The Game’s Afoot


They’re off!

We’re now into the election of the next Conservative party leader. It feels a bit like the Grand National. Time, perhaps, to have a wee flutter. Which ones of them will fall as they jump the hurdles? And will the favourite win?

We’ve had a quick look at the “odds” on Paddy Power here where you too can lose a few quid on betting on who is going to be the next PM. As of writing this post, the odds given are:

Boris Johnson 8/13

Jeremy Hunt 9/2

Andrea Leadsom 9/1

Michael Gove 14/1

Boris Johnson on odds of 8/13 is currently seen as the favourite and the two “runners up” are Jeremy Hunt (9/2) and Andrea Leadsom (9/1). A bit of a change for Michael Gove who had been on 9/1 prior to his “admission” of partaking in cocaine and who has fallen to 14/1. But there you go. There’s probably a lot of potential movement in the odds, which are driven both by the weight of money and the likelihood of the contender actually making it first to the finishing line.

What interests us as economists in our dark pit here is how this might pan out when we consider it as a game. You know, that branch of the dismal science called game theory which can throw some light on the coming struggle.

This is a “run-off” election. The first round, to stay in the game, the contenders have to get eight votes. In the second round it goes up to 17 votes, then 33 votes. It will be the same voters each time. So those who voted for eliminated candidates get to switch to those who will—or maybe won’t—go forward to be voted on by the membership.

Now, there’s a fair degree of partisanship. In particular, there’s a “stop Boris faction” and a “we are for Boris faction” who, presumably, have endorsed him for PM. But unless he miraculously gets over half of MPs voting for him in the first round, he becomes vulnerable to tactical voting. It will all depend on how these voters switch their votes. This is where the “vote Boris” and “stop Boris” preferences come in.

By the way we haven’t a clue as to what might drive these preferences (seriously, we don’t)—and it isn’t the purpose here to try and put up reasons for these—but what we want to show that even with overwhelming betting odds in favour of Boris is whether he’ll get the keys to Number Ten.

We’ll use a simple example. We’ve 13 voters (A to M) who have the following preferences for the top four candidates, which we’ll consider fixed. Taht is the ranking won’t change as the votes are taken. Note that it doesn’t really distract much to leave out the “wild card” candidates, though in practice they aren’t completely out of the picture. So here is our fictitious electorate of MPs and their rank ordering of their preferences for future PM (one being the preferred candidate; four being their least wanted future PM):

Voter Boris Johnson Jeremy Hunt Andrea Leadsom  Michael Gove
A 1 3 2 4
B 1 2 4 3
C 1 3 2 4
D 1 3 4 2
E 4 1 2 3
F 4 1 2 3
G 4 1 3 2
H 3 2 1 4
I 4 2 1 3
J 4 3 2 1
K 3 2 1 4
L 4 3 2 1
M 3 1 2 4


Note the important piece here is the distribution of preferences. That is, how MPs intend to vote if their preferred candidate isn’t still in the running. So the ranking here matters a lot.

So how does this stack up. Adding up the number of preferences for each candidate, we have the following:

Voter preferences Boris Johnson Jeremy Hunt Andrea Leadsom  Michael Gove
1st p
2nd p
3rd p
4th p


When we consider the 2nd preferences, it doesn’t look quite so good for Boris Johnson if a large proportion of the electorate would rather vote for anyone else than him.

In our hypothetical election for prime minister, we’ll have the least successful vote winner (Michael Gove) eliminated as we go into the second round. This is how the votes pan out. All vote preferences remain the same except they now relate to the three remaining candidates:

Voter Boris Johnson Jeremy Hunt Andrea Leadsom
A 1 3 2
B 1 2 3
C 1 3 2
D 1 2 3
E 3 1 2
F 3 1 2
G 3 1 2
H 3 2 1
I 3 2 1
J 3 2 1
K 2 2 1
L 3 2 1
M 2 1 3



The race is going against Boris because he lacks those important “second preference” votes and we find that the “stop Boris” voters who would have voted for Michael Gove turn to the other remaining candidates and Boris is eliminated! Oops!

Voter preferences Boris Johnson Jeremy Hunt Andrea Leadsom
1st p
2nd p
3rd p


While the analysis is hypothetical and doesn’t quite capture the complexities of the voting process, we can see that under plausible scenarios Boris, even with the commanding lead of endorsements he’s received to date, isn’t quite the shoe-in that some commentators expect. We’ll know a lot more after the second round of voting where the lead that Boris is expected to command in the first round gets eroded—or there’s a nice bandwagon effect of MPs getting behind him in the expectation he’ll emerge triumphant. It all depends crucially on those pesky second preferences.

So do you still want to place a bet on who’s going to be the next leader of the Conservative party?

And who at the back of the class said economics is dull?



You might like to point your browser and have a look at my comic fantasy novel

The Sorcerer’s Lackey

which is available as an e-book here.

Go on, you know you want to.