Friday, August 22, 2008

Medal Orderings

Social choice concerns how to arrive at a social ordering of alternative policies or states of affairs based on individual preference orderings. Notoriously, Condorcet showed that majority voting over pairs of alternatives need not produce a transitive result - for instance, if there are three people (Alf, Betty and Charles) and three options (X, Y and Z) it may be that Alf ranks X, Y, Z; Betty ranks Y, Z, X; and Charles ranks Z, X, Y. In this case, X > Y (2:1), Y > Z (2:1), but Z > X (2:1)...

It has generally been assumed that we must make do with ordinal comparisons. That is, we only know the order in which each individual places the alternatives - we cannot say, for example, how much Alf prefers X to Z. The Borda count attempts to infer such cardinal information from the ranking of other laternatives (I've written about this here), but this method is notoriously vulnerable to manipulation. For example, if X seems the closest rival to Z then Charles may strategically misrepresent his preferences as Z, Y, X - giving Y more points at the expense of X, to ensure Z wins.

There's a lot of debate over whether there's any non-arbitrary way of ranking preferences, and I think it's interesting that something similar is exhibited in the Olympic medal tables - as brought out by this BBC feature.

The standard table exhibits what is known as 'lexical dominance', in which more golds trump any number of bronze and silver medals and more silvers trump any number of bronzes. This is sometimes compared to the ordering of words in a dictionary. AZZZZ comes before BAAAA because the first letter dominates the others. Another common instance is a football league table - points lexically dominate goal difference, since a superiority in the latter can never compensate for fewer points.

The Olympic table would therefore rank (1,0,0) ahead of (0,50,50), which seems distinctly suspect to me. I don't agree that one gold should make such a difference. (Even though, as pointed out in the BBC article, Britain's current success is largely based on golds - our nearest rivals have more silver/bronze).

On the other hand, it seems pretty silly to count all medals equally, as the totals in the first 'alternative' table do. (This puts Australia level with GB - unfortunately the BBC's story about this bet is contradictory as to whether what matter is the official table or total number of medals).

Perhaps a sensible compromise is to adopt something like the Borda count, as the BBC do in their second table. The question is, why rank gold = 3, silver = 2, bronze = 1. You may, for example, say points ought to extend beyond top three finishers or that, like in formula one, the increments ought to be spread out more - for instance, gold = 5, silver = 3 and bronze = 1.

Ultimately, it seems that any comparison is somewhat arbitrary. The problem is not simply one of measuring - for we know exactly how many golds, silvers and bronzes each country has - but one of how to compare gold to silver, or (10, 10, 10) to (9, 12, 9).

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