# Conditional probability and election odds

It’s often said that people have trouble processing information about probabilities. Election odds are a perfect example.

For instance, all three of these claims are true:

1. Trump is overwhelmingly favored to win the GOP nomination.
2. If Trump is nominated, he’s expected to win the general election.
3. The Democrats are expected to win the 2020 general election.

How can that be? The simplest answer is that there’s about a 20% chance that someone other than Trump will be the GOP nominee, and in that case the Dems are strong favorites to win (as they should be in my view, as a non-Trump candidate would imply a civil war within the GOP.)

Thus Warren would be likely to win if she gets the nomination, but she’d be likely to lose if both she and Trump are nominated.  The Dems would do better with the second tier (Biden, Sanders, Buttigieg, Yang, etc.—but not Hillary.)

On the other hand, I would not take conditional probabilities of minor candidates too seriously, as the data may not be statistically significant when probabilities are low.  Nonetheless, I believe that a candidate from that group of four men would be more likely to win than Warren, even over Trump.  Warren looks like a weak candidate, which is why I expect the Dems to pick her.  They are dumb enough to fall into Trump’s trap, taking the Biden corruption allegations seriously.

PS.  Bill Weld as the GOP nominee?  LOL.  Who took that bet?

PPS.  If I weren’t so lazy I’d put money on Trump to be the GOP nominee.

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22 Responses to “Conditional probability and election odds”

1. sty.silver
7. October 2019 at 01:37

Good post. I realize I’ve previously always thought about conditional odds as being odds against Trump, but of course, you’re correct in pointing out that Trump isn’t guaranteed to be the GOP’s candidate, and the democrats are more likely to win if he isn’t. This is more relevant now that Trump dipped from ~90% to ~80%.

> The Dems would do better with the second tier (Biden, Sanders, Buttigieg, Yang, etc.—but not Hillary.)

And not Kamala. Also, Buttigieg’s high odds are a bit of an outlier, I believe, I think he’s below Warren most of the time. The conditional odds aren’t that stable for the lower-tier candidates (which is certainly evidence of imperfect calibration) – taking the mean over a short time span (like a week) would probably make the number more informative.

2. Christian List
7. October 2019 at 03:16

Thus Warren would be likely to win if she gets the nomination, but she’d be likely to lose if both she and Trump are nominated.

I’d like to believe you, the result is fine. But I guess you can easily calculate Warren’s chances by dividing her presidential chances by her nomination chances, say 0.25 / 0.5, then you get 50%. With the current odds it’s even 0.28 / 0.53, so more like 53%. Unfortunately, this also fits to the recent polls Warren vs. Trump.

In addition, Biden and Sanders seem to continue to perform better against Trump than Warren. So let’s really hope Warren gets nominated. The Democrats are on the right track so far.

a non-Trump candidate would imply a civil war within the GOP

I might remember it incorrectly, but a few days ago it seemed that you thought the GOP would have better chances without Trump. The opposite is true, it would be their sure defeat. It’s nice that you realized this now as well.

3. sty.silver
7. October 2019 at 04:12

Christian, you’re missing the point. Dividing presidential chances by nomination chances gives you P(win presidency | win nomination). What Scott has pointed out is that P(win presidency | win nomination) > 0.5 > P(win presidency | win nomination AND Trump wins nomination).

You can’t actually compute the latter directly, but from the fact that P(Trump wins presidency | Trump wins nomation) > 50% > P(republicans win presidency), we know that the chances for Democrats to win go down if we condition on Trump winning the nomination. Since Warren is only slightly above 50% before conditioning (52.7% as we speak), she’s probably below 50% after doing so.

4. stoneybatter
7. October 2019 at 04:30

With regards to your PPS, the reason that Trump is at 80% not 100% is entirely due to impeachment odds. Predictit suggests around a 19% chance that he is removed from office this year (first link). If you factor in the implied 71% chance that the House votes to impeach, it suggests that there’s a 27% chance that the Senate would vote to remove him from office if forced to vote. Surprisingly high, in my view.

https://www.predictit.org/markets/detail/5914/Will-the-Senate-convict-Donald-Trump-on-impeachment-in-his-first-term

https://www.predictit.org/markets/detail/3537/Will-Donald-Trump-be-impeached-in-his-first-term

5. Mike Sandifer
7. October 2019 at 05:41

Scott,

You’re underestimating Warren in a general election. “Fightin’ Grandma from Oklahoma”, to use Republican strategist Mike Murphy’s nickname, could actually have pretty strong populist appeal. She touches on a lot of the same issues Trump does, but from a non-crazy(though still ill-informed) perspective.

I think most voters will want something considerably different from Trump in terms of style and intelligence, but someone who will continue to punish DC and Wall Street elites.

It will considerably help Warren that she raises her money from small donors because the effects of the Citizens United ruling, while overstated by critics concerning the influence it grants corporate donors, has nonetheless erased nearly all confidence in the system of raising money from big donors generally. Most, if not all of the elites are seen as corrupt, and Warren will strike quite a contrast. She’s enthusiastically earnest, if nothing else.

She’s the last Democrat I’d want to have to support, but I won’t underestimate her. She’s also becoming a much better campaigner over time, while Biden, Sanders, and Trump are frozen in time.

6. Ram
7. October 2019 at 06:18

Important to understand that P(Candidate X Wins | Candidate X Nominated) is not the same thing as P(Candidate X Wins | do(Candidate X Nominated)), to use Pearl’s notation. In English, the probability that a candidate wins given that we *observe* them winning the nomination is different from the probability that a candidate wins given that we *intervene* so that they win the nomination.

Take Yang for example. The first of these probabilities is 0.034 / 0.049 = 0.694. One might conclude that he is therefore very electable, and that we should all coalesce around him so that the Dems win back the White House. In fact, what this probability really means is that *in the type of world where Yang becomes the nominee, he is also likely to win the election*. In other words, if something crazy happens, causing Yang to win the nomination, it is also likely to cause him to win the general election. On the other hand, if we infer electability from this probability, and thereby make him the nominee, this crazy thing wouldn’t have happened, so there’s no reason to think he’d be in a strong position to win the general election.

Macroeconomists should recognize this as a generalization of the Lucas Critique. The association between winning a nomination and winning the general *under observation* is not the same as the association between winning a nomination and winning the general *under intervention*. For the electability question, what we really want is the second probability, but this cannot directly be computed from the listed prices. We would need to observe plausibly exogenous variation in a candidate’s chances of winning the nomination, and see how their probability of winning the general responds, and daily fluctuations in these things are not exogenous.

I realize this is not a point being made in the post, but I see this confusion a lot. While it’s true that these markets are thin, particularly for the long-shot candidates, the other reason their conditional probabilities may seem wacky is that it’s only in wacky scenarios that they become the nominee in the first place. In such a wacky scenario, it wouldn’t be all that much more wacky for them to go on to win the White House.

7. sty.silver
7. October 2019 at 06:37

@Ram, thanks for that post. Do you think the difference between P(x|y) and P(x|do(y)) is a good reason to mistrust relative comparisons between two candidates with similar values for P(x|y)?

8. Cameron Blank
7. October 2019 at 06:48

I remember being in a graduate economics class in 2011 when one student opined that the InTrade republican nomination market shouldn’t be taken too seriously because it gave a certain candidate a 1% chance to win even though that candidate clearly had NO chance to win. The person he was talking about? Donald Trump (and this was just odds to win the nomination!). While he didn’t win it in 2012, but I don’t think 1% was high considering.

Doubting betting markets is usually a bad idea.

(Just remember conditional probabilities should reflect current bid/ask prices and potential profits (fees, forgone interest, etc) rather than last matched price as you see on the electionbettingodds site. That makes them pretty messy, and there isn’t much profit to be had betting against Trump for president and for Trump as nominee or vice versa.)

9. Ram
7. October 2019 at 07:11

@sty.silver:

In a word, yes. It’s the difference between correlation and causation. Correlation is a good guide to causation when the orientation of the causal arrow is clear, and when the correlation is unconfounded. Here I think it’s clear that being nominated affects being elected, and not the other way around. However, I would say that almost everything which affects the prospects of nomination also affects the prospects of election. For example, if a candidate puts out a new proposal that shifts them to the left in the eyes of the electorate, this might increase their chances of being nominated, and also decrease their chances of being elected. As such it will induce a spurious association between nomination and election which will render the conditional probability causally meaningless. You can think of a million other examples like this.

What would be useful is something like the following: Say that the party unexpectedly changes its primary rules, and this causes a shift in the distribution over candidates for nomination. It’s plausible that the only way this would affect the election outcome is *by way of* its effect on nomination. As such, you can exploit this exogenous variation to identify a causal effect like a LATE. The problem is that this LATE will vary by the type of exogeneity you exploit, and will vary over the course of the election season, and so any one such event is unlikely to give a lot of insight into electability either.

10. E. Harding
7. October 2019 at 07:21

“Thus Warren would be likely to win if she gets the nomination, but she’d be likely to lose if both she and Trump are nominated.”

No; Warren will win.

The biggest problem for Biden is that he’s corrupt (Ukraine is just the tip of the iceberg). The biggest problem for Warren is that she’s an Indian. Generally, ethnic self-misdiagnosis is a lesser obstacle to becoming president than corruption. Just stop pretending you actually think Biden is more electable and admit you just don’t like Warren because she’s a woman.

Prediction markets have certain biases which are easy to exploit. It’s best not to get in them before December, however.

11. E. Harding
7. October 2019 at 07:25

I find your blase dismissal of Biden’s corruption deeply disturbing. Where has your soul gone?

12. ssumner
7. October 2019 at 08:05

Christian, You said:

“I might remember it incorrectly”

Yeah, that’s pretty common with you.

Stoneybatter, That’s why I said I’d take the bet, although in fairness there may also be a modest health risk, given Trump’s age. Maybe a 2% to 4% health risk?

Mike, I’m not sure the public is enthused about losing their private health insurance.

Ram, Yes, that’s an excellent point. Although I was aware of it, I overlooked that perspective when writing this post.

BTW, I noticed the Yang ratio, and suspect that’s just market inefficiency.

Cameron, I agree, although I’m sticking with my Weld incredulity. 🙂

Harding, You are getting ever more deranged, day by day. I preferred Hillary to Sanders in 2016. And there’s no evidence that Biden is corrupt. He may well be (it wouldn’t surprise me at all) but there’s no evidence. Trump’s charges against Biden are laughable lies.

You said:

Says the man who approves of death threats against family members of reporters who criticize Trump.

13. Mike Sandifer
7. October 2019 at 08:25

Scott,

I disagree with you about voters and how they view their health insurance too. There have been dramatic increases in the cost of private insurance plans over the past few years, and an increasing number of people are at or near a financial breaking point. This is even true of many with employer-sponsored plans. In many states, premiums have risen 10-40%-per-year, for the last few years, while deductibles increase.

Warren is showing herself to be a good communicator and if she can communicate her Medicare for All idea effectively, I think people will gladly give up private insurance. Most voters aren’t familiar with this issue yet and so are naturally reluctant to give up what they have. But, they also know something has to change soon.

Warren will communicate her ideas effectively, I predict, and Medicare for All will be widely popular. Pointing out that, while taxes will have to increase, they will be somewhat offset by employees getting more cash compensation, while not having their coverage tied to a specific employer and never having to worry about lacking upfront money to pay for treatment, Warren will present a solution (suboptimal) to the problems tens of millions face. I think she can also sell the idea, even if incorrect, that under her plan Medicare will negotiate down prices and save money on net, even saving most taxpayers money, on net. She can so argue that virtually no one else will have to change healthcare providers, because all providers will have to accept Medicare, unless they want to leave the country or only cater to the rich.

It’s no coincidence that so many developed countries have single player healthcare. It’s very popular.

14. E. Harding
7. October 2019 at 09:38

Says the man who approves of death threats against family members of reporters who criticize Trump.

Sumner, I’m not pro-Trump. I might be misremembering, but I certainly don’t remember saying anything like that.

Trump’s charges against Biden are laughable lies.

So what sort of evidence will you accept here?

I preferred Hillary to Sanders in 2016

Sumner, I’m not basing my claim off what you said in 2016. I’m basing it off of what you say now.

Warren is a slippery character, and a smooth political operator. She knows when to back off the less savory aspects of M4A.

15. E. Harding
7. October 2019 at 09:44

Also, Sumner, if Donald Trump did exactly the same as Joe Biden (pressured a Ukrainian prosecutor investigating the company his son was receiving an exorbitant compensation for being on the board on to resign), what would you be saying?

16. Cameron Blank
7. October 2019 at 09:50

Scott,

I agree. Weld is far too respectable and wise to win the republican nomination. I hope he tries for the libertarian nomination so he can damage Trump in the general.

17. Mike Sandifer
7. October 2019 at 10:30

My impression is that Weld would make a fine President. It really says something about the Republican Party that he will get no serious consideration at all, and in fact, the primary contests for some states have been cancelled.

18. sty.silver
7. October 2019 at 10:33

I’m pretty sure the Yang odds aren’t market inefficiency (or due to the difference between conditional odds and the do operator). There seems to be solid evidence that he’s uniquely appealing to Trump voters, and certainly to libertarians. He does things like talk about the suffering of white males.

And every Trump voter who switches is worth 2 people who vote instead of staying at home.

19. E. Harding
7. October 2019 at 11:48

Yang’s ranking there is simple:

1. Most devoted base of any candidate

2. Plenty of room to grow

3. Inoffensive to the Democratic base

20. Ben
7. October 2019 at 13:14

Wait, I thought you believe in the EMH?

What makes betting markets any different from financial markets?

21. Lorenzo from Oz
7. October 2019 at 17:25

Nice teaching moment.

22. Craig
16. October 2019 at 10:46

I’ve spent too much time thinking about low-probability options on PredictIt.

I actually bet against “Will Michelle run in 2020?” when it was at 9%. It’s currently at 7%.

Say it’s actually 0%. Here are the limits to your profitability if you only want to focus on this one contract:

1. There’s a 10% tax on trading or realization profits. Thus, we’re now at expected gain of 6.3%.

2. There’s a 5% fee on withdrawal. Now, 1.3%.

3. lock up money between now and (?). It’s not clear when it will end, but maybe not till Nov. 2020.

Knock it down to 1.2% because you have to wait over a year. A 1-year CD can be had for >2%, thus it’s not a good play.

Of course, #2 can be reduced if you perform a series of these bets. I think a reasonable strategy would be to parlay these against all such contracts over a decade to reduce the impact of the 5% withdrawal fee. You’re still going to be bound by the time-value of money for how long they have to be locked up. Note, there’s a limit to how much you can scale it up, because you can only invest \$850 per contract. Maybe a decent strategy would be to look for bargains, buy in, then publicize, and maybe move the market and cash in quicker.

Thus my general hypothesis: prediction markets would be more accurate if you subtracted off a time-value of money percentage for both sides and re normalized.

If the prediction said 50%, wouldn’t change: 48 / (48 + 48). But for some dramatic, 3%: 1 / (1 + 95) ~= 1%.