Rank Robustness of Composite Indices: Dominance and Ambiguity
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Rank Robustness of Composite Indices:
Dominance and Ambiguity
James E. Foster George Washington University & OxfordMark McGillivray Ausaid Suman Seth Vanderbilt University
Composite IndicesMany multidimensional indices take the form
C(x;w) = w·xwhere
x = (x1,x2,…,xD) is a vector of dimensional achievements and
w = (w1,w2,…,wD) is a vector of weights satisfying
w1,…,wD 0 and w1+···+wD = 1
“Composite index”
Human Development Index or HDI (UNDP) Multidimensional Povery Index Mα (Alkire/Foster) Gender Empowerment Index (UNDP) Global Peace Index (Inst. Econ Peace) Environmental Sustainability Index (Yale) Child Well-being Index (Inst. Child Dev.) US News College Ranking
Why this Form? Natural Easy to understand Statistical properties
Key challenge…how to choose weights?
Examples
Methods Normative Statistical e.g. Principal component analysis Equal weights
“Our argument for equal index weights is based on the premise that no objective mechanism exists to determine the relative importance of the different aspects of environmental sustainability.”
Environmental Sustainability Index “…we have no reliable basis for doing [otherwise]”
Mayer and Jencks, 1989
Note: Analogous to “Principle of insufficient reason”
Choice of Weights
Choice of WeightsNote
Inevitable arbitrariness in choice of weightsArrow’s critique of HDI’s weights w = (1/3,1/3,1/3)
Why important?Comparison could be ambiguous or robust
C(x;w0) > C(y;w0)at initial w0 and yet
C(x;w) < C(y;w)at some other reasonable w
“Ambiguous”
Example: Ambiguous Comparison
w = (1/2, 1/4, 1/4)w0 = (1/3, 1/3, 1/3)
Country HDIIreland (x) 0.956
Canada (y) 0.950
Country HDIIreland (x) 0.937
Canada (y) 0.942
C(x;w0) > C(y;w0)at initial w0 and
C(x;w) > C(y;w)at all reasonable w
“Robust”
Example: Robust ComparisonCountry HDIAustralia (x) 0.957
Sweden (y) 0.951
Same ranking for all w
How to discern?Clearly,
Ranking C(x;w0) > C(y;w0) reveals nothing about the robustness or ambiguity of the comparison
The following look the same – but are different
Goal: Provide intuitive criteria for distinguishing between robust and ambiguous comparisons
Country HDI
Ireland (x) 0.956
Canada (y) 0.950
Country HDI
Australia (x) 0.957
Sweden (y) 0.951
Our ApproachDefinitions
D Simplex of dimension D
w0 Initial weighting vectorW Set of “reasonable” weighting vectors around
w0
W is non-empty
(1,0,0)
(0,0,1) (0,1,0)
w0
Our Approach
Notation For any a, b in D, a b denotes ad bd for all d; a > b denotes ad bd with a ≠ b for all d; and a >> b denotes ad > bd for all d
X D: non-empty set of alternatives to be ranked
Define weak robustness relation RW on X by x RW y if and only if C(x;w) C(y;w) for all w W
Characterize RW among all binary relation R on X Based on Bewley (1986)
Our ApproachCharacterization of RW Axioms of R
Quasiordering (Q): R is transitive and reflexive Monotonicity (M): (i) If x > y then x R y; (ii) if x >>
y then y R x cannot hold. Independence (I): Let x, y, z, y', z' X where y' =
x +(1–)y and z' = x + (1–)y' for 0 < < 1. Then y R z if and only if y' R z'.
Continuity (C): The sets {x X | x R z} and {x X | z R x} are closed for all z X.
Our ApproachCharacterization of RW Theorem 1 Suppose that X is closed, convex, and
has a nonempty interior. Then a binary relation R on X satisfies axioms Q, M, I, and C if and only if there exist a non-empty, closed, and convex set W D such that R = RW
Interesting interpretation Maximin criterion of Gilboa and Schmeidler (1989) for multiple priors
Our Approach
Robustness of rankingx CW y if and only if C(x;w0) > C(y;w0) and x RW y
Analogous constructs Knightian uncertainty (Bewley, 1986)Partial comparability (Sen, 1970)Poverty ordering (Foster-Shorrocks, 1988)
Q/ Which W?A/ We use nested sets De where e [0,1]
Epsilon-Contamination model of ambiguity (Ellsburg, 1961)
The C0 Relation
Suppose W = {w0} = D0 Denote resulting relation by C0 so that
x C0 y if and only if C(x;w0) > C(y;w0)Original complete ordering of the composite index
Interpretation of D0
Supremely confident in choice of initial weightings - offers no robustness test at all
Implausible
HDI Top Ten According to C0
Rank Country HDI1 Norway 0.965
2 Iceland 0.960
3 Australia 0.957
4 Ireland 0.956
5 Sweden 0.951
6 Canada 0.950
7 Japan 0.949
8 United States 0.948
9 Switzerland 0.947
10 Netherlands 0.947
Human Development Report 2006, UNDP
Coun. Nor Ice Aus Ire Swe Can Jap USA Swit NethRank 1 2 3 4 5 6 7 8 9 10
Nor 1Ice 2 C0
Aus 3 C0 C0
Ire 4 C0 C0 C0
Swe 5 C0 C0 C0 C0
Can 6 C0 C0 C0 C0 C0
Jap 7 C0 C0 C0 C0 C0 C0
USA 8 C0 C0 C0 C0 C0 C0 C0
Swit 9 C0 C0 C0 C0 C0 C0 C0 C0
Neth 10 C0 C0 C0 C0 C0 C0 C0 C0 C0
Complete Ordering
Column Dominates Row
The C1 Relation
Suppose W = S = D1 Denote resulting ranking by C1 so that
x C1 y if and only if C(x;w0) > C(y;w0) and x R1 y
where x R1 y denotes C(x;w) C(y;w) for all w S
Interpretation of D1
No confidence in choice of initial weightings – offers full robustness test
Stringent requirement
Characterization of C1
e1 = (1,0,0)
e2 = (0,1,0) (0,0,1) = e3
Simplex D
Let C(x;ei) = xi; i = 1,2,3
w0
Theorem 2 Let x, y X. Then (i) x R1 y if and only if x ≥ y and (ii) x C1 y if and only if x > y
Example: Robust Comparison
e3 = (0,0,1)
e1 = (1,0,0)
e2 = (0,1,0)
Swe x3 : 0.949Swe x1 : 0.922
Swe x2 : 0.982
Swe HDI : 0.951
Aus x1 : 0.925Aus HDI : 0.957
Aus x3 : 0.954
Aus x2 : 0.993
x C1 y if and only if x > y
Aus C1 Swe
Coun. HDI x1 x2 x3
Aus 0.957 0.925 0.993 0.954
Swe 0.951 0.922 0.982 0.949
w0
Example: Ambiguous Comparison
e3 = (0,0,1)
e1 = (1,0,0)
e2 = (0,1,0)
Can x3 : 0.959
Can x1 : 0.919
Can x2 : 0.970
Can HDI : 0.950
Ire x1 : 0.882 Ire HDI : 0.956Ire x3 : 0.995
Ire x2 : 0.990
Coun. HDI x1 x2 x3
Ire 0.956 0.882 0.990 0.995
Can 0.950 0.919 0.970 0.959
Ireland Canada ranking is not fully robust.
e1 = (1,0,0)
e2 = (0,1,0) (0,0,1) = e3
Can
Ire
w0 w0
Recall C0 Ranking
Column Dominates Row
Coun. Nor Ice Aus Ire Swe Can Jap USA Swit Neth
Rank 1 2 3 4 5 6 7 8 9 10
Nor 1
Ice 2 C0
Aus 3 C0 C0
Ire 4 C0 C0 C0
Swe 5 C0 C0 C0 C0
Can 6 C0 C0 C0 C0 C0
Jap 7 C0 C0 C0 C0 C0 C0
USA 8 C0 C0 C0 C0 C0 C0 C0
Swit 9 C0 C0 C0 C0 C0 C0 C0 C0
Neth 10 C0 C0 C0 C0 C0 C0 C0 C0 C0
Fully Robust Ranking C1
Column Dominates Row
Coun. Nor Ice Aus Ire Swe Can Jap USA Swit Neth
Rank 1 2 3 4 5 6 7 8 9 10
Nor 1
Ice 2
Aus 3
Ire 4
Swe 5 C1
Can 6 C1
Jap 7
USA 8
Swit 9 C1
Neth 10 C1
Partial Ordering Ce
Consider any e satisfying 0 e 1Define
De = {w' S : w' = (1 – e)w0 + ew for some w S}
Interpretation1 – e degree of confidence in initial weighting w0
Ellsburg (1961)
e “size” of resulting set for checking robustnessEx
D1 = D lowest degree of confidence in w0, largest setD0 = {w0} highest degree of confidence, smallest setDe = intermediate
Partial Ordering Ce
e1
De
(1-e)w0 + ee1 = ve1
w0
1-e
e
Partial Ordering Ce
Suppose W = De Denote resulting ranking by Ce so that
x Ce y if and only if C(x;w0) > C(y;w0) and x Re ywhere x Re y denotes C(x;w) C(y;w) for all w De
w0De
e1
e3e2
Characterization of Ce
Denoteve
d = (1- e)w0 + eed
xed = ve
dx Value of x at ved
xe = (xe1, …,xe
D) Vector of these values
w0De
ve1
e1
e3e2
ve2
ve3
Characterization of Ce
Theorem 3 Let x, y X. Then (i) x Re y if and only if xe ≥ ye and (ii) x Ce y if and only if xe > ye
w0De
ve1
e1
e3e2
ve2
ve3
Partial Ordering Ce
e3 = (0,0,1)
e1 = (1,0,0)
e2 = (0,1,0)
Recall Canada/Ireland example
Can
Ire
Recall Fully Robust Ranking C1
Column Dominates Row
Coun. Nor Ice Aus Ire Swe Can Jap USA Swit Neth
Rank 1 2 3 4 5 6 7 8 9 10
Nor 1
Ice 2
Aus 3
Ire 4
Swe 5 C1
Can 6 C1
Jap 7
USA 8
Swit 9 C1
Neth 10 C1
Ce for e = 1/4
Coun. Nor Ice Aus Ire Swe Can Jap USA Swit Neth
Rank 1 2 3 4 5 6 7 8 9 10
Nor 1
Ice 2
Aus 3 Ce
Ire 4 Ce
Swe 5 Ce Ce Ce
Can 6 Ce Ce Ce
Jap 7 Ce Ce
USA 8 Ce Ce Ce
Swit 9 Ce Ce Ce
Neth 10 Ce Ce Ce Ce Ce
Column Dominates Row
Measure of Robustness
IdeaHow robust is a given comparison?
DenoteA = C(x,w0) - C(y,w0) difference in HDI’s
B = maxwS{C(y,w) - C(x,w)} max dimensional departure
Definer = A/(A+B) Measure of robustness
Measure of Robustness
Theorem 4 Suppose that x C0 y for x, y X and let r be the robustness level associated with this comparison. Then the e-robustness relation x Ce y holds if and only if e ≤ r.
Measure of Robustness
Size of the largest sub-simplex is the measuure of robustness r
e1 = (1,0,0)
e2 = (0,1,0)
Can
Ire
e3 = (0,0,1)
Robustness Calculation
Coun. HDI x1 x2 x3
Ire 0.956 0.882 0.990 0.995
Can 0.950 0.919 0.970 0.959
Difference 0.006 -0.037 0.02 0.036
A = 0.006B = 0.037r = 0.006/(0.006 + 0.037) = 0.139
Robustness Calculation
Coun. HDI x1 x2 x3
Aus 0.957 0.925 0.993 0.954Swe 0.951 0.922 0.982 0.949Difference 0.006 0.003 0.011 0.005
A = 0.006 B = - 0.003 r = 1
Note that both comparisons have same A
Yet very different robustness levels!
Coun. Nor Ice Aus Ire Swe Can Jap USA Swit NethRank 1 2 3 4 5 6 7 8 9 10
Nor 1Ice 2 20 Aus 3 35 19 Ire 4 86 14 4 Swe 5 53 94 100 11 Can 6 61 100 60 14 14 Jap 7 28 34 23 9 7 2 USA 8 77 28 17 67 5 3 1 Swit 9 49 100 41 16 17 20 6 2 Neth 10 100 68 57 47 25 13 4 7 1
Measure of Robustness (%)
Column Dominates Row
Conclusion
Propose a tool for measuring robustness of ranking The idea is motivated by partial orderings, epsilon-
contamination, and Knightian uncertainty The tool can be extended to the case of general
means (e.g., HPI) A measure of robusteness is proposed Robustness Vs. Redundancy (McGillivray 1991)
Thank you
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