# Fixing Election Markets

One year from now the US will elect a new president, almost surely either a Republican R or a Democrat D. If there are US voters for whom politics is about policy, such voters should want to estimate post-election outcomes y like GDP, unemployment, or war deaths, conditional on the winning party w = R or D. With reliable conditional estimates E[y|w] in hand, such voters could then support the party expected to produce the best outcomes.

Sufficiently active conditional prediction markets can produce conditional estimates E[y|w] that are well-informed and resistent to biases and manipulation. One option is to make bets on y that are called off if w is not true. Another is to trade assets like  “Pays $y if w” for assets like “Pays$1 if w.” A basic problem this whole approach, however, is that simple estimates E[y|w] may reflect correlation instead of causation.

For example, imagine that voters prefer to elect Republicans when they see a war looming. In this case if y = war deaths then E[y|R] might be greater than E[y|D], even if Republicans actually cause fewer war deaths when they run a war. Wolfers and Zitzewitz discuss a similar problem in markets on which party nominees would win the election:

It is tempting to draw a causal interpretation from these results: that nominating John Edwards would have produced the highest Democratic vote share. …The decision market tells us that in the state of the world in which Edwards wins the nomination, he will also probably do well in the general election. This is not the same as saying that he will do well if, based on the decision market, Democrats nominate Edwards. (more)

However, this problem has a solution: conditional close-election markets — markets that estimate post-election outcomes conditional not only on which party wins, but also on the election being close. This variation not only allows a closer comparison between candidates’ causal effects on outcomes, but it is also more relevant to an outcome-oriented voter’s decision. After all, an election must be close in order for your vote to influence the election winner.

To show that conditional close markets estimate causality well, I’ll need to get technical. And use probability math. Which I do now; beware.

First let me introduce some notation. Here are some relevant variables:

x = context before the election
v = sum of votes, each +1 or -1,  in election
w = R if v>0, D if v<0, the election winner
y = an outcome influenced by election winner

Assume ties v=0 are decided by a coin flip. Let the estimates of a consistent market reflect consensus beliefs given by a joint probability distribution p(y,w,v,x). Assume traders know that this joint must satisfy a causality relation:

p(y,w,v,x) = [R(y|vx)*(1[v>0]+½*1[v=0]) +

D(y|vx)*(1[v<0]+½*1[v=0])]*q(vx)

where R(y|v,x), D(y|v,x) describe expected causal results of parties R,D on outcome y, which may depend on context v,x, and where q(v,x) describes expectations for that context. (The form 1[claim] is 1 if claim is true, else 0.)

Let us approximate v as being distributed continuously.  If so, here are integral expressions for naive conditional estimates, the ones that simple conditional prediction markets would give:

E[y|R] = ∫_{v>0} y R(y|vx) q(vx) dydvdx /

∫_{v>0} R(y|vx) q(vx) dydvdx

E[y|D] = ∫_{v<0} y D(y|vx) q(vx) dydvdx /

_{v<0} D(y|vx) q(vx) dydvdx

Note that while the difference between E[y|R] and E[y|D] does reflect differences between causal effects R(y|vx) and D(y|vx), it can also give a misleading comparison as these expressions integrate over quite different ranges of v.

Consider in contrast outcome estimates conditional on an exactly tied election, where everyone’s vote matters:

E[y|R,v=0] = ∫ y R(y|0x) q(0x) dydx /

R(y|0x) q(0x) dydx

E[y|D,v=0] = ∫ y D(y|0x) q(0x) dydx /

∫ D(y|0x) q(0x) dydx

Since both these expressions integrate over exactly the same range for all parameters, a comparison between these estimates gives a direct comparison between the causal effects R(y|vx) and D(y|vx) of the different parties.

Of course prediction markets may not give meaningful for very unlikely conditions like a tie v=0. A reasonable compromise with practicality would be to condition on close elections, won by e or fewer votes:

E[y|R,|v|<e] = ∫_{v in [0,+e]} y R(y|vx) q(vx) dydvdx /

∫_{v in [0,+e]} R(y|vx) q(vx) dydvdx

E[y|D,|v|<e] = ∫_{v in [-e,0]} y D(y|vx) q(vx) dydvdx /

∫_{v in [-e,0]} D(y|vx) q(vx) dydvdx

If we assume that averages of R(y|vx), D(y|vx) over yx are continuous in v, then these close election estimates must approach the ideal tied election estimates in the limit as the allowed vote margin e goes to zero.

In six of the 57 US presidential elections where the public voted, the election was won by less than 1% of the vote. (They were: Bush II, Nixon, Kennedy, Harrison, Clevenland, Garfield.) So prediction markets on post election outcomes that are conditional both on a particular party winning, and on a 1%-close election, should have a roughly 5% chance of paying off. That seems feasible, at least given sufficient market subsidies.

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• I’m not sure how meaningful it is to predict the effect of a Republican president on GDP or unemployment without disentangling the effect of who controls congress. I think you’d want to have eight separate contracts for the configurations of president / house / senate control. Not that the system isn’t already complicated enough.

• David C

Except that Congress simply doesn’t have that much power anymore because they ceded most of it to the President. There’s a historically strong relationship over the last 50 years between the party that controls the White House and budget deficits, tax rates, and GDP growth. There is no such relationship for Congress or any other major macro-level variables of consequence.

• Since both these expressions integrate over exactly the same range for all parameters, a comparison between these estimates gives a direct comparison between the causal effects R(y|vx) and D(y|vx) of the different parties.

No.

What this analysis misses is that the election being close (v being small) is itself a special state of x. For example, suppose President R is elected by a large majority in 2020, and the variable we are interested in is GDP. If the 2024 election is close, then that implies that GDP probably did not perform well in the 2020-4 period, and hence that, to the extent that he can control it, the President is below average at managing the economy.

Hence, conditional on the 2024 election being close, I would expect GDP to be higher under President D in 2024-8 than if we gave President R a second chance. But this is merely a correlation with the close election, rather than genuine causation.

• Ben, while one may be able to make more accurate forecasts by conditioning on more context, the meaning of unconditional forecasts is quite clear.

Salem, yes close elections may represent atypical states. Even so, they are the states where your vote might matter, and in such states the different estimates associated with the different parties will reliably reflect their causal differences.

• But collective votes always matter; we don’t know in advance which elections are likely to be close; and you suggest prediction markets as a way of informing voters, which leads to cascades.

In my above example, if voters collectively pay attention to the prediction markets, they will turn out President R even if he’s doing a good job. In other words, prediction markets don’t scale.

• David

Initially parsed that as (Fixing Election) Markets…

• Assuming the purpose of these prediction markets is to guide voting behavior and that voting takes place in a very short time-frame, these markets only need to be active for a short period of time, right? You could open them up a week or two before an election; opening them earlier wouldn’t have additional value for informing voters. It seems like this alone should isolate the causal relationship; the correlation-distortion applies only when there’s time between the existence of the market at the election, right?

• Ilya Shpitser

Robin,

If you are interested in using prediction markets to estimate causal quantities, and these markets already predict conditional probabilities well, you are in business.

You can simply use interventionist notion of causal effect (what Judea Pearl calls p(y|do(x) the effect on y of fixing/randomizing x), and use assumptions to express these effects in terms of conditional probabilities.

For example, if you have a sufficient number of covariates C to adjust for, we can argue p(y|do(x)) = \sum_c p(y | x, c) p(c).

All you would have to do is get the markets to estimate p(y | x, c) (or E[y | x, c] if we are just interested in the mean).

• Ben, yes this problem is reduced, but not eliminated, just before the election.

Illya, I don’t think you know how to apply that abstract conception to an actual election.

• ThePenileFamily

In my estimation the most important part of this entire ordeal (math and all) is that elections can be and sometime are incredibly close. This puts much more emphasis on any one vote.

I haven’t yet personally figured out a way to logically ascertain the importance or non-importance of democracy.

• Anonymous

It’s not strictly serious, but I still think I’ll provide this link here:

http://www.smbc-comics.com/index.php?db=comics&id=2418#comic

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