Steve Alpern, Thomas Lidbetter, Alec Morton, and Katerina Papadaki Patrolling a pipeline

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1 Steve Alpen, Thomas Lidbette, Alec Moton, and Kateina Papadaki Patolling a pipeline Book section Oiginal citation: Oiginally published in Alpen, Steven, Lidbette, Thomas, Moton, Alec and Papadaki, Kateina (016) Patolling a pipeline. In: Quanyan, Zhu, Alpcan, Tansu, Panaousis, Emmanouil, Tambe, Milind and Casey, William, (eds.) Decision and Game Theoy fo Secuity: 7th Intenational Confeence, GameSec 016, New Yok, NY, USA, Novembe -4, 016, Poceedings. Lectue Notes in Compute Science, Spinge Intenational Publishing, London, UK, pp ISBN Spinge Intenational Publishing AG This vesion available at: Available in LSE Reseach Online: Decembe 016 LSE has developed LSE Reseach Online so that uses may access eseach output of the School. Copyight and Moal Rights fo the papes on this site ae etained by the individual authos and/o othe copyight ownes. Uses may download and/o pint one copy of any aticle(s) in LSE Reseach Online to facilitate thei pivate study o fo non-commecial eseach. You may not engage in futhe distibution of the mateial o use it fo any pofit-making activities o any commecial gain. You may feely distibute the URL ( of the LSE Reseach Online website. This document is the autho s submitted vesion of the book section. Thee may be diffeences between this vesion and the published vesion. You ae advised to consult the publishe s vesion if you wish to cite fom it.

2 Patolling a Pipeline Steve Alpen 1, Thomas Lidbette, Alec Moton 3, and Kateina Papadaki 1 Wawick Business School, Univesity of Wawick, Coventy, CV4 7AL, United Kingdom Depament of Mathematics, London School of Economics, Houghton Steet, London, WCA AE, United Kingdom 3 Depatment of Management Science, Univesity of Stathclyde, 199 Cathedal Steet, Glasgow, G4 0QU, United Kingdom steve.alpen@wbs.ac.uk {t..lidbette,k.p.papadaki}@lse.ac.uk alec.moton@stath.ac.uk Abstact. A pipeline netwok can potentially be attacked at any point and at any time, but such an attack takes a known length of time. To counte this, a Patolle moves aound the netwok at unit speed, hoping to intecept the attack while it is being caied out. This is a zeo sum game between the mobile Patolle and the Attacke, which we analyze and solve in cetain cases. Keywods: patolling, zeo-sum game, netwoks 1 Intoduction A game theoetic model of patolling a gaph was ecently intoduced in [1], in which an Attacke chooses a node of a gaph to attack at a paticula time and a Patolle chooses a walk on the nodes of the gaph. The game takes place in discete time and the attack lasts a fixed numbe of time units. Fo given mixed stategies of the playes, the payoff of the game is the pobability that the attack is intecepted by the Patolle: that is, the pobability that the Patolle visits the node the Attacke has chosen duing the peiod in which the attack takes place. The Patolle seeks to maximize the payoff and the Attacke to minimize it, so the game is zeo-sum. In [1], seveal geneal esults of the game ae pesented along with solutions of the game fo some paticula gaphs. This wok is extended in [5], in which line gaphs ae consideed. The game is supisingly difficult to solve on the line gaph, and the optimal policy fo the Patolle is not always, as one might expect, the stategy that oscillates to and fo between the teminal nodes. Rathe, depending on the length of time equied fo the attack to take place, it may be optimal fo the Patolle to stay aound the two ends of the line with some positive pobability. In this pape we pesent a new continuous game theoetic model of patolling, in a simila spiit to [1], but on a continuous netwok, so that the attack may

3 Patolling a Pipeline take place at any point of the netwok (not just at nodes). We also model time as being continuous, athe than discete. This is a bette model fo a situation in which a pipeline may be disupted at any point. At fist glance, this might appea to be a moe complicated game to analyze. Howeve, it tuns out that continuity simplifies mattes, and we ae able to solve the game fo Euleian netwoks (Section 3) and fo line netwoks (Section 4). The solution of the game on the line netwok is consideably easie to deive than fo the discete analogue, and we also show that the value oof the latte game conveges to that of the fome as the numbe of nodes of the gaph appoaches infinity. A game theoetical appoach to patolling poblems has been successful in eal life settings, fo example in [6] and [7]. Othe wok on game theoetic models of patolling a netwok include [] and [4]. Definition of the game We stat by defining a continuous time patolling game, whee the Patolle moves at unit speed along a netwok Q with given ac lengths, and the Attacke can attack at any point of the netwok (not just at nodes). In this section we define the game fomally and descibe each of the playes stategy spaces. The netwok Q can be viewed as a metic space, with d(x, y) denoting the ac length distance, so we can talk about the midpoint of an ac and othe metic notions. We assume that the game has an infinite time hoizon and that a Patolle pue stategy is a unit speed (Lipshitz continuous) path w : [0, ) Q, in paticula, one satisfying d (w (t), w (t )) t t, fo all t, t 0. Fo the Attacke, a pue stategy is a pai [x, I], whee x Q and I [0, ) is an inteval of length. It is sometimes useful to identify I with its midpoint y, whee I = I y = [y /, y + /]. Thus y [/, ). The payoff function, taking the Patolle as the maximize, is given by P (w, {x, y}) = { 1 if w (t) = x fo some t Iy, 0 othewise. (1) Hence the value, if it exists, is the pobability that the attack is intecepted. Note that in this scenaio the pue stategies available to both playes ae uncountably infinite, so the von Neuman minimax theoem no longe applies. Futhemoe, the payoff function is not continuous (in eithe vaiable), so minimax theoems using that popety also don t apply. Fo example, if w is the constant function x, then P (w, [x, I]) = 1, howeve an abitaily small petubation of w o x can have P (w, [x, I]) = 0. Howeve, in the examples we study in this pape we show that the value exists by explicitly giving optimal stategies fo the playes.

4 Patolling a Pipeline 3 3 Geneal esults We stat by giving uppe and lowe bounds fo the value of the game fo geneal netwoks. Fist, we define the unifom attack stategy. Definition 1. The unifom attack stategy chooses to attack in the time inteval [0, ] at a unifomly andom point on Q. Moe pecisely, the pobability the attack takes place in a egion A of the netwok is popotional to the total length of A. We use the unifom attack stategy to deduce a simple lowe bound on the value of the game. We denote the total length of Q by µ. Lemma 1. The unifom attack stategy guaantees that the pobability P of inteception is no moe than /µ. We also define a natual stategy fo the Patolle. Recall that a Chinese Postman Tou (CPT) of the netwok Q is a minimum length tou that contains evey point of Q. We denote the length of a CPT by µ. It is well known [3] that thee ae polynomial time algoithms (polynomial in the numbe of nodes of the netwok) that calculate µ. It is easy to see that µ µ, since doubling each ac of the netwok esults in a new netwok whose nodes all have even degee and theefoe contains an Euleian tou. Definition. Fix a CPT, w : [0, ) Q that epeats with peiod µ. The unifom CPT stategy w : [0, ) Q fo the Patolle is defined by w(t) = w(t + T ), whee T is chosen unifomly at andom fom the inteval [0, µ]. In othe wods, the Patolle chooses to stat the CPT at a andom point along it. This stategy gives an uppe bound on the value of the game. Lemma. The unifom CPT stategy guaantees that the pobability P of inteception is at least / µ. Lemmas 1 and give uppe and lowe bounds on the value of the game. If the netwok is Euleian (that is, the netwok contains a tou that does not epeat any acs) then µ = µ and Lemmas 1 and imply that the value of the game is /µ = / µ. We sum this up in the theoem below. Theoem 1. The value V of the game satisfies µ V µ. If the netwok is Euleian then both bounds ae tight, V = /µ = / µ, the unifom attack stategy is optimal fo the Attacke and the unifom CPT stategy is optimal fo the Patolle.

5 4 Patolling a Pipeline Witing P fo the pobability the unifom CPT stategy intecepts the attack, we note that since it is tue fo any netwok that µ µ, we have V ( ) µ µ = P. This shows that the value of the game is no moe than twice the inteception pobability guaanteed by the unifom CPT stategy. 4 Solution on the line netwok We now give a complete solution to the game on a line of unit length, that is the closed unit inteval [0, 1]. The Attacke picks a point x [0, 1] and an inteval I [0, ) of length. The Patolle picks a unit speed walk w on the unit inteval, w : R + [0, 1]. The attack is intecepted if w(t) = x, fo some t I. We assume 0, othewise the Patolle can always intecept the attacks by oscillating between the endpoints of the unit inteval. 4.1 The Case > 1 We begin by assuming the attack inteval is elatively lage compaed to the size of the line, in paticula when > 1. We shall see that the following stategies ae optimal. Definition 3. Let the diametical Attacke stategy be defined as follows: choose y unifomly in [0, 1] and attack equipobably at one of the endpoints x = 0 o 1 duing the time inteval I = [y, y + ]. Fo the Patolle, the oscillation stategy is defined as the stategy whee the Patolle andomly picks a point x on the unit inteval and a andom diection and oscillates fom one endpoint to the othe. We note that the oscillation stategy is simply the unifom CPT stategy as defined in Definition, and thus ensues a pobability P / µ = / of inteception, by Lemma. We can show that the diametical stategy ensues the attack will not be intecepted with pobability any geate than /. Lemma 3. If 1 and the Attacke adopts the diametical stategy then fo any path w the attack is intecepted with pobability P /. We have the following coollay: Theoem. The diametic Attacke stategy and the oscillation stategy ae optimal stategies and give value V = /. Poof. This follows diectly fom Lemma and Lemma 3.

6 Patolling a Pipeline 5 4. The Case 1 Now we conside the case of 1. In this case is small compaed to 1 (the size of the unit inteval), thus the Patolle stays at the end with some pobability and oscillates between the endpoints of the unit inteval with the emaining pobability. Let q be the quotient and ρ the emainde when divides 1. Thus 1 = q +ρ, whee q is an intege and 0 ρ <. Let k = + ρ. We fist define the Attacke stategies. Definition 4. Conside the following Attacke stategy, which we call -attack stategy, that is pefomed at a andom point in time, hee we stat it at time 0: 1. Attack at points E = {0,,,..., (q 1), 1}, stating attacks equipobably between times [0, ], each with total pobability. We call these the extenal attacks.. Attack at the midpoint of (q 1) and 1, which is the point 1 +ρ = 1, stating the attack equipobably between times [ ρ, +ρ ] with total pobability. We call this the intenal attack. ρ The attacks ae shown in Figue 1. The hoizontal axis is time and the vetical axis is the unit inteval. Fig. 1. The -attack stategy is shown. The stating points of the attacks ae shown in ed. Let f(t) be the pobability of inteception at an extenal attack point if the Patolle is pesent thee at time t. Let g(t) be this pobability fo the intenal

7 6 Patolling a Pipeline Fig.. The pobability of inteception at each point in time t is shown both fo extenal attacks, f(t), and fo intenal attacks, g(t), fo the -attack stategy. attack point. These pobability functions fo the -attack stategy ae shown in Figue. The functions f and g ae as follows: g(t) = f(t) = t t, t [0, ], t [, ] 0, t [, ) 0, t [ 0, ρ ], t [ ρ, +ρ ] ρ, t [ ] +ρ +ρ, t, t [ +ρ ] ρ, 0, t [ ρ, ) t ρ ρ We now define some Patolle stategies. Definition 5. Conside the Patolle stategies whee the Patolle plays a mixtue of oscillations of the inteval [0, 1] (the big oscillations) with pobability 1, and oscillations of the intevals [ ] [ 0, and 1, 1] (the small oscillations) with pobability of () on each. We call this mixed-oscillation stategy. () (3) The mixed oscillation stategy is shown in Figue 3. Note that the small oscillations have peiod and thus intecept all attacks in the espective intevals. By attacking at 0 o 1 the Attacke secues () + 1 =, since the big oscillation intecepts attacks at the endpoints with pobability. Any attacks in the open intevals ( ( 0, ) and 1, 1), ae dominated by attacks at endpoints.

8 Patolling a Pipeline 7 Attacking in [, 1 ] secues an inteception pobability of 1 =, since at points in [, 1 ], the big oscillation in each of its peiod time intevals of length, it intecepts attacks that stat at two time intevals each of length. Hence, V. Fig. 3. The mixed oscillation stategy, whee the hoizontal axis is time and the vetical axis is the unit inteval. Theoem 3. If 1, then the -attack stategy and the mixed-oscillation stategy ae optimal and the value of the game is V =. 4.3 Relation to Discete Patolling Game The discete analogue of ou game, intoduced in [1] was solved fo line gaphs in [5]. It is inteesting (and eassuing) to find that the value of the discete game conveges to the value of the continuous game as the numbe of nodes tends to infinity. We biefly descibe the set-up of the discete game. The game is played on a line gaph with n nodes in a discete time hoizon T = {1,,..., T }. The Attacke chooses an attack node at which to attack and a set of m successive time peiods in T, which is when the attack takes place. The Patolle chooses a walk on the gaph. As in the continuous case, the payoff of the game, which the Attacke seeks to minimize and the Patolle to maximize, is the pobability that the Patolle visits the attack node while the attack is taking place. The value of the game depends on the elationship between n and m, and the solution divides into 5 cases (see Theoem 6 of [5]). We ae inteested in fixing the atio = m/n and letting n tend to infinity, theefoe the solution of two of the cases of the game fom [1] ae ielevant: in paticula the case when m =, and the case when n = m + 1 o n = m +. The case n < (m + )/ (coesponding to the case in the continous case) is also uninteesting, since then the value is 1. Theefoe we ae left with two cases, whose solutions we summaize below.

9 8 Patolling a Pipeline Theoem 4 (Fom Theoem 6 of [5]). The value V of the discete patolling game on the line is 1. V = m/(n ) if (m + 1)/ n m + 1, and. V = m/(n + m 1) if n m + 3, o n = m + and m 3 is odd. We now conside the behaviou of the value of the discete game as n, assuming that the atio = m/n is fixed. In the fist case of Theoem 4, as n, the condition (m + 1)/ n m + becomes 1 and we have V = m n = /n, as n. This coesponds to the solution of the continuous game as given in Theoem. In the second case of Theoem 4, as n, the condition on m becomes 1 and we have V = m n + m 1 = 1 + 1/n 1 +, as n. Again, this coesponds to the solution of the continuous game as given in Theoem 3. 5 Conclusion We have intoduced a new game theoetic model of patolling a continuous netwok in continuous time, analagous to the discete patolling game intoduced in [1]. We have given geneal bounds on the value of the game and solved it in the case that the netwok is Euleian o if it is a line. We ae optimistic that ou esults on the line netwok can be extended to a lage class of netwoks, such as stas o tees, and we conjectue that the value of the game is / µ fo any tee netwok with diamete D such that D µ, whee µ is the length of a CPT of the netwok. Refeences 1. Alpen, S., Moton, A., Papadaki, K.: Patolling Games, Ope. Res. 59(5), (011). Basilico, N., Gatti, N., Amigoni, F.: Patolling secuity games: Definition and algoithms fo solving lage instances with single patolle and single intude, Atif. Intell., 184:78 13 (01) 3. Edmonds J., Johnson E.L.: Matching, Eule tous and the Chinese postman. Math. Pogam. 5(1), (1973) 4. Lin, K.Y., Atkinson, M.P., Chung, T.H., Glazebook, K.D.: A gaph patol poblem with andom attack times, Ope. Res. 61(3): (013) 5. Papadaki, K., Alpen, S., Lidbette, T., Moton, A., Patolling a Bode: Ope. Res. (in pess) (016)

10 Patolling a Pipeline 9 6. Pita, J., Jain, M., Maecki, J., Odóñez, F., Potway, C., Tambe, M., Westen, C., Pauchui, P., Kaus, S.: Deployed ARMOR potection: the application of a game theoetic model fo secuity at the Los Angeles Intenational Aipot. In: Poceedings of the 7th intenational joint confeence on Autonomous agents and multiagent systems: industial tack, pp Intenational Foundation fo Autonomous Agents and Multiagent Systems (008) 7. Yang, R., Fod, B., Tambe, M., Lemieux, A.: Adaptive esouce allocation fo wildlife potection against illegal poaches. In: Poceedings of the 014 intenational confeence on Autonomous agents and multi-agent systems, pp Intenational Foundation fo Autonomous Agents and Multiagent Systems (014) Appendix: Omitted poofs Poof of Lemma 1 The attack must be taking place duing the time inteval [0, ]. Let A be the set of points that the Patolle intecepts in this time inteval. Then clealy A must have length no geate than and so the pobability the attack takes place at a point in A is /µ. It follows that P /µ. Poof of Lemma Suppose the attack stats at time t 0 at some point x Q. Then the attack is cetainly intecepted if w is at x at time t 0. Let t x [0, µ] be such that w(t 0 + t x ) = x, so that the attack is intecepted by w if T = t x. Let A be the set of times t [0, µ] such that t x t t x o t t x + µ, so if T A, then the attack is intecepted by w. But the measue of A is, so the pobability that T is in A is / µ and hence P / µ. Poof of Lemma 3 Take a Patolle path w. We can assume that w stats at an endpoint, othewise it is weakly dominated by a stategy that does. To see this, suppose the Patolle stats at an inteio point befoe taveling diectly to an endpoint, aiving thee at time t < 1. Now conside the Patolle stategy that is the same but in the time inteval [0, t] the Patolle emains at the endpoint. Then clealy the second stategy intecepts the same set of attacks as the fist one. Without loss of genealization we assume w stats at x = 0. We only need to conside the path in the time inteval [0, 1 + ], afte which time the attack has been completed with pobability 1. Since < the walk cannot go between the two ends moe than twice, so thee ae thee possibilities. The fist is that w stays at x = 0 fo the whole time, in which case the pobability the attack is intecepted is P = 1/ /. The second possibility is that w stays at x = 0 fo time t 1, then goes to x = 1 and stays thee fo time t. We can assume it takes the Patolle time 1 to go between the endpoints since any path taking longe than that would be dominated, so t 1 + t =. The attack is intecepted at x = 0 if it stats sometime duing [0, t 1 ], which has pobability (1/)t 1. It is intecepted at x = 1 if it ends sometimes duing [1 + t, 1 ], which has pobability (1/)t. Hence P = (1/)(t 1 + t ) = /. The final possibility is that w stays at x = 0 fo time t 1, then goes diectly to x = 1 fo time t, then goes diectly back to x = 0 fo time t 3, in which case

11 10 Patolling a Pipeline we must have t 1 + t + t 3 = 1. This time the attack is intecepted at x = 0 in the case of eithe of the two mutually exclusive events that it stats in [0, t 1 ] o ends in [1 + t 3, 1 ], which have total pobability (1/)(t 1 + t 3 ). If the attack takes place at x = 1, it must be taking place duing the whole of the time inteval [1, ]. But w must each x = 1 sometime duing this time inteval, since it must have time to tavel fom x = 0 to x = 1 and back again, and hence intecepts the attack with pobability 1. So the oveall pobability the attack is intecepted is (1/)(t 1 + t 3 ) + 1/ (1/)(t 1 + t + t 3 ) + 1/ = /. Poof of Theoem 3 We aleady showed that /(1 + ) is a lowe bound fo the value and now we show that it is also an uppe bound. Now, suppose that the Attacke plays the -attack stategy. The Patolle could: 1. Stay at any attack point but will not win with pobability geate than.. Tavel between consecutive extenal attacks and if possible ty to each the intenal attack: Suppose the Patolle is at point 0 up to time t: If t [0, ] and then leaves fo point, she will each point at times in the ange [, ]. This gives total inteception pobability f(t) + f(t + ) = t + (t+) =. Note that if the Patolle continues to the next attack along the unit inteval, if it is the intenal attack she will each it at times geate than + +ρ = ρ, when the intenal attack has been completed, and if it is an extenal attack she will each it at time geate than, whee all extenal attacks have been completed. If t [, ] then all attacks at point 0 have been intecepted but the Patolle aives at point afte all attacks. have been completed, which gives inteception pobability of 3. Tavel between last two extenal attacks, cossing intenal attack in the middle (this is the same as doing a oundtip fom one of the last extenal attacks to the intenal attack and back): Suppose the Patolle leaves point (q 1) at time t, towad the intenal attack point and the last extenal attack point 1: If t [0, ρ], she will each the intenal attack point at times [ +ρ, ρ + +ρ ] [ = +ρ ] +ρ, s, and she will each the extenal attack at point 1 at times [ + ρ, ]. This sums to a pobability of f(t) + g ( t + +ρ ) +f(t++ρ) = t + ρ + (t++ρ) =. If t [ ρ, ], she will each the intenal attack point at times [ +ρ ] ρ,, and the extenal attack point 1 at times geate than. This sums to a pobability of f(t) + g ( t + +ρ ) ρ +ρ = t (t+ ) + =. Finally, if t [, ], the Patolle will intecept all attacks at point (q 1) and will not make it in time fo the intenal attack no the attack at point 1, this gives the desied pobability.