On Wednesday this week I had the pleasure of attending a party to celebrate the twentieth anniversary of the Isaac Newton Institute.
I think I was invited because I directed a programme there, on Combinatorics and Statistical Mechanics, in 2008. But in fact I have two other connections with the history of the Institute, neither of them amounting to much, and neither officially noted in the records. When Tim Gowers was appointed to the Scientific Steering Committee, he was in Princeton, and asked me to stand in for him at a couple of meetings of the Committee. The other connection will be touched on later.
Apart from a special birthday cake bearing a picture of Newton, and a glass of bubblly to celebrate Martin Rees’ appointment as a Fellow of the Institute, we were entertained with two lectures: one by Peter Goddard on the prehistory of the Institute, and one by Fields Medallist Wendelin Werner on some new developments in probability theory. It turned out that there was a remarkable synergy between the two talks, which I will describe. I have greatly simplified what Wendelin Werner said in his talk, even though he had already watered it down after discovering at a late stage that he was not addressing an audience of mathematicians.
In an ordinary election, it is virtually never the case that any one voter can say honestly, “my vote determined the outcome”. Assume the simplest model, where there are two candidates and the one with the most votes wins. As long as the margin of victory is greater than one, no voter is in the position that if (s)he had voted differently the result would have been different.
But now consider the following model. In the simplest case, three voters decide, let us say independently at random with equal probability, between two alternatives A and B. The result is determined by a simple majority. Clearly each of A and B has a probability 1/2 of winning. But out of the four cases in which A wins, three of them split the votes 2 to 1; if either of the two people voting for A changed their mind, the overall result would go the other way.
Now suppose that the three voters, instead of choosing at random, decide on the basis of three subsidiary voters who choose at random. The final outcome is exactly the same, since the effect of the subsidiary voters is that each of the original three voters chooses with probability 1/2 and the choices are independent.
This can be extended to any number of levels. Take a ternary tree with n levels, so that there are 3n vertices at level n. Suppose that each vertex at level n makes up its mind independently with probability 1/2 between two alternatives A or B. Now each parent is bound by the majority vote of its three children, and this propogates up the tree, so that the root chooses either A or B, and clearly each alternative has probability 1/2.
Now mark the nodes in the tree according to the following rule. First, the root is marked. Then, recursively, a node is marked if the following conditions hold:
- its parent is marked;
- its parent got its choice from a 2 to 1 majority vote of its children;
- the node in question was one of the majority.
It follows from the theory of Galton–Watson trees that the expected number of marked nodes on level n is (3/2)n: that is, exponentially many nodes are marked, but they form an exponentially small fraction of the total number of nodes on this level.
Now we come to the main point. Each one of the exponentially many marked nodes on level n can legitimately say to itself, “If I had voted differently, the overall result would have been different”. For suppose that outcome A was accepted. This means that every marked node voted for A. If a given marked node changed its vote to a B, then its parent would have been bound by a 2-to-1 majority for B, and so would change; and so on up the tree to the root.
As we saw, this is very different from first-past-the-post voting.
The point of Werner’s talk is that this process has a limit, which is a 1-dimensional process very different from Brownian motion (in some sense the limit of first-past-the-post voting), and that both have 2-dimensional analogues. The 2-dimensional analogue of the limit of the tree-like process was constructed fairly recently by Schramm and Smirnov, and was the main topic of Werner’s talk.)
Which of the two competing models is closer to “reality”? This was a question that Werner deliberately didn’t address. Roughly, in a typical situation, does no individual choice make a difference to the outcome, or are there many individual choices which would change the outcome if made differently?
Peter Goddard’s story of the history of the Isaac Newton Institute suggested (to me at least) the latter. He was introduced by John Toland as the person who “knows where the bodies are buried”, and he said that, when he found out that the talk was being recorded, he decided to leave out the more scurrilous stories. But nevertheless, there was a clear sense that there were many points along the way where just a small change would have given a completely different outcome, not least the builder going bust shortly after the completion of the building.
At a certain point, once the site and some initial funding had been secured from the University of Cambridge and St John’s and Trinity Colleges, it was realised that this would not be enough, and the begging bowl would have to be taken to the research council. In those days, the research council was called SERC, and it had subject committees; I was on the mathematics committee. In the interests of fairness, SERC decided that they couldn’t simply accept this bid; there would have to be a competition. Desipte the tight timetable, many entries were received, including one from the University of London, masterminded by my first boss, Paul Cohn, who proposed using space in Senate House for a mathematical institute. But I can say, from the inside, that this is not a point where a small change could have made things work out differently. It was clear that Cambridge were so far ahead of the game, both in the planning they had already done, and in the offers of site and money they had already obtained, that there was really no chance that they wouldn’t emerge as winners.
The two talks should be available on the Newton Institute website soon.