Praesidium

Saturday, November 12, 2005

Chaos

Explain the McKelvey-Schofield (chaos) theorems

Riker employs McKelvey and Schofield’s chaos theorems to argue that once there is no unique stable optimum or transitive ordering, election results are effectively arbitrary and meaningless. As he puts it, “not only is a Condorcet winner unlikely, but also, when one does not exist, anything can happen” . If there is no equilibrium, then we can move away from any status quo in any direction, and all possible points seem to be included in a possible cycle.

Mackie, however, criticises the realism of the assumptions in these models. “In the absence of friction most initial states result in nonstationary orbits or cycles that would continue forever in disequilibrium… it is a mistake to argue that the counterfactual world of no friction somehow reveals a more fundamental truth about the world of friction” .

It takes only a small amount of homogeneity to produce stability. Rather than assuming an ‘impartial culture’, if just 5% of voters have identical preferences, the Condorcet efficiency of many voting procedures is greatly increased . Presumably this also increases stability. Further, super-majority rules can ensure stability. If there is only one dimension, >50% ensures stability. This can be generalised to d/(d+1), where d is the number of dimensions; thus if there are two dimensions, a 2/3rds majority suffices for equilibrium.

Certainly empirical evidence suggests that while somewhat unexpected outcomes can emerge – for example, the USA’s recent repeal of estates tax – normally political outcomes are fairly predictable. Knowing just a little about voters/decision makers’ preferences, we usually feel we have a good idea of the kind of outcomes they might arrive at. While strategic manipulation may swing results more in favour of one party than another, we certainly don’t expect any outcome to be possible. For example, while we may not be greatly shocked by a deviation from the median voter, we would be extremely surprised if the outcome was to the left (or right) of the furthest left (or right) voter.

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