The document discusses paired comparison analysis, a method used for decision-making by comparing units against each other to rank preferences, particularly in situations with multiple alternatives. It highlights the historical context, applications in ranking systems such as college football, and critiques of the Borda count method for having flaws related to irrelevant alternatives and integer rankings. The document advocates for using paired comparison analysis and a prioritization matrix for more effective decision-making and vendor selection.

1. +
Paired Comparison
Analysis
In paired comparison experiments, the
worth or merit of a unit is measured
through comparisons against other units.
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2. We rank movies
We rank bands
We rank everything

3. +
Plurality Choice is Simple
 Picking 1 of 2 candidates is what we do in major elections
 But what about picking between more than 2
 Which puppy to do want?
 Which car should we buy?
 Which girl friend should I take to the prom? (nice to be 17 uh?)
 Which vendor “best” meets provide the capabilities in a PPM
system?
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4. +
Too Many Choices Make for Bad
Decisions
 Simple importance ranking cannot be the source for making
the right choice
 Weighted ranking hides underlying importance between
individual selection elements
 Paired Comparison Analysis is an approach that deals with
multi-selection decision making
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5. +
History of Paired Comparison
 Introduced nearly 150 years ago:
 Method of paired comparison is perhaps the most straightforward
way of presenting items for comparative judgment.
 With the method items are presented in pairs to one or more
judges:
 For each pair, the judge selects the item that best satisfies the
specified judgment criterion.
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6. +
Rank Ordering in College Football
 Before 2006 the “best teams” were
chosen using a formula by the BCS
 In 2001 Nebraska was soundly
defeated by CU in the final regular
season game, but not in a
Conference championship game
 BCS ranked Nebraska above CU,
who went on to be soundly defeated
by Miami
“…division 1A college football uses what may be the most
complicated monstrosity on the planet.” – Alissa Bauer, 2004
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7. +
College Football Rank Ordering
circa 2003
 The BCS system raised or lowered the weight put
on votes cast by coaches, sports writers, and
even the computers
 All these individual ranks are assembled into the
BCS ranking
 This is called Borda Ranking, from Jean-Charles
de Borda, 1781Jean-Charles de Borda
1733 – 1799
Born in the city of Dax,
in 1756 Borda wrote
Mémoire sur le
mouvement des
projectiles, a product
of his work as a
military engineer. For
that, he was elected to
the French Academy
of Sciences in 1764.
Jean-Charles de Borda
1733 – 1799
Born in the city of Dax,
in 1756 Borda wrote
Mémoire sur le
mouvement des
projectiles, a product
of his work as a
military engineer. For
that, he was elected to
the French Academy
of Sciences in 1764.
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8. +
The Borda Method
 Give a certain number of points for each 1st
place ranking, 2nd
place ranking, etc.
 If there are n alternatives to be ranked, a 1st
place vote is worth
n – 1 points, a 2nd
place vote is worth n – 2 points, and so on all
the way to an nth
place vote is worth 0 points.
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9. +
The Borda Count
 The approach works sensibly in many situations
 When all the voters agree on the rankings for all the
alternatives
 This “sum of ranks” works:
 for the BCS
 for elections in the US
 for the Academy Awards
 Although there are bitter squabbles over the results
at times, there is rarely commentary that there is a
fundamental flaw in the Borda Methodology
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10. +
The Fundamental Flaw is the
presence of Irrelevant Alternatives
 An Irrelevant alternative effects the outcome
 This is what happen when Millie Vanilli was
eliminated from the Grammy’s for lip-syncing
 The next ranked group should have won, but they
didn’t
 Because in Borda, the reordering of the
remaining top three are inverted
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11. +
Irrelevant Alternatives
A proper selection principle holds that the ranking
of two alternatives should not be influenced by the
placement of other alternatives
Suppose your friend and his family have received a gift certificate
for a new car, a Toyota or a Honda. Focused on those two
alternatives, the desirability of all other makes should be
irrelevant. You’re are discussing this in the dealer's parking lot
and the friend says out loud, “We prefer the Toyota to Honda. We
will go in to pick that one.” You’re sure all the work is done, so you
go home. Later, the friend drives up to in a shiny new Honda. You
say, “what happened? You preferred Toyota!” The friend says,
“While we were in line, my son heard that Cadillac‘s have great
durability. So we changed from Toyota to Honda!”
Could the outcome be more ridiculous? It can not possibly make
sense to have the choice between a Honda and a Toyota depend
on the road performance of a Cadillac. And yet, that can happen
with the Borda count.
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12. +
The Borda Winner is the Loser
 The Borda method could produce a
winner that would lose in a head-to-head
comparison against other alternative
choices
 When Borda proposed his election
system, Marquis de Condorcet pointed
the fatal flaw:
 A Borda selection can be defeated in a head-
to-head contest with other selections in the
same population of choices
 This head-to-head selection process is
the Pair Wise Selection we’re after
Marquis de Condorcet
1743 – 1794
a French philosopher,
mathematician, and
early political scientist
who devised the
concept of a
Condorcet method.
Marquis de Condorcet
1743 – 1794
a French philosopher,
mathematician, and
early political scientist
who devised the
concept of a
Condorcet method.
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13. +
The Guts of the Flaw
 Using integer rankings is the culprit
 If the choice x is better than choice y and
choice y is better than z
 The ranking is 1, 2, 3 for x, y, z
 The magnitude of the preference is ignored
 But this approach does not make the
comparison between x and z
 The same problem occurs in single and
double elimination tournaments
 And of course our beloved BCS rankings
of Big 12 Football
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14. +
Some Math Behind This Approach
The probability that an object j is judged to have more of an
attribute that object i is:
Where is the scale location for object i.
{ }Pr 1 ,
1
j i
j i
ji
e
X
e
δ δ
δ δ
−
−
= =
+
iδ
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Getting out of the dilemma of Integer Ranking,
Borda Methods, and all the attendant problems
means using a Round Robin Tournament, where
every team plays every other team
This is Paired Comparison Analysis

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It's a Two Step Approach to Vendor
Selection
 Use Paired Comparison Analysis to Rank the capabilities of the
desired system
 Used a weighted comparison to build the Pareto Chart of the
resulting rankings
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First Rank the Capabilities
Rank each of the desired capabilities using this tool. The result
may be surprising, but it represents how the capabilities are in
fact ranked in order of importance using the “round robin”
approach
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Then Construct a Prioritization
Matrix
 This approach “sorts” items into their order of importance using
Paired Comparison to:
 Prioritize complex or unclear issues, where there are multiple criteria
for deciding importance
 when there is data available to help score criteria and issues
 Gain agreement on priorities and key issues
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Use the Ranking Tool to compare
the Vendors
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This tool
provides the rank
order selection
for each “design
element” of
Capabilty
compared for
each vendor. It
allows the
display of
ranking and
ordering of both
the capabilities
and the vendors

20. +
Step By Step Instructions
 Build a list of items to be
prioritized
 Identify the list of criteria used
to judge how well each item
meets he criteria
 Allocate weighting to each
criteria
 Select actual criteria to be used
for prioritization
 Score each item against the
criteria
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