What Decision to Make In a How Interval . . . Conflict Situation - PowerPoint PPT Presentation

How Conflict . . . An Algorithm for . . . Need for Parallelization Need to Take . . . What Decision to Make In a How Interval . . . Conflict Situation under Need for a More . . . Analysis of the Problem Interval Uncertainty: What Is Known: . . . Algorithm for Solving . . . Efficient Algorithms for the Home Page Hurwicz Approach Title Page ◭◭ ◮◮ Bart� lomiej Jacek Kubica 1 , Andrzej Pownuk 2 , and Vladik Kreinovich 2 ◭ ◮ 1 Department of Applied Informatics, Warsaw University of Life Sciences ul. Nowoursynowska 159 02-776 Warsaw, Poland Page 1 of 19 bartlomiej.jacek.kubica@gmail.com 2 Computational Science Program, University of Texas at El Paso Go Back El Paso, TX 79968, USA, ampownuk@utep.edu, vladik@utep.edu Full Screen Close Quit

How Conflict . . . An Algorithm for . . . 1. How Conflict Situations Are Usually Described Need for Parallelization • In many practical situations – e.g., in security – we Need to Take . . . have conflict situations. How Interval . . . Need for a More . . . • For example, a terrorist group wants to attack one of Analysis of the Problem our assets, while we want to defend them. What Is Known: . . . • For each possible pair of strategies ( i, j ), let u ij be the Algorithm for Solving . . . gain of the first side (negative if this is a loss). Home Page • Let v ij be the gain of the second side. Title Page • A conflict situation is when we cannot improve v with- ◭◭ ◮◮ out worsening u . ◭ ◮ • Example: zero-sum games , when v ij = − u ij . Page 2 of 19 • While zero-sum games are a useful approximation, they Go Back are not always a perfect description of the situation. Full Screen Close Quit

How Conflict . . . An Algorithm for . . . 2. Describing Conflict Situations (cont-d) Need for Parallelization • For example, the main objective of the terrorists may Need to Take . . . be publicity, so: How Interval . . . Need for a More . . . – a small attack in the country’s capital may not Analysis of the Problem cause much damage but bring media attention, What Is Known: . . . – a serious attack in a remote are may be more dam- Algorithm for Solving . . . aging but not as media-attractive. Home Page • To take this difference into account, we need, for each Title Page pair of strategies ( i, j ), to describe both: ◭◭ ◮◮ – the gain u ij of the first side and ◭ ◮ – the gain v ij of the second side. Page 3 of 19 • In general, we do not necessarily have v ij = − u ij . Go Back Full Screen Close Quit

How Conflict . . . An Algorithm for . . . 3. It Is Often Beneficial to Act Randomly Need for Parallelization • If we only one security person and two objects to pro- Need to Take . . . tect, then we can: How Interval . . . Need for a More . . . – post this person at the first objects and Analysis of the Problem – post him/her at the second object. What Is Known: . . . • If we follow one of these strategies, then the adversary Algorithm for Solving . . . will attack the other (unprotected) object. Home Page • It is thus more beneficial to assign the security person Title Page to one of the objects at random. ◭◭ ◮◮ • This way, for each object of attack, there will be a 50% ◭ ◮ probability that this object will be defended. Page 4 of 19 • In general, the first side’s strategy can be described by Go Back the probabilities p 1 , . . . , p n of selecting an arrangement: n Full Screen � p i = 1 . Close i =1 Quit

How Conflict . . . An Algorithm for . . . 4. Toward Precise Formulation of the Problem Need for Parallelization • Similarly, the second side selects probabilities Need to Take . . . m How Interval . . . q 1 , . . . , q m for which � q j = 1 . j =1 Need for a More . . . • The expected gains of the two sides are: Analysis of the Problem n m n m What Is Known: . . . � � � � g 1 ( p, q ) = p i · q j · u ij and g 2 ( p, q ) = p i · q j · v ij . Algorithm for Solving . . . i =1 j =1 i =1 j =1 Home Page • Once the 1st side selects the probabilities p i , the 2nd Title Page side knows them – simply by observing the past history. ◭◭ ◮◮ • So, the 2nd side selects a strategy q for which its gain ◭ ◮ is the largest possible: Page 5 of 19 g 2 ( p, q ( p )) = max g 2 ( p, q ) . q Go Back • Similarly, the 2nd side select a strategy q for which Full Screen def g 2 ( p ( q ) , q ) → max q , where p ( q ) = arg max g 1 ( p, q ) . Close p Quit

How Conflict . . . An Algorithm for . . . 5. Towards an Algorithm for Solving this Problem Need for Parallelization • Once the strategy p is selected, the 2nd side selects q Need to Take . . . that maximizes g 2 ( p, q ). How Interval . . . Need for a More . . . • The expression g 2 ( p, q ) is linear in terms of q j . Analysis of the Problem • Thus, g 2 ( p, q ) is the convex combination of gains What Is Known: . . . corr. to deterministic strategies: Algorithm for Solving . . . m n Home Page def � � g 2 ( p, q ) = q j · q 2 j ( p ) , where g 2 j ( p ) = p i · v ij . Title Page j =1 i =1 ◭◭ ◮◮ • So, the largest possible gain is attained when q is a ◭ ◮ deterministic strategy. Page 6 of 19 • The j -th strategy is selected it is better than others: Go Back n n � � p i · v ij ≥ p i · v ik for all k � = j. Full Screen i =1 i =1 Close Quit

How Conflict . . . An Algorithm for . . . 6. Towards an Algorithm (cont-d) Need for Parallelization • For strategies p for which the second side selects the Need to Take . . . n How Interval . . . � j -th response, the gain of the 1st side is p i · u ij . i =1 Need for a More . . . • Among all strategies p with this “ j -property”, we select Analysis of the Problem the one with max expected gain of the 1st side. What Is Known: . . . Algorithm for Solving . . . • This can be found by optimizing a linear function under Home Page constraints which are linear inequalities. Title Page • It is known that for such linear programming problems, ◭◭ ◮◮ there are efficient algorithms. ◭ ◮ • Then, we find j for which the gain is the largest. Page 7 of 19 Go Back Full Screen Close Quit

How Conflict . . . An Algorithm for . . . 7. An Algorithm for Solving the Problem Need for Parallelization • For each j from 1 to m , we solve the following linear Need to Take . . . programming (LP) problem: How Interval . . . Need for a More . . . n p ( j ) � · u ij → max under the constraints Analysis of the Problem i p ( j ) i =1 i What Is Known: . . . Algorithm for Solving . . . n n n p ( j ) = 1 , p ( j ) p ( j ) p ( j ) � � � ≥ 0 , i · v ij ≥ i · v ik for all k � = j. Home Page i i i =1 i =1 i =1 Title Page � � • We then select p ( j ) = p ( j ) 1 , . . . , p ( j ) , 1 ≤ j ≤ m for ◭◭ ◮◮ n n p ( j ) ◭ ◮ which the value � · u ij is the largest. i i =1 Page 8 of 19 • Comment. Solution is simpler in zero-sum situations, Go Back where we only need to solve one LP problem. Full Screen Close Quit

How Conflict . . . An Algorithm for . . . 8. Need for Parallelization Need for Parallelization • When each side has a small number of strategies, the Need to Take . . . corresponding problem is easy to solve. How Interval . . . Need for a More . . . • However, e.g., when we assign air marshals to different Analysis of the Problem international flights, the number of strategies is huge. What Is Known: . . . • Then, the only way to solve the problem is to perform Algorithm for Solving . . . at least some computations in parallel. Home Page • Good news: all m linear programming problems can Title Page be solved on different processors. ◭◭ ◮◮ • Not so good news: programming problems are P-hard, ◭ ◮ i.e., provably the hardest to parallelize efficiently. Page 9 of 19 Go Back Full Screen Close Quit

How Conflict . . . An Algorithm for . . . 9. Need to Take Uncertainty into Account Need for Parallelization • In practice, we rarely know the exact gains u ij and v ij . Need to Take . . . How Interval . . . • At best, we know the bounds on these gains, i.e., we Need for a More . . . know: Analysis of the Problem – the interval [ u ij , u ij ] that contains the actual (un- What Is Known: . . . known) values u ij , and Algorithm for Solving . . . – the interval [ v ij , v ij ] that contains the actual (un- Home Page known) values v ij . Title Page • It is therefore necessary to decide what to do in such ◭◭ ◮◮ situations of interval uncertainty. ◭ ◮ Page 10 of 19 Go Back Full Screen Close Quit

How Conflict . . . An Algorithm for . . . 10. How Interval Uncertainty is Taken into Ac- Need for Parallelization count Now Need to Take . . . • In the above description of a conflict situation, we men- How Interval . . . tioned that: Need for a More . . . Analysis of the Problem – when we select the strategy p , What Is Known: . . . – we maximize the worst-case situation, i.e., the Algorithm for Solving . . . smallest possible gain min g 1 ( p, q ). Home Page q • It seems reasonable to apply the same idea to the case Title Page of interval uncertainty. ◭◭ ◮◮ • So, we maximize the smallest possible gain g 1 ( p, q ): ◭ ◮ – over all possible strategies q of the 2nd side and Page 11 of 19 – over all possible values u ij ∈ [ u ij , u ij ]. Go Back • Efficient algorithms for such worst-case formulation Full Screen have indeed been proposed. Close Quit