Equality by lot

One fascinating aspect of sortition is that it treats all groups in the population fairly: If a group constitutes x% of the population, the group’s share in the panel will be x% in expectation (that is, on average over many random panels). Furthermore, it is unlikely that, in a random panel, this percentage will deviate much from x%; this event becomes ever less likely the larger the panel is. Unfortunately, there are practical limits to how large sortition panels can get, which means that a certain variance remains. Had the Irish citizens’ assembly been sampled without consideration for gender, for example, a gender imbalance of at most 45 women against at least 54 men would have happened in about 15% of random panels.¹

One way around this problem is stratified sampling. For example, one could fill half of the seats with random women and half of them with random men. As Yoram wrote in an earlier post on this blog, one can still guarantee that every person is selected with equal probability, and thus, that every group will get its fair share of the panel in expectation. Stratification by gender will obviously ensure accurate representation to the genders. But what happens to the representation of other groups?

My collaborators — Gerdus Benadè and Ariel Procaccia — and I studied this question in a paper that we recently presented at the ACM Conference on Economics and Computation. As Mueller, Tollison, and Willett argued as early as 1972,² stratification can greatly reduce the variance in representation for groups that highly correlate with the feature we stratified on. This is good news since correlation is everywhere; for example, stratifying by gender will help to represent opinion groups related to military intervention, gun control, and healthcare.³ We show that there is no real downside to stratification: Even in the worst case, stratification cannot increase the variance of another group by more than a negligible amount. These results hold up to very fine stratifications, where each seat is filled by a random member of a dedicated stratum. This suggests that we should indeed make extensive use of stratification. In a case study on a real-world dataset, we show that stratification can reduce the variance in an opinion group’s representation by a similar amount as an increase of panel size by multiple seats — even if the stratifier does not know the opinion in question!

The main technical difficulty in the paper is working with indivisibilities. For instance, if we split the 99 seats of the Irish citizens’ assembly proportionally by gender, women should get around 50.66 seats. To ensure that every person is still selected with equal probabilities, we need to randomly “round” the seat assignments, giving women sometimes 50 and sometimes 51 seats. This process is somewhat delicate — rounding introduces new variance, which might lead to some unfortunate group becoming much less accurately represented than without stratification. If one uses the rounding procedure suggested in our paper, this is not the case.

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