May 2004

## Relationships: solution

Here is the table of decisions after Dell and Tracy have added their "No's":

Decorate house? Tidy garden? Buy TV?
Ali Yes Yes No
Bob No Yes Yes
Carla Yes No Yes
Old Majority
decision
YES YES YES
Dell No No No
Tracy No No No
Overall
decision
NO NO NO

An untidy garden? What a paradox!

We see that their negative votes have tipped the scales on each question. Now there is a majority of 3 to 2 not to decorate the house, not to tidy the garden, and not to buy a TV. But more striking is the fact that a majority (each of Ali, Bob, and Carla) are on the losing side of the vote on two of the three issues. These are the issues on which they don't vote "No". So Ali is on the losing side over the house and garden, Bob on the losing side over the garden and TV, and Carla is on the losing side over the house and the TV. Hence a majority of the people (three out of five) were on the losing side on a majority of the issues (two out of three)! This is called the Anscombe Paradox.

• G.E.M. Anscombe, On the frustration of the majority by fulfillment of the majority's will, Analysis 36, 161-8 (1976).
• D.G. Saari, Decisions and Elections: explaining the unexplained, Cambridge University Press (2001).
• For All Practical Purposes: Mathematical Literacy in Today's World, fifth edition by the COMAP collaboration, W.H. Freeman, (2000), see chapters on voting and elections. This book is especially recommended; see the review in issue 13 of Plus.

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