From: Joe Marshall Subject: Re: Using physics to solve the "social choice" problem Date: 9 May 2000 08:53:46 GMT Newsgroups: sci.physics.research Summary: Arrow's Impossibiilty Theorem (voting paradox) Alan Johnsrud writes: > Very few physicists know about an ancient problem (first described at > about the time of the French revolution) having to do with voting. It > is sometimes called the problem of social choice and, for some reason, > it has fallen into the domain of the economics profession. Economists > work on problems related to choice in many contexts (auctions, > bargaining, maximization strategies, etc.) so this was probably a > natural placement for social choice. > > The problem is about finding a fair way of processing votes in an > election when voters can express their preferences among 3 or more > candidates. Certain criteria must be satisfied for a method to be > considered a fair one. A Nobel Prize in economics was awarded (1972) > for a proof that it is impossible to devise any vote-aggregation > method which will satisfy all of the fairness criteria. Work since > then has been in describing ways to best handle various election > situations, no way being perfect. It would be helpful to enumerate the conditions proposed for `fairness': 1. Universality: The voting method should provide a complete ranking of all alternatives from any set of individual preference ballots. 2. Monoticity: If one set of preference ballots would lead to an overall ranking of alternative X above alternative Y, and if some preference ballots are changed in such a way that the only alternative that has a higher ranking on any preference ballots is X, then the method should still rank X above Y. 3. Criterion of independence of irrelevant alternatives: If one set of preference ballots would lead to an overall ranking of alternative X above alternative Y, and some preference ballots are changed without changing the relative rank of X and Y, then the method should still rank X above Y. 4. Citizen Sovereignty: Every possible ranking of alternatives can be achieved with some set of individual preference ballots. (The Condorcet criterion and the majority criterion are combined here: 4a. Majority criterion: If X ranks above Y by a majority of voters, then X ranks above Y overall. 4b. Condorcet criterion: If X ranks above all Y's in a two-way contest, it will still rank above Y in an N-way contest. 5. Non dictatorship: There should not be one specific voter whose preference ballot is always adopted. Kenneth Arrow proved that no voting system satisfies all these criteria. Any voting method will violate one of these criteria, so it is simply a question of which one to throw out. All these criteria seem reasonable, so it is difficult to decide which one to toss (perhaps we should vote on it?) > Well, it turns out that there is a fair way of processing an > n-preference election and, instead of vote aggregation, it utilizes an > algorithm derived from a physical analog model. The "impossibility > theorem" no longer applies, since vote aggregation is not used. > Furthermore, the method can be used to solve many other problems > involving value judgments, none of them previously solved. I do not doubt that you have come up with a method that satisfies 4 out of the above 5 criteria; you may have a persuasive argument that the criteria you are eliminating is less reasonable than the other 4. It would be helpful if you would identify which criterion you are eliminating, and why. > The authorities in the field of social choice are all economists or > mathematicians working in economics. None of the six authorities I > have approached (all professors and one a Nobel Prize winner) have > ventured an opinion for or against this physical method. (This > silence is itself an indication of something, I think.) Certainly an indication of something.... > A mathematics > professor outside the field (and therefore not an authority) did go so > far as to tell me that he could find nothing wrong with the method. > > It may be that this problem belongs not in economics, but in physics > -- even though the physics part of it is very elementary. Economists > may be uncomfortable with imaginary physical models, while physicists > use them all the time to illustrate points of theory. What makes people uncomfortable is the fact that each of the critiria seem reasonable and desirable; consider the alternatives. If we toss out universality, then there are some things we simply may not vote upon. If we toss out dictatorship, then one person's vote decides all. If we toss out monoticity, then we will have situations where voting your preference decreases the likelyhood of getting it. If we toss out independence of irrelevant alternatives, then we will have situations where voting on the local dogcatcher has implications in national elections. If we toss out citizen sovereignty, then we will have situations where the favorite candidate won't win, even though everyone prefers him to the alternatives. ============================================================================== From: jthorn@galileo.thp.univie.ac.at (Jonathan Thornburg) Subject: Re: Using physics to solve the "social choice" problem Date: Fri, 12 May 2000 03:04:58 GMT Newsgroups: sci.physics.research Summary: further reference on Arrow's theorm Keywords: voting systems, mathematics Alan Johnsrud describes... > an ancient problem (first described at > about the time of the French revolution) having to do with voting. It > is sometimes called the problem of social choice and, for some reason, > it has fallen into the domain of the economics profession. Economists > work on problems related to choice in many contexts (auctions, > bargaining, maximization strategies, etc.) so this was probably a > natural placement for social choice. > > The problem is about finding a fair way of processing votes in an > election when voters can express their preferences among 3 or more > candidates. Certain criteria must be satisfied for a method to be > considered a fair one. A Nobel Prize in economics was awarded (1972) > for a proof that it is impossible to devise any vote-aggregation > method which will satisfy all of the fairness criteria. [[...]] There's a nice discussion of the theorem in question ("Arrow's theorem"), its assumptions, and some of its implications, at http://www.csc.vill.edu/faculty/bartlow/html/mat1220/arrowthm.html -- -- Jonathan Thornburg http://www.thp.univie.ac.at/~jthorn/home.html Universitaet Wien (Vienna, Austria) / Institut fuer Theoretische Physik "There can be no doubt, I think, that the possession of money causes people to take a more favorable view of this world in comparison to the next." -- John Kenneth Galbraith [Moderator's note: We seem to be getting away from physics here... Followups should be elsewhere unless they have something to do with physics. -MM]