Thursday, October 30, 2008

Our next government?

The markets today are saying 70% National, 30% Labour as next PM. Let's have a look at the underlying numbers.

At high bid for each party, assuming that Peters does not return to Parliament, Hide takes Epsom, Dunne and Anderton return to Parliament and the Maori Party take 6 electorates, we're looking at a 123 seat Parliament arrayed as follows:

GRN: 9
PRG: 1
LAB: 45
MAO: 6
UF: 1
NAT: 57
ACT: 4

62 seats are needed for a majority. National + Act + UF are at 62; Labour + Green + Progressive + Maori are at 61.

The markets also are saying there's a 17% chance that National's true vote share will be above 50%. If National's price is normally distributed around the true mean, though, that means there's also a 17% chance that National's true vote share will be below 43.3%. Again, we can take the combination of the prices in VOTE.NATIONAL and MAJORITY.NAT to work out the standard deviation of VOTE.NATIONAL: it's currently at 3.5. Let's assume that the standard deviation of VOTE.LABOUR is the same. We'll benchmark minor party vote share standard deviation by that of NZ First, which we can work out from the combination of VOTE.NZFIRST, TAURANGA.PETERS and MP.PETERS. At current prices, that suggests that the minor party standard deviation is 1.15.

Ok. National's coalition's expected vote share is 50.65%, getting it 61 seats plus Dunne. While the vote share point estimate is 50.65, the standard deviation will be the sum of the standard deviations of the underlying components: 3.5 + 1.15 = 4.65. National's seat share is then expected to be 62, with standard deviation 5.58 seats (as 1% of the vote gets you 1.2 seats).

Labour's coalition's expected vote share is 49.37%, getting it 59 seats plus 1 Maori overhang plus 1 Anderton. The standard deviation will be (as a rough cut) the same as that for National's coalition because we don't need to worry as much about the variance of the Maori party vote if they're set anyway for overhang, so we're really only worried about variance in Green and Labour. So, 4.65. Labour's seat share is then expected to be 61, with standard deviation 5.58 seats.

Now, what's the probability that the realized seat share for the National coalition really lies above the realized seat share for the Labour coalition? A cheap way of checking this is just to check the probability that National lies at 61 seats or less. Our standard normal table tells us the chances of this are 42.86% as only 7.14% of the distribution lies between 62 and 61 seats.

So, National's chances of getting more seats than Labour look more like 57% than like 70%. And those chances get much worse in the 26% probability case that Peters returns to Parliament. I'm not going to bother working them out at this point though.

The market has something wrong here or my calculations are way out. It could be that the normality assumption is just wrong and that National has much more upwards risk than downwards risk. I've made tons of simplifying assumptions along the way, but I can't see that they'd push me that far out. Or, the prices are out somewhere: either Vote.Labour (and partners) is overpriced and Vote.National (and partners) is underpriced, or PM.Labour is underpriced and PM.National is overpriced. Either is rather plausible. There's not much candle in moving the VOTE.Party markets if they're out by a point or two. But similarly folks could just be underestimating the effects of overhang on the PM markets: National gets +1 with overhang but Labour gets +2, which matters in a tight race. Finally, Maori could decide to go with National. If you think that's reasonably likely, then the current prices might not be too bad.

Full disclosure: I'm currently short PM.National and long PM.Labour, with positions taken after having done a rough eyeball version of the above a few days back in the comments section of a prior post. I'm going to further short PM.National and further purchase PM.Labour as result.

Addendum Since posting, PM.National has gone up a couple points, PM.Labour has dropped a couple points, and NOBODY'S vote share has moved. How does the probability of one party or the other winning change when the underlying vote shares don't move? Are folks upweighting the likelihood of Maori going with National?


JC said...

Is it possible people are influenced by the lack of large gains to be made on the party vote stocks (as they tend to not fluctuate much) rather than the PM stocks which have the potential of a $1 payout

Anonymous said...

please forgive me for asking this but what do you mean by "overhang"

malcolm said...

I don't think sampling theory applies to prediction markets. They are not a random sample, and they are not an indication of the distribution of individual voting preference. The proportion is aggregate from individual ratings (reservation prices) rather than from binary preferences, as is usual with political polls.

Sampling error relies on generalizability to a population, but the fulfillment of intentions relies on the reliability of the measurement scale, exogenous events and regression to the mean (this can be formally modelled through a modified beta-binomial distribution). Also voting intentions are subject to serious demographic bias, differential turnout, late deciders, and switchers.

So sorry, but I think you need some new statistics to calculate confidence intervals around Ipredict means! I suggest a Marsden fund application.

Crampton said...

JC: I think you're right. That was my "not much candle" suggestion.

Anon: overhang is the situation where a party gets more electorate seats than it is entitled to on the basis of the party vote. So, if the Maori Party wins 4% of the vote and are entitled to 5 seats, but win 6 electorate seats, the size of Parliament is increased by 1.

Malcolm: you're entirely right that I'm doing very rough cut stats here. I don't think that they'd put me out by more than a few points either way though. So, for example, with National pegged to get 46% of the popular vote (in VOTE.NATIONAL), there has to be a distribution around that point estimate. We can get some kind of handle on that distribution by looking at the price in MAJORITY.NAT (currently 19%). If the distribution is symmetric around the price estimate, then the distribution I use is only going to have second order effects on the analysis above. I'm thinking less about sampling error and more about the probability density function that the marginal trader assigns to various outcomes. The bigger problem is going to be if the probability density function is skewed. I don't think that differential turnout, etc, is going to matter at all here: that all ought to be factored into the price estimates.

I will agree with you that Marsden ought to fund this project though!

Anonymous said...

A follow up.

The belief that national will win despite the seat allocation is not unfounded. There's a long series of polls with National miles ahead. How could all the polls be wrong?

Perhaps national being at 70% is more a reflection of the belief built up by the pattern of polling, rather than a seat-by-seat breakdown.

Ask the same question in two different ways, and different answers spring to mind. Based on polling and mood of the nation (media), who do you think will will the election? 70% to National. Based on seat by seat predictions, who do you think will win the next election? 57% to National.

The 70% number is easy to come by, but how many traders made the calculations to arrive at the 57% number?

A new theory: if you think that national will win at 57% odds, but you don't think traders on ipredict will do the calculations, then there's no reason to believe that the odds will move down from 70%.

Once the odds you believe cross the 50% threshold, aren't you now taking positions against how you think ipredict will move, rather than the true outcome?

Crampton said...

Anon: I'm more than happy for the polls to be right; if they are, though, VOTE.NATIONAL should be trading at 0.49-0.51, not 0.46.

It strictly doesn't matter whether I think other traders will agree with me. If it really is only 60% likely that National wins, and I can sell a contract paying off at $1 in that event for $0.70, then I'm expectationally up $0.10 per contract regardless of what other traders think. This isn't a beauty contest: it benchmarks against a real outcome.