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Entropy: Measure ofDiversity?
David Lee Baker
David E. Booth
William Acar
Management & Information Systems Working Paper MIS2007-08:1(Do not cite without permission)
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EntropyIn Management Strategy I
Managers are interested in knowing howdiversified are the firms lines of business
It has been considered (and aptlydiscussed and hotly debated) that morediverse businesses are more profitable
May not always be the case though
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EntropyIn Management Strategy II
In diversification we need to consider related(similar to the firms core business) versusunrelated (dissimilar) diversification
In related diversification the firms several linesof business, even though distinct, still possesssome kind of fit
In unrelated diversification there is no common
linkage or element of fit among the firms lines ofbusiness In this sense unrelated diversification may be
considered as pure diversification
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DiversityAn Example
If we take a beaker of water (H2O) and a veryconcentrated solution of red food coloring andwe then add a drop of the coloring to the water
we will see the red color diffusing throughout thewater and thus, we go from concentration todiversification
Economists, as well as chemists and physicists,
want to define concentration vs. diversification
Concentration and diversification are two endsof the spectrum
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EntropyBackground & History I
Introduction of a mysterious entity calledthe H-function, or statistical entropy, by
Ludwig Boltzmann (1896) Defined as the mean value of the logarithm ofa probability density function (p.d.f.)
Measured the amount of uncertainty about thepossible states of a physical system
Because there was still disagreement about theexistence of atoms his statistical entropygenerated much debate
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EntropyBackground & History II
Claude E. Shannon (1948) generalized Boltzmannsentropy to information theory and proved that it hadthe properties that allow it to be taken as theaverage amount of information conveyed by a
discrete random variable about another Mathematicians have further refined the Shannonentropy, and new tools, such as the relative orconditional entropies have been developed
Norbert Wiener and Claude E. Shannon along with
others extended Boltzmanns earlier theories tomore general cases Shannon had studied under Wiener at MIT in the late 1930s,
graduating with both a masters and doctorate in mathematics
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EntropyBackground & History III
Mathematically this reduces to the amountof uncertainty contained in a probabilistic
experimentA as measured by thefunction:
Hm (p1, . . . , pm) = i = 1 m pi log pi,
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EntropyBackground & History IV
The Entropy (inverse) measure of industryconcentration weights each pi by the
logarithm (log) of 1/pi, e.g.:
E =i = 1 n pi log 1/pi,
Notice that we have replaced H by E andused the fact that log(A)=log(1/A)
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Herfindhahls Measure
Herfindahls contribution to diversificationmeasures was the suggestion that the
share of each firm be weighted by itself,i.e..: (using H for Herfindahl)
H = i = 1 n pi pi
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Decomposability of Entropy
Entropy is a decomposable measure(Khinchins Decomposition Theorem)
Herfindahl is decomposable because in fact,
Herfindahl is an approximation to Entropy
2 & 4 digit SIC code is compatible withthese decompositions but is NAICS?
Further research is needed
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Diversification-Score AnomaliesBased on Entropy Decomposition proposed in this paper.
Probably violates the Decomposition Theorem
Note that columns 17 & 18 do not sum to column 16.
Source: Ragunathan (1995), Journal of Management, 21(5), J une, excerpts of p. 992.
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Corporate DiversificationCorrect Totals
Note that columns 8 & 9 add up to column 7, as they should.
Source: J acquemin & Berry (1979),The Journal of Industrial Economics, 27(4), J une, p. 362.
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Triangular Numbers I
Source: http://www.mathematische-basteleien.de/triangularnumber.htm
These are the first 100 triangular numbers:
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Triangular Numbers II
You can illustrate the name triangular number by thefollowing drawing:
Source: http://www.mathematische-basteleien.de/triangularnumber.htm
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Sample Triangular Load
DistributionGraphTriangular Load1; 0.3333
2; 0.2667
4; 0.1333
5; 0.0667
3; 0.2000
0.0000
0.0500
0.1000
0.1500
0.2000
0.2500
0.3000
0.3500
0 1 2 3 4 5 6
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Triangular DistributionsExamples
Position in Pascal's Triangle top
... ...
Pascal's triangle makes a contribution tomany fields of the number theory.
The red numbers are triangle numbers.You even can find the sum of thetriangular numbers easily.Example: 1+3+6+10+15=35
You can express the triangular numbers as binomial coefficients
Source: http://www.mathematische-basteleien.de/triangularnumber.htm
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Triangular DistributionsAnalyses I
Triangular
Samples,
Figure # n
3.1 5/15, 4/15, 3/15, 2/15, 1/15 5
3.2 10/55, 9/55, 8/55, 7/55, . . . 1/55 10
Range of Values
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Triangular DistributionsAnalyses II
TRIANGULAR
SAMPLES
Figure # n Uncalibrated
Herfinda
hl
Calibrated
Herfinda
hl
Uncalibrated
Entropy
Calibrated
Entropy
Calibrated
A1,
Acar-Bhatnagar
Calibrated
A2,
Acar-Troutt
Single-S
umF
ormula
3.1 5/15, 4/15, 3/15, 2/15, 1/15 5 0.75556 0.94444 1.48975 0.92563 0.50000 0.66667
3.2 10/55, 9/55. 8/55, 7/55, . . . 1/55 10 0.87273 0.96970 2.15128 0.93429 0.51279 0.66667
Range of Values
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Concluding Remarks
and Future Directions II In their seminal article, J acquemin and
Berry (1979) have specified how the
decomposition of the Entropy measurecan be related to SIC codes by breakingdown the diversity measurement betweenthe 2-digit and the 4-digit codes. We now
need to see if that is still true for NAICS We will be following up on their work and
further examining statistical properties
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Triangular Number Theory
A triangular numberis the sum of the n natural numbers from 1 to n.Triangular numbers are so called because they describe numbers ofballs that can be arranged in a triangle. The nth triangular number isgiven by the following formula:
Tn =k=1n k = 1+2+3+ . . . +(n-2)+(n-1)+n = n(n+1) = n2+1 = (n+1)2 2 ( 2 )
As shown in the rightmost term of this formula, every triangular numberis a binomial coefficient: the nth triangular is the number of distinctpairs to be selected fromn + 1 objects. In this form it solves the'handshake problem' of counting the number of handshakes if eachperson in a room shakes hands once with each other person.
The sequence of triangular numbers (sequenceA000217 in OEIS) forn =1, 2, 3... is:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
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Thermodynamics
The first law of Thermodynamics which statesthat energy is neither created or destroyeddirects us to a world where energy is lost
The second law says that entropy always tendsto increase in a closed system, forecasting auniverse that is constantly winding down
The tension between the first and second lawsruns like a recurring theme between turn-of-thecentury cultural formations
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NAICS vs. SIC Codes
The North American Industry ClassificationSystem (NAICS) has replaced the U.S.Standard Industrial Classification (SIC)
system. NAICS will reshape the way weview our changing economy.
NAICS was developed jointly by the U.S.,Canada, and Mexico to provide newcomparability in statistics about businessactivity across North America.
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NAICS
The official 2007 US NAICS Manual NorthAmerican Industry Classification System--UnitedStates, 2007 includes definitions for eachindustry, tables showing correspondence
between 2007 NAICS and 2002 NAICS forcodes that changed, and a comprehensiveindex--features also available on this web site.
To order the 1400-page 2007 Manual, in print,call NTIS at (800) 553-6847 or (703) 605-6000,
or check the NTIS web site. The 2002 Manual,showing correspondence between 2002 NAICSand 1997 NAICS, and the 1997 Manual,showing correspondence between 1997 NAICSand 1987 SIC, are also available.