Towards Selecting the Best Abstraction for a Patrolling Game

Transcription

1 University of Teas at El Paso Departmental Technical Reports CS) Department of Computer Science -05 Towards Selecting the Best bstraction for a Patrolling Game njon Basak University of Teas at El Paso, abasak@miners.utep.edu Chris Kiekintveld University of Teas at El Paso, cdkiekintveld@utep.edu Vladik Kreinovich University of Teas at El Paso, vladik@utep.edu Follow this and additional works at: Part of the Mathematics Commons Comments: Technical Report: UTEP-CS-5-93 To appear in Journal of Uncertain Systems, 07, Vol.. Recommended Citation Basak, njon; Kiekintveld, Chris; and Kreinovich, Vladik, "Towards Selecting the Best bstraction for a Patrolling Game" 05). Departmental Technical Reports CS). Paper This rticle is brought to you for free and open access by the Department of Computer Science at DigitalCommons@UTEP. It has been accepted for inclusion in Departmental Technical Reports CS) by an authorized administrator of DigitalCommons@UTEP. For more information, please contact lweber@utep.edu.

2 Towards Selecting the Best bstraction for a Patrolling Game njon Basak, Christopher Kiekintveld, and Vladik Kreinovich Department of Computer Science University of Teas at El Paso 500 W. University El Paso, TX 79968, US abasak@miners.utep.edu, cdkiekintveld@utep.edu, vladik@utep.edu bstract When the number of possible strategies is large, it is not computationally feasible to compute the optimal strategy for the original game. Instead, we select our strategy based on an approimate approimate description of the original game. The quality of the resulting strategy depends on which approimation we select. In this paper, on an eample of a simple game, we show how to find the optimal approimation, the approimation whose use results in the best strategy. Formulation of the Problem Need for abstraction. Ideally, in a conflict situation, we should select a strategy which is optimal in some reasonable sense. For eample, in a zero-sum game, it makes sense to select a strategy that minimizes the worst-case loss. The more strategies we need to consider, the more computations we need to perform to find the optimal strategy. When the number of strategies becomes large, it is often not feasible to eactly compute the optimal strategy. In such situations, to produce a reasonable strategy, it makes sense to approimate the actual difficult-to-analyze game with a simpler-to-analyze game that has fewer strategies. One of the natural ways to come up with such an approimation is abstraction, when we divide all possible strategies into a small number of groups, so that strategies within each group lead to approimately the same consequences. Then, in the abstracted simplified) game, we just select a group instead of selecting an actual strategy); see, e.g., [, 3]. Which abstraction is the best? Of course, since we are using only an

3 approimate description of the actual game situation to come up with a strategy, the resulting strategy is only approimately optimal. The quality of the resulting strategy depends on the choice of the abstraction. It is therefore desirable to find an abstraction that results in the best possible performance. What we do in this paper. In this paper, we solve the problem of selecting the best abstraction on the eample of a simple patrolling game. -D Patrolling Game: Brief Description Description of the game. Let us assume that have a -D area that we want to protect against an adversary: e.g., a border segment. Points along this area will be marked by a parameter that takes values between and. For eample, for border protection, is the distance along the border. We also assume that for each point, we know the utility gain u) that the adversary will get if his penetration at this point succeeds. This value may differ from one location to another. For eample, in case of border protection, a successive smuggling in an area with a smooth terrain will enable the adversary to smuggle a larger amount than in an area with a harsher terrain and thus, gain more utility. To prevent the adversary s attacks, we place agents along the -D area. The closer the agent to the adversary, the less probable is the adversary s success. For simplicity, let us assume that there is a threshold distance r such that the attack is prevented if and only if the distance between the adversary and the agent does not eceed this threshold h. We randomly allocate agents to different points. Let ρ a ) be the probability density that describes the corresponding agents distribution. This means that for a small distance, the probability of finding an agent at locations between and + is equal to ρ a ). Overall, we have n a agents, so we must have ρ a ) d = n a. ) We assume that the adversary knows our strategy, i.e., knows the values ρ a ). This is a reasonable assumption, since the adversary can observe agents at different locations and thus, determine the frequencies with which agents are posted to different locations. Such games in which the adversary knows the agent s strategy are known as Stackelberg games; see, e.g., [6]. Based on his knowledge, the adversary selects an attack location at which his epected utility gain is the largest possible. We assumed that we know the adversary s gain in the situation when the adversary s attack at a point is successful: this gain is equal to u). Under our assumptions, the probability of being unsuccessful is equal to the probability to find an agent within a distance h from the point, i.e., in the interval [ h, +h] of width = h. Based on the probability density ρ a ) describing the agents appearance, we can therefore estimate the probability of thwarting an attack as the product h ρ a ). Thus,

4 the probability of a successful attack is h ρ a ), and so, the epected gain of the adversary is equal to the product h ρ a )) u). For each selection of the agent distribution ρ a ), the adversary selects an attack location at which his epected utility gain h ρ a )) u) is the largest possible. Thus, the adversary s gain is equal to ma h ρ a )) u). ) We select the agent distribution ρ a ) for which this gain is as small as possible: ma h ρ a )) u) min 3) ρ a ) under the constraint that ρ a ) d =. 4) Known optimal strategy for the patrolling game. Let u 0 denote the desired maimum ma h ρ a)) u). Then, we have h ρ a )) u) u 0 5) for all u. For all the points at which ρ) = 0, the gain is equal to u), so we have u) u 0. One can easily prove that in the optimal agent distribution, h ρ a )) u) = u 0 6) for all the points [, ] for which ρ a ) > 0. Indeed, if at some of such points 0, we have h ρ a 0 )) u 0 ) < ma h ρ a )) u), 7) then we can decrease the density in the vicinity of this point 0 without changing the maimum, and distribute these probabilities among all the points where the product h ρ a )) u) attains its maimum. This will guarantee that the maimum adversary s gain decreases. So, the optimal solution is to select some threshold u 0 and take ρ) = 0 for all values for which u) < u 0 and h ρ a )) u) = u 0 for all other. This equality implies that h ρ a ) = u 0 u), 8) hence h ρ a ) = u 0 u) 9) 3

5 and ρ a ) = h u ) 0. 0) u) The threshold u 0 should be determined by the condition that ρ a ) d = n a, i.e., that :u) u 0 h u ) 0 = n a. ) u) The case that we will consider in this paper. In general, we may have points which are not worth defending, i.e., for which the possible loss u) is so low, that we do not even bother sending agents there. In many realistic important patrolling problems, however, we cannot ignore any of the points, so we have ρ a ) > 0 for all. This is the case that we consider in this paper. In this realistic case, we have u) u 0 for all, hence the equality ) takes the form Thus, in this case, we have h u ) 0 = u) h u 0 u) = n a. ) u 0 = ) n a h h. 3) u) d Thus, the optimal strategy is to select the agent density 0), where the parameter u 0 is determined by the formula 3). 3 pproimations to the -D Patrolling Game: Description How to describe an approimation? s we have mentioned earlier, an approimation means that instead of considering all possible values, we divide the interval [, ] into a small number of sub-zones, and, when planning the agent distribution, only decide on the sub-zone so that agents are uniformly distributed within each sub-zone. t some important locations, we may have a denser concentration of these sub-zones, in other locations, we may have fewer sub-zones. Let us describe the distribution of sub-zones by the sub-zone density ρ s ) that describes how many sub-zones per unit length we have at a location. The overall number of sub-zones should be equal to the the total number of sub-zones n s that our computational abilities allows us: ρ s ) d = n s. 4) 4

6 How to select a solution based on the approimate game? In the original game, for each location, we take into account the adversary s gain u). In the approimate game, instead of a single value, we need to consider the whole sub-zone. By definition of the sub-zone density, a unit interval into ρ s ) sub-zones, thus the width w of each sub-zone is equal to w = ρ s ). 5) So, the distance from the midpoint to each of the endpoints of the zone is equal to half of this width, and thus, instead of a single point, we have a sub-zone [ S = ρ s ), + ], 6) ρ s ) where is the midpoint of this sub-zone. For different points within this sub-zone, we have different values of adversary s gain u) and thus, different values of our loss u)). So, instead of a single value u), we now only know that the actual unknown) value u) belongs to the interval [u, u] = {u) : S} 7) of possible values from this sub-zone. How can we make a decision in situations with such interval uncertainty? The way to make such decisions was developed by a Nobelist Leo Hurwicz in the early 950s [, 4, 5]: he argued that in such situations, we should optimize a weighted combination of the best-case gain and the worst-case gain, with weights adding up to and with the weight α at the best-case gain determined by the decision maker s optimism-pessimism. In our case, the best-case situation is when we have the smallest possible loss u and the worst-case situation is when we have the largest possible loss u. So, Hurwicz s recommendation means that in our decision making, we should use the function u ) = α u + α) u = α min S which is constant on each sub-zone. u) + α) ma u) 8) S Formulation of the problem. By definition of an approimate game, we select the agent distribution ρ a) based not on the actual utility function u), bur rather based on the piece-wise constant approimation u ). The adversary knows this selection ρ a), so he selects an attack point for which his epected gain h ρ a)) u) is the largest possible. s a result, for each approimation density, the adversary s gain is equal to ma h ρ a)) u). 9) 5

7 Since we used an approimation, the resulting gain is, in general, larger, than in the ideal situation when we have the optimal agent density. It is therefore desirable to select the sub-zone distribution ρ s ) for which the adversary s gain is the smallest possible: ma h ρ a)) u) min ρ s ). 0) 4 nalysis of the Problem: General Case Let us first come up with eplicit formulas for u ). In this paper, we assume that the approimation is reasonable, i.e., that the sub-zones are narrow. When the sub-zone is narrow, we can epand the value of u) on each zone [ w, +w] into Taylor series and only keep linear terms in this epansion and safely ignore terms proportional to terms which are quadratic or of higher order with respect to w: u) = u ) + u ), where def =. In the sub-zone, the difference takes all possible values between w and w. In this approimation, the function u) is linear on each sub-zone, and it is easy to find its smallest and largest values: and u = u u w ) u = u + u w. ) Indeed, when the derivative u is positive, the maimum is attained when attains its largest possible value = w, 3) and the minimum is attained when attains its smallest possible value = w. 4) When the derivative is negative, locations of the maimum and minimum switch places. In both cases, we get the desired formula. So, we get u = α u u w ) + α) u + u w ) = u α ) u w. 5) 6

8 Our agent allocation is based on this approimate piece-wise constant utility function u ). What is the agent allocation ρ a) based on this approimate utility function u ). ccording to formulas 8) and 3), we have h ρ a) = u 0 u ), 6) where u 0 = ) n a h. 7) h u ) d How much the adversary gains. On each zone, the adversary selects a point at which the epected gain h ρ a)) u) is the largest. The value h ρ a)) is constant within the zone, so within each zone, the adversary selects the point at which u) is the largest, i.e., the point at which u) = u = u)+ u ) w. For this point, the adversary s gain is equal to h ρ a)) u. From 6), we conclude that thus the adversary s gain is equal to Here, and thus h ρ a)) = u 0 u ), 8) u 0 u) u ). 9) u) = u) + u ) w u ) = u) α ) u ) w w u) u) + u ) u ) = u) α ) u ) w = + u ) u) w α ) u ) u) 30) 3), w. 3) For narrow w, when we can ignore terms which are quadratic and higher order in terms of w, this ratio is equal to u) u ) = + α) u ) u) w. 33) 7

9 Since w =, the gain is thus equal to ρ s ) ) u 0 + α) u ) u). 34) ρ s ) This value characterizes the adversary s gain for each sub-zone. Thus, the adversary will select a sub-zone at which this value is the largest possible. The resulting gain is equal to u 0 + α) ma Here, due to 3) and 7), we have u 0 = u 0 u ) ). 35) u) ρ s ) u) d u ) d. 36) Due to 5), we have u ) = u) α ) u ) u) w ) = u) α u ) ). 37) u) ρ s ) Thus, since the width w = is small, we have ρ s ) u ) = u) + α u ) ) = u) ρ s ) Hence, u) + α u ) u ) ρ s ). 38) and u ) d = u ) d = α d + u) u) d + α 8 u ) u d, 39) ) ρ s ) u ) u ) ρ s ) d u) d. 40)

10 Thus, hence u ) d u) d u) d u ) d So, due to 36), we have = + α = α u 0 = u 0 α u ) u ) ρ s ) d u) d u ) u ) ρ s ) d u) d, 4). 4) u ) u ) ρ s ) d u) d. 43) Substituting the epression 43) into the formula 35) and taking into account that terms proportional to i.e., to width of sub-zones) are small, we ρ s ) conclude that the adversary s gain is equal to where G def = α) ma u 0 + G) 44) u ) u) ρ s ) α u ) u ) ρ s ) d u) d. 45) Resulting constraint optimization problem. To minimize the adversary s gain, we need to find the sub-zone distribution function ρ s ) that minimizes the epression 45) under the constraint 4), that epresses the fact that the overall number of sub-zones is equal to n s : ρ s ) d = n s. How we will solve this problem. The minimized function 45) is the sum of two terms: the maimum term and the integral terms. To solve the corresponding minimization problem, we first consider the cases when one of these terms is equal to 0. In each of the corresponding two cases, the minimized function is easier-to-describe and thus, the corresponding minimization problem is easier to solve. fter that, we will show how the solutions to these two simplified cases can be combined to produce a solution to the general case. 9

11 5 First Special Case: α = 0.5 Let us first consider the case when α = 0.5, i.e., when we consider a midpoint of an interval in case of uncertainty. In this case, the integral term in the formula 45) is equal to 0, and thus, the minimized function 45) takes the simplified form G = 0.5 ma u ) u) ρ s ). 46) Similarly to Section, we can conclude that the optimal sub-zone distribution corresponds to the case when the maimized ratio u ) u) ρ s ) 47) is constant, i.e., when This makes sense: ρ s ) = c u ) u). 48) Since and When the derivative u ) is small, i.e., when the gain u) practically does not change, we can have fewer sub-zones of larger size, since for all from this sub-zone we will have, in effect, the same gain value u). On the other hand, when the derivative u ) is large, we need narrower sub-zones if we want to accurately represent the utility function u) and thus, we need more frequent sub-zones. the product u ) u) = d lnu)) 49) d = w, 50) ρ s ) u ) u) ρ s ) 5) is equal to d lnu)) w, 5) d i.e., in the first approimation, to the absolute value of the difference between the values of ln u w )) 53) and ln u + w )) 54) 0

12 on the two sides of the sub-zone. For each sub-zone, this difference should be the same. This difference is the logarithm of the ratio u + w ) u w ). 55) Thus, we should select zones in which the ratio of utilities at the endpoints is a constant C >. So, for the case when α = 0.5, we arrive at the following optimal location of sub-zones i.e., optimal approimation): We start with the leftmost location =. This location is the left-end point of the first zone. Its right-hand point is the value at which u ) = C u ). Then, we take 3 at which u 3 ) = C u ), etc. When we reach the maimum, we start decreasing the utility value by the factor of C. In short, the endpoints of the sub-zones are points at which the utility u) take values from a geometric progression u), C u), C u), etc. 6 Second Special Case: α = In the case when α =, i.e., when we only consider the worst-case scenarios when making a decision, the maimum term in the formula 45) disappears, and thus, the minimized function 45) takes the simplified form G = In this epression, the denominator u ) u ) ρ s ) d u) d. 56) d 57) u) does not depend on our choice of the sub-zone distribution function ρ s ), so minimizing the epression 56) is equivalent to maimizing the integral u ) u d 58) ) ρ s )

13 under the constraint ρ s ) d = n s. 59) To solve this constraint maimization problem, we can use the Lagrange multiplier technique and optimize the epression ) u ) u ) ρ s ) d + λ ρ s ) d n s 60) for an appropriate Lagrange multiplier λ. Differentiating the unconstrained objective function 60) with respect to ρ s ) and equating derivatives to 0, we conclude that u ) u ) ρ + λ = 0, 6) s) hence ρ s ) = c u ) u) 6) for an appropriate constant c. This constant can be easily determined from the condition 59). It is worth mentioning that the resulting formula 6) is similar to the formula 48) corresponding to the case α = 0.5: in both cases, the density ρ s ) of sub-zones is inverse proportional to the utility u), and in both case, the density ρ s ) increases as the utility s non-homogeneity u ) increases. The difference between these two formulas is that: when α = 0.5, the density of sub-zones is proportional to the non-homogeneity itself: ρ s ) u ), while when α =, the density of sub-zones grows slower, proportionally to the square root of non-homogeneity: ρ s ) u ). Let us now show how these two special cases can be combined into a generic case. 7 General Cases: 0 α nalysis of the problem. We would like to find the sub-zone density ρ s ) that minimizes the loss function 45). This loss function is the sum of two terms: the maimum term and the integral term. Let us denote the optimal value of the maimum term by c ma : c ma def = ma u ) u) ρ s ). 63)

14 For the locations at which this maimum is attained, i.e., at which we thus have u ) u) ρ s ) = c ma, 64) ρ s ) = C ma u ) u), 65) where C ma def = c ma. For the locations at which the maimum is not attained, i.e., at which u ) u) ρ s ) < c ma 66) and where small changes in ρ s ) do not affect the maimum part of the objective function, we need to select the values of ρ s ) at these locations so as to minimize the integral term α u ) u ) ρ s ) d u) d 67) under the constraint ρ s ) = n s. Let us denote the set of all locations that satisfy the inequality 66) by X. Since we are only looking for the values ρ s ) for X, this means that we need to minimize the corresponding part of the integral term, i.e., the term under the constraint α X u ) u ) ρ s ) d X u) d 68) ρ s ) = const. 69) Similarly to the case α =, we can use the Lagrange multiplier method to solve this constraint optimization problem and thus conclude that for X, we have ρ s ) = C int u ) u) 70) for an appropriate constant C int. 3

15 These values must satisfy the inequality 66), so we must have and thus, C int ρ s ) > C ma u ) u) u ) u) 7) > C ma u ) u). 7) Multiplying both sides of this inequality by u) and dividing by u ), we get an equivalent inequality squaring which we get C int > C ma u ), 73) u ) < Cint C ma ). 74) This is thus the threshold for allocating locations either to the set X or to the set of points where the maimum is attained: when the absolute value u ) of the derivative eceeds the threshold, the maimum is attained and thus, we get the formula 65); on the other hand, when the absolute value of the derivative is smaller than the threshold, we use the formula 70). So, we arrive at the following solution to our optimization problem. Main result for the -D case: optimal abstraction. Let us assume that we know the gain function u) on the interval [, ]. Instead of using the original gain function u), we consider an abstraction, i.e.: we divide the interval [, ] into n s sub-zones, and we only consider sub-zones when deciding on the best patrolling strategy. In this situation, the optimal sub-zone density ρ s ) i.e., the sub-zone density for which the epected losses are the smallest can be obtained if we select two parameters C int and C ma and take ) u ) ρ s ) = C int when u Cint ) < ; 75) u) C ma Cint ρ s ) = C ma u ) when u ) u) C ma The parameters C int and C ma should be selected in such a way that ). 76) ρ s ) d = n s. 77) Under this constraint, we must select the values C int and C ma that minimizes the functional 45), in which the optimism-pessimism parameter α describes the decision maker s degree of optimism. 4

16 8 Multi-D Patrolling Game Description of the game. Let us now consider a more general case, when the agents need to patrol a multi-d area. For each location =,..., d ) within this area, we know the value u) that the adversary gains if he succeeds in attacking at this point. Similarly to the -D case, we assume that the adversary succeeds if there is no agent within a certain distance h from the attack point. Our objective is to find the distribution of the agents ρ a ) that will minimize our epected loss, under the constraint that the overall number of agents is n a : ρ a ) d = n a. 78) For each distribution, the adversary s gain is equal to ma V d h d ρ a )) u), 79) where V d is the d-dimensional volume of a unit ball B = { : }: for d =, we have V =, so the length of an interval of radius h is equal to h; for d =, we have V = π, so the area of a circle of radius h is equal to π h ; for d = 3, we have V 3 = 4π 3, so the volume of a ball of radius h is equal to 4π 3 h3 ; etc. Optimal strategy for the multi-d patrolling game. Similarly to the - D case, in situations in which there are no not-worth-defending locations, the optimal patrolling strategy can be determined from the condition that i.e., V d h d ρ a ) = u 0 u), 80) ρ a ) = V d h d u ) 0, 8) u) where the threshold u 0 is determined by the condition 78), as where V is the volume of the area. u 0 = V n a V d h d, 8) V d h d u) d How to describe approimations. Instead of considering individual locations, we divide the area into rectangular boes [, ]... [ d, d ]. 83) 5

17 To describe the distribution of such boes, we now need to have d different density functions: the function ρ ) describes how many intervals [, ] we have per unit length at location ; the function ρ ) describes how many intervals [, ] we have per unit length at location ; etc. The -width of each bo is thus equal to ρ ), the -width of each bo is equal to, etc. Thus, the volume V ) of each bo 83) is equal to the ρ ) product of these widths: d V ) = ρ i ). 84) In a unit volume, we can thus place i= d V ) = ρ i ) 85) such boes. The overall number of boes can be obtained by adding up these numbers for all locations, i.e., as an integral of this product over the area. This overall number should be equal to the largest number of n a of boes that we can computationally handle when selecting a strategy, so we must have i= i= d ρ i ) d = n b. 86) nalysis of the problem. Similarly to the -D case, we can represent the i-th coordinate i of each point in a bo as i = i + i, where i is the midpoint of the i-th interval [ i, i ] and i w i, where w i = is the ρ i ) width of this interval. In the linear approimation, we have u,..., n ) = u,..., n ) + d i= i i. 87) The largest possible value u of the gain function on this bo and its smallest possible value u thus have the form u = u + D ), u = u D ), 88) where D ) def = d i ρ i ). 89) i= 6

18 So, when we select a patrolling strategy, we use the following Hurwicz optimismpessimism combination: u = α u + α) u = u) α ) D). 90) Based on this approimate gain function u ), we select the agent allocation ρ a) for which where V d h d ρ a) = u 0 u ), 9) u 0 = V n a V d h d. 9) V d h d u ) d On each zone, the adversary selects a point at which the epected gain V d h d ρ a)) u) 93) is the largest. The value V d h d ρ a)) is constant within the zone, so within each zone, the adversary selects the point at which u) is the largest, i.e., the point at which u) = u) = u) + D). Similarly to the -D case, we thus conclude that the adversary s gain is equal to u 0 + G), 94) where G = α) ma α u ) d i ρ i ) u) d i ρ i ) d. 95) u) d i= i= We need to minimize this gain based on the condition that d ) ρ i ) d = n b. 96) i= For each location, we need to select the values of d functions ρ ),..., ρ d ). The only constraint that comes from 96) is the constraint on their product: d ρ i ) = c 97) i= 7

19 for some constant c. The only effect of all these d values ρ ),..., ρ d ) on the objective function G is via the combination d i ρ i ) i= 98) Thus, for each, we need to optimize the combination 98) under the constraint 97). For this constraint optimization problem, the Lagrange multiplier method leads to the need to optimize the following objective function: d i ρ i ) + λ i= d ) ρ i ) c. 99) Differentiating this epression with respect to ρ i ) and equating the derivative to 0, we conclude that hence where i= i ρ i ) + λ ρ i ) d ρ j ) = 0, 00) j= ρ i ) = c) i, 0) c) def = λ d ρ j ) j= does not depend on i. For the functions 0), we have d i i= ρ i ) = thus the objective function G takes the form G = α) d ma c) u) α d 0) d c), 03) c) u ) d. 04) u) d In terms of the new unknown c), the constraint 96) takes the form d ) c d ) i d = n b. 05) i= For the locations at which the maimum is attained, i.e., at which c) u) = c def ma = ma c) u), 06) 8

20 we get i.e., c) = c ma u), 07) c) = C ma u), 08) def where C ma =. For the set X of locations at which the maimum is not c ma attained, Lagrange multiplier method leads to optimizing the combination α d X c) u ) d + λ u) d c d ) d ) ) i d n b. 09) Differentiating this epression with respect to c) and equating the derivative to 0, we conclude that hence and const c ) u ) + λ d cd ) i= d i = 0, 0) i= c d+ const ) =, ) d u ) i C int i= c) = d ) u)) /d+) /d+) ) i for some constant C int. For this value, the fact that the maimum is not attained means that c) < C ma, i.e., that u) C int i= d ) u)) /d+) /d+) < C ma u). 3) i i= Raising both sides by the power d +, we get an equivalent inequality C int ) d+ d u ) i i= < C ma) d+, 4) u)) d+ 9

21 i.e., equivalently, d i= i u)) d < Cint C ma ) d+. 5) Thus, we arrive at the following formulas for the optimal abstraction. Main result for the multi-d case: optimal abstraction. Let us assume that we know the gain function u) on the d-dimensional area. Instead of using the original gain function u), we consider an abstraction, i.e.: we divide the area into boes and [, ]... [ d, d ], 6) we only consider boes when deciding on the best patrolling strategy. To describe the boes, we use d density functions ρ ),..., ρ d ): the function ρ ) describes how many intervals [, ] we take per unit intervals in the vicinity of a given location, the function ρ ) describes how many intervals [, ] we take per unit intervals in the vicinity of a given location, etc. In this situation, the optimal densities ρ i ) have the form ρ i ) = c) i. 7) To find the function c), we need to select two parameters C int and C ma and take d c) = C ma u) when i= i ) d+ u)) d Cint ; 8) C int C ma c) = d ) u)) /d+) /d+) otherwise. 9) i i= The parameters C int and C ma should be selected in such a way that d ) d ) ρ i ) d = c)) d i d = n b. 0) i= Under this constraint, we must select the values C int and C ma that minimizes the functional 04), in which the optimism-pessimism parameter α describes the decision maker s degree of optimism. i= 0

22 cknowledgments This work was supported in part by the National Science Foundation grants HRD and HRD-4 Cyber-ShRE Center of Ecellence) and DUE References [] L. Hurwicz, Optimality Criteria for Decision Making under Ignorance, Cowles Commission Discussion Paper, Statistics, No. 370, 95. [] C. Kiekintveld, J. Marecki, and M. Tambe, pproimation methods for infinite Bayesian Stackelberg games: modeling distributional payoff uncertainty, Proceedings of the Tenth International Conference on utonomous gents and Multiagent Systems MS 0, Taipei, Taiwan, May 6, 0, Vol. 3, pp [3] C. Kiekintveld, J. Marecki, and M. Tambe, pproimation methods for infinite Bayesian Stackelberg games: modeling distributional payoff uncertainty, in [6], pp [4] V. Kreinovich, Decision making under interval uncertainty and beyond), In: P. Guo and W. Pedrycz eds.), Human-Centric Decision-Making Models for Social Sciences, Springer Verlag, 04, pp [5] R. D. Luce and R. Raiffa, Games and Decisions: Introduction and Critical Survey, Dover, New York, 989. [6] M. Tambe ed.), Security and Game Theory: lgorithms, Deployed Systems, Lessons Learned, Cambridge University Press, 0.