Tuesday, March 28, 2006

Football, Procedures and Transitivity

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[1]. 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)[2], 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[3], 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.

[1] MacKay (1980) Arrow’s Theorem: The Paradox of Social Choice. A Case Study in the Philosophy of Economics pp.21-4.
[2] Rawls (1999 [1971]) A Theory of Justice. Revised Edition. pp.74-5.
[3] 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.


At 7:14 pm, Blogger Rob Jubb said...

I don't actually think that jury trials are imperfect procedures, because I suspect that they don't solely aim at convicting the guilty and pardoning the innocent. I think there are at least two other aims at work in jury trials, one of which indicates pure procedural ambitions. The more obvious of the two, perhaps, is ensuring that the law as administered is basically socially acceptable - that is, a jury can refuse to convict someone if they think that what legally is a crime should not be, as they have done in some cases to be with suppression of criticism of governments.

The second, and more important I think because it explains and justifies the first, is that jury trials operate as an important mechanism of political accountability. They assure that no-one, even if known to be guilty, can be imprisoned by the state without public justification of that imprisonment. I don't think - and I might be wrong about this - that jury trials are merely an epistemic device, simply because they're not always a very good epistemic device, which means that there should, if we want to defend jury trials, be some other purpose they serve. It's difficult to construct an argument which demonstrates the role I think they play because we naturally associate good public reasons for guilt with guilt, but in the event that they did come apart, I suspect that we would want, at least in part, good public reasons for guilt to trump guilt. Maybe, as I suggested earlier, the periodical acquital of defendants who are technically guilty but are not regarded as having done anything culpable offers an example, but it'd have to be worked at.

To perform something of an instrumentalist reduction on it, the idea is that rather than having a purely epistemic criterion for the success of juries, we have a quasi-moral criterion: that they require that guilt is not merely known, but proved. The difference, I suppose, between the instrumentalist reduction and the original statement, is that proved in the original statement does not just mean the distribution of knowledge on the same terms as it was previously possessed, but places requirements on the grounds for that knowledge. I am rambling now, I realise, so I'll stop.

At 9:44 pm, Blogger Ben said...

Thanks Rob,

I use the jury example because it's Rawls', but you raise some good points there not all of which I'd considered.

Maybe it can be incorporated into footnote 3 and largely set aside for my present purposes, because I'm only holding a (caricatured) jury up to illustrate an extreme.

On the other hand, one might be able to use it towards and attack on the pure/perfect/imperfect procedure distinction, perhaps arguing nothing is ever solely one. (Personally I think the perfect/imperfect is rather blurred - they're the same in kind, differing merely in degree)

At 6:57 pm, Anonymous John Lawrence said...

I think transitivity is overrated. It is one of Arrow's 2 basic axioms, and, effectively, precludes a tie solution since a tie is, by definition, intransitive. If you have a third of the voters voting aPbPc and a third voting bPcPa and a third voting cPaPb, logically the social choice would be a tie between the 3 profiles aPbPc, bPcPa and cPaPb, but Arrow insists, by virtue of his transitivity axiom, that you pick one and only one profile. That leads to the voting paradox. In the tie solution aPbPc AND bPcPa AND cPaPb which violates transitivity.

Note, however, that a tie solution does not violate Arrow's Axiom 1. He says you can have both aPb AND bPa.

At 12:21 am, Blogger Ben said...

Thanks John.

I don't think Arrow sees a cycle as a tie though. He sees aPbPcPa as aPb (no tie), bPc (no tie) and cPa (no tie) - and the overall result merely irrational.

He doesn't, to my knowledge, ever suggest the result is anything like aIbIc, which is what he'd consider a tie. (I know you have some difficulty with the tie/indifference thing - I still haven't made my mind up about that)

However you understand a tie - and I'll here represent it by indifference - I don't see that it's necessarily intransitive. Take aIb and bIc. It follows that aIc - that's transitivity.

The cycle case is, of course, intransitive, and that's the problem. I'm slightly uneasy saying 'treat as if tied' (because I do see a difference), but I think I'd regard a lottery as a fair way of resolving either a tie or a cycle.

And surely Arrow doesn't say you can have aPb and bPa? - He deals in the R relation, and you can have aRb and bRa, which implies aIb.

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