Consensus Building

Beware of voting paradoxes

Al R. Vilcius

Getting agreement by consensus on business issues in a democratic manner, frequently encounters difficulties, particularly in an "association" environment or in meetings among peers. In some circumstances, there are scientific explanations for this. I will illustrate using the following example.

A Board of Directors consisting of 9 members is presented with 3 alternatives: A, B, C

Only one alternative can be accepted, so the members vote for their preferences(1):

From this result, we have by reading down the first column:

Decision 1: Accept alternative A with C second choice and eliminate B.

Is this correct? Let's reconsider.

Hence accepting A makes the majority unhappy.

Therefore A should be eliminated, leaving the choice between B and C.

Now 6/9=2/3 (or over 66%) preferred B to C , so B is the clear winner.

Decision 2: Accept alternative B and eliminate A.

Both decisions:

  1. Accept A and eliminate B
  2. Accept B and eliminate A

are supportable from the same vote.

Unfortunately, these two decisions are contradictory!

Conclusion: Consensus needs to be achieved by carefully establishing clear decision criteria, and beware of voting paradoxes.

These are well known and explained by chaos theory, as in Saari [1] and [2].

I used a variation of this example in 1991 to illustrate to senior management reasons for difficulties in selecting deals at CIBC credit committee.

For this example given above, the recommended action is to adjourn the board meeting and decide what restaurant to go to: there are three choices … … can you imagine what happens next?


(1): The notation  A  >  B  or “A is greater than B”  means here that if some metric (utility function) is used for preference, "A" would score higher than "B".  Furthermore, the relation is intended to be transitive, meaning that A  >  B  >  C  implies  A > C , which would be consistent with the use of a numerical metric.


Recommended Further Reading

These titles can be purchased on-line from

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Paperback | 300 Pages | ISBN 3540600647 1st edition Published in 1995 by Springer - Verlag [DS'95]

Donald G. Saari
"Basic Geometry of Voting"
Springer - Verlag , 1995
link to purchase [KC'98]

K. C. Cole
"The Universe & the Teacup: The Mathematics of Truth & Beauty"
Harcourt Brace & Company, 1998

 Hardcover | 232 Pages | ISBN 0151003238

Hardcover | 493 Pages | ISBN 047161842X Published in 1989 by John Wiley & Sons Canada, Limited [JC'89]
John L. Casti
" Alternate Realities:Mathematical Models of Nature & Man"
John Wiley & Sons Canada, Limited, 1989



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