Issues in Quantifying "Electibility"
In honor of Pete Buttigieg's apparent victory in the Iowa Caucus (though I believe, in terms of delegates, Sanders tied him), I thought I'd write some thoughts on the concept of electability, though only partially my own. Greg Mankiw wrote a blog post months ago (http://gregmankiw.blogspot.com/2019/04/bayes-likes-mayor-pete.html) in which he argued, more or less, that Pete Buttigieg appeared to be the most electable Democratic candidate. Admittedly, he didn't use that word, but I think he implied this. Basically, Buttigied had the highest conditional probability of winning the general election given that he won the primary. Later, Andrew Gelman criticized the post in his blog, though mainly because he didn't think using betting odds as genuine, stable probabilities of winning an election was a good idea. But it was in the comments section that there was a discussion of what I think was a more interesting point: even if the betting odds are correct, they still are not necessarily a good indicator of electability, due to the possible nonindependence of the two events in question: winning the primary, and winning the general election.
To use an example, suppose candidate A has a 5% change of winning the primary if there isn't a recession, and a 75% chance of winning the primary if there is a recession. And let's also suppose that he has a 20% chance of winning the general election if there isn't a recession, and an 80% chance if there is a recession. He holds some policy position that, let's say, that is fairly unpopular in normal times, but would become broadly popular across party lines in the event of a recession. If there's a 10% chance of a recession occurring, then candidate A's probability of winning both the primary and the general is: 0.9*(0.05*0.2) + 0.1*(0.75*0.8) = 0.069. His overall probability of winning the primary is 0.9*0.05 + 0.1*0.75 = 0.12. This means the conditional probability of A winning the general given that he wins the primary is 0.69/0.12 = 0.575. Not bad, right? Better chance that average. If the other candidates' probabilities of winning the general given they win the primary are all about 50%, then the conditional probabilities alone would suggest that A's party should rally behind A to maximize its chance of winning.
Of course, that doesn't really follow. A's probability of winning the general given winning the primary is so high because the same rare event - a recession - that drives up his chance of winning the primary also drives up his chance of winning the primary. But that event is, again, unlikely. If, leaving everything else the same, we managed to convince primary voters of A's electability and got him the nomination, his actual chance of winning the general is 0.2*(0.9) + 0.8*(0.1) = 0.26. Not good.
The core problem here is, assuming the betting odds are taking all possible eventualities into account, then if the set of eventualities in which candidate A is most likely to win the primary tend strongly overlap with the set of eventualities in which candidate A is most likely to win the general (given that he wins the primary), then he may end up with a deceptively high probability of winning the general given he wins the primary. If we define electabality as the probability that a candidate will win the general if we were to 'fiat' that this candidate wins the primary, then the conditional probability doesn't give us a good indication of electabality. The conditional probability doesn't necessarily have any 'strategic' implications at all. It merely tells us that, in whatever event this candidate has a good chance of winning the primary, he will also have a very good chance of winning the general. But if that event (or those events) are very rare, then trying to induce him to win the primary despite said event(s) not happening may leave one with a candidate that actually has a very poor chance of winning the general.
The take home lesson being: betting odds, even if entirely accurate, don't necessarily tell us whether, assuming nothing else changes, shifting support to one candidate from another will increase the likelihood of winning the general. This may be one case in which polling is decidedly more useful than betting odds, since, whatever the faults of polls, we can actually ask someone, without knowing what will happen between now and election time, if you knew for sure A was going to win the primary, would you vote for him in the general? The latter question gets at the issue of electability, whereas the conditional probability we get from the betting odds does not necessarily do so. As such, perhaps people should be cautious about using such conditional probabilities to try to influence voters
toward an electable candidate. On the other hand, one might argue that it is unlikely that a candidate will have a high conditional probability for reasons like the recession scenario I use in my example, and that, most often, it will mainly reflect 'actual' electability. That may be true, but I don't think we can be certain that it is. I can imagine a candidate's chances of winning - especially a fringe candidate - being closely tied to rare events.
Finally, I should note, I think Mankiw, though wrong in general regarding conditional probability as a measure of electability, is probably right in particular, and Buttigieg probably is one of the most likely to win the general election given that he wins the primary. But I think cross-referencing the conditional probability with polling data on whether voters would vote for him given that he wins the primary before reaching that conclusion.
To use an example, suppose candidate A has a 5% change of winning the primary if there isn't a recession, and a 75% chance of winning the primary if there is a recession. And let's also suppose that he has a 20% chance of winning the general election if there isn't a recession, and an 80% chance if there is a recession. He holds some policy position that, let's say, that is fairly unpopular in normal times, but would become broadly popular across party lines in the event of a recession. If there's a 10% chance of a recession occurring, then candidate A's probability of winning both the primary and the general is: 0.9*(0.05*0.2) + 0.1*(0.75*0.8) = 0.069. His overall probability of winning the primary is 0.9*0.05 + 0.1*0.75 = 0.12. This means the conditional probability of A winning the general given that he wins the primary is 0.69/0.12 = 0.575. Not bad, right? Better chance that average. If the other candidates' probabilities of winning the general given they win the primary are all about 50%, then the conditional probabilities alone would suggest that A's party should rally behind A to maximize its chance of winning.
Of course, that doesn't really follow. A's probability of winning the general given winning the primary is so high because the same rare event - a recession - that drives up his chance of winning the primary also drives up his chance of winning the primary. But that event is, again, unlikely. If, leaving everything else the same, we managed to convince primary voters of A's electability and got him the nomination, his actual chance of winning the general is 0.2*(0.9) + 0.8*(0.1) = 0.26. Not good.
The core problem here is, assuming the betting odds are taking all possible eventualities into account, then if the set of eventualities in which candidate A is most likely to win the primary tend strongly overlap with the set of eventualities in which candidate A is most likely to win the general (given that he wins the primary), then he may end up with a deceptively high probability of winning the general given he wins the primary. If we define electabality as the probability that a candidate will win the general if we were to 'fiat' that this candidate wins the primary, then the conditional probability doesn't give us a good indication of electabality. The conditional probability doesn't necessarily have any 'strategic' implications at all. It merely tells us that, in whatever event this candidate has a good chance of winning the primary, he will also have a very good chance of winning the general. But if that event (or those events) are very rare, then trying to induce him to win the primary despite said event(s) not happening may leave one with a candidate that actually has a very poor chance of winning the general.
The take home lesson being: betting odds, even if entirely accurate, don't necessarily tell us whether, assuming nothing else changes, shifting support to one candidate from another will increase the likelihood of winning the general. This may be one case in which polling is decidedly more useful than betting odds, since, whatever the faults of polls, we can actually ask someone, without knowing what will happen between now and election time, if you knew for sure A was going to win the primary, would you vote for him in the general? The latter question gets at the issue of electability, whereas the conditional probability we get from the betting odds does not necessarily do so. As such, perhaps people should be cautious about using such conditional probabilities to try to influence voters
toward an electable candidate. On the other hand, one might argue that it is unlikely that a candidate will have a high conditional probability for reasons like the recession scenario I use in my example, and that, most often, it will mainly reflect 'actual' electability. That may be true, but I don't think we can be certain that it is. I can imagine a candidate's chances of winning - especially a fringe candidate - being closely tied to rare events.
Finally, I should note, I think Mankiw, though wrong in general regarding conditional probability as a measure of electability, is probably right in particular, and Buttigieg probably is one of the most likely to win the general election given that he wins the primary. But I think cross-referencing the conditional probability with polling data on whether voters would vote for him given that he wins the primary before reaching that conclusion.
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