Modelling lame duck prime ministers

David Clark in the Guardian is pointing out that people are thoroughly bored with the Tony Blair versus Gordon Brown show. So I thought I’d settle the matter once and for all by setting a date for Tony’s departure, based on quantitative economics rather than all this nebulous political stuff. I am taking my modelling strategy from David Clark’s observations that what really matters here is 1) the non-Blairites perception of whether Tony is staying or going and 2) how vindictive a victorious Gordon is planning to be to the Blairites. So let’s make a model.

We start by assuming that there is a single “representative” MP, that all he wants is a “prize” (say a ministerial job or some other sort of preferment) and the only way he can get a prize is by grovelling. He has to decide whether to grovel to Tony or to Gordon. We model this thing starting from day 1 of the current Parliament.

At some date between 6 May 2005 and 5 May 2010, Tony will leave. Before this date D, Tony can give prizes and after this date, Gordon can award prizes to those who don’t have them, and take prizes away from those who do have them. So, assuming that the MP’s objective is to own a prize on 6 May 2010 (on which day we assume the game ends), he has to choose the grovelling strategy which maximises his chances of having a prize at the end, subject to the following rules of the game (a “grovelling strategy” is a calendar for the next 3651 days specifying who he plans to grovel to on each day. We simplify the game mightily by only considering strategies of the form “grovel x days to Tony and afterwards grovel to Gordon”, but I don’t think this is unrealistic; you only really get one chance at regicide). Set up the game with the following rules:

1. The likelihood of getting a prize from Tony is increasing in X; the more days you grovel to Tony, the more likely he is to give you a prize.
2. Gordon hates Tony, so the more days you grovel to Tony, the less likely you are to get a prize from Gordon, and the more likely you are to lose the prize that you got.
3. However, people who have a prize are more likely to keep it than people who don’t have a prize are to get one.
4. The representative agent starts the game grovelling to Tony. When he switches to grovelling to Gordon, Tony has to resign and grovelling to him is no longer any use.
5. If you get a prize from Tony, then grovelling to Gordon will neither help nor hinder you; you will either keep or lose your prize, depending on whether you’re on Gordon’s shit-list. (This is a bit arbitrary but makes the maths much simpler)
6. When Gordon gets in, the shit-list is only relevant to people who got prizes under Tony. People who didn’t get prizes under Tony have the slate wiped clean and their chances of getting a prize from Gordon only depend on the amount of grovelling to Gordon that they did.

We can make this a bit more specific by introducing three functions:
Gt(x) is Tony’s grovelling function; the probability of getting a prize from Tony expressed as a function of the number of days spent grovelling to Tony
Gg(x) is Gordon’s grovelling function; the probability of getting a prize from Gordon expressed as a function of the number of days spent grovelling to Gordon (which, of course, equals 3651 days minus the number of days spent grovelling to Tony).
V(x) is Gordon’s vindictiveness function, the probability of losing a prize given to you by Tony, expressed as a function of the days spent grovelling to Tony.

To make the functions specific, let’s say that on every day you spend grovelling to Tony, you have a probability p of being given a prize by Tony and a probability q of going onto Gordon’s shit-list of people who are to have their prizes taken away. Once Tony is gone, every day spent grovelling to Gordon (assuming you don’t have a prize) gives you a probability r of getting a prize. This means:

Gt(x) = 1-(1-p)^x
Gg(x) = 1-(1-r)^(3651-x)
V(x) = 1-(1-q)^x

And your probability of having a prize at the end of the game is equal to the chance of getting a prize from Tony and not losing it, plus the chance of not getting a prize from Tony but getting it from Gordon. I’ll call that Y, for no real reason:

Y(x) = Gt(x)(1-V(x)) + (1-Gt(x))(1-Gg(x))

I am assuming a zero discount rate for the sake of this game, by the way. The obvious thing to do now would be to differentiate with respect to x and find the value which maximises Y(x). However, you can’t actually do that, because the problem changes over time; for every day that passes before you make the decision, you’ve burned a Tony-grovelling day and the supply of potential Gordon-grovelling days has diminished. So we need a dynamic programming approach.

Basically, (and fear not; there is no way that I am going to wheel out the Bellman equations here; this is the prelude to a crappy spreadsheet exercise), the correct way to solve this problem is to start with May 4, 2010 and work backwards. You calculate (conditional on the assumption that the representative agent hadn’t made the switch by then), what the probability of getting a prize would be if you switched on the last day, and if you didn’t. This is pretty easy. You then go back one day and model the decision on the last day but one. On this day, you either switch to Gordon, or you keep with Tony for that day and on the next day, you adopt whichever of the strategies for that day had the highest value. You can then roll back to day T-3 and so on, recursively. If you are trying to solve real problems in closed form, this is a real bugger to do, but if you are solving toy problems like this one you can just put together a spreadsheet and actually calculate the values for every day.

Which is what I did, working on the assumption that there are 100 players of this game, that Tony can potentially give prizes to 78 of them over five years (roughly two reshuffles) and that Gordon can have one reshuffle (34 prizes) in which no more than half of his moves can be to screw Blairites (the shitlist has 17 spaces). I ended up calculating the thing on the basis of 60 months rather than 3560 days, so I got a value for p of 1.3%, q = 0.28% and r = 0.57%. Solving it recursively gave me the following three interesting model results: (by the way, I promise you that I didn’t rejig the model calibration a hundred times to come up with these, much)

1. The value of “switch to Gordon now” is higher than “stay with Tony for one more period” starting in month 12, which would be April 2006 ie now. This would be the point at which a dynamic programming solution to the optimisation problem would say that the switch would occur, but of course in a real world model with all sorts of other things going on it might not. But I deduce from this that there is some fundamental basis for the rather nebulous feeling that “the time has come” among Labour MPs.
2. The positive gap between the values is at its largest in month 27, which is July 2007. This might represent the point at which the speculation will reach its height if Tony hasn’t gone by then, although it is a recess month.
3. Starting in month 41 (if the model ever reaches that point), the value of “stick with Tony” goes above “switch now” again, and stays above it for the rest of the model. The idea here is that because the shit-list chance is low and there are not enough potential Gordon-grovelling days left to make a difference, someone with 41 months worth of grovelling invested in Tony might as well stick with it. So if Tony stays on until September 2008, he will most likely stay on until the next election.