Formulation of the . . . Changes Cannot Be . . . What Is Our Objective Since Changes Are . . . A Bad Plan Is Better Than Formulation of the . . . No Plan: A Theoretical What Does “No Plan” . . . What We Mean by a Plan Justification of an Empirical What We Mean by a . . . Conclusion: the . . . Observation Home Page Songsak Sriboonchitta 1 and Vladik Kreinovich 2 Title Page ◭◭ ◮◮ 1 Faculty of Economics, Chiang Mai University, Thailand songsakecon@gmail.com ◭ ◮ 2 University of Texas at El Paso, El Paso, Texas 79968, USA vladik@utep.edu Page 1 of 23 Go Back Full Screen Close Quit
Formulation of the . . . Changes Cannot Be . . . 1. Formulation of the Problem What Is Our Objective • In his book “Zero to One”, software mogul Peter Thiel Since Changes Are . . . lists the lessons he learned from his business practice. Formulation of the . . . What Does “No Plan” . . . • Most of these lessons make intuitive sense, with one What We Mean by a Plan exception. What We Mean by a . . . • This exception is his observation that a bad plan is Conclusion: the . . . better than no plan. Home Page • At first glance, this empirical observation seems to be Title Page counterintuitive. ◭◭ ◮◮ • In this paper, we provide a possible theoretical expla- ◭ ◮ nation for this empirical observation. Page 2 of 23 Go Back Full Screen Close Quit
Formulation of the . . . Changes Cannot Be . . . 2. Describing the Problem in Precise Terms What Is Our Objective • We decide between different plans of action. Since Changes Are . . . Formulation of the . . . • There may be many parameters that describe possible What Does “No Plan” . . . actions. What We Mean by a Plan • For example, for the economy of a country: What We Mean by a . . . – the central bank can set different borrowing rates, Conclusion: the . . . Home Page – the government can set different values of the min- imal wage and of unemployment benefits, etc. Title Page ◭◭ ◮◮ • For a company, such parameters include: ◭ ◮ – percentage of revenue that goes into research and development, Page 3 of 23 – percentage of revenue that goes into advertisement, Go Back etc. Full Screen Close Quit
Formulation of the . . . Changes Cannot Be . . . 3. Describing the Problem in Precise Terms What Is Our Objective (cont-d) Since Changes Are . . . • In general, let us denote the number of such parameters Formulation of the . . . by n , and the parameters themselves by x 1 , . . . , x n . What Does “No Plan” . . . What We Mean by a Plan • From this viewpoint, selecting an action means select- What We Mean by a . . . ing the appropriate values of all these parameters. Conclusion: the . . . • In mathematical terms, we select a point x = Home Page ( x 1 , . . . , x n ) in the corresponding n -dimensional space. Title Page • Let x (0) 1 , . . . , x (0) denote the values of the parameters n ◭◭ ◮◮ corresponding to the current moment of time t 0 . ◭ ◮ • Our goal is to select parameters at future moments of Page 4 of 23 time t 1 = t 0 + h , t 2 = t 0 + 2 h , . . . , t T = t 0 · T · h . Go Back Full Screen Close Quit
Formulation of the . . . Changes Cannot Be . . . 4. Changes Cannot Be Too Radical What Is Our Objective • It is very difficult to make fast drastic changes. Since Changes Are . . . Formulation of the . . . • There is a large amount of inertia in economic systems. What Does “No Plan” . . . • Therefore, we will only consider possible actions What We Mean by a Plan x 1 , . . . , x n which are close to the initial state: What We Mean by a . . . x i = x (0) Conclusion: the . . . + ∆ x i for small ∆ x i . i Home Page • Changes x ( t j +1 ) − x ( t j ) from one moment of time t j to Title Page the next one t j +1 are even more limited. ◭◭ ◮◮ • Let b be the upper bound on such changes: ◭ ◮ � n � Page 5 of 23 ( x i ( t j +1 ) − x i ( t j )) 2 ≤ b. � � � x ( t j +1 ) − x ( t j ) � = � Go Back i =1 Full Screen Close Quit
Formulation of the . . . Changes Cannot Be . . . 5. Changes Cannot Be Too Radical (cont-d) What Is Our Objective • That we talk about changes means that we are not Since Changes Are . . . completely satisfied with the current situations. Formulation of the . . . What Does “No Plan” . . . • The more changes we undertake at each moment of What We Mean by a Plan time, the faster we will reach the desired state. What We Mean by a . . . • The size of each change is limited by the bound b . Conclusion: the . . . Home Page • Within this limitation, the largest possible changes are changes of the largest possible size b . Title Page • Thus, we assume that all the changes from one moment ◭◭ ◮◮ of time to the next one are of the same size b : ◭ ◮ � n � Page 6 of 23 ( x i ( t j +1 ) − x i ( t j )) 2 = b. � � � x ( t j +1 ) − x ( t j ) � = � Go Back i =1 Full Screen Close Quit
Formulation of the . . . Changes Cannot Be . . . 6. What Is Our Objective What Is Our Objective • We discuss which plans are better. Since Changes Are . . . Formulation of the . . . • This assumes that we have agreed on how we gauge What Does “No Plan” . . . the effect of different plans. What We Mean by a Plan • So, we have agreed on a numerical criterion y that What We Mean by a . . . describes: Conclusion: the . . . Home Page – for each possible action, – how good is the result of this action. Title Page ◭◭ ◮◮ • The value of this criterion depends on the action: y = f ( x 1 , . . . , x n ) for some function f ( x 1 , . . . , x n ). ◭ ◮ • In some cases, we may know this function. Page 7 of 23 • However, in general, we do not know the exact form of Go Back this function. Full Screen Close Quit
Formulation of the . . . Changes Cannot Be . . . 7. Since Changes Are Small, We Can Simplify the What Is Our Objective Expression for the Objective Function Since Changes Are . . . • We are interested in the objective function Formulation of the . . . f ( x 1 , . . . , x n ) in the small vicinity of the original What Does “No Plan” . . . state x (0) = ( x (0) 1 , . . . , x (0) n ): What We Mean by a Plan f ( x 1 , . . . , x n ) = f ( x (0) What We Mean by a . . . 1 + ∆ x 1 , . . . , x (0) n + ∆ x n ) . Conclusion: the . . . • Since the deviations ∆ x i are small, we keep only linear Home Page terms in Taylor expansion: Title Page n f ( x (0) � 1 + ∆ x 1 , . . . , x (0) ◭◭ ◮◮ n + ∆ x n ) = a 0 + a i · ∆ x i , i =1 ◭ ◮ = ∂f def def = f ( x (0) 1 , . . . , x (0) where a 0 n ) and a i ∂x i | x = x (0) . Page 8 of 23 Go Back • Maximizing this expression is equivalent to maximizing n Full Screen � the linear part a i · ∆ x i . Close i =1 Quit
Formulation of the . . . Changes Cannot Be . . . 8. Formulation of the Problem What Is Our Objective • Let us denote the deviations ∆ x i by u i . Since Changes Are . . . Formulation of the . . . • Then, we arrive at the following problem: What Does “No Plan” . . . • We start with the values What We Mean by a Plan u (0) = ( u 1 , . . . , u n ) = (0 , . . . , 0) . What We Mean by a . . . Conclusion: the . . . • At each moment of time, we change the action by a Home Page change of a given size b : Title Page � n � ◭◭ ◮◮ ( u i ( t j +1 ) − u i ( t j )) 2 = b. � � � u ( t j +1 ) − u ( t j ) � = � ◭ ◮ i =1 Page 9 of 23 • We want to gradually change u i so as to maximize Go Back n � a i · u i . Full Screen i =1 Close Quit
Formulation of the . . . Changes Cannot Be . . . 9. What Does “No Plan” Mean What Is Our Objective • An intuitive understanding of what “no plan” means Since Changes Are . . . that: Formulation of the . . . What Does “No Plan” . . . – at each moment of time, we undertake a random What We Mean by a Plan change, What We Mean by a . . . – uncorrelated with all the previous changes. Conclusion: the . . . • So, u ( t j +1 ) − u ( t j ) is a vector of length b with a random Home Page direction. Title Page • The resulting trajectory u ( t ) is thus an n -dimensional ◭◭ ◮◮ random walk (= n -dimensional Brownian motion). ◭ ◮ Page 10 of 23 Go Back Full Screen Close Quit
Formulation of the . . . Changes Cannot Be . . . 10. What We Mean by a Plan What Is Our Objective • Intuitively, a plan means that we have a systematic Since Changes Are . . . change u ( t ). Formulation of the . . . What Does “No Plan” . . . • We consider local planning, for a few cycles t 1 , t 2 , . . . , def What We Mean by a Plan for which the difference ∆ t = t − t 0 is small. What We Mean by a . . . • Thus, we can expand u ( t ) = u ( t 0 + ∆ t ) into Taylor Conclusion: the . . . series and keep only linear terms in this expansion: Home Page = du def u ( t ) = u ( t 0 + ∆ t ) = u ( t 0 ) + v · ∆ t, where v . dt | t = t 0 Title Page • By definition of the deviation u ( t ), we have u ( t 0 ) = 0, ◭◭ ◮◮ and thus, u ( t ) = v · ∆ t . ◭ ◮ • So, the change between each moment of time and the Page 11 of 23 next one takes the form Go Back u ( t j +1 ) − u ( t j ) = v · ( t j +1 − t j ) = v · h. Full Screen • If we see that the objective functions decreases, we Close abandon the plan and select a new one. Quit
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