Differences in the shower
[This is how my latest article for Nature’s Muse column started out. Check out also a couple of interesting papers in the latest issue of Phys. Rev. E: a study of how ‘spies’ affect the minority game, and a look at the value of diversity in promoting cooperation in the spatial Prisoner’s Dilemma.]


A company sets out to hire a 20-person team to solve a tricky problem, and has a thousand applicants to choose from. So they set them all a test related to the problem in question. Should they then pick the 20 people who do best? That sounds like a no-brainer, but there situations in which it would be better to hire 20 of the applicants at random.

This scenario was presented four years ago by social scientists Lu Hong and Scott Page of the University of Michigan [1] as an illustration of the value of diversity in human groups. It shows that many different minds are sometimes more effective than many ‘expert’ minds. The drawback of having a team composed of the ‘best’ problem-solvers is that they are likely all to think in the same way, and so are less likely to come up with versatile, flexible solutions. “Diversity”, said Hong and Page, “trumps ability.”

Page believes that studies like this, which present mathematical models of decision-making, show that initiatives to encourage cultural diversity in social, academic and institutional settings are not just exercises in politically correct posturing. To Page, they are ways of making the most of the social capital that human difference offers.

There are evolutionary analogues to this. Genetic diversity in a population confers robustness in the face of a changing environment, whereas a group of almost identical ‘optimally adapted’ organisms can come to grief when the wind shifts. Similarly, sexual reproduction provides healthy variety in our own genomes, while in ecology monocultures are notoriously fragile in the face of new threats.

But it’s possible to overplay the diversity card. Expert opinion, literary and artistic canons, and indeed the whole concept of ‘excellence’ have become fashionable whipping boys to the extent that some, particularly in the humanities, worry about standards and judgement vanishing in a deluge of relativist mediocrity. Of course it is important to recognize that diversity does not have to mean ‘anything goes’ (a range of artistic styles does not preclude discrimination of good from bad within each of them) – but that’s often what sceptics of the value of ‘diversity’ fear.

This is why models like that of Hong and Page bring some valuable precision to the questions of what diversity is and why and when it matters. That issue now receives a further dose of enlightenment from a study that looks, at face value, to be absurdly whimsical.

Economist Christina Matzke and physicist Damien Challet have devised a mathematical model of (as they put it) “taking a shower in youth hostels” [2]. Among the risks of budget travel, few are more hazardous than this. If you try to have a shower at the same time as everyone else, it’s a devil of a job adjusting the taps to get the right water temperature.

The problem, say Matzke and Challet, is that in the primitive plumbing systems of typical hostels, one person changing their shower temperature settings alters the balance of hot and cold water for everyone else too. They in turn try to retune the settings to their own comfort, with the result that the shower temperatures fluctuate wildly between scalding and freezing. Under what conditions, they ask, can everyone find a mutually acceptable compromise, rather than all furiously altering their shower controls while cursing the other guests?

So far, so amusing. But is this really such a (excuse me) burning issue? Challet’s previous work provides some kind of answer to that. Several years ago, he and physicist Yi-Cheng Zhang devised the so-called minority game as a model for human decision-making [3]. They took their lead from economist Brian Arthur, who was in the habit of frequenting a bar called El Farol in the town of Santa Fe where he worked [4]. The bar hosted an Irish music night on Thursdays which was often so popular that the place would be too crowded for comfort.

Noting this, some El Farol clients began staying away on Irish nights. That was great for those who did turn up – but once word got round that things were more comfortable, overcrowding resumed. In other words, attendance would fluctuate wildly, and the aim was to go only on those nights when you figured others would stay away.

But how do you know which nights those are? You don’t, of course. Human nature, however, prompts us to think we can guess. Maybe low attendance one week means high attendance the next? Or if it’s been busy three weeks in a row, the next is sure to be quiet? The fact is that there’s no ‘best’ strategy – it depends on what strategies other use.

The point of the El Farol problem, which Challet and Zhang generalized, is to be in the minority: to stay away when most others go, and vice versa. The reason why this is not a trivial issue is that the minority game serves as a proxy for many social situations, from lane-changing in heavy traffic to choosing your holiday destination. It is especially relevant in economics: in a buyer’s market, for example, it pays to be a seller. It’s unlikely that anyone decided whether or not to go to El Farol by plotting graphs and statistics, but market traders certainly do so, hoping to tease out trends that will enable them to make the best decisions. Each has a preferred strategy.

The maths of the minority game looks at how such strategies affect one another, how they evolve and how the ‘agents’ playing the game learn from experience. I once played it in an interactive lecture in which push-button voting devices were distributed to the audience, who were asked to decide in each round whether to be in group A or group B. (The one person who succeeded in being in the minority in all of several rounds said that his strategy was to switch his vote from one group to the other “one round later than it seemed common sense to do so.”)

So what about the role of diversity? Challet’s work showed that the more mixed the strategies of decision-making are, the more reliably the game settles down to the optimal average size of the majority and minority groups. In other words, attendance at El Farol doesn’t in that case fluctuate so much from one week to the next, and is usually close to capacity.

The Shower Temperature Problem is very different, because in principle the ideal situation, where everyone gets closest to their preferred temperature, happens when they all set their taps in the same way – that is, they all use the same strategy. However, this solution is unstable – the slightest deviation, caused by one person trying to tweak the shower settings to get a bit closer to the ideal, sets off wild oscillations in temperature as others respond.

In contrast, when there is a diversity of strategies – agents use a range of tap settings in an attempt to hit the desired water temperature – then these oscillations are suppressed and the system converges more reliably to an acceptable temperature for all. But there’s a price paid for that stability. While overall the water temperature doesn’t fluctuate strongly, individuals may find they have to settle for a temperature further from the ideal value than they would in the case of identical shower settings.

This problem is representative of any in which many agents try to obtain equal amounts of some fixed quantity that is not necessarily abundant enough to satisfy them all completely – factories or homes competing for energy in a power grid, perhaps. But more generally, the model of Matzke and Challet shows how diversity in decision-making may fundamentally alter the collective outcome. That may sound obvious, but don’t count on it. Conventional economic models have for decades stubbornly insisted on making all their agents identical. They are ‘representative’ – one size fits all – and they follow a single ‘optimal’ strategy that maximizes their gains.

There’s a good reason for this assumption: the models are very hard to solve otherwise. But there’s little point in having a tractable model if it doesn’t come close to describing reality. The static view of a ‘representative’ agent leads to the prediction of an ‘equilibrium’ economy, rather like the equilibrium shower system of Matzke and Challet’s homogeneous agents. Anyone contemplating the current world economy knows all too well what a myth this equilibrium is – and how real-world behaviour is sure to depend on the complex mix of beliefs that economic agents hold about the future and how to deal with it.

More generally, the Shower Temperature Problem offers another example of how difference and diversity can improve the outcome of group decisions. Encouraging diversity is not then about being liberal or tolerant (although it tends to require both) but about being rational. Perhaps the deeper challenge for human societies, and the one that underpins current debates about multiculturalism, is how to cope with differences not in problem-solving strategies but in the question of what the problems are and what the desired solutions should be.


References

1. Hong, L. & Page, S. E. Proc. Natl Acad. Sci. USA 101, 16385 (2004).
2. Matzke, C. & Challet, D. preprint http://www.arxiv.org/abs/0801.1573 (2008).
3. Arthur, B. W. Am. Econ. Assoc. Papers & Proc. 84, 406. (1994).
4. Challet, D. & Zhang, Y.-C. Physica A 246, 407 (1997).