After much thinking about social choice theory, such as Arrow, I've come to the conclusion that a lot of it is actually talking at cross purposes to my thesis. Arrow wants a Social Welfare Function (decision-making procedure) to produce some sort of social preference ordering. This goal seems wrong-headed to me. I think it is a mistake to understand a decision-making procedure as either determining, or attempting to directly report, what is better.
Let me compare another case, where we may be mistaken about what our procedure is doing – a sports contest. MacKay considers scoring a polyathlon contest as an attempt to resolve who is the best all-round athlete
. In other words, he takes it for granted that there is an objective truth (it is a fact that one competitor is the best), and the procedure is what Rawls would call an ‘imperfect’ one (like a jury trial)
, that aims to best uncover that truth.
This may be true of athletic contests. We imagine that the athletes are striving for perfection, and we want the winner of the contest to be the best athlete. It is not, however, true of all sports. In an ‘entertainment’ sport, like football for instance, it does not seem we always want the best team to win. Maybe we may say ‘let the best team win’, but all we mean is the best team on the day – i.e. the one that plays best should win, rather than the results being decided by chance or a referee’s decision. We don’t, however, mean the team that is in fact the best should win all their matches. The thrill – particularly in a Cup competition – comes from upsets, times when a Premiership ‘giant’ loses to a lower division opponent. We don’t lament the fact that the best team lost, we celebrate the ‘David and Goliath’ achievement.
Of course, who wins a given game of football is influenced by who is the better team. Quite obviously, the better team is more likely to win. However, in a one-off match, the better team doesn’t always win, and that’s part of the excitement. If we wanted to establish which of two teams was better, a more accurate (more perfect) way would be for the two of them to play a series of games, and then compare total wins or aggregate scores. We don’t do this, as we’re not directly concerned with which of any two teams is better. All a particular match determines is the winner.
Of course, this isn’t quite so true of the league table. We generally want and expect the league table to reflect the objective quality of the teams – that is, for better teams to finish above worse ones, and for the best team to win. This usually happens, because various chance events (freak results, refereeing decisions, injuries, suspensions, etc) average out over the season. Having every team play every other twice, and then adding up points for each win and draw seems a reasonably accurate way of resolving who is the best team. Thus, looking at the Premiership table, it suggests Chelsea are the best team, which is probably what we’d expect from looking at them on paper. The fact that Middlesbrough beat Chelsea 3-0
(11th February 2006) doesn’t disturb this judgement. It doesn’t mean that Middlesbrough are a better team than Chelsea, it simply means that on that day a weaker team beat a stronger one. Nor, therefore, should we be bothered by the fact that Aston Villa had just previously beaten Middlesbrough 4-0
(4th February 2006) and had themselves drawn with Chelsea 1-1
(1st February 2006). If we were taking each result as demonstrating ‘betterness’ then we would seemingly have an intransitivity: Aston Villa > Middlesbrough, Middlesbrough > Chelsea, and yet Aston Villa = Chelsea. As I have suggested, however, this is by no means the case. The simple fact that Middlesbrough beat Chelsea does not mean they are the better team – and the resultant intransitivity should be a reductio ad absurdem
of that understanding of the results.
Who wins a given football match is not like either a jury trial or tossing a coin. It is not like tossing a coin, because the odds aren’t equal. We know, going into the matches against both Middlesbrough and Aston Villa, that Chelsea were probably favourites. The fact Chelsea didn’t win either of these matches doesn’t tell against that, and I’m sure they’d still be favourites in the next meeting with either club. A coin-toss gives each side an equal chance, whereas a football match favours the better side. But it doesn’t follow that the football match is like the jury trial either.
In the jury trial, there is an independently right answer, the aim of the trial is to establish this, and we assume the trial is more likely to than not – that is, that it is more likely to find the right answer than tossing a coin (‘heads you’re guilty’). We want the jury trial to produce the right answer
, and we would take necessary steps to improve its epistemic quality, e.g. giving the jury access to expert testimonies. If the trial produces an objectively wrong outcome – condemning an innocent – we regard it as an injustice. This isn’t analogous to the worse team winning a football match – as I said, we often celebrate this. It’s because the aim of the football match isn’t – at least simply – to establish the best team, but also to entertain, and unexpected upsets are exciting. We don’t think the procedure is faulty because it has produce the ‘wrong’ answer – indeed, we don’t even think of it as a wrong answer, because we aren’t asking the ‘who’s better?’ question.
It may seem we’ve been taken a long way from collective decision-making, but we haven’t. Football matches are, I think, relevantly like coin-tossing in that what they resolve (who wins) isn’t the same as a betterness ordering. Hence, I have argued, we shouldn’t be concerned by an intransitivity in results, such as that reported earlier. Now, I want to suggest we should think of decision-procedure outcomes in a similar way. That x beats y in a vote doesn’t mean x is better, merely that x is chosen.
MacKay (1980) Arrow’s Theorem: The Paradox of Social Choice. A Case Study in the Philosophy of Economics pp.21-4.
Rawls (1999 ) A Theory of Justice. Revised Edition. pp.74-5.
Actually, while this is true, the example is complicated slightly by the fact we’d rather let a guilty person go free than condemn an innocent. Hence ‘beyond all reasonable doubt’ clauses, and the consequent use of super-majorities. I leave aside these difficulties.