aa r X i v : . [ m a t h . O C ] M a r Continuous Patrolling and Hiding Games
Tristan Garrec ∗ November 8, 2018
Abstract
We present two zero-sum games modeling situations where oneplayer attacks (or hides in) a finite dimensional nonempty compactset, and the other tries to prevent the attack (or find him). The firstgame, called patrolling game, corresponds to a dynamic formulationof this situation in the sense that the attacker chooses a time and apoint to attack and the patroller chooses a continuous trajectory tomaximize the probability of finding the attack point in a given time.Whereas the second game, called hiding game, corresponds to a staticformulation since both the searcher and the hider choose simultane-ously a point and the searcher maximizes the probability of being atdistance less than a given threshold of the hider.
To ensure the security of vulnerable facilities, a planner may deploy eitherdynamic or static security devices. The dynamic case includes, but is notlimited to: soldiers or police officers patrolling the streets of a city, robotspatrolling a shopping mall, drones flying above a forest to detect fires, ornaval radar systems signaling the detection of an enemy ship. On the otherhand, security guards positioned in the rooms of a museum, security camerasscrutinizing subway corridors, or motion detectors placed in a house are someexamples of static security devices. Note that these real-world situationsinclude both human and electronic agents. The paradigm we adopt is the oneof an adversarial threat, hence we propose a game theoretical approach tothese security problems. Some game theoretic security systems are already inuse, for example in the Los Angeles international airport (Pita et al. (2008))and in some ports of the United States (Shieh et al. (2012)).Motivated by the examples given above, we study two zero-sum gamesin which a player (the patroller or searcher) aims to detect another player(the attacker or hider).
Patrolling games model dynamic security devices. ∗ Toulouse School of Economics, Universit´e Toulouse 1 Capitole. E-mail address:[email protected]
1n these games, the patroller moves continuously in a search space withbounded speed. The attacker chooses a point in the search space and a timeto attack it. The attack takes a certain duration to be successful (thinkof a terrorist needing time to set off a bomb). The patroller wins if andonly if he detects the attack before it succeeds. The case of static securitydevices is modeled by hiding games , in which both the searcher and the hidersimultaneously deploy at a particular point in a search space, the searcherwins if and only if the hider lies within his detection radius. We providelinks between patrolling and hiding games, and show how patrolling gamesreduce to hiding games if the attack duration is zero, that is the patrollerhas to detect the attack at the exact time it occurs, or if the patroller isalerted of the attack point when the attack begins.
For patrolling games, we prove that the value always exists and obtain ageneral upper bound. We then study patrolling games on networks. Inparticular we compute the value as well as optimal strategies for the classof Eulerian networks. The special network composed of two nodes linked bythree parallel arcs is also examined and bounds on the value are computed.Lastly, we study patrolling games on R and obtain an asymptotic expressionfor the value as the detection radius of the patroller goes to 0.For hiding games, we first focus on a particular class of strategies forboth the searcher and the hider called ”equalizing”. These strategies havethe property that if one exists, then it is optimal for both players. Our mainresult regarding hiding games is an asymptotic formula for the value of hidinggames on a compact set with positive Lebesgue measure. A counterexamplebased on a Cantor-type set showing that this last result cannot extend tocompact sets with zero Lebesgue measure is also presented.Finally, we discuss some basic properties of monotonicity and continuityof the value function of continuous patrolling and hiding games. Continuous patrolling and hiding games belong to the literature of searchand security games, consult Hohzaki (2016) for a survey. These games havetheir source in search theory, a field of operations research whose originscan be found in the works of Koopman (1956a,b, 1957). The use of gametheory in a search and security context goes back to the famous book ofMorse and Kimball (1951).
Patrolling games were introduced by Alpern et al. (2011) in a discrete set-ting, that is the patroller visits nodes of a graph, where the attacker can2trike, at discrete times. A companion article Alpern et al. (2016a) is dedi-cated to the resolution of patrolling a discrete line. The idea of investigatinga continuous version of patrolling games is suggested in Alpern et al. (2011).In Alpern et al. (2016b), the authors solve the continuous patrolling gameplayed on the unit interval. Several papers in the field of search games havedealt with the transcription of discrete models to a continuous ones, consultRuckle and Kikuta (2000) and Ruckle (1981).Other models of games involving a patroller and an attacker can befound in Basilico et al. (2012, 2015), in which the authors design algorithmsto solve large instances of Stackelberg patrolling security games on graphs.Lin et al. (2013, 2014) use linear programming and heuristics to study alarge class of patrolling problems on graphs, with nodes having differentvalues. Zoroa et al. (2012) study a patrolling game with a mobile attackeron a perimeter.Continuous patrolling games are closely related to search games with animmobile hider, introduced in the seminal book of Isaac (1999) and devel-oped in the monographs of Gal (1980) and Alpern and Gal (2003). In thesegames, a searcher intends to minimize the time necessary to find a hider.Search games have been extensively studied, let us mention Gal (1979);Alpern et al. (2008); Dagan and Gal (2008) for search games on a network.In particular, see Bostock (1984); Pavlovic (1993), for the special networkconsisting of two nodes linked by three parallel arcs, for which the solutionis surprisingly complicated.
The first published example of a hiding game goes back to Gale and Glassey(1974). The proposers gave a solution to the problem of hiding in a discwhen the detection radius is r = 1 /
2. Later, Ruckle (1983) considered in hisbook several examples of hiding games (hiding on a sphere, hiding in a disc,among others). Computing the value of hiding games is in general a verydifficult task. Danskin (1990) improved substantially the resolution of thehiding game played on a disc, he called the cookie-cutter game. Howeverthe solution is not complete for small values of r and no progress has beenmade since then, see also Alpern et al. (2013) and Washburn (2014). Hidinggames in a discrete setting, when the search space is a graph, have also beenstudied, see Bishop and Evans (2013).Games in which the payoff is the distance between the two points se-lected by the players, introduced by Karlin (1959), have been extensivelystudied, consult Ibragimov and Satimov (2012) and references therein. Al-though these games resemble hiding games to a certain extent, the lack ofcontinuity of the payoff function in hiding games makes their analysis muchmore involved.Finally, ambush games can be seen as hiding games in which the players3elect not one, but several points in the unit interval, consult Zoroa et al.(1999); Baston and Kikuta (2004, 2009) for more details. The paper is organized as follows. In section 2 the models of patrolling andhiding games are formally presented. Section 3 is dedicated to patrollinggames, and section 4 is dedicated to hiding games. Finally in section 5 wegive some basic properties of the value function of continuous patrolling andhiding games. The proofs that are not included in the body of the paperare postponed to the appendix section 6.
In all the article, R n is endowed with a norm denoted k · k , which induces ametric d . For all x ∈ R n and r >
0, the closed ball of center x with radius r is denoted B r ( x ) = { y ∈ R n | k x − y k ≤ r } . For all Lebesgue measurableset B ⊂ R n , λ ( B ) denotes its Lebesgue measure. Finally, λ ( B r ) denotes theLebesgue measure of any ball of radius r .Let X be a topological space, the set of Borel probability measures on X is denoted ∆( X ), and the set of probability measures on Y with finitesupport is denoted ∆ f ( Y ). In a patrolling game two players, an attacker and a patroller, act on a set Q called the search space, which is assumed to be a nonempty compactsubset of R n . An example of this could be a metric network, as defined insection 3.3 below. The attacker chooses an attack point y in Q and a timeto attack t in R + . The patroller walks continuously in Q with speed at most1. When the attack occurs at time t and point y , the patroller has a timelimit m ∈ R + to be at distance at most r ∈ R + of the attack point y . In thiscase he detects the attack and wins, and otherwise he does not. Thus, m represents the time needed for an attack to be successful, and r representsthe detection radius of the patroller.A patrolling game is thus a zero-sum game given by a triplet ( Q, m, r ).The attacker’s set of pure strategies is A = Q × R + . An element of A iscalled an attack. The patroller’s set of pure strategies is W = { w : R + → Q | w is 1-Lipschitz continuous } . An element of W is called a walk. W isendowed with the topology of compact convergence (consult the proof ofproposition 1 in the appendix section 6 and Munkres (2000) for details).4he payoff to the patroller is given by g m,r ( w, ( y, t )) = ( d ( y, w ([ t, t + m ])) ≤ r , where w ([ t, t + m ]) = { w ( τ ) | τ ∈ [ t, t + m ] } . In a hiding game two players, a searcher and a hider, act on a set Q , whichis again assumed to be a nonempty compact subset of R n . An example ofthis could be the unit interval, as considered in example 5, or a Cantor-typeset, as in example 4. Both players choose a point in Q . The searcher has adetection radius r ∈ R + . He finds the hider if and only if the two points areat distance at most r .Hence, a hiding game is a zero-sum game given by a couple H = ( Q, r ).The set of pure strategies of both players, the searcher and hider, is Q . Thepayoff to the searcher is given by h r ( x, y ) = ( k x − y k ≤ r . Hiding games can be interpreted as two possible variants of patrolling games.In the first variant, hiding games are considered as a particular class ofpatrolling games in which the attack duration m is taken equal to 0. Indeed,consider a hiding game H = ( Q, r ) and a patrolling P = ( Q, , r ). In P , forall w ∈ W and ( y, t ) ∈ A the payoff to the patroller is g ,r ( w, ( y, t )) = ( k w ( t ) − y k ≤ r . A strategy x ∈ Q of the searcher in the hiding game H is mapped in thepatrolling game P to the constant strategy w ∈ W equal to x . Similarly,a strategy y ∈ Q of the hider in H is mapped to the strategy ( y, ∈ A in P . Any quantity guaranteed by the searcher in H is thus guaranteed bythe patroller in P . Conversely, any quantity guaranteed by the hider in H is guaranteed by the attacker in P . Thus, since H and P have a value (seepropositions 1 and 3), the values of these two games are the same.The second interpretation is as follows. Alpern et al. (2011) suggestthe study of patrolling games in which the patroller may be informed ofthe presence of the attacker. Suppose that the patroller is informed of theattack point when the attack occurs. Suppose also the search space Q is5onvex. The detection radius r is taken equal to 0 for simplicity. The payoffof this game is g m, ( w, ( y, t )) = ( y ∈ w ([ t, t + m ])0 otherwise . This patrolling game with signal is denoted P ′ . In P ′ , if the patroller’sstrategy is to choose a point and not move until the attack, then go tothe attack point in straight line when he is alerted, the attacker is time-indifferent. In particular, the attacker has a best reply in the set of attacksoccurring at time 0. Symmetrically, if the attack occurs at time 0, thepatroller has a best reply consisting in choosing a starting point in Q andgoing directly to the attack point when he is informed of the attack.Thus, with the same mappings of strategies in the hiding game H ′ =( Q, m ) to strategies in P ′ as before, any quantity guaranteed by the searcherin H ′ is guaranteed by the patroller in P ′ . Conversely, any quantity guaran-teed by the hider in H ′ is guaranteed by the attacker in P ′ . Thus the valuesof these two games are the same. The first result is the existence of the value of a patrolling games. We denoteit V Q ( m, r ). In addition, we prove that the patroller has an optimal strategyand the attacker has an ε -optimal strategy with finite support. The fact thatthe patroller has an optimal strategy means that he can guarantee that theprobability of detecting the attack is at least V Q ( m, r ), no matter what theattacker does. Similarly, the attacker can guarantee that the probability ofbeing caught is at most V Q ( m, r ), up to ε , no matter what the patroller does.Hence, in patrolling games, the value represents the probability (up to ε ) ofthe attack being intercepted when both the patroller and the attacker play( ε -)optimally. Proposition 1.
The patrolling game ( Q, m, r ) played with mixed strategieshas a value denoted V Q ( m, r ) .Moreover the patroller has an optimal strategy and the attacker has an ε -optimal strategy with finite support, i.e., there exists µ ∈ ∆( W ) such thatfor any ( y, t ) ∈ A Z W g m,r ( w, ( y, t )) dµ ( w ) ≥ V Q ( m, r ) , and for every ε > there exists ν ∈ ∆ f ( A ) such that for any w ∈ W Z A g m,r ( w, ( y, t )) dν ( y, t ) ≤ V Q ( m, r ) + ε. .2 A general upper bound Our goal is now to obtain a general upper bound for the value of patrollinggames. As in Alpern and Gal (2003), let us introduce the maximum rate atwhich the patroller can discover new points of Q . Definition 1.
The maximum discovery rate is given by ρ = sup w ∈W ,t> λ ( w ([0 , t ]) + B r (0)) − λ ( B r ) t . The next remark shows that in R endowed with the Euclidean norm,the maximum discovery rate is 2 r , that is the sweep width of the patroller.Similarly, in R and R endowed with the Euclidean norm, the maximumdiscovery rate is respectively 1 and πr . Remark . Let (
Q, m, r ) be a patrolling game. If Q has nonempty interiorin R endowed with the Euclidean norm, then ρ = 2 r .Indeed, since Q has nonempty interior let x ∈ Q and s > B s ( x ) ⊂ Q . Define w ( t ) = x + t ! for t ∈ [0 , s ] (and arbitrarily such that w ∈ W for t > s ). Then λ ( w ([0 , s ]) + B r (0)) − λ ( B r ) s = 2 rss = 2 r, and it is clear that this is the maximum.Let us now give an upper bound for patrolling games whose search spacehave nonzero Lebesgue measure. This upper bound is rather powerful andwill be extensively used in the remaining of the paper. It is in particular theupper bound used to prove theorems 1, 2 and 3. Proposition 2.
Let Q be a search space such that λ ( Q ) > . Then V Q ( m, r ) ≤ mρ + λ ( B r ) λ ( Q ) . To prove proposition 2, we define a strategy for the attacker called uni-form. It corresponds to the strategy for which the attacker uniformly choosesan attack point in Q , and attacks this point at time 0. Intuitively, a bestreply of the patroller is to cover as much points in Q as possible betweentime 0 and time m . Definition 2.
Let Q be a search space such that λ ( Q ) >
0. The attacker’suniform strategy on Q , denoted a λ , is a random choice of the attack point a at time 0 such that for all measurable sets B ⊂ Q , a λ ( B ) = λ ( B ) λ ( Q ) . roof of proposition 2. For all w ∈ W , the payoff to the patroller when theattacker plays a λ is Z A g m,r ( w, ( y, t )) da λ ( y, t ) = λ (cid:0) w ([0 , m ]) + B r (0) (cid:1) λ ( Q ) ≤ mρ + λ ( B r ) λ ( Q ) . We now investigate the particular case of patrolling games on a network.As in a network the patroller can manage to have a nonzero probability ofwalking through the exact attack point, his detection radius r is set to 0. We follow the construction of a network of Fournier (2016). Let (
V, E, l ) bea weighted undirected graph, V is the finite set of nodes and E the finiteset of edges whose elements e ∈ E are associated to a length l ( e ) ∈ R + . Anedge e ∈ E linking the two nodes s and t is also denoted ( s, t ).We identify the elements of V with the vectors of the canonical basis of R | V | . The network generated by ( V, E ) is the set of points N = { ( s, t, α ) | α ∈ [0 ,
1] and ( s, t ) ∈ E } , where ( s, t, α ) = αs + (1 − α ) t .A network N is endowed with a natural metric d as follows. Let u and u be two points of the same edge ( s, t ). There exist α , α ∈ [0 ,
1] suchthat u = ( s, t, α ) and u = ( s, t, α ). The distance d ( u , u ) is given by d ( u , u ) = l ( s, t ) × | α − α | .If u and v are not in the same edge, consider the set of paths P ( u, v )between u and v as the set of all sequences ( u , . . . , u n ), n ∈ N ∗ such that u = u , u n = v and such that for all i ∈ { , . . . , n − } , u i and u i +1 belongto the same edge. The distance d ( u, v ) is then defined as: d ( u, v ) = inf ( u ,...,u n ) ∈ P ( u,v ) n − X i =1 d ( u i , u i +1 ) . Finally, we define the Lebesgue measure on N . Let u = ( s, t, α ) and u = ( s, t, α ), suppose α < α . The set[ u , u ] = { ( s, t, α ) | α ∈ [ α , α ] } is called an interval. An interval [ u , u ] can be isometrically identified withthe real interval [ α l ( s, t ) , α l ( s, t )]. As a subset of N can be identified witha finite union of subsets of intervals, the Lebesgue measure on N is definedas a natural extension of the Lebesgue measure on a real interval.8 .3.2 Eulerian networks For a particular class of networks called Eulerian, it is possible to computethe value and optimal strategies of the game. Note that we use for net-works a similar vocabulary to the one used in graph theory, see for exampleBondy and Murty (1982).As stated in the next definition, an Eulerian tour is a closed path in N visiting all points and having length λ ( N ). Definition 3.
Let u ∈ N and π = ( u , u , . . . , u n − , u n ) ∈ P ( u, u ). If S n − k =1 [ u k , u k +1 ] = N then π is called a tour.Moreover, if P n − k =1 λ ([ u k , u k +1 ]) = λ ( N ) , then π is called an Euleriantour.A network N is said to be Eulerian if there exists an Eulerian tour in N . Example 1.
Figure 1 and figure 2 give two examples of networks. N is anEulerian network with Eulerian tour π = ( u , u , u , u , u , u , u , u , u ).In contrast, N is not an Eulerian network. u u u u u u u Figure 1: The network N isEulerian u u u u Figure 2: The network N isnot EulerianOur objective is now to define the uniform strategy of the patroller forEulerian networks. This strategy is optimal for Eulerian networks. First,we need to define a parametrization of the network. Definition 4.
Let N be an Eulerian network. A continuous function w from [0 , λ ( N )] to N such thati) w (0) = w ( λ ( N )),ii) w is surjective.iii) ∀ t , t ∈ [0 , λ ( N )] λ ( w ([ t , t ])) = | t − t | (the speed of w is 1).is called a parametrization of N .Moreover such a w can be extended to a λ ( N )-periodic function on R which is still denoted w . 9 emma 1. Let N be an Eulerian network, then there exists a parametriza-tion of N . It is now possible to define the uniform strategy of the patroller. Theidea behind this strategy is that the patroller uniformly chooses of a startingpoint in N , and then follows a parametrization as defined in definition 4above. Definition 5.
Suppose N is an Eulerian network. Let w be a parametriza-tion of N . Denote ( w t ) t ∈ [0 ,λ ( N )] the family of λ ( N )-periodic walks suchthat w t ( · ) = w ( t + · ) . The patroller’s uniform strategy is given by theuniform choice of t ∈ [0 , λ ( N )].The next theorem is the main result on patrolling games for networks.It gives a simple expression of the value of a patrolling games played on anyEulerian network. The result relies on the fact that for such networks, thepatroller can achieve the upper bound of proposition 2 using his uniformstrategy. Theorem 1. If N is an Eulerian network, then V N ( m,
0) = min (cid:18) mλ ( N ) , (cid:19) . Moreover the attacker’s and the patroller’s uniform strategies are optimal.
For non Eulerian networks, it may be difficult to compute the value of thecorresponding patrolling game. In the next example, we compute bounds onthe value of a patrolling game played over the network with three parallelarcs, which is not Eulerian. For some values of the attack duration m , thesebounds are not tight. Example 2.
We consider again the network N represented in figure 2. Inthis example, we take l ( u , u ) = l ( u , u ) = l ( u , u ) = l ( u , u ) = 1 / l ( u , u ) = 1. Notice that λ ( N ) = 3. We compute the following bounds onthe value of ( N , m, V N ( m, = m if m ≤ ∈ (cid:20) m − m +2) , − (cid:16) − m (cid:17) (cid:21) if m ∈ h , i ∈ (cid:20) − m − m ) , − (cid:16) − m (cid:17) (cid:21) if m ∈ h , i = 1 if m ≥ . These bounds are plotted on figure 3 below.10 V N ( m,
0) 2 412 / m First case: m ≤ . Recall that by proposition 2, V N ( m, ≤ m for all m ≥ π =( u , u , u , u , u , u ) and π = ( u , u , u , u , u , u ) be two paths. π and π naturally induce two walks on [0 ,
3] at speed 1, respectively denoted w and w . For all u ∈ N \ { u , u } and all i ∈ { , } there exists a unique t iu ∈ [0 ,
3] such that w i ( t iu ) = u . Now for all u ∈ N \ { u , u } and all t ∈ R + ,define w u ( t ) = w ( t + t u ) if t ∈ (cid:2) , − t u (cid:3) w (cid:0) t − (3(2 k + 1) − t u ) (cid:1) if t ∈ (cid:0) k + 1) − t u , k + 2) − t u (cid:3) w (cid:0) t − (3(2 k + 2) − t u ) (cid:1) if t ∈ (cid:0) k + 2) − t u , k + 3) − t u (cid:3) for all k ∈ N . The walk w u starts at t u and alternates between following w and w . The walk w u is defined analogously: switch the superscripts 1and 2 in the definition above. Denote µ the uniform choice of a walk in( w iu ) i ∈{ , } u ∈N \{ u ,u } .It is not difficult to check that µ guarantees m/ µ yields a payoff of m/ y, t ) ∈ A ). Hence V N ( m,
0) = m . Second case: < m < . We detail the computation for m = 3. Thewalks w , w and w hereafter can be adapted and similar strategies can beused to derive the bounds for all m ∈ (2 , π , π and π as in figure 4, 5 and 6 respectively.That is, π = ( u , u , u , u , u , u , u , u ), π = ( u , u , u , u , u , u , u , u )and π = ( u , u , u , u , u , u , u , u , u ). Where u = ( u , u , / u =( u , u , / u = ( u , u , / u = ( u , u , / u = ( u , u , /
4) and u = ( u , u , / π , π and π naturally induce three 3-periodic walks at speed 1, de-noted respectively w , w and w . These are such that for i ∈ { , , } , w i intercepts any attack on w i ([0 , u u u u u Figure 4: The path π u u u u u u Figure 5: The path π u u u u u u Figure 6: The path π With a slight abuse of notation, for y ∈ [0 , / y the point( u , u , y ) and 1 − y the point ( u , u , − y ). By symmetry it is enoughto consider attacks occurring at y . Moreover, µ , w , w and w make thepatroller time indifferent, hence we only consider attacks at time 0. µ intercepts the attack ( y,
0) with probability 1 − − y = + y . Indeed,only the walks w u , such that u belongs to the open interval { ( u , u , α ) | α ∈ ( y, − y ) } do not intercept the attack. Finally, define e µ = ( δ w + δ w + δ w ) + µ , where δ w is the Dirac measure at w ∈ W .At any time, an attack at y ≤ is intercepted by e µ with probability115 · (cid:18)
56 + y (cid:19) ≥
315 + 45 ·
56 = 1315 . An attack at y > / e µ with probability115 · (cid:18)
56 + y (cid:19) ≥
215 + 45 (cid:18)
56 + 112 (cid:19) = 1315 . Hence V N (3 , ≥ . Define the following attack e a : choose uniformly a point in N × [0 , u , u , u , u , u , u , u , u , u , u , u ) induces a 6-periodic walk w which is a best reply for the patroller. Moreover g , ( w , e a ) = 11 /
12. Hence V N (3 , ≤ . Third case: m ≥ . The tour ( u , u , u , u , u , u , u ) induces a 4-periodic walk which guarantees 1 to the patroller. Hence V N ( m,
0) = 1 . R In this section, we are interested in patrolling games in R for a large classof search spaces called simple. To introduce this class of search spaces, wefirst need to recall the notion of bounded variation of a function f .12 efinition 6. Let a >
0. Let f : [0 , a ] → R n be a continuous function.Then the total variation of f is the quantity: T V ( f ) = sup ( n X i =1 k f ( t i ) − f ( t i − ) k (cid:12)(cid:12)(cid:12)(cid:12) n ∈ N ∗ , t < t < · · · < t n = a ) . If T V ( f ) < + ∞ , then f is said to have bounded variation.The next definition introduces a classical assumption on the boundaryof a search space in R . This is a weak assumption already made in Gal(1980) and Alpern and Gal (2003). Definition 7.
Let a >
0, let f and f be two continuous functions from[0 , a ] to R such that f ≥ f , f = f , and f and f have bounded variation.Then the nonempty compact set { ( x, t ) ∈ [0 , a ] × R | f ( x ) ≤ t ≤ f ( x ) } iscalled an elementary search space.Let Q be the finite union of elementary search spaces such that any twohave disjoint interiors. If Q is path-connected, then it is called a simplesearch space.The next theorem is the main result on patrolling games on simple searchspaces. It gives a simple asymptotic expression of the value as the detec-tion radius r goes to 0. The result relies on the fact that the patroller canuse a uniform strategy in the spirit of what has been done in the previ-ous section for Eulerian networks. This strategy yields a lower bound thatasymptotically matches the upper bound of proposition 2.As one would expect, the value goes to 0 as r goes to 0. It is interestingto note that due to the movement of the patroller the convergence is linearin r and not quadratic. Indeed, the relevant parameter is the sweep widthof the patroller and not the area of detection. Theorem 2. If Q is a simple search space endowed with the Euclideannorm, then V Q ( m, r ) ∼ rmλ ( Q ) , as r goes to . Recall that the value of a hiding game is equal to the value of a patrollinggame with time limit m equal to 0. Hiding games have a value which repre-sents the probability (up to ε ) that the searcher and the hider are at distanceless that r when they play ( ε -)optimally. Proposition 3.
The hiding game ( Q, r ) played in mixed strategies has avalue denoted V Q ( r ) . Moreover the searcher has an optimal strategy and thehider has an ε -optimal strategy with finite support. .1 Equalizing strategies We now study particular strategies called equalizing, these have been in-troduced in Bishop and Evans (2013) when the search space is a graph (seedefinition 7.3 and proposition 7.3 therein). We adapt those considerationsto our compact setting.A strategy is equalizing if the induced payoff does not depend on thestrategy of the other player. The interest of equalizing strategies lies in thefact that if such a strategy exists, then it is optimal for both players.
Definition 8.
Let Q be a search space. A strategy µ ∈ ∆( Q ) is said to beequalizing if there exists c ∈ R + such that µ ( B r ( y ) ∩ Q ) = c for all y ∈ Q . Proposition 4.
Let µ ∈ ∆( Q ) . Then µ is an equalizing strategy (withconstant payoff c ) if and only if µ is optimal for both players (and in thatcase V Q ( r ) = c ). The following game is an example of a hiding game with finite searchspace without equalizing strategies.
Example 3.
Let r = 1 and Q = { x , x , x , x , x } be the finite subset of R such that x = (0 , x = (0 , x = (1 , x = (1 ,
0) and x = (1 / , i ∈ { , . . . , } Q i = { j ∈ { , . . . , } | k x i − x j k ≤ r } . Thatis Q = { , , , } , Q = { , , } , Q = { , , } , Q = { , , , } and Q = { , , } .The game ( Q, r ) admits an equalizing strategy if and only if the followingsystem of equations admits a solution p = ( p i ) ≤ i ≤ : p i ≥ i ∈ { , . . . , } P i =1 p i = 1 P i ∈ Q p i = P i ∈ Q j p i for all j ∈ { , . . . , } . (1)It is easy to verify that this system does not admit a solution, hence thegame ( Q, r ) does not have an equalizing strategy.
The next theorem is the main result on hiding games. For any search space Q ⊂ R n with positive Lebesgue measure, it gives a simple asymptotic ex-pression of the value when the detection radius goes to 0. In this staticsetting the value is equivalent, as r gos to 0, to the ratio of the volume ofthe ball of radius r over the volume of Q . This result relies on the fact thatthe searcher has a strategy that yields a lower bound which asymptoticallymatches the upper bound of proposition 2.14 heorem 3. Let Q be a compact subset of R n . Suppose λ ( Q ) > . Then V Q ( r ) ∼ λ ( B r ) λ ( Q ) as r goes to . A consequence of theorem 3 is that for a compact set Q included in R n such that λ ( Q ) > V Q ( r ) ∼ r n λ ( B ) λ ( Q ) as r goes to 0. When λ ( Q ) = 0, it isnot always the case that V Q admits an equivalent of the form M r α , with α and M positive, as r goes to 0, as it is shown in example 4. Example 4.
Let Q ⊂ [0 ,
1] be the following Cantor-type set. Define C =[0 , n ∈ N ∗ C n = C n − ∪ (cid:16) + C n − (cid:17) . Finally, let Q = T n ∈ N C n . Q is compact and λ ( Q ) = 0.The value of the hiding game played on Q is given by the followingformula: V Q ( r ) = n if r ∈ h n , n (cid:17) , n − if r ∈ h n , n − (cid:17) , n ∈ N ∗ . Indeed, let Σ = { , } and for all n ∈ N ∗ \{ } let Σ n = Σ n − ∪ (cid:16) + Σ n − (cid:17) .For n ∈ N ∗ , consider the following strategy σ n : choose uniformly a point inΣ n , that is with probability | Σ n | = n . Let n ∈ N ∗ suppose r ∈ h n , n (cid:17) .Then for all q ∈ Q there is exactly one point s in Σ n such that | q − s | ≤ r .Hence σ n is an equalizing strategy which guarantees n to both players.Let Σ ′ = { } and for all n ∈ N ∗ \{ } let Σ ′ n = Σ ′ n − ∪ (cid:16) − Σ ′ n − (cid:17) . For n ∈ N ∗ consider the following strategy σ ′ n : choose uniformly a point in Σ ′ n ,that is with probability | Σ ′ n | = n − . Suppose now that r ∈ h n , n − (cid:17) .Then for all q ∈ Q there is exactly one point s in Σ ′ n such that | q − s | ≤ r .Hence σ ′ n is an equalizing strategy which guarantees n − to both players.In particular, for all n ∈ N ∗ V Q (cid:18) n − (cid:19) = V Q (cid:18) n (cid:19) = 12 n . Let ( r n ) n ∈ N ∗ = (cid:16) n (cid:17) n ∈ N ∗ and let α >
0. Then for all n ∈ N ∗ V Q ( r n − )( r n − ) α = 12 α (2 α − n and V Q ( r n )( r n ) α = 2 (2 α − n . Thus we havelim n → + ∞ V Q ( r n − )( r n − ) α = + ∞ if α > / √ if α = 1 /
20 if α < / n → + ∞ V Q ( r n )( r n ) α = + ∞ if α > /
21 if α = 1 /
20 if α < / . Hence r V Q ( r ) does not admit an equivalent of the form r M r α , with α and M positive numbers, as r goes to 0.15 Properties of the value function of patrollingand hiding games
In this section we give some elementary properties of the function V Q forpatrolling and hiding games. Proposition 5.
Let Q be a search space. The function V Q : R + × R + → [0 , m, r ) V Q ( m, r ) is i) non decreasing in m and r ,ii) upper semi-continuous in r for all m ,iii) upper semi-continuous in m for all r . Example 5 in the next section shows that in general, for fixed m , V Q ( · , m )is not lower semi-continuous. Remark . Let m, r ≥
0, and Q , Q be two search spaces, it is clear that if Q ⊂ Q then the attacker is better off in Q hence V Q ( m, r ) ≥ V Q ( m, r ). Recall that the value of a hiding game is equal to the value of a patrollinggame with time limit m equal to 0. Hence, the negative results presented inthis section also hold for patrolling games when m = 0.The following simple example of a hiding game on the unit interval wasfirst solved by Ruckle (1983). It shows that in general, V Q is not lowersemi-continuous. Example 5.
Let Q be the [0 ,
1] interval, then V Q ( r ) = min (cid:16) ⌈ r ⌉ − , (cid:17) if r >
00 otherwise . Indeed, it is clear when r equals 0 and r ≥ /
2. Let n ∈ N ∗ and suppose r ∈ h n +1) , n (cid:17) . Then the patroller guarantees n +1 by choosing equiprob-ably a point in n k n +1) o ≤ k ≤ n . And the attacker, choosing equiprobably apoint in n (2+ ε ) k n +1) o ≤ k ≤ n , with 0 < ε ≤ /n , also guarantees n +1 .16he next proposition disproves the somehow intuitive belief that thevalue of hiding games is continuous with respect to the Haussdorff metricbetween nonempty compact sets. Proposition 6.
Let r ≥ . The function which maps any search space Q to V Q ( r ) is in general not continuous with respect to the Hausdorff metricbetween nonempty compact sets.Proof. Let D s = { x ∈ R | k x k ≤ s } be the Euclidean disc of radius s > V D s (1) = s ∈ [0 , π arcsin (cid:16) s (cid:17) if s ∈ (cid:16) , √ i . Hence lim s → ,s> V D s (1) = < . The intuition is the following: it is clear that when s equals 1 the searcherguarantees 1 by playing x = (0 , s equals 1 + ε . Thenthe searcher covers almost all the area of the disc but less than half of itscircumference. Hence the hider guarantees 1 / D s . Subsection 3.1
Proof of proposition 1.
Let us first define a metric on the set W inducingthe topology of compact convergence. For n ∈ N , define K n = [0 , n ]. Then D : W × W → R + ( f, g ) ∞ P n =1 12 n sup x ∈ K n k f ( x ) − g ( x ) k . is a metric on W .We recall the following fact about the topology of compact convergencetopology. Proposition 7 (Application of theorem 46.2 in Munkres (2000)) . Let Q be a search space. A sequence f n : R + → Q of functions converges to thefunction f in the topology of compact convergence if and only if for eachcompact subspace K of R + , the sequence f n | K converges uniformly to f | K . The following corollary follows from Sion’s theorem, Sion (1958).
Corollary 1 (Proposition A.10 in Sorin (2002)) . Let ( X, Y, g ) be a zero-sumgame such that: X is a compact metric space, for all y ∈ Y , the function g ( · , y ) is upper semi-continuous. Then the game (∆( X ) , ∆ f ( Y ) , g ) has avalue and player 1 has an optimal strategy.
17e are now able to complete the proof. By Ascoli’s theorem (applica-tion of theorem 47.1 in Munkres (2000)), W is compact for the topology ofcompact convergence. Moreover, for all ( y, t ) ∈ A the function g m,r ( · , ( y, t ))is upper semi-continuous. The conclusion follows from corollary 1. Subsection 3.3
Proof of lemma 1.
Let π = ( u , u , . . . , u n − , u n ), u = u n = u ∈ V be anEulerian tour. Without loss of generality, suppose l ( u i , u i +1 ) = 0 for all i ∈ { , . . . , n − } . The parametrization is constructed in the following way.If t ∈ [0 , l ( u , u )] then w ( t ) = (cid:18) u , u , tl ( u , u ) (cid:19) . Else, suppose n ≥
3. For all k ∈ { , . . . , n − } if t ∈ k − X i =1 l ( u i , u i +1 ) , k X i =1 l ( u i , u i +1 ) then w ( t ) = u k , u k +1 , t − P k − i =1 l ( u i , u i +1 ) l ( u k , u k +1 ) ! . It is not difficult to verify that such w is appropriate. Proof of theorem 1. If m ≥ λ ( N ), the patroller guarantees 1 by playing aparametrization of N . Suppose that m < λ ( N ). Let ( y, t ) ∈ N × R + bea pure strategy of the attacker and let w be a in definition 4. There exists t y ∈ [0 , λ ( N )] such that w ( t y ) = y . Now let t ∈ [ t y − t − m, t y − t ]. Then w t ( t y − t ) = w ( t y ) = y . And t y − t ∈ [ t, t + m ]. Thus y ∈ w t ([ t, t + m ]) . Hence under the patroller’s uniform strategy P ( y ∈ w t ([ t, t + m ])) ≥ P ( t ∈ [ t y − t − m, t y − t ]) = mλ ( N ) . The other inequality follows from proposition 2 since in this case, ρ equals1. Subsection 3.4
To prove theorem 2 we first need some preliminary definition and lemmas.
Definition 9.
Let Q be a search space. A continuous function L : [0 , → Q such that L (0) = L (1) is called an r -tour if for any x ∈ Q there exists l ∈ L ([0 , d ( x, l ) ≤ r .The next lemma shows that when the radius of detection r is small, onecan find in Q an r -tour with length not exceeding λ ( Q ) / r , up to some ε .18 emma 2 (Lemma 3.39 in Alpern and Gal (2003)) . Let Q ⊂ R be a simplesearch space. Endow Q with the Euclidean norm. Then for any ε > thereexits r ε > such that for any r < r ε there exists an r -tour L : [0 , → Q such that T V ( L ) ≤ (1 + ε ) λ ( Q )2 r . The next lemma gives a parametrization of L ([0 , Lemma 3.
Let L be an r -tour as in lemma 2. Then for all ε ′ > thereexists w : [0 , T V ( L ) + ε ′ ] → L ([0 , continuous such that:i) w (0) = w ( T V ( L ) + ε ′ ) , ii) w is surjective,iii) w is -Lipschitz continuous,iv) T V ( w ) = T V ( L ) .w is extended to a ( T V ( L )+ ε ′ ) -periodic function on R , which is still denoted w .Proof of lemma 3. Let ε ′ > f : [0 , → [0 , T V ( L ) + ε ′ ] s T V ( L | [0 ,s ] ) + ε ′ s. The function f is increasing and continuous on [0 , f is an homeo-morphism. Define w as L ◦ f − on [0 , T V ( L ) + ε ′ ]. It is not difficult to provethat such w verifies the condition of the lemma.We are now able to prove theorem 2. Proof of theorem 2.
Let L and w be as in lemma 2 and lemma 3 respectively.For all t ∈ [0 , T V ( L ) + ε ′ ] define w t ( · ) as w ( t + · ).Let ( l, t ) ∈ L ([0 , × R + . By lemma 3 ii), there exists t l ∈ [0 , T V ( L )+ ε ′ ]such that w ( t l ) = l . Now let t ∈ [ t l − t − m, t l − t ]. Then w t ( t l − t ) = w ( t l ) = l . And t l − t ∈ [ t, t + m ]. Hence l ∈ w t ([ t, t + m ]) . Suppose t is chosen uniformly in [0 , T V ( L ) + ε ′ ]. By lemma 3 iii) thisis an admissible strategy for the patroller. Let ( y, t ) ∈ A be a pure strategyof the attacker. Then if l ∈ L ([0 , d ( y, l ) ≤ r , P ( d ( y, w t ([ t, t + m ])) ≤ r ) ≥ P ( l ∈ w t ([ t, t + m ])) , ≥ P ( t ∈ [ t l − t − m, t l − t ])= mT V ( L ) + ε ′ . By lemma 2, this last quantity is grater than or equal to m (1+ ε ) λ ( Q )2 r + ε ′ . Hencethe patroller guarantees m (1+ ε ) λ ( Q )2 r + ε ′ for all ε ′ >
0, that is V Q ( m, r ) ≥ rm (1 + ε ) λ ( Q ) ∼ rmλ ( Q )19s r goes to 0.In this context, proposition 2 yields V Q ( m, r ) ≤ rm + πr λ ( Q ) ∼ rmλ ( Q ) as r goes to 0 (see remark 1). Subsection 4.1
Proof of proposition 4.
Suppose µ ∈ ∆( Q ) is an equalizing strategy. If thesearcher plays µ , then for all y ∈ Q µ ( B r ( y ) ∩ Q ) = c , hence V Q ( r ) ≥ c .Symmetrically, if the hider plays µ , then for all x ∈ Q µ ( B r ( x ) ∩ Q ) = c ,hence V Q ( r ) ≤ c , and V Q ( r ) = c .Conversely, suppose µ ∈ ∆( Q ) is optimal for both players. Then thesearcher guaranties V Q ( r ) that is for all y ∈ Q µ ( B r ( y ) ∩ Q ) ≥ V Q ( r ), andthe hider guaranties V Q ( r ) that is for all x ∈ Q µ ( B r ( x ) ∩ Q ) ≤ V Q ( r ). Hencefor all y ∈ Q µ ( B r ( y ) ∩ Q ) = V Q ( r ) . Subsection 4.2
To prove theorem 3 we first need to introduce a technical lemma.Denote B r ( x ) = { y ∈ R n | k x − y k ≤ r } the closed ball of center x withradius r for the Euclidean norm, and ∂B r ( x ) = { y ∈ R n | k x − y k = r } the sphere of center x with radius r for the Eucliean norm.The intuition behind lemma 4 below is the following. We consider theballs B ε (0) and B r ( x ) with x on the boundary of B ε (0). When r goes tozero, the ratio between the volume of the ball B r ( x ) and the ball B r ( x )intersected with the ball B ε (0) goes to 2. Lemma 4 gives an upper boundto this ratio, as r goes to 0, for a non necessary Euclidean ball B r ( x ). Lemma 4.
Let k·k be a norm on R n and c , c > be such that c k·k ≤k·k ≤ c k·k . Then for all x ∈ ∂B ε (0)lim sup r → λ ( B r ) λ ( B ε (0) ∩ B r ( x )) ≤ (cid:18) c c (cid:19) n . Proof of lemma 4.
Let x ∈ ∂B ε (0), let ε >
0. Denote I the regularizedincomplete Beta function: for a, b > < z < I z ( a, b ) = B ( z ; a,b ) B ( a,b ) .Where B ( z ; a, b ) = R z t a − (1 − t ) b − dt and B ( a, b ) = B (1; a, b ) is the Betafunction. Then we have, see Li (2011), λ (cid:16) B ε (0) ∩ B r ( x ) (cid:17) = π n/ n + 1) r n I − ( r ε ) (cid:18) n + 12 , (cid:19) + ε n I ( rε ) (cid:16) − ( r ε ) (cid:17) (cid:18) n + 12 , (cid:19) ! . t t n − (1 − t ) − / is integrable over [0 , I − ( r ε ) (cid:18) n + 12 , (cid:19) = R − ( r ε ) t n − (1 − t ) − / dtB ( n − , ) → , as r goes to 0. And, I ( rε ) (cid:16) − ( r ε ) (cid:17) (cid:18) n + 12 , (cid:19) = R ( rε ) (cid:16) − ( r ε ) (cid:17) t n − (1 − t ) − / dtB ( n − , )which, since 1 ≤ (1 − t ) − / when t ∈ [0 , n +1 (cid:0) rε (cid:1) n +1 (cid:16) − (cid:0) r ε (cid:1) (cid:17) n +12 B ( n − , ) = 2 r n +1 ( n + 1) ε n +1 B ( n − , ) + o ( r n +2 )when r goes to 0. Hence we have λ (cid:16) B ε (0) ∩ B r ( x ) (cid:17) ≥ π n/ n + 1) ( r n + o ( r n ))as r goes to 0. Moreover since B ε (0) ∩ B r ( x ) = { y ∈ R n | k y k ≤ ε and k x − y k ≤ r }⊃ { y ∈ R n | k y k ≤ ε and k x − y k ≤ c r } , and B r (0) ⊂ B c r (0), we have λ (cid:0) B ε (0) ∩ B r ( x ) (cid:1) ≥ λ (cid:0) B ε (0) ∩ B c r ( x ) (cid:1) , and c n λ ( B r ) ≥ λ ( B r ) . Finally, dividing by λ (cid:0) B ε (0) ∩ B r ( x ) (cid:1) and takingthe lim sup, since λ ( B r ) = π n/ r n Γ ( n +1 )lim sup r → λ ( B r ) λ ( B ε (0) ∩ B r ( x )) ≤ lim sup r → c n λ ( B r ) λ (cid:0) B ε (0) ∩ B c r ( x ) (cid:1) ≤ (cid:18) c c (cid:19) n . We are now able to prove theorem 3.
Proof of theorem 3.
Let ε > r ∈ (0 , ε ). We regularize the boundaryof Q by defining Q ε = Q + B ε (0) , and I ε ( r ) = { y ∈ Q ε | B r ( y ) ⊂ Q ε } . Define as well λ ε min ( r ) = min y ∈ Q ε λ ( B r ( y ) ∩ Q ε ) . Finally define µ ∈ ∆( Q ε )such that for all B ⊂ Q ε measurable µ ( B ) = λ ( B ∩ I ε ( r )) λ ε min ( r ) + λ ( B ∩ ( Q ε \ I ε ( r )) λ ( B r ) λ ( I ε ( r )) λ ε min ( r ) + λ ( B r ) λ ( Q ε \ I ε ( r )) . λ ( B r ) ≥ λ ε min ( r ), for all x ∈ Q ε µ ( B r ( x ) ∩ Q ε ) ≥ λ ε min ( r ) λ ( B r ) λ ( I ε ( r )) λ ε min ( r ) + λ ( B r ) λ ( Q ε \ I ε ( r )) . Because the hider can play in ( Q ε , r ) as he would play in ( Q, r ), V Q ε ( r ) ≤ V Q ( r ). By proposition 2, λ ε min ( r ) λ ( B r ) λ ( I ε ( r )) λ ε min ( r ) + λ ( B r ) λ ( Q ε \ I ε ( r )) ≤ V Q ε ( r ) ≤ V Q ( r ) ≤ λ ( B r ) λ ( Q ) . Dividing by λ ( B r ) /λ ( Q ) ,λ ε min ( r ) λ ( Q ) λ ( I ε ( r )) λ ε min ( r ) + λ ( B r ) λ ( Q ε \ I ε ( r )) ≤ V Q ( r ) λ ( Q ) λ ( B r ) ≤ . (2)Let us show that for all ε > S r> I ε ( r ) = ˚ Q ε . Indeed, let y ∈ S r> I ε ( r ). There exists r > y ∈ I ε ( r ). Thus there exists r > B r ( y ) ⊂ Q ε . Conversely, let y ∈ ˚ Q ε . There exists r ′ > B ′ r ′ ( y ) ⊂ ˚ Q ε , where B ′ r ′ ( y ) = { x ∈ R n | k x − y k < r ′ } . Take0 < r < r ′ , then B r ( y ) ⊂ ˚ Q ε hence y ∈ I ε ( r ).For all r , r > r > r one has I ε ( r ) ⊂ I ε ( r ). Hencelim r → λ ( I ε ( r )) = λ ( ˚ Q ε ). Dividing by λ ε min ( r ) and letting r go to 0 inequation (2), by lemma 4 one has, since the minimum in λ ε min ( r ) is reachedon the boundary of a Euclidean ball, λ ( Q ) λ ( ˚ Q ε ) + 2 (cid:16) c c (cid:17) n λ ( ∂Q ε ) ≤ lim inf r → V Q ( r ) λ ( Q ) λ ( B r ) ≤ lim sup r → V Q ( r ) λ ( Q ) λ ( B r ) ≤ . (3)Let us show that T ε> ˚ Q ε = T ε> Q ε = Q. Indeed, let y ∈ T ε> Q ε .For all ε > z ∈ Q k y − z k ≤ ε , hence y ∈ Q . Conversely, for all ε > Q ⊂ ˚ Q ε hence Q ⊂ T ε> ˚ Q ε . Moreover for all ε , ε > ε < ε one has Q ε ⊂ Q ε . Hence lim ε → λ ( ˚ Q ε ) = λ ( Q ), lim ε → λ ( Q ε ) = λ ( Q ) and λ ( ∂Q ε ) = λ ( Q ε ) − λ ( ˚ Q ε ) so lim ε → λ ( ∂Q ε ) = 0 . Letting ε → λ ( Q ) λ ( Q ) ≤ lim r → V Q ( r ) λ ( Q ) λ ( B r ) ≤ . Subsection 5.1
Proof of proposition 5.
Since i) is direct we only prove ii). For all ( m, r ) ∈ R , V Q ( m, r ) = max µ ∈ ∆( W ) inf ( y,t ) ∈A Z W g m,r ( w, ( y, t )) dµ ( w )= inf ( y,t ) ∈A Z W g m,r ( w, ( y, t )) dµ ∗ ( w ) , µ ∗ ∈ ∆( W ) is an optimal strategy of the patroller. Let ( y, t ) ∈ A and m ≥
0. For all w ∈ W , the function r g m,r ( w, ( y, t )) is upper semi-continuous, as the indicator function of a closed set. Let r n → r , then byFatou’s lemma,lim sup n Z W g m,r n ( w, ( y, t )) dµ ∗ ( w ) ≤ Z W lim sup n g m,r n ( w, ( y, t )) dµ ∗ ( w ) ≤ Z W g m,r ( w, ( y, t )) dµ ∗ ( w ) . Thus the function r R W g m,r ( w, ( y, t )) dµ ∗ ( w ) is upper semi-continuous.Hence V Q ( m, · ) : r inf ( y,t ) ∈A Z W g m,r ( w, ( y, t )) dµ ∗ ( w )is upper semi-continuous.Since for all w ∈ W , the function m g m,r ( w, ( y, t )) is upper semi-continuous, as the indicator function of a closed set, the proof of iii) isstrictly analogous. Acknowledgments
The author wishes to express his gratitude to his Ph.D. advisor J´erˆomeRenault, as well as Marco Scarsini for his help in improving the generalpresentation of the paper.
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