Scaling Invariance and a Characterization of Linear Objective Functions. Saša Pekeč

Transcription

1 Scaling Invariance and a Characterization of Linear Objective Functions Saša Pekeč Decision Sciences The Fuqua School of Business Duke University pekec@duke.edu ADT 2011 October 27, 2011

2 A Generic Discrete Choice Problem A finite set of alternatives [n] := {1, 2,..., n} Info about alternatives given by w = (w 1,..., w n ) R n Value of choice S, S [n], determined by f S : R n R: f S (w) The set of all feasible choices S: H 2 [n] The decision-maker chooses the "best" S H: max S H f S(w) Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

3 A Generic Discrete Choice Problem max f S (w) S H f S can have different functional forms Example: a simple unique functional form can be represented as f S (w) = i S w i max w x Ax b x {0, 1} n 0-1 integer programming problem Note: linear objective function w x is a modeling choice. Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

4 Data representation What do values w 1,..., w n typically represent? money, time, physical measurement (e.g., volume, length, etc.),... What is the effect of the unit of measurement convention to the optimal choice? Should optimal choice be different if money is measured in EUR or USD or GBP? Should optimal choice be different if time is measured in seconds or minutes? A modest scaling invariance requirement: S is an optimal choice for w if and only if S is an optimal choice for λw, λ > 0 Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

5 Invariance to Linear Scaling Invariance to Linear Scaling (ILS) w R n, λ > 0 : f S (w) = max{f S (w) : S H} f S (λw) = max{f S (λw) : S H} minimal invariance requirement: same scale for all w i concept of meaningfulness from Measurement Theory linear objective w x (f S (w) = i S w i) satisfies (ILS) What constraints, if any, does (ILS) impose on the choice of the objective function {f S }? Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

6 (ILS) and linear objectives Note that there are 2 n 1 objective functions f S. (ILS) is a single-parameter linear scaling property. Still, we show that (ILS) severely limits plausible objective functions. Under "reasonable" conditions, all f S have to be linear for (ILS) to hold "Reasonable conditions"??? Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

7 Conditions (L) Locality (N) Normality (C) Completeness (S) Separability Theorem (Preview) If (ILS), (L), (N), (C), (S) all hold, then all f S are linear. Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

8 (L) Locality It is reasonable to assume that the value f S (w) only depends on the weights corresponding to the elements from S. Changing the weight w j corresponding to any element j S, will not change the value of f S. (L) Locality Family of functions {f S } has property (L) if S [n], j S : f S w j = 0 Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

9 (N) Normality The weights w should (in a transparent way) indicate the value of f S for all singletons S. (N) Normality Family of functions {f S } has property (N) if, i, w R n, f {i} (w) = w i The property (N) should not be considered restrictive: If {f S } were not normalized, the problem could be reformulated by introducing new weights w defined by w i := f {i} (w i ). Of course, all other f S would need to be redefined: f S ( w) := f S (w). Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

10 (C) Completeness An unbounded change in w should result in unbounded change in f S (w). In fact, we require that f S (R n ) = R. (C) Completeness Family of functions {f S } has property (C) if every f S is surjective. (The main result holds with a considerably weaker condition.) Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

11 (S) Separability The rate of change of f S (w) with respect to changing w i should depend only on w i (and not on the values of w j, j i). Furthermore, this dependence should be "smooth". (S) Separability A function f has property (S) if for any i [n], there exists a function g i : R R, g i C 1 (R), such that f w i (w) = g i (w i ). Family of functions {f S } has property (S) if every function in the family has property (S). The separability is restrictive from the modeling point of view. Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

12 (S) Separability (S) Separability A function f has property (S) if for any i [n], there exists a function g i : R R, g i C 1 (R), such that f w i (w) = g i (w i ). Family of functions {f S } has property (S) if every function in the family has property (S). The separability is restrictive from the modeling point of view. It excludes: norms: f S (w) = w S p, with p 1 OWA operators: f S (w) = c i w r(i), where r is the rank ordering permutation of w i on S (max and min are special cases) fuzzy measure-based operators ( e.g. Choquet capacity idea) Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

13 The Main Theorem Proposition {f S } satisfies (ILS) for any H if and only if S, T [n], w R n, λ R + : f S (w) f T (w) f S (λw) f T (λw) Theorem Suppose that {f S } satisfies (L), (N), (C), and (S). Then {f S } has property (ILS) if and only if every f S is linear, that is, if and only if for every S [n] there exist constants C S,i, i S, such that f S (w) = i S C S,i w i. Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

14 Proof of the Main Theorem Def. f : R n R is r-homogeneous if λ > 0, w, f (λw) = λ r f (w) If (L), (C), and (ILS) hold, then for S, T [n], f S is r-homogeneous f T is r-homogeneous. Euler s homogeneity relation (Eichhorn, 1978). For r-homogeneous, differentiable f : rf (w) = f (w) w 1 + w 1 f (w) f (w) w w n. w 2 w k For r-homogeneous f that satisfies (S), C i so that n f (w 1,..., w n ) = C i wi r. i=1 (N) implies 1-homogeneity for {f S } Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

15 Main Theorem Variations (N) and (C) are not necessary. Theorem. Suppose that {f S } satisfies (L), and (S) and that at least one f S is r-homogeneous. Also, suppose that S, T, f S (R n ) = f T (R n ). Then (ILS) holds if and only if all f S are linear. Note that (S) is critical: f S = [ (w i ) p] 1/p i S with p > 1 and odd, satisfies all conditions except (S). Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

16 Optimal Aggregation Conditions Aggregating different subsets of {w 1, w 2,..., w n } Each subset S has its own aggregation function f S Want (ILS): w, λ > 0: Properties. f S (w) f T (w) f S (λw) f T (λw) (U) Unanimity. S, w R, f S (w, w,..., w) = w. Note that (U) (N). (Y) Symmetry. Let Π S = {π Π n : π(s) = S} Then for any w and any π Π S f S (w 1, w 2,..., w n ) = f S (w π(1), w π(2),..., w π(n) ) Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

17 ILS and the Arithmetic Mean Theorem Suppose that {f S } satisfies (L), (C), (S), (U), and (Y). Then (ILS) holds if and only if every f S is the arithmetic mean of {w i : i S}. Every f S is linear by the main theorem (and (U) (N)). By (Y), linear coefficients in f S are equal. By (U), linear coefficients are 1/ S. Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

18 Recap Parameters of the decision/optimization problem rarely have unique representation. Representation invariance constrains modeling choices. The simplest case of invariance to linear scaling (money, time, etc.) together with Separability, dictates that any "well-behaved" objective function must be linear. (The Main Theorem.) Symmetry and Unanimity further imply that the arithmetic mean is the only sensible aggregation choice. (Opt. Aggregation Thm.) Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

19 Comments Separability condition without differentiability? Ordinal scales and OWA operators characterization? Issues with meaningfulness: only order statistics survive (i.e., degenerate weight with w i 0 for only one i). If data representation has more degrees of freedom (e.g., interval scales, ordinal scales), constraints are imposed on the structure of the set of feasible solutions. What if the feasible set H depends on parameter representation? Saša Pekeč (Duke University) Invariance to Linear Scaling ADT / 20

20 Scaling Invariance and a Characterization of Linear Objective Functions Saša Pekeč Decision Sciences The Fuqua School of Business Duke University pekec@duke.edu ADT 2011 October 27, 2011