve in Fig. 1. No agent withdraws when n/(n + 1) < k < n/(n  1), which is the area between the highest and lowest curves in Fig. 1 (on both sides of the middle curve).

Fig. 5. Defender moves first: four regions in parameter space for competition and withdrawal when C1/R1 = 1. The horizontal axis shows the ratio of attacker 2’s unit attack cost and asset valuation. The vertical axis shows the ratio of the defender’s unit defense cost and asset valuation, c/r.

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Fig. 6. Defender moves first: efforts and profits as functions of the ratio of attacker 2’s unit attack cost and asset valuation, C2/R2, when c/r = C1/R1 = r = Ri = 1 for i = 1, 2.

Fig. 7. Defender moves first: efforts and profits as functions of the ratio of attacker 2’s unit attack cost and asset valuation, C2/R2, when c/r = 1.5 and C1/R1 = 0.5, r = Ri = 1 for i = 1, 2.

Fig. 5 shows the four regions in parameter space when the defender moves first and C1/R1 = 1. Compared with Fig. 2 for the simultaneous game, the region involving competition between all three agents is smaller when the defender moves first. Hence, the defender enjoys the first-mover advantage of more easily deterring one pffiffiffi of the two attackers. In Fig. 6, where c/r = C1/R1 = r = Ri = 1 for i = 1, 2, this advantage causes attacker 2 to be deterred when C 2 =R2 > 2 (instead of C2/R2 > 2 in the simultaneous game; see Fig. 3); in Fig. 7, c/r = 1.5 and C1/R1 = 0.5, r = Ri = 1 for i = 1, 2, attacker 2 is deterred when C2/R2 > 1.83 (instead of C2/R2 > 2 in the simultaneous game, see Fig. 4). 5. Two-period game when attackers move first 5.1. General analysis When the attackers move first, the defender’s first-order condition for the second period is given by the first equation in (3), while the second equation in (3) is no longer valid. This gives

P @u r ni¼1 T i ¼ 2  c ¼ 0 P @t t þ n Ti i¼1

)

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n n X ur X t¼t Ti  Ti: c i¼1 i¼1

ð21Þ

Inserting Eq. (21) into Eq. (2) and simplifying gives the first-period profits of attacker i as

Ui ¼

rffiffiffi c Ti qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ri  C i T i : r Pn T

ð22Þ

j¼1 j

Differentiating Ui with respect to Ti to determine the first-order conditions for an interior solution gives

@U i ¼ @T i

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  rffiffiffi 2 Pn T  T i j¼1 j c 3=2 Ri  C i ¼ 0 Pn r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 j¼1 j

)

vffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi13=2 rffiffiffi 0v uX uX u n u n r C i @t T i ¼ 2t Tj  2 TjA : c Ri j¼1 j¼1

ð23Þ

Summing up the Ti for the n attackers gives n X i¼1

vffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi13=2 0v rffiffiffi n u n u n X uX r X C i @u t t T i ¼ 2n Ti  2 TiA ; c i¼1 Ri i¼1 i¼1

ð24Þ

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which can be solved to yield n X

Ti ¼

i¼1

ð2n  1Þ2 c=r P 2 : n Ci 4 i¼1 Ri

ð25Þ

Inserting Eq. (25) into Eqs. (21) and (23) gives

 P  ð2n  1Þ 2 ni¼1 CRi  ð2n  1Þ cr i t¼ ; P 2 n Ci 4 i¼1 R i

 P  C ð2n  1Þ2 cr 2 nj¼1 Rj  ð2n  1Þ CRi j i Ti ¼ ; P 3 n Cj 4 j¼1 R

tþ

n X i¼1

2n  1 T i ¼ Pn C 2 i¼1 Ri

j

ð26Þ

i

and inserting the results into Eq. (2) gives

u¼

2

Pn

Ci i¼1 Ri

 ð2n  1Þ cr Pn C i 2 i¼1 R

!2 r;

i

 P 2 C ð2n  1Þ cr 2 nj¼1 Rjj  ð2n  1Þ CRii Ui ¼ Ri : P 3 n Cj 4 j¼1 R

ð27Þ

j

Using these general results for the levels of defense and attack effort at the sub-game perfect equilibrium enables us in the next proposition to specify the conditions for when an agent is deterred in games where the attackers move first. 2 Proposition 7. Conditions for deterrence in the sequential game when the attackers move first: The defender gives up its asset if cr P 2n1 Pn Cj 2 Attacker i ceases attacking if CRii P 2n3 for i ¼ 1; 2 . . . n. j¼1;j–i Rj

Pn

Ci i¼1 Ri

.

Proof. Follows from Eq. (26), the non-negativity of t and the Ti (i.e.,  Pt P 0 and Ti P 0),and the fact that 2/(2n  1) < 1/(n  1). For example, from Eq. (26), we will have t > 0 when the numerator ð2n  1Þ 2 ni¼1 CRi  ð2n  1Þ cr > 0. Dividing through by (2n  1) and rearranging i Pn C i 2 terms, we get cr < 2n1 i¼1 Ri . When this inequality does not hold, then we will not have an interior solution for t, so we must have t = 0; i.e., the defender gives up its asset. h Comparison with Propositions 1 and 4 reveals that the defender gives up its asset more easily when the attackers move first than in the simultaneous game or when the defender moves first. Thus, Proposition 7 illustrates the possibility of a second-mover disadvantage for the defender. Observe that each attacker’s decision to withdraw depends not at all on the defender characteristics, but only on the characteristics of the other attackers. This means that attackers may be willing to attack as first movers even when the defender is extremely advantaged (in contrast to their strategies in the simultaneous game and when the defender moves first). 5.2. Special cases when attackers move first Beginning with the case of a single attacker, inserting n = 1 and C/R = c/(k r) into Eqs. (26) and (27) gives

t¼

ð2  kÞ ; 4c=kr

T¼

k ; 4c=kr

u¼

 2 2k r; 2

U¼

k R: 4

ð28Þ

When k > 1, the (single) attacker invests more than the defender does, and earns a higher profit. In particular, when k is at least 2, the defender is so disadvantaged that it ceases defending, gives up its asset, and earns zero profit. This stands in contrast to the results of the simultaneous game in Eq. (6), in which neither the defender nor the attacker ever withdraws from the game. The attacker, on the other hand, always attacks. This is the same result as for the simultaneous game, but stands in contrast to the twoperiod game where the defender moves first and the attacker is sufficiently disadvantaged. For example, observe in Eq. (19) (for the sequential game where the defender moves first) that the attacker does not attack and earns zero profit when disadvantaged with k 6 1/2. For n attackers, we let Ci/Ri = C/R = c/(k r). In this case, Eqs. (26) and (27) give

t¼

ð2n  1Þ½2n  kð2n  1Þ ; 4n2 c=kr

Ti ¼

ð2n  1Þ2 4n3 c=k

2

r

;

u¼

 2 2n  kð2n  1Þ r; 2n

Ui ¼

ð2n  1Þ Ri : 4n3 =k

ð29Þ

Just as for the results of the simultaneous game shown in Eq. (7), but unlike the results for the two-period where the defender moves first in Eq. (20), the attackers never withdraw from attacking, regardless of how many attackers there are. Using these more specific results for the levels of defense and attack effort at equilibrium for the case of n homogeneous attackers where the attackers move first enables us in the next proposition to compare the defender’s and attackers’ investments and profits, and how the agents’ profits depend on the number of attackers. Proposition 8. Equilibrium solutions for the sequential game when the attackers move first: When C/R = c/r and the attackers move first, they invest (2n  1)/n times more than the defender, Ti/t = (2n  1)/n, and earn (2n  1)/n times more profit, Ui/u = (2n  1)/n. The defender’s profit is inversely proportional to n2, while the profit of any given attacker decreases more slowly than 1/n2 when there are multiple attackers (since 2 n  1 > n when n > 1).

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Fig. 8. Attackers move first: four regions in parameter space for competition and withdrawal when C1/R1 = 1. The horizontal axis shows the ratio of attacker 2’s unit attack cost and asset valuation. The vertical axis shows the ratio of the defender’s unit defense cost and asset valuation, c/r.

2

ð2n1Þ 2n1 Proof. Follows from Eq. (29) when k = 1. For example, substituting k = 1 into Eq. (29) gives t ¼ 4n 2 c=r and T i ¼ 4n3 c=r , leading to Ti/ t = (2n  1)/n. h

This is in contrast to the results of the simultaneous game given in Eq. (7) when k = 1, where when C/R = c/r and R = r, we have Ti/t = 1 and Ui/u = 1, and both Ui and u are inversely proportional to (n + 1)2. Thus, moving first confers an advantage on the attackers, at least when there is more than one attacker. However, the profit earned by any one attacker decreases as the number of attackers increases. The defender withdraws from investing when k P 2n/(2n  1), which may be only slightly greater than one when the number of attackers n is large. This scenario is maximally detrimental for the defender, since the defender is disadvantaged both by k and by being the second mover. In particular, if k = 2, then the defender will always withdraw from investing, since the number of attackers always satisfies n P 1. This is in contrast to the results of the simultaneous game in Eq. (7) and the two-period game with the defender moving first in Eq. (20). In both of those cases, the defender ceases investing only when facing at least two attackers. Also, the attackers never withdraw in this case. This stands in contrast to the result in Eq. (20) (for the sequential game where the defender moves first), where the attackers withdraw from attacking when k 6 n/(n + 1). Based on the equilibrium levels of defense and attack effort for n homogeneous attackers in games where the attackers move first, the next proposition specifies how disadvantaged the defender and the attackers must be (in terms of c/r compared with C/R) in order to be deterred. Proposition 9. Conditions for deterrence in the sequential game with homogeneous attackers when the attackers move first: Assume that all attackers have equivalent characteristics, Ci/Ri = C/R. If the defender is disadvantaged with a ratio c/r that is more than 2n/(2n  1) times as high as C/R for the n attackers, then the defender ceases investing, gives up the asset, and earns zero profit, and the attackers never withdraw. Proof. Follows from Eq. (29), and the fact that t and Ti must be non-negative; i.e., t P 0 and Ti P 0. For example, setting t > 0 in Eq. (29) gives (2n  1) [2n  k (2n  1)] > 0. Dividing through by (2n  1) and rearranging terms gives k < 2n/(2n  1). When that inequality does not hold, then we cannot have an interior solution for t, and hence must have t = 0. h The defender withdraws when k P 2n/(2n  1), which is the area above the middle curve in Fig. 1. Fig. 8 shows the four regions in parameter space when the attackers move first and C1/R1 = 1. Compared with Fig. 2, the region where the defender gives up its asset is larger. Hence, the attackers enjoy the first-mover advantage of more easily deterring the defender. In Fig. 9, where c/r = 1, this advantage causes the defender to give up its asset when C2/R2 < 0.5 (as opposed to never giving up in the simultaneous game), and in Fig. 10, the defender gives up its asset for all values of C2/R2 (instead of only when C2/R2 < 1 in the simultaneous game). That n equivalent attackers in Proposition 9 never withdraw when moving first stands in contrast to the defender (facing n equivalent attackers) withdrawing when moving first in the case when k P n/(n  1) in Proposition 6. The asymmetry comes from the fact that there are at least n = 2 attackers, but only one defender. Note also that the inequality k P n/(n  1) is more easily satisfied as n increases, so that the defender is more likely to withdraw when facing a larger number of attackers. 6. Comparing the three games We first consider the case of a single attacker, n = 1. Inserting k = 1 in Eqs. (6), (19) and (28) reveals that the solutions for the three games are equivalent when the attacker and the defender are equally advantaged; i.e., when C/R = c/r. Such equivalence no longer holds when k differs from one, nor does it hold when k = 1 but there is more than one attacker. Fig. 11 illustrates the conditions under which the attacker and the defender withdraw in the three games for the case of a single attacker, as a function of k = (c/r)/(C/R). The defender can deter the attacker altogether when moving first, if it is at least twice as advantaged as the attacker. Analogously, the attacker can deter the defender altogether when moving first, if it is at least twice as advantaged as the defender (in terms of unit cost divided by asset value). In the intermediate range, where neither the attacker nor the defender is twice as advantaged as the other agent, neither agent withdraws. Let us then consider n attackers. We begin by comparing Proposition 7 with Propositions 1 and 4. First, the difference 1/(n  1)  2/ (2n  1) = 1/(2n2  3n + 1) is positive and decreasing in n. Hence, there is a range of values of c/r where the defender gives up its asset when

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Fig. 9. Attackers move first: efforts and profits as functions of the ratio of attacker 2’s unit attack cost and asset valuation, C2/R2, when c/r = C1/R1 = r = Ri = 1 for i = 1, 2.

Fig. 10. Attackers move first: efforts and profits as functions of the ratio of attacker 2’s unit attack cost and asset valuation, C2/R2, when c/r = 1.5 and C1/R1 = 0.5, r = Ri = 1 for i = 1, 2.

Fig. 11. Agent withdrawal in the three games with one attacker, as a function of the ratio of the defender’s unit defense cost and asset valuation, c/r, divided by the ratio of the attackers’ unit attack cost and asset valuation, C/R.

moving second or simultaneously with the attackers, but not otherwise. Second, the requirement for an attacker to cease attacking when moving first is independent of c/r, and depends only on the unit cost of attack and the value of the asset to the attacker. Hence, attackers attack when moving first even when the defender is advantaged with an extremely low ratio of unit defense cost to asset value. We formulate this as follows. Pn C i c Pn C i 2 1 Proposition 10. (a) When 2n1 i¼1 Ri 6 r 6 n1 i¼1 Ri , the defender gives up its asset when moving second, but not otherwise. (b) When c/r is below the appropriate thresholds in Propositions 4 and 7, respectively, one or more attackers will cease attacking in the simultaneous game and when moving second, but not when moving first. Proof. Follows from comparing Propositions 1, 4, and 7. h Fig. 1 shows the conditions under which the various agents withdraw in each of the three games. 7. Conclusions In the model considered here, one defender defends an asset, and multiple heterogeneous attackers attack it. We show how the defense and attack efforts and the profits of the attackers depend on the unit costs of defense and attack, and on the agents’ valuations of the asset. Three scenarios are considered: first, a simultaneous game in which all agents move simultaneously; second, a two-period game in which the defender moves first; and third, a two-period game in which the attackers move first.

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Advantages of analyzing two-period games are that these enhance our understanding and can be more realistic than simultaneous games, at least when one type of agent clearly chooses its strategy before the other. In particular, games where the defender moves first are frequently encountered in practice, since defenses often involve capital assets that are readily observable, and cannot easily be reallocated in response to an observed attack. Infrastructures and their defense are usually built up over time, which exemplifies the case in which the defender moves first. A typical form of the solution is for the defender to invest sufficiently to deter the weaker attackers, while limiting the damage from those attackers that are not deterred. By contrast, attackers could be viewed as moving first if, for example, the defender gains intelligence about the attackers’ resource commitments before deciding on a defensive strategy. Throughout, we specify how the unit costs, asset evaluations, and number of attackers influence the conditions under which an agent withdraws from the contest, as well as the agents’ investments and profits. Agents with high unit costs of attack or defense (divided by the asset valuation) withdraw, but two agents always remain in the simultaneous game-either one attacker and one defender, or two attackers. As a special case, we show that if all attackers have equivalent characteristics, the defender withdraws when sufficiently disadvantaged, but the attackers always attack. The reason for this result is that one attacker always remains in the game if the defender withdraws; because all attackers have equivalent characteristics, all attackers remain in the game, since there is no mechanism for distinguishing between them. For the sequential game, it is possible for only one agent to remain in the contest, and enjoy the entire value of the asset; for the simultaneous game, at least two agents always remain. We also show that the defender gives up its asset more easily when the attackers move first than in the simultaneous game or when the defender moves first. Interestingly, the results show that the presence of one particularly strong attacker can cause other attackers to withdraw from the contest, if the advantaged attacker appropriates so much of the defender’s asset that it is no longer sufficiently attractive to interest other attackers. Thus, a nation or company under severe attack by one adversary may no longer be an attractive target to others. In such cases, the defender can focus exclusively on the contest with the strong attacker. As long as the other attackers remain sufficiently disadvantaged to refrain from attacking, the optimal allocation of defensive resources no longer depends on the characteristics of those attackers who withdraw. Future research should test whether this result is due specifically to the rent-seeking nature of the problem formulation, and whether results would differ if the defender’s asset is a good that has to be produced, e.g. a public good that can be shared among multiple attackers, allowing for free riding. Moreover, a sufficiently advantaged defender will be able to deter all attackers by moving first, while at least one attacker will always remain in the contest in a simultaneous game, or when the attackers move first. Thus, even a highly advantaged defender will continue to suffer from attacks if not playing as the first mover. This demonstrates the importance of proactive defense, especially for strong defenders. 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