[Hum] "Democracy is mathematically arbitrary" proof wins Nobel Prize.

Summary: Kenneth Arrow won the 1972 Nobel Prize in economic for his proof of the "Impossibility Theorem" which demonstrates that you can't aggregate individual preferences to define a group preference between multiple options.

Link: http://www.udel.edu/johnmack/frec444/444voting.html

Article Text

FREC 444 Economics of Environmental Management
Voting Theory

Our democratic process mostly uses a one person/one vote, majority-rules system to elect people and pass legislation, and the two dominant parties each use primary systems to select single candidates for office in the general election.. But there are many other possible voting systems. Here's a theoretical example lifted from J.A. Paulos' Beyond Numeracy (Alfred A. Knopf, NY, 1991), which Paulos borrowed from W.F. Lucas.

Suppose there are 5 candidates, A, B, C, D and E, and 55 voters with the following preferences:

18 voters prefer A > D > E > C > B  12 voters prefer B > E > D > C > A  10 voters prefer C > B > E > D > A   9 voters prefer D > C > E > B > A   4 voters prefer E > B > D > C > A   2 voters prefer E > C > D > B > A  

If the outcome is determined by plurality, candidate A wins, having the most (18) first-place votes even though the 37 other voters all think he is the absolute worst candidate.

If the outcome is determined by a runoff between the two candidates receiving the most first-place votes, candidate B beats A 37 to 18.

If the outcome is determined by iteratively eliminating the candidate with the fewest first-place votes and moving the remaining candidates up in the preference orders accordingly, we eliminate E (4+2=6 first-place votes, above), then D (9 first-place votes, below left), then B (12+4=16 first-place votes, below center), then candidate C emerges with 37 first-place votes versus A's 18:

18   A > D > C > B      A > C > B      A > C  12   B > D > C > A      B > C > A      C > A  10   C > B > D > A  ->  C > B > A  ->  C > A   9   D > C > B > A      C > B > A      C > A   4   B > D > C > A      B > C > A      C > A   2   C > D > B > A      C > B > A      C > A  

If the outcome is determined by Borda count, so that a first-place vote represents 5 points, a second-place vote represents 4 points, etc., then candidate D wins with 191 points:

       A   B   C   D   E  18  x  5   1   2   4   3  12  x  1   5   2   3   4  10  x  1   4   5   2   3   9  x  1   2   4   5   3   4  x  1   4   2   3   5   2  x  1   2   4   3   5       127 156 162 191 189  

If the outcome is determined by pairwise preferences, candidate E wins by being pairwise-preferred to every other candidate.

Since the outcome depends on which voting system we use, we might vote to choose a voting system first, but voting on the voting system doesn't really solve the problem, particularly where each candidate advocates the system that favors him or her. The outcome ultimately depends on some arbitrary choice of system. And there are many more possible systems: we could iteratively eliminate the candidate with the most last-place votes, or use some other Borda count system, or allocate each voter multiple votes to allocate between the candidates as he wishes, or let voters vote for as many candidates as they approve of. Approval voting is a particularly interesting system: it tends to favor centrist politics, since candidates with similar views don't suffer from split votes. Several states have considered legislation to institute approval voting systems.

The basic conclusion from this analysis is that democracy is mathematically arbitrary.

Kenneth Arrow won the 1972 Nobel Prize in economic for his proof of the "Impossibility Theorem" which demonstrates that you can't aggregate individual preferences to define a group preference between multiple options without violating at least one of the following basic conditions:

1. If and individual or group prefers A to B and B to C, then A is preferred to C (transitivity).
2. The preferences must be restricted to the complete set of options.
3. If each individual prefers A to B, then the group must also.
4. No individual's preferences can necessarily dictate group preferences.
5. The group's pairwise preference ordering is independent of irrelevant alternatives, i.e. determined solely by individual's pairwise preference orderings.

Arrow basically proved that the first two conditions are logically inconsistent with the latter three.