We’ve talked about uncertainty in polls (see Margin of Error, Total Margin of Error, Total Margin of Error II) and we’ve talked about ranked data (see exploded logit !). A new paper, Rosenman & Liang 2026, looks at uncertainty in ranked choice voting (RCV) polls.

Recall the multinomial logit model that Train (2009) Chapter 7 calls the exploded logit:

P[ranking Other then Left then Right] = exp(f_Other) / sum_c’ exp(f_c’)   *   exp(f_Left) / (exp(f_Left) + exp(f_Right))

Without covariates, it has only 3 parameters: f_Other, f_Left, f_Right. It makes the independence from irrelevant alternatives (IIA) assumption to go from these 3 parameters to rank probabilities.

In contrast, the multinomial model in Rosenman & Liang 2026 does not make the IIA assumption and has 14 parameters, one for each of 15 possible rankings minus one so they sum to 1:

P[ranking Other then Left then Right] = pi_{Other, Left, Right}

Rosenman & Liang 2026 note that in RCV the election outcome is not expressable as one parameter. Instead, the winner is determined by instant runoff:

1. If a candidate wins >50% of first choice votes, they win.
2. Otherwise, the candidate with the least first choice votes is eliminated, and each ballot counts for its top remaining choice. Return to step 1.

Say you use polling data to estimate rank probabilities pi_j for each ranking j. These estimates differ from the true probabilities due to many sources of error (see our favorite Figure 2.5 from Groves et al. shown in quantity vs quality and is a mismeasured X better than none at all ?). Rosenman & Liang 2026 focus on sampling error.

How can we propagate uncertainty about the rank probabilities pi_j to uncertainty about the RCV winner ? If you have draws from the posterior of pi_j, you can do instant runoff on each to get a winner for that draw. This gives win probabilities according to your model and data.

To see the importance of uncertainty in RCV, let’s look at their 2022 Alaska House special election example. With 3 candidates, RCV is determined by 5 margins (see their Lemma 1). Most of these margins are well-identified by the data, but 2 were quite close: Palin vs Begich first choice margin and Peltola vs Palin pairwise margin. They plot these 2 margins in the right panel of Figure 1. The true outcome is the black dot, with sampling uncertainty shown as ellipses around it. For small sample sizes (the biggest ellipse), we see that a plurality of the mass falls into green, where point estimates would declare that Begich wins. Uncertainty quantification would help put this in context, giving all candidates win probabilities around 20-40%, showing the race is difficult to call with such small data.

For details, see Rosenman & Liang 2026.