hibbert@arisia.Xerox.COM (Chris Hibbert) (01/07/89)
In article jeff@stormy.atmos.washington.edu (Jeff L. Bowden) writes: >Didn't some Nobel Prize winning economist show some result wherin it is >impossible to have a "fair" election between more than two candidates (where, >of course, the definition of "fair" is included in the proof - as is par for >the course in economics)? Yes, Jeff, Ken Arrow (I'm not sure about the Nobel, but he is a prominent economist) proved something now referred to as Arrow's Impossibility Theorem. The following is written while referring to my college micro-economics text "Micro Economic Theory: Basic Principles and Extensions, 2nd ed.", by Walter Nicholson, copyright 1978 The Dryden Press. I've seen much better explanations than the one in this book so I'll try to improve it somewhat while leaving its accuracy unaffected. You might also try looking in Arrow's "Social Choice and Individual Values" (New Haven, Conn.: Yale University Press, 1951) which probably says it better than I will. The theorem assumes that individuals preferences are transitive. [The text presents the theorem as if it only applies to two individuals and 3 choices. I'm positive that that is an oversimplification, and that the theorem holds for larger groups and many choices.] It shows that there are no possible voting systems that have all of the following seemingly usefull properties: 1. No ties are allowed. All possible outcomes must be ordered in the ranking produced by the voting function. 2. The ranking must be transitive. 3. The ranking must be positively related to individual preferences. If all [both] voters prefers outcome a to outcome b, the outcome a must be preferred in the ranking. 4. The outcome must be independent of irrelevant alternatives. That is, adding more choices shouldn't affect the rankings of the existing choices. 5. The outcome shouldn't be independant of voters' choices. (Decision based on custom are an example where voters' preferences don't matter 6. There shouldn't be a single person (the dictator) whose choices overrule all the others. There's a reference in a footnote that might also be interesting. on page 550, Nicholson says "Some interresting voting models are examined in J. M. Buchanan and G. Tullock, 'The Calculus of Consent' (Ann Arbor, Michigan: University of Michigan Press, 1962). Buchanan did just win the Nobel prize, so maybe this was the work Jeff meant. So, yes, economists have shown that it's impossible to come up with one mechanism for voting that will satisfy a pretty reasonable set of criteria. That's why we're so often able to show that particular outcomes are ludicrous given people's expressed preferences, and why there are so many seemingly useful schemes that we haven't adopted. Chris