Published online 18 January 2008 | Nature | doi:10.1038/news.2007.352

Column: Muse

The maths behind group showers

A simple model of the tribulations of hostel shower systems highlights the benefits of human diversity, says Philip Ball.

Human variety is more than the spice of life — it seems to be an essential part of what makes our communities function. From the boost that genetic diversity gives to populations in the face of a changing environment, to the fact that we’re not all trying to fly to the same Pacific island for a bit of winter sunshine, differences in character and behaviour are vital to the smooth running of our lives. That’s now been illustrated in a mathematical model of what happens when everyone in a youth hostel decides to take a shower at the same time1.

This might not sound like a burning issue (forgive the pun). But the study has something more general to say about the benefits of diverse behaviour in optimizing group decisions.

This hot topic has been explored before. Imagine a company setting out to hire a 20-person team to solve a tricky problem, with a thousand applicants to choose from. Should they set them a test and pick the 20 who do best? That sounds like a no-brainer, but there are situations in which it would be better to hire 20 of the applicants at random.

So said social scientists Lu Hong and Scott Page of the University of Michigan in Ann Arbor four years ago2. Their work 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 the group is less likely to come up with a versatile, flexible solution. “Diversity”, say Hong and Page, “trumps ability”.

Busy night at the El Farol

Physicist Damien Challet from the Institute for Scientific Interchange in Torino, Italy, and one of the authors of the shower study, has previously explored the role of diverse decision-making behaviour. Several years ago, he and his colleague Yi-Cheng Zhang devised the so-called minority game as a model for human decision-making3. They took their lead from economist Brian Arthur, who used to frequent the El Farol bar in Santa Fe, New Mexico4. The bar hosted an Irish music night on Thursdays that was so popular that the place was often 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, and vice versa: in other words, to be in the minority.

But how do you know which nights will be busy, and which won’t? 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 others use.

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.”)

Challet and Zhang’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 wouldn't fluctuate so much from one week to the next, and would usually be close to capacity.

The minority game serves as a proxy for many social situations, from changing lanes in heavy traffic to choosing your holiday destination. It is especially relevant in economics: it pays to be a seller when the majority is buying. 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, convinced that they can deduce the ‘best’ strategy from the trends and fluctuations.

Back in the shower

The Shower Temperature Problem, which Challet has now devised in collaboration with economist Christina Matzke from the University of Bonn in Germany, is a different sort of collective challenge. It’s common for budget hostels to have water systems that can’t cope with lots of demand: if many people shower at the same time, one person changing their 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 oscillate unpredictably 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?

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In principle, everyone gets closest to their preferred temperature if everyone sets their taps in the same way — that is, they all use the same strategy. (In contrast, achieving the optimum in the minority game relies on the use of different strategies.) 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 variations in temperature as others respond.

When there is instead a diversity of strategies — some upping the hot madly and others turning down the cold in dribbles — then these oscillations are suppressed and the system converges more reliably on an acceptable temperature for all. But there’s a price paid for that stability. Although 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.

Rational sharing

This problem exemplifies any in which many agents try to obtain equal amounts of some fixed quantity — 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 can 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 traders, all following a single, rationally devised ‘optimal’ strategy.

There’s a good reason for this: 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 hostellers. Anyone contemplating the current world economy knows all too well what a myth this equilibrium is.

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 political correctness, nor even about being liberal or tolerant (although it tends to require both), but about being rational. 

  • References

    1. Matzke, C. & Challet, D. preprint at http://www.arxiv.org/abs/0801.1573 (2008).
    2. Hong, L. & Page, S. E. Proc. Natl Acad. Sci. USA 101, 16385-16389 (2004). | Article | PubMed | ChemPort |
    3. Arthur, B. W. Am. Econ. Assoc. Papers & Proc. 84, 406 (1994).
    4. Challet, D. & Zhang, Y.-C. Physica A 246, 407 (1997). | Article |
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