Eisenhower’s Observation Rational Decision . . . In These Terms, What . . . Plans Are Worthless but Options Estimating L 0 Planning Is Everything: Estimating L 1 Conclusions A Theoretical Explanation Acknowledgments of Eisenhower’s Observation Home Page Title Page Angel F. Garcia Contreras, Martine Ceberio, and Vladik Kreinovich ◭◭ ◮◮ ◭ ◮ Department of Computer Science University of Texas at El Paso Page 1 of 14 500 W. University El Paso, Texas 79968, USA Go Back afgarciacontreras@miners.utep.edu, Full Screen mceberio@utep.edu@utep.edu, vladik@utep.edu Close Quit
Eisenhower’s Observation 1. Eisenhower’s Observation Rational Decision . . . In These Terms, What . . . • Dwight D. Eisenhower was: Options – the Supreme Commander of the Allied Expedi- Estimating L 0 tionary Forces in Europe during WW2 Estimating L 1 – and later the US President. Conclusions Acknowledgments • He emphasized that his war experience taught him that Home Page “plans are worthless, but planning is everything”. Title Page • At first glance, this sounds paradoxical: if plans are worthless, why bother with planning at all? ◭◭ ◮◮ ◭ ◮ • In this paper, we show that this Eisenhower’s observa- tion has a meaning: Page 2 of 14 – while following the original plan in constantly Go Back changing circumstances is often not a good idea, Full Screen – the existence of a pre-computed original plan en- Close ables us to produce an almost-optimal strategy. Quit
Eisenhower’s Observation 2. Rational Decision Making: a Brief Reminder Rational Decision . . . In These Terms, What . . . • According to decision making theory: Options – decisions by a rational decision maker Estimating L 0 – can be described as maximize the value a certain Estimating L 1 function known as utility. Conclusions Acknowledgments • E.g., in financial situations, when a company needs to Home Page make a decision, the overall profit can be used as utility. Title Page • To describe a possible action x , we usually need to describe the values of several quantities x 1 , . . . , x n . ◭◭ ◮◮ • E.g., a decision about a plant involves selecting ◭ ◮ amounts x i of manufactured gadgets of different type. Page 3 of 14 • Similarly, we need several quantities a 1 , . . . , a m to de- Go Back scribe a situation. Full Screen • Let u ( x, a ) denote the utility that results from perform- Close ing action x in situation a . Quit
Eisenhower’s Observation 3. In These Terms, What Is Planning Rational Decision . . . In These Terms, What . . . • Let � a describe the original situation. Options • Based on this situation, we come up with an action � x Estimating L 0 that maximizes the corresponding utility: Estimating L 1 Conclusions u ( � x, � a ) = max u ( x, � a ) . x Acknowledgments Home Page • Computing this optimal action � x is what we usually call planning . Title Page ◭◭ ◮◮ • When we need to start acting, the situation may have changed to a � = � a . ◭ ◮ def • Let us denote the corresponding change by ∆ a = a − � a , Page 4 of 14 then a = � a + ∆ a . Go Back Full Screen Close Quit
Eisenhower’s Observation 4. Options Rational Decision . . . In These Terms, What . . . • One possibility is to simply ignore the change, and ap- Options ply the original plan � x to the new situation a = � a +∆ a . Estimating L 0 • This plan is, in general, not optimal for the new situ- Estimating L 1 ation. Conclusions • The actually optimal plan is x opt for which Acknowledgments Home Page u ( x opt , � a + ∆ a ) = max u ( x, � a + ∆ a ) . Title Page x ◭◭ ◮◮ • In comparison with the optimal plan, we lose the def = u ( x opt , � amount L 0 a + ∆ a ) − u ( � x, � a + ∆ a ) . ◭ ◮ • Why cannot we just find the optimal solution for the Page 5 of 14 new situation? Go Back • Optimization is NP-hard, so, it is not possible to find Full Screen the exact optimum in reasonable time. Close Quit
Eisenhower’s Observation 5. Options (cont-d) Rational Decision . . . In These Terms, What . . . • What we can do is: Options – try to use some feasible algorithm – e.g., solving a Estimating L 0 system of linear equations, Estimating L 1 – to modify the plan � x into � x + ∆ x . Conclusions • Due to NP-hardness, this feasibly modified plan is, in Acknowledgments Home Page general, not optimal. def Title Page = u ( x opt , � • We hope that the resulting loss L 1 a +∆ a ) − u ( � x + ∆ x, � a + ∆ a ) is much smaller than L 0 . ◭◭ ◮◮ • In this paper, we show that indeed L 1 ≪ L 0 ; so: ◭ ◮ – even if L 0 is so large that the original plan is worth- Page 6 of 14 less, Go Back – the modified plan may leads to a reasonably small Full Screen loss L 1 ≪ L 0 . Close • This explains Eisenhower’s observation. Quit
Eisenhower’s Observation 6. Estimating L 0 Rational Decision . . . In These Terms, What . . . • We assume that the difference ∆ a is reasonably small. Options def • So, the corresponding difference in action ∆ x opt = Estimating L 0 x opt − � x is also small. Estimating L 1 Conclusions • We can therefore expand L 0 in Taylor series and keep Acknowledgments only terms linear and quadratic in ∆ x : Home Page a + ∆ a ) − u ( x opt − ∆ x opt , � L 0 = u ( x opt , � a + ∆ a ) = Title Page n � ∂u a + ∆ a ) · ∆ x opt ( x opt , � i + ◭◭ ◮◮ ∂x i i =1 ◭ ◮ n n � � ∂ 2 u 1 a +∆ a ) · ∆ x opt · ∆ x opt ∂x i ∂x i ′ ( x opt , � i ′ + o ((∆ a ) 2 ) . 2 · Page 7 of 14 i i =1 i ′ =1 Go Back • By definition, the action x opt maximizes u ( x, � a + ∆ a ). Full Screen • Thus, we have ∂u ( x opt , � a + ∆ a ) = 0. Close ∂x i Quit
Eisenhower’s Observation 7. Estimating L 0 (cont-d) Rational Decision . . . In These Terms, What . . . • So, the above expression for L 0 takes the simplified Options form Estimating L 0 n n � � ∂ 2 u L 0 = 1 a +∆ a ) · ∆ x opt i · ∆ x opt ∂x i ∂x i ′ ( x opt , � i ′ + o ((∆ a ) 2 ) . Estimating L 1 2 · Conclusions i =1 i ′ =1 Acknowledgments • ∆ x opt can be estimated from the condition: i Home Page ∂u a + ∆ a ) = ∂u ( x opt , � x + ∆ x opt , � ( � a + ∆) = 0 . Title Page ∂x i ∂x i ◭◭ ◮◮ x , so ∂u • For a = � a , u is max when x = � ( � a ) = 0. x, � ◭ ◮ ∂x i • Expanding the equation in Taylor series in ∆ x i and Page 8 of 14 ∆ a j and taking ∂u ( � a ) = 0 into account, we get: x, � Go Back ∂x i Full Screen � n � m ∂ 2 u ∂ 2 u a ) · ∆ x opt ∂x i ∂x i ′ ( � i ′ + ( � a ) · ∆ a j + o (∆ x, ∆ a ) = 0 . x, � x, � Close ∂x i ∂a j i ′ =1 j =1 Quit
Eisenhower’s Observation 8. Estimating L 0 (final) Rational Decision . . . In These Terms, What . . . • Thus, the first approximation ∆ x i to the values ∆ x opt i Options satisfies a system of linear equations: Estimating L 0 n m � � ∂ 2 u ∂ 2 u Estimating L 1 ∂x i ∂x i ′ ( � a ) · ∆ x j = − ( � a ) · ∆ a j . x, � x, � ∂x i ∂a j Conclusions i ′ =1 j =1 Acknowledgments • A solution to a system of linear equations is a linear Home Page combination of the right-hand sides. Title Page • Thus, the values ∆ x i are a linear function of ∆ a j . ◭◭ ◮◮ • Substituting these linear expressions into the formula ◭ ◮ for L 0 , we conclude that L 0 is quadratic in ∆ a j : Page 9 of 14 � m � m Go Back k jj ′ · ∆ a j · ∆ a j ′ + o ((∆ a ) 2 ) for some k jj ′ . L 0 = Full Screen j =1 j ′ =1 Close Quit
Eisenhower’s Observation 9. Estimating L 1 Rational Decision . . . In These Terms, What . . . • The 1st approximation ∆ x to the difference ∆ x opt can Options be obtained by solving a system of linear equations. Estimating L 0 • How much do we lose if we use x lin = � x + ∆ x ? Estimating L 1 • Here, ∆ x opt = ∆ x + δx , where δx is of 2nd order in ∆ x Conclusions and ∆ a : δx = O ((∆ a ) 2 ). Acknowledgments • The loss L 1 of using x lin = x opt − δx instead of x opt is: Home Page Title Page L 1 = u ( x opt , � a + ∆ a ) − u ( x lin , � a + ∆ a ) = a + ∆ a ) − u ( x opt − δx, � ◭◭ ◮◮ u ( x opt , � a + ∆ a ) . ◭ ◮ • If we expand this expression in δx , we get: � n Page 10 of 14 ∂u ( x opt , � L 1 = a + ∆ a ) · δx i + ∂x i Go Back i =1 � n � n Full Screen ∂ 2 u 1 ∂x i ∂x i ′ ( x opt , � a + ∆ a ) · δx i · δx i ′ + o (( δx ) 2 ) . 2 · Close i =1 i ′ =1 Quit
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