Mathematical conversation-starters

It comes up, from time to time, in discussions we have with each other. You're chatting with that long-suffering creature, the Intelligent Layperson, and you feel the urge to explain your professional interests to him/her. And, it has been established that n×n chessboards just don't cut it, or even 100×100 chessboards.

It's time to identify some topics that really work - here's one I tried recently. Consider the following quote from the beginning of the paper Minimal Subsidies in Expense Sharing Games by Meir, Bachrach and Rosenschein, to appear in SAGT.

Three private hospitals in a large city plan to purchase an X-ray machine. The standard type of such machines cost $5 million, and can fulfill the needs of up to two hospitals. There is also a more advanced machine which is capable of serving all three hospitals, but it costs $9 million. The hospital managers understand that the right thing to do is to buy the more expensive machine, which will serve all three hospitals and cost less than two standard machines, but cannot agree on how to allocate the cost of the more expensive machine among the hospitals. There will alway be a pair of hospitals that (together) need to pay at least $6 million, and would then rather split off and buy the cheaper machine for themselves.

The question you ask your audience is, what will be the outcome of the negotiation between the hospitals? Hopefully, someone will begin by saying that 2 hospitals will share a $5M machine, and with any luck, someone else will suggest that the 3rd hospital will offer to share a $5M machine with one of the first two, and pay more than 50%. At this stage, you are in good shape.

Now, you might object that this has nothing to do with computational complexity, which is sort of true, however you can introduce some later on in the discussion if you feel the urge (non-constant number of hospitals or machines). What makes this a nice mathematical topic is that - assuming your audience start to consider a sequence of offers and counter-offers - it raises the issue of making a proper mathematical model of the negotiation (so, if 2 hospitals make an agreement, is it meant to be binding? Presumably not if the 3rd one can "attack" it with a more attractive offer. But if it's not binding, how can the process come to an end?) Finally, despite the fact the problem is ill-posed, there is still a cute answer that is analogous to the answer to this ill-posed problem: when asked what the outcome should be, you say "by symmetry, the hospitals will share a $9M machine". (Actually, don't use the phrase "by symmetry".)

Research funding, pensions

Quite a flurry of genuinely important material for the Times Higher. The story that the REF will be delayed a year to resolve the dispute over "impact" is good news; regardless of whether this aspect of the REF is a good thing, any delay is a win, simply because of the enormous expense (mainly on academics' time) of national research assessment. Lots of comments follow the article, with Philip Moriarty in good form. (added 12.7.10: 2 more interesting links: This article (5.12.09) discusses the "James Ladyman index" and attracted a lot of comments; this one (1.7.10) has a comment by Ladyman)

The outcome of the USS meeting I mentioned previously is that probably the USS will switch to career-average earnings to compute pensions. In an attempt to make that fact sound interesting, let me put it this way: In the future, your pension contributions statement will not state the number of years of service you have accrued, but it will state your pension entitlement as a sum of money. Oh well, if that bores you, maybe you are wiser than I am.

This article reports on a speech by David Willetts on the case for science funding. So, he believes in it, that's a good thing. From the article:

Mr Willetts said he could not talk about spending commitments until the Comprehensive Spending Review is published this autumn, but warned that the UK could not afford to emulate the examples of the US, Canada and France, which had reacted to the recession by spending more on science.

Full marks, at any rate, for admitting that these other countries are raising spending on science.

Finally, sort of a weak forecast: maybe full economic costing of research grants is not going to collapse under the weight of its own stupidity, as I thought it would. Here's the argument. The Government wants to cut spending on universities but at the same time, wants to protect the strong against the weak. Now, one way to do that is to identify specific universities for preferential treatment, and I guess that's sort of what they're doing in Germany, but it's delicate, to say the least. Alternatively, you can identify characteristics of "strong" and show favoritism to universities having those characteristics. And research grants are quite handy, for that purpose. So, artificially inflate the value of all research grants, and the Matthew effect is duly enforced.

USS changes

The Times Higher is a good magazine to read if you're feeling a bit euphoric, there's an unaccountable spring in your step, and you need to simmer down a bit. Last week's issue raised the alarm about the sustainability of the Universities Superannuation Scheme (USS). The article notes a forthcoming meeting (7th July) between representatives of the University and College Union (UCU) and USS to negotiate changes. The UCU's question-and-answer page on changes to the USS, argues that no change is needed. Trouble is, we all know that very few final-salary pension scheme still exist; in the private sector most of then closed down to new member during the past 10 years. In a recent blog post, Robert Peston gives an overview of cuts to the benefits from the BBC's pension scheme (which like the USS and the private sector, is also a funded pension scheme.) From Peston's post:

Even those on relatively high salaries, who don't expect their earnings to rise, will have to think twice about whether it makes sense to continue with contributions.

This presents scheme managers with a dilemma - they have to keep the deal sweet enough to retain members and attract new ones, a problem that becomes much more urgent when a scheme is in deficit. When that happens, the whole thing is at risk of being sustained by promises that managers cannot ensure they will be able to keep.

To be honest, I think the UCU is obstructing the necessary dose of bad news that will be necessary to ensure the USS's sustainability. One problem that's come up: the USS's 3rd-largest equity holding is BP, with a market value of 693.7M (the list I got that information from is updated only every 3 months, so BP is likely to be lower-down by now.) USS has a total of about 22Bn in invested capital. (Not that I blame them for owning some BP. It looked like a good bet a few months ago.)

Here's a quote from that UCU question-and-answer page.

Q: Aren't all public sector pension funds going to have to make changes to save money. Whys should academics have special treatment?

A: USS is not a public service pension scheme, so it has to meet the same targets as other funded occupational pension schemes in the private sector. Its big advantages over such schemes are that the HE sector as a whole is less likely to go bust than any individual group of companies, so that it can afford to take a longer view. That does not mean that the politics of envy will not give traction to attacks on USS; but that is no good reason for the preemptive cringe that is part of what characterises the stance of the employers' negotiators.

Full marks for emphasizing that the USS is not public-sector, but the usage of "politics of envy" made me cringe. Finally, here's a fragment of the USS investments page to show that in some respects they have impeccable taste.

(It's the Go game that caught my eye.)