Abstract
The beverage brouhaha, which initiated the season of dissent for the hypothetical department of the fable, started with the winer’s discovery that the department’s plurality ranking conflicts with their rankings of pairs of beverages. The radical disagreement raises interesting theoretical questions. How does a majority vote ranking of a pair relate to its relative ranking within a plurality outcome? Can anything go wrong with pairwise rankings?
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Notes
Not all binary relationships are transitive. For instance, define the binary relationship s1@@@s2 to mean that s1 loves s2. Now, it may be that “John @@@ Katri”, and “Katri @@@ Erik,” but it need not be true that “John @@@ Erik.”
Suppose each of the three members A, B, C of a condo organization has S100. For a fee, the manager, M, offers to assess some member $5; one dollar (say, in services) goes to each of the two unassessed members while the remaining $3 is the manager’s fee. With the advantage to the majority, one can see how a money pump cycle would be set up. At the end of the first vote where B, C voted against A, we have A-895, B-$101, C-$101, M-$3; at the end of the second vote A, C vote down B to obtain A- 896, B- $96, C- $102, M-S6; and so forth.
This conflict between individual/group behavior is central to statistics and the social sciences. The essence of the arguments developed here extends to other aggregation processes.
As Diana Richards and I noted, a tacit assumption in the literature is the transitivity of strategic actions. As the procedure can’t detect rationality, the space of strategic actions is much richer than commonly believed.
“Intensity” is closely related to “conditional probability” where the likelihood of events A and B can change when the comparison is subject to event C occurring.
For readers familiar with group theory, notice that the Condorcet profiles are orbits of a Z3 action and that the reduced profile involves a natural quotient. For other algebraic structures, consider Prob. 11 from an algebraic perspective.
Conclusions based on these restrictions are valid only should the voters carefully select preferences in the specified manner.
These coordinates are attributed to R. Des Cartes (1595-1650). As true with most discoveries, other philosophers, such as I. M. Des Horst, experimented with related concepts. While I have not found reliable publication dates for Des Horst’s weighty contributions, I doubt that Des Horst came before Des Cartes.
These cones are useful indicators, but they do not provide accurate values.
Examples abound in professional and academic circles and anywhere else where committee members are afflicted with a driving need to pontificate upon their decisions.
If each of 60 voters has t votes and I have 120 votes, then as long as t ∈ [0,2), I am the de facto dictator. Letting t → 0, defines the usual dictator. Mathematically, this means that these procedures belong to the same homotopy class.
This assertion, of course, applies to all aggregation procedures even those from, say, economics or statistics.
We can use less than the pairwise rankings and their intensity levels, but, to avoid stilted methods, this is close to being a minimal requirement.
In Example 3.4.1, Lillian and I altered rankings of our assigned pairs independent of what the other person believed about that pair. To avoid the same impossibility difficulty, a procedure may constrain Eric’s influence in terms of Alan’s choices.
The interested reader should consult [S14].
See [S14] for more general conditions that ensure a possibility or impossibility theorem.
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© 1995 Springer-Verlag Berlin Heidelberg
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Saari, D.G. (1995). The Problem with Condorcet. In: Basic Geometry of Voting. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57748-2_3
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DOI: https://doi.org/10.1007/978-3-642-57748-2_3
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