Peter Lohmander Professor SLU Umea, SE-901 83, Sweden, Lohmander

1. A Stochastic Differential (Difference) Game Model With an LP Subroutine for Mixed and Pure Strategy OptimizationINFORMS International Meeting 2007, Puerto Rico Peter Lohmander Professor SLU Umea, SE-901 83, Sweden, http://www.Lohmander.com Version 2007-09-29

2. Abstract:  This paper presents a stochastic two person differential (difference) game model with a linear programming subroutine that is used to optimize pure and/or mixed strategies as a function of state and time. In ”classical dynamic models”, ”pure strategies” are often assumed to be optimal and deterministic continuous path equations can be derived. In such models, scale and timing effects in operations are however usually not considered. When ”strictly convex scale effects” in operations with defence and attack (”counter air” or ”ground support”) are taken into consideration, dynamic mixed OR pure strategies are optimal in different states. The optimal decision frequences are also functions of the relative importance of the results from operations in different periods. The expected cost of forcing one participant in the conflict to use only pure strategies is determined. The optimal responses of the opposition in case one participant in the conflict is forced to use only pure strategies are calculated. Dynamic models of the presented type, that include mixed strategies, are highly relevant and must be used to determine the optimal strategies. Otherwise, considerable and quite unnecessary losses should be expected.

3. Inspiration: • Isaacs R. (1965) Differential Games – A mathematical theory with applications to warfare and pursuit, control and optimization, Wiley, 1965 (Also Dover, 1999) • Washburn, A.R. (2003) Two-person zero-sum games, 3 ed., INFORMS, Military applications society, Topics in operations research

4. Flexibility Some resources may be used in different ways. This is true and important in the air force, in the army and in the enterprises in the economy.

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9. Y t Y X X

14. Special case 1: Special case 2:

15. In Model 1:

16. In Model 2:

17. In Model 3 and Model 4:

18. Model 1

19. Y t Y X X

20. Model 1 !TAW_070611; !Peter Lohmander; Model: sets: xset/1..3/:; yset/1..3/:; nset/1..3/:; D1set/1..3/:; D2set/1..3/:; xyset(xset,yset):; xynset(xset,yset,nset):V; xyD1set(xset,yset,D1set):GS1, CA1; xyD2set(xset,yset,D2set):GS2, CA2; xynD1s(xset,yset,nset,D1set):xdec; xD1D2s(xset, D1set, D2set):; xD1D2Xs(xset,D1set, D2set, xset):TPX; yD2D1s(yset,D2set,D1set):; yD2D1yset(yset,D2set,D1set,yset):TPY; endsets

21. max = Z; Z = @sum(xyset(x,y):V(x,y,3)); !Terminal condition: No more activities occur when n <= 1 . ; @for(xyset(x,y):V(x,y,1) = 0);

22. ! The last day when activities occur, n=2, all resources should be used for GS. ; @for(xyset(x,y):V(x,y,2) = x-y);

23. ! Day n >= 3, you may use some resources for GS and some for CA. The rows and columns represent the resources used for GS. ; @for(xyD1set(x,y,D1)|D1 #LE# x: GS1(x,y,D1) = D1-1 ); @for(xyD1set(x,y,D1)|D1 #GT# x: GS1(x,y,D1) = 0 ); @for(xyD1set(x,y,D1): CA1(x,y,D1) = x - 1 - GS1(x,y,D1) ); @for(xyD2set(x,y,D2)|D2 #LE# y: GS2(x,y,D2) = D2-1 ); @for(xyD2set(x,y,D2)|D2 #GT# y: GS2(x,y,D2) = 0 ); @for(xyD2set(x,y,D2): CA2(x,y,D2) = y - 1 - GS2(x,y,D2));

24. @for(xynset(x,y,n): @sum(D1set(D1):xdec(x,y,n,D1)) = 1); @for(xynD1s(x,y,n,D1)| D1#GT#x: xdec(x,y,n,D1) = 0);

25. !Sum (Transition probability) = 1; @for(xD1D2s(x1,GS1,CA2): @sum(xset(x2): TPX(x1,GS1,CA2,x2) ) = 1 ); @for(yD2D1s(y1,GS2,CA1): @sum(yset(y2): TPY(y1,GS2,CA1,y2) ) = 1 );

26. !********************************************; ! Transition probability calculations for x ; !********************************************; ! Transition probabilities for x if x1 = 0; @for(xD1D2s(x1,GS1P1,CA2P1)|x1#EQ#1 : TPX(x1,GS1P1,CA2P1,x1) = 1 ); ! Transition probabilities for x if CA2 = 0; @for(xD1D2s(x1,GS1P1,CA2P1)|CA2P1 #EQ# 1 : TPX(x1,GS1P1,CA2P1,x1) = 1 ); ! Transition probabilities for x if GS1 = 0; @for(xD1D2s(x1,GS1P1,CA2P1)|GS1P1 #EQ# 1 : TPX(x1,GS1P1,CA2P1,x1) = 1 );

27. ! Transition probabilities for x if GS1 >= 1 and CA2 = 1; @for(xD1D2s(x1,GS1P1,CA2P1)|x1#GE#2 #AND# GS1P1#GE#2 #AND# CA2P1#EQ#2 : TPX(x1,GS1P1,CA2P1,x1) = 1/2 ); @for(xD1D2s(x1,GS1P1,CA2P1)|x1#GE#2 #AND# GS1P1#GE#2 #AND# CA2P1#EQ#2 : TPX(x1,GS1P1,CA2P1,x1-1) = 1/2 );

28. ! Transition probabilities for x if GS1 = 1 and CA2 = 2; @for(xD1D2s(x1,GS1P1,CA2P1)|x1#GE#2 #AND# GS1P1#EQ#2 #AND# CA2P1 #EQ# 3 : TPX(x1,GS1P1,CA2P1,x1) = 1/4 ); @for(xD1D2s(x1,GS1P1,CA2P1)|x1#GE#2 #AND# GS1P1#EQ#2 #AND# CA2P1 #EQ# 3 : TPX(x1,GS1P1,CA2P1,x1-1) = 3/4 );

29. ! Transition probabilities for x if GS1 >= 2 and CA2 = 2; @for(xD1D2s(x1,GS1P1,CA2P1)|x1#GE#3 #AND# GS1P1#GE#3 #AND# CA2P1 #EQ# 3 : TPX(x1,GS1P1,CA2P1,x1) = 1/4 ); @for(xD1D2s(x1,GS1P1,CA2P1)|x1#GE#3 #AND# GS1P1#GE#3 #AND# CA2P1 #EQ# 3 : TPX(x1,GS1P1,CA2P1,x1-1) = 1/2 ); @for(xD1D2s(x1,GS1P1,CA2P1)|x1#GE#3 #AND# GS1P1#GE#3 #AND# CA2P1 #EQ# 3 : TPX(x1,GS1P1,CA2P1,x1-2) = 1/4 );

30. !*******************************************; ! Transition probability calculations for y ; !*******************************************; ! Transition probabilities for y if y1 = 0; @for(yD2D1s(y1,GS2P1,CA1P1)|y1#EQ#1 : TPY(y1,GS2P1,CA1P1,y1) = 1 ); ! Transition probabilities for y if CA1 = 0; @for(yD2D1s(y1,GS2P1,CA1P1)|CA1P1 #EQ# 1 : TPY(y1,GS2P1,CA1P1,y1) = 1 ); ! Transition probabilities for y if GS2 = 0; @for(yD2D1s(y1,GS2P1,CA1P1)|GS2P1 #EQ# 1 : TPY(y1,GS2P1,CA1P1,y1) = 1 );

31. ! Transition probabilities for y if GS2 >= 1 and CA1 = 1; @for(yD2D1s(y1,GS2P1,CA1P1)|y1#GE#2 #AND# GS2P1#GE#2 #AND# CA1P1#EQ#2 : TPY(y1,GS2P1,CA1P1,y1) = 1/2 ); @for(yD2D1s(y1,GS2P1,CA1P1)|y1#GE#2 #AND# GS2P1#GE#2 #AND# CA1P1#EQ#2 : TPY(y1,GS2P1,CA1P1,y1-1) = 1/2 );

32. ! Transition probabilities for y if GS2 = 1 and CA1 = 2; @for(yD2D1s(y1,GS2P1,CA1P1)|y1#GE#2 #AND# GS2P1#EQ#2 #AND# CA1P1#EQ#3 : TPY(y1,GS2P1,CA1P1,y1) = 1/4 ); @for(yD2D1s(y1,GS2P1,CA1P1)|y1#GE#2 #AND# GS2P1#EQ#2 #AND# CA1P1#EQ#3 : TPY(y1,GS2P1,CA1P1,y1-1) = 3/4 );

33. ! Transition probabilities for y if GS2 >= 2 and CA1 = 2; @for(yD2D1s(y1,GS2P1,CA1P1)|y1#GE#3 #AND# GS2P1#GE#3 #AND# CA1P1 #EQ# 3 : TPY(y1,GS2P1,CA1P1,y1) = 1/4 ); @for(yD2D1s(y1,GS2P1,CA1P1)|y1#GE#3 #AND# GS2P1#GE#3 #AND# CA1P1 #EQ# 3 : TPY(y1,GS2P1,CA1P1,y1-1) = 1/2 ); @for(yD2D1s(y1,GS2P1,CA1P1)|y1#GE#3 #AND# GS2P1#GE#3 #AND# CA1P1 #EQ# 3 : TPY(y1,GS2P1,CA1P1,y1-2) = 1/4 );

34. In Model 1:

35. In Model 3 and Model 4:

36. !**********************; ! The game problems ; !**********************; @for(xyset(x,y): @for(D2set(D2)|D2 #LE# y: [ydec] V(x,y,3) <= @sum(D1set(D1): (GS1(x,y,D1)/(1+CA2(x,y,D2)) - GS2(x,y,D2)/(1+CA1(x,y,D1)) + @sum(xyset(x2,y2): TPX(x,GS1(x,y,D1)+1,CA2(x,y,D2)+1,x2)*TPY(y,GS2(x,y,D2)+1,CA1(x,y,D1)+1,y2)* V(x2,y2,2) )) * xdec(x,y,3,D1) ) ); );

37. @for(xynset(x,y,n): @FREE(V(x,y,n))); end

38. In Model 3 and Model 4:

39. Model 3; (x,y,t) = (2,2,T-1)

40. Model 3; (x,y,t) = (2,2,T-1)

45. Results from Model 1

46. Table 1.1 V(x,y,3) with Model 1 (This (n=3) means that t =T-1 since we use backward recursion and period T+1 is defined as n=1. )

47. Y t Y X X

48. Table 1.2 (GS1*, GS2*)at n=3 with Model 1

49. Table 1.2 (GS1*, GS2*)at n=3 with Model 1

50. Y t Y X X