When will the 2020 Election "Start"?

Pundits are already speculating about who will run for president on the Democratic ticket for 2020, which begs the question: what does the data suggest when the first person will declare? How many candidates can we expect? And what is the rate at which candidates will enter the race?

Shut up and tell me the answers

Assuming the Democratic race will be like the Republican 2012 and 2016 race, apparently candidates declaring their entrance follows a Poisson Distribution, as the interval between announcements follows a Exponential Distribution; with 95% probability, candidates announce their intent to run anywhere between 7.62 days on the low end, and 17.54 on the high end, with an expected rate of about 11 days between each announcement.

Consequently, since the first serious contender announced Sunday 11 November 2018, we can expect, with 95% probability, anywhere between 21 to 48.55 candidates to make the announcement, with likelihood maximized at 33.6 candidates to be the Democratic nominee in the 2020 cycle.

Rate at which Candidates Declare

According to Wikipedia, the 2012 Republican candidates declared in the following order:

Candidate	Date Declared
Newt Gingrich	May 11, 2011
Ron Paul	May 13, 2011
Herman Cain	May 21, 2011
Mitt Romney	June 2, 2011
Rick Santorum	June 6, 2011
Jon Huntsman, Jr.	June 21, 2011
Michele Bachmann	June 27, 2011
Rick Perry	Aug. 13, 2011

Similarly for the 2016 Republican primary race, again from wikipedia, we have the following:

Candidate	Date Declared
Ted Cruz	Mar. 23, 2015
Rand Paul	Apr. 7, 2015
Marco Rubio	Apr. 13, 2015
Ben Carson	May 4, 2015
Carly Fiorina	May 4, 2015
Mike Huckabee	May 5, 2015
Rick Santorum	May 27, 2015
George Pataki	May 28, 2015
Lindsey Graham	June 1, 2015
Rick Perry	June 4, 2015
Jeb Bush	June 15, 2015
Bobby Jindal	June 24, 2015
Chris Christie	June 30, 2015

The interval between each candidate declaring his or her candidacy for Republicans in 2012 is (in days): 8, 12, 4, 15, 6, 47. This has a mean of 15.33333 days, and a variance of 256.66666 days. (Observe that the variance is approximately the square of the mean.)

The intervals between each candidate for Republicans in 2016 is (in days): 6, 21, 1, 22, 1, 4, 3, 11, 9, 6. This has a mean of 8.4 days, and a variance of 57.822222 days (the fractional part is 37/45). (Again, observe how the mean squared is approximately the variance.)

When we combine this data together, the concatenated dataset has a mean of 11 days, and a variance of 132.26666 days (observe the square of the mean is approximately the variance).

This property (the variance is approximately the square of the mean) supports the hypothesis that this dataset is described by a Random Variable following an Exponential Distribution with its rate parameter approximately 1/λ ≈ 11 days.

Well, really, 95% of the posterior distribution lies between 0.05702248 < λ < 0.1312337 and peaks at 1/λ ≈ 11 days; the high density interval thus described is shaded in blue in the following plot:

When first candidate will declare

We should probably ask the question How many days before election day November 3, 2020 will the first candidate announce his or her bid for presidency?

The data for the Republican candidates in 2016, dataset (of days before the election the candidate announced) is: 497, 503, 512, 523, 526, 530, 531, 553, 554, 554, 575, 581, 596. Observe this occurred within a span of 199 days.

Similarly, the data for the 2012 election: 451, 498, 504, 519, 523, 535, 543, 545. Also observe this occurred within the span of 94 days (roughly half the length of the 2016 election).

Observe that the 2016 election had 5 candidates declare earlier than the earliest nominee in 2012, so it stands to reason to suppose that the greater the number of candidates, the earlier the first bid, and vice-versa:

Herd-Size Conjecture: the number of candidates is directly proportional to the earliest candidate's declaration date (as measured by days before the election).

(This is not an unreasonable conjecture, since candidates declare at intervals which are described by an exponential distribution, and there is a hard deadline to declare your candidacy.)

Assuming every candidate wants to run in every primary, South Carolina requires filing for primary candidates by September 30, 2019 (assuming it is like the 2016 primary — the deadlines for the 2016 primaries may be found here). This is 400 days before the election, everyone must file before then.

There is actually a fairly decent correlation between the total funds raised, total spent, and total left on-hand (just add up the quarterly books) and the length of a primary campaign. For the nominee, I consider the "end date" to be election day.

I had to work with 2012 data because it's far neater than the 2016 data. The R² = 0.8546118, and R²_adj = 0.6607609; the model is:

(number of days) 
= 115.53788314621659
  - 11.278698441397473×(total raised in $Mn)
  + 22.78101502109625×(total spent in $Mn)
  - 10.49938850575515×(total left on-hand in $Mn)

Supposing this model also holds for the Democrats, all we have to do is estimate how much money is "out there" for candidates. For the Republicans in 2012, all candidates raised a total sum of $337,615,860

Corollary: If the Herd-Size Conjecture holds, then number of candidates is directly proportional to total funds raised by the candidates.

Proof sketch: There are several steps in the proof.

1. Since the length of a campaign is directly proportional to the funds raised, and the Herd-Size Conjecture says the earliest candidate's declaration (i.e., the start of the campaign) is directly proportional to the number of candidates, it follows the candidate's declaration date is proportional to the funds raised.
2. Since the candidates declare at a fairly steady rate following an exponential distribution, it follows the earlier the first candidate declares, the greater the number of candidates will declare.
3. The greater the funds raised, the earlier the candidate's start date tends to be (by step 1), and hence the greater the number of candidates (by step 2). ■

Of course, this just punts the problem (of determining who will declare first) to the much more difficult problem of how much money will be in the Democratic 2020 primaries, and how will it be partitioned.

Something to consider is that the amount may be approximately the total amount of money raised by the party in the House from the previous midterm election (2010 Republicans raised a total of $353Mn in the House, according to OpenSecrets). If this is a good approximation, then there will be roughly $649Mn raised by the Democratic candidates in the primary, and the total amount spent will vary between $421Mn to $649Mn (topic for future post!). But if $443Mn is spent, we could expect to see primary announcements as early as December 4th, 2018.

Animals and Rational Behaviour

Having introduced a notion of Instrumental Rationality, perhaps a good question to ask is "What is an example of a rational agent which is not human?"

It might seem that animals behave rationally at times; for example, worker bees give up their own production in favor of other offspring of the queen, surely this must give some benefit to the donor. This idea (generalizing the bee situation) is known as Hamilton's Rule, and it doesn't really work. For example, it is neither testable nor does it make predictions.

With the bees, the altruist (i.e., worker bee) cooperates by giving a benefit b to the recipient (another offspring) at a cost c to itself. Both b and c are measured in terms of fitness, specifically the expected number of offspring. Naively one might expect b > c to suffice, but Hamilton's major insight was that relatedness ("degree of kinship") r between donor and recipient must enter into the equation, giving us Hamilton's rule br > c.

Game theorists are overjoyed to hear this can be derived from utility maximization, and one might expect it to have a status similar to Newton's laws in physics. However, Nowak, Tarnita, and Wilson have argued that Hamilton's rule almost never holds. In short, simple game theoretic models here fail to describe the biological situation.

Decades ago, evolutionary biologists would have treated Hamilton's rule as an "iron law". That no longer seems to be the case. For more on this, see de Vladar and Szathmáry's "Beyond Hamilton's Rule".

Agents are Instrumentally Rational

Game Plan: We'll introduce the notion of "instrumental rationality" as an ordering of alternatives with some technical condition. Then we'll discuss measures of "preference" via utility functions. Then we conclude by discussing maximizing utility under uncertainty.

Loosely put, individuals who are instrumentally rational have preferences over various "things" (e.g., baby-back ribs are preferred to chicken, and chicken is preferred to bread). Such individuals are deemed "rational" for picking actions which satisfy those preferences. The only constraint is that preferences are ordered in some suitably "weakly coherent" way (e.g., ribs are still preferred to bread if there is no chicken).

The convention is to call these "things" preferred as "Alternatives".

Definition 1. Let an actor be choosing between countably many possible different alternatives x₁, x₂, x₃, …. An actor is called Instrumentally Rational if the actor has preferences satisfying the following conditions:

1. Reflexivity: No alternative x_i is less preferred than itself.
2. Completeness: For any two alternatives x_i and x_j, either (1) x_i is strictly preferred over x_j, (2) x_j is strictly preferred over x_i, or (3) the actor is indifferent between the two alternatives.
3. Transitivity: For any alternatives x_i, x_j, x_k, if x_i is no less desired than x_j, and if x_j is no less desired than x_k, then x_i is no less desired than x_k.
4. Continuity: For any alternatives x_i, x_j, x_k, if x_i is (strictly) preferred to x_j, and if x_j is (strictly) preferred to x_k, then there exists some "composite" of x_i and x_k (call it y) which is equally as desired as x_j.

Remark 1 (On Continuity). There are two ways to interpret the continuity axiom. The first perspective is to think of y as a "basket" containing "bits" of x_i and "bits" of x_k. For example, if x_i is "18 ribs", x_j is "half a roasted chicken", and x_k is "10 rolls", then there is some composite ("9 ribs and 5 rolls") which is equally as desirable as half a chicken.

The other perspective is to think of y as a lottery, where the actor obtains x_i with probability p (0 < p < 1) and x_k with probability 1 − p. The continuity axiom then says there is some p for which the actor is indifferent between the lottery y and the alternative x_j.

Remark 2 (Ordering, Utility Functions). The first three axioms taken together implies the actor has a well-defined preference ordering (in the mathematical sense). When the continuity axiom is added, the preference ordering may be "represented" by a utility function (i.e., a function assigning to each alternative x_i some real number U(x_i) reflecting the "utility" or "desire" for that alternative). An actor making choices to satisfy his or her preference ordering can be viewed "as if" maximizing his or her utility function.

Now, discussions of "utility" of an alternative should not be confused with the philosophy of Utilitarianism. A utility function just assigns some numbers such that the ordering induced by it is the same as the actor's preference relation. That is to say, U(x_i) > U(x_j) if and only if x_i is strictly preferred to x_j. The numbers represented by U(x_i) are measured in utils, which is Agent-dependent and measures that Agent's preference for the given alternative.

Ordinal Utilities, Cardinal Utilities, Maximizing Expected Utility

Definition 2. If we assign utility "arbitrarily" but in a manner consistent with the preference ordering (e.g., for any alternatives X and Y such that X is preferred to Y, we assign the utilities such that U(X) > U(Y) but otherwise the quantities remain arbitrary), then we call such utility the Ordinal Utility.

Here we must stress again there are two important points of assigning utility in a manner which captures only the preference ordering (and nothing else).

First, ordinal utility does not describe the agent's "intensity of desire" for an alternative. The "strength of preference" is not captured by this notion. So how much more I want ribs than chicken is not adequately described by this notion, just the fact that I really want ribs right now (and not chicken, much less bread).

Second, ordinal utility cannot be compared "across agents". The ordinal utility I assign to a full rack of baby-back ribs cannot be compared to anyone else's ordinal utility for, say, Lasagna. We can only compare my ordinal utility for baby-back ribs against my ordinal utility for Lasagna.

Dealing with Uncertainty

My local BBQ joint smokes 1 pig per day, and when it's all sold, there's no more. If I am hungry, should I go before the lunch rush or afterwards?

Here we must talk of Prospects, outcomes and their associated probabilities.

For our particular situation, there is a decision I must make (go before the lunch rush or after) and two outcomes (there is food left, or they ran out of food). One prospect is given by the possible outcomes to a given choice of going before the lunch rush (go before lunch AND they have food, go before lunch AND no more food; p, 1 − p). The other prospect is given by the decision to go after the lunch rush (go after lunch AND they have food, go after lunch AND no more food; q, 1 − q).

Observe, each decision has different possible outcomes, but the probabilities for the outcomes on a given decision must sum to 100%: something must happen when I take a decision.

We now need to consider preference ordering over prospects.

Definition 3. Suppose a person must choose between actions with uncertain outcomes, in the sense that: each action has various possible outcomes associated with it, each with some probability. We call this action a Prospect and represent it by a pairing of the possible outcomes with their respective probabilities (y₁, y₂, ...; p₁, p₂, ...) where the outcome y_i occurs with probability p_i, and the probabilities sum to 1 = p₁ + p₂ + ... (since there must be an outcome to the action).

Remark. There is a "nested structure" to prospects, in the sense that y_i might be an "atomic outcome" (e.g., "there will be food", "there will be no food", "it will rain", "the world will end", etc.) or another prospect (imagine "I flip a coin; if it is heads, then I do this action, but if it is tails then I do some other action").

An actor's Preferences over Prospects are called Consistent if the preference satisfies axioms (1), (2), and (3) of Definition 1, and:

4. Continuity: Consider three prospects y_i, y_j, and y_k, and suppose the first is preferred to the second and the second is preferred to the third. Then there exists some probability p such that the prospect (y_i, y_k; p, 1 − p) is equally as preferable to y_j (compare to the second interpretation forwarded in Remark 1).
5. Preference increasing with probability: If y_i is preferred to y_j, letting y_m = (y_i, y_j; p₁, 1 − p₁) and y_n = (y_i, y_j; p₂, 1 − p₂), then y_m is preferred to y_n only if p₁ > p₂.
6. Independence: For any three prospects y_i, y_j, and y_k, if y_i is preferred to y_j, then there exists a probability p such that the prospect (y_i, y_j; p, 1 − p) is no less desired than (y_i, y_k; p, 1 − p)

Given this notion of "consistent preferences over uncertain prospects", how can we develop a notion of instrumental rationality?

Maximizing Expected Utility

The first step is to introduce the notion of Cardinal Utility, which assigns to a given outcome y_i the intensity for an agent's preference for that outcome u(y_i).

The second step is to consider the Expected Utility of a Prospect y = (y₁, y₂, ...; p₁, p₂, ...) as the sum E_u[y] = u(y₁)p₁ + u(y₂)p₂ + ..., which is the expected value of the "random variable".

Now, an agent with cardinal utility u(-) is considered instrumentally rational if it picks the action whose prospect has the maximum expected utility.

Example 1. If I go to my favorite BBQ restaurant before the lunch rush, the prospect looks like ("get food", "no food"; 0.95, 0.05). If I leave after the lunch rush, the prospect looks like ("get food", "no food"; 0.1, 0.9).

My cardinal utility function looks like u(get food) = 10, u(no food) = −30.

The expected utility for going before the lunch rush is then

E[before]	=	u(get food)×0.95 + u(no food)×0.05
	=	10×0.95 − 30×0.05
	=	9.5 − 1.5
	=	8

The expected utility for going after the lunch rush is then

E[after]	=	u(get food)×0.1 + u(no food)×0.9
	=	10×0.1 − 30×0.9
	=	1 − 27
	=	−26

Since 8 > −26, it is rational to go before the lunch rush to try to get food.

Next time, we'll discuss flaws with this notion of instrumental rationality, both logical and empirical.

References

• Shaun Hargreaves Heap and Yanis Varoufakis, Game Theory: A Critical Introduction. Second ed., Routledge. (This is the axiomatization scheme I am following.)
• John Searle, Rationality in Action. MIT Press, 2001. (This provides a different set of axioms for rational behaviour, equivalent to the axioms of game theory, and discusses implicit assumptions & its flaws.)