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1 Tactical7_030812 (Marginal spelling corrections 030813) Optimal decentralized tactical decisions of isolated Blue units behind Red lines LOHMANDER Peter, SUAS, Sweden, The paper has been presented as an invited paper at EURO/Informs Istanbul 2003, Military Operations Research III. Keywords: Military Operations Research (46), Game Theory (30) Abstract: The tactical decision problems of isolated blue units (IBU) with limited supplies behind enemy (Red) lines are analysed with mixed strategy game theory. Red has a limited number of mobile units for randomised protection and defence of potential Red targets. The Nash equilibrium mixed attack and defence strategies are determined. The sensitivity of the Nash equilibrium to possible parameter changes is determined in general form. The optimal Blue response to non-optimal Red strategies is described. The dynamics of adaptive strategy frequency changes is analysed. Easily used INTERNET software has been developed for simple decentralized optimisation of the mixed strategies. 2 Optimal decentralized tactical decisions of isolated Blue units behind Red lines LOHMANDER Peter, SUAS, Sweden, Keywords: Military Operations Research (46), Game Theory (30) Abstract: The tactical decision problems of isolated blue units (IBU) with limited supplies behind enemy (Red) lines are analysed with mixed strategy game theory. Red has a limited number of mobile units for randomised protection and defence of potential Red targets. The Nash equilibrium mixed attack and defence strategies are determined. The sensitivity of the Nash equilibrium to possible parameter changes is determined in general form. The optimal Blue response to non-optimal Red strategies is described. The dynamics of adaptive strategy frequency changes is analysed. Easily used INTERNET software has been developed for simple decentralized optimisation of the mixed strategies. www.Lohmander.com 3 Briefing The enemy already controls large parts of our territory. The enemy has located the headquarters of one army brigade to our area and has started using our railroad, transporting troops and armed vehicles. Most enemy resources are being transported to other battle fields. In our neighbourhood, there is one enemy tank, used for local protection of the brigade staff and the railroad. The enemy tank commander has optimized a randomized patrol schedule. We constitute one ranger platoon and have very limited contacts with higher units. We have sufficient supplies for one qualified mission, including explosives, ammunition, food and water. After such a mission, we will have to try to retire to friendly units behind our own lines. 4 The decision problem How should we act? How will the enemy act? What result can we expect if we act optimally? These problems have been well described and analyzed in classical game theory. In this paper, we will start by defining the mixed strategy Nash equilibrium. Then, we will use comparative statics applied to the mixed strategy Nash equilibrium. We will investigate the qualitative effects of changes in the revenues and costs, represented by the elements of the game matrix, on the optimal frequences of our strategy selections. We will also investigate the qualitative effects of changes in one game matrix element on the expected value of the game under optimal play. 5 If the "value" of one particular enemy target increases (our expected net gain from attacking that target increases if it is not protected by the enemy tank), then it will be shown that our optimal frequency of attacking that particular enemy decreases (ceteres paribus). In some cases, if the value (the element in the game matrix) of one particular enemy target increases (if not protected) then the value also increases if it happens to be protected. These two value changes both affect the optimal Nash equilibrium. In different cases, the increasing target values imply that our optimal attack frequences increase, are unchanged or decrease. It will be shown that the optimal frequency of one strategy may increase, be unchanged of decrease if the value of one enemy target increases. The direction of change is a function of all the elements in the game matrix. Hence, it is hard or even impossible to construct general tactical guidelines concerning target selection that really are optimal in different situations. 6 Ranger platoon commanders need to estimate the elements of the game matrix in the field, using locally available information together with target value parameters sent from the headquarters. Then, optimal local platoon level decisions have to be based both on the locally obtained information and on the centrally provided information. 7 Analytical solution with comparative statics 11 12 21 22 Assumption A1. 11 12 11 21 22 12 22 21 A1. guarantees that: 11 22 12 21 0 Probability that Blue attacks Red target T1. Probability that Blue attacks Red target T2. Probability that the Red tank protects T1. Probability that the Red tank protects T2. x (1-x) y (1-y) 8 Expected net gain of Blue: 11 xy 12 x(1 y) 21 (1 x) y 22 (1 x)(1 y) 11xy 12 x 12 xy 21 y 21 xy 22 22 x 22 y 22 xy 22 (12 22 ) x (21 22 ) y (11 12 21 22 ) xy The Nash equilibrium is determined by these equations: (12 22 ) (11 12 21 22 ) y 0 x ( 21 22 ) (11 12 21 22 ) x 0 y The Nash equilibrium is: 22 21 x 11 12 21 22 22 12 * y 11 12 21 22 * 9 0 x* 1 A1 * 0 y 1 Comparative statics: In the following section, a number of derivatives are determined via the quotient rule. In all cases, f denotes the nominator and g denotes the denominator of the expression in question. This simplified notation is used in order to make all steps obvious to the reader and at the same time to save space. A1 g (11 22 12 21 ) 0 x f ´g fg´ ( 22 21 ) 0 2 2 11 (g) (g) * y f ´g fg´ ( 22 12 ) 0 2 2 11 (g) (g) * 10 x f ´g fg´ ( 22 21 )(1) 0 2 2 12 (g) (g) * y* f ´g fg´ g ( 22 12 )(1) 2 2 12 (g) (g) y* (11 12 21 22 ) ( 22 12 ) 12 ( g )2 y 11 12 21 22 22 12 12 ( g )2 * y 21 11 0 2 12 (g) * 11 x* f ´g fg´ (1)(11 12 21 22 ) ( 22 21 )(1) 2 21 (g) ( g )2 x* 11 12 21 22 22 21 2 21 (g) x 12 11 0 2 21 (g) * y* f ´g fg´ ( 22 12 )(1) 2 21 (g) ( g )2 y ( 22 12 ) 0 2 21 (g) * x* f ´g fg´ (11 12 21 22 ) ( 22 21 ) 2 22 (g) ( g )2 12 x* 11 12 21 22 22 21 22 ( g )2 x* 11 12 0 2 22 (g) y* f ´g fg´ (11 12 21 22 ) ( 22 12 ) 2 22 (g) ( g )2 y 11 21 0 2 22 (g) * 13 Summary of the comparative statics analysis: dx* 0 d11 0 * dy 0 dx* 0 d12 0 * dy 0 dx* 0 d 21 0 * dy 0 dx* 0 d 22 0 * dy 0 14 Case #1 #2 #3 #4 Optimal Blue Response The net gain of Blue from Attack T1 with attacking red target T1 (if T1 higher is defended by Red) probability. increases. (Attack T2 with lower probability.) Optimal Red Response Allocate more defence resource time to T1. The net gain of Blue from Attack T1 with attacking red target T1 (if T1 lower probability. is not defended by Red) increases. (Attack T2 with higher probability.) Allocate more defence resource time to T1. The net gain of Blue from Attack T2 with attacking red target T2 (if T2 higher is defended by Red) probability. increases. (Attack T1 with lower probability.) Allocate more defence resource time to T2. The net gain of Blue from Attack T2 with attacking red target T2 (if T2 lower probability. is not defended by Red) increases. (Attack T1 with higher probability.) Allocate more defence resource time to T2. (Allocate less defence resource time to T2.) (Allocate less defence time resources to T2.) (Allocate less defence resource time to T1.) (Allocate less defence time resources to T1.) 15 Expected Blue net gains from game matrix element changes: x y ij ij x ij y ij * * * 0 However, in Nash equilibrium, x . y Hence, * x y (0) (0) ij ij ij ij ij ij * * * 16 Hence, in the two by to case, x* y * 0 11 * x* (1 y* ) 0 12 * * (1 x* ) y* 0 21 * * (1 x )(1 y ) 0 22 * We conclude that: 0 ij * (i, j ) 17 Optimal Blue (and Red) response(s) to non-optimal Red (and Blue) strategies (12 22 ) (11 12 21 22 ) y 0 x ( 21 22 ) (11 12 21 22 ) x 0 y The dynamics of adaptive strategy frequency changes x sgn( ) sgn( x) sgn( ) t x y sgn( ) sgn( y ) sgn( ) t y 18 A quantitative solution to the decision problem In order to make it possible for the ranger platoon commander to optimize the decisions based on local information, a "user friendly" tactical game optimizer has been constructed. 19 Results: The tactical decision problems of isolated (Blue) units with limited supplies behind enemy (Red) lines have been analysed with mixed strategy game theory. The Nash equilibrium mixed attack and defence strategies have been determined in general form. The sensitivity of the Nash equilibrium to possible parameter changes has been determined in general form. If the value of an enemy target increases, it may be optimal to increase, leave unchanged or to decrease the probability of attack. Detailed analysis is necessary. The optimal Blue (and Red) response to non-optimal Red (and Blue) strategies has been described. The dynamics of adaptive strategy frequency changes has been analysed. Easily used INTERNET software has been developed for simple decentralized optimisation of the mixed strategies. 20 References: Chiang, A.C., Fundamental Methods of Mathematical Economics, 2 ed., McGraw-Hill, 1974 Dresher, M., Games of Strategy, Theory and Application, Prentice-Hall, 1961 Försvarsmakten, Armereglemente del 2 Taktik, Stockholm, Sverige, M7741-100612, 1995 Försvarsmakten, Militärstrategisk doktrin, Stockholm, Sverige, M7740-774002, 2002 Isaacs, R., Differential Games, A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, Wiley, 1965 Lohmander, P., The constrained probability orbit of mixed strategy games with marginal adjustment: General theory and timber market application, Swedish Univerity of Agricultural Sciences, Faculty of Forestry, Dept. of Forest Economics, WP-134, 1991 Lohmander, P., Expansion dynamics and noncooperative decisions in stochastic markets Theory and pulp industry application, Helles F. & Linddal M. (editors), Scandinavian Forest Economics, No. 35, 1994 Lohmander, P., The constrained probability orbit of mixed strategy games with marginal adjustment: General theory and timber market application, SYSTEMS ANALYSIS - MODELLING - SIMULATION, Vol. 29, 27-55, 1997 Lohmander, P., Orienterande föreläsning om spelteori – virkesköp m.m., (Web-sida): http://www.sekon.slu.se/~plo/spel5.htm Nash, J.F., Equilibrium points in n-person games, Proceedings of the National Academy of Sciences, U.S.A., 36, 48-49. 1950 von Neumann, J., A numerical method to determine optimum strategy, Naval Research Logistics Quarterly, 1, 1954 Smedberg, M., Om stridens grunder, Page One Publishing AB, Stockholm, Sweden, ISBN 91-7125-028-X, 1994 Washburn, A.R., Two-person zero-sum games, Informs, Topics in operations research, second edition, December 1994 21 Optimize tactical decisions online! www.Lohmander.com Click on "Other options"!