A computationally useful algebraic representation of nonlinear disjunctive convex sets using the perspective function

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It took 10 years (mostly my slacker fault), but it’s finally submitted! Pre-print available at Optimization Online.

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Abstract: Nonlinear disjunctive convex sets arise naturally in the formulation or solution methods of many discrete-continuous optimization problems. Often, a tight representation of the disjunctive convex set is sought, with the tightest such representation involving the characterization of the convex hull of the disjunctive convex set. In the most general case, this can be explicitly expressed through the use of the perspective function in higher dimensional space – the so-called extended formulation of the convex hull of a disjunctive convex set. However, there are a number of challenges in using this characterization in computation which prevents its wide-spread use, including non-differentiability issues that arise because of the functional form of the perspective function. In this paper, we propose an explicit algebraic representation of a fairly large class of nonlinear disjunctive convex sets using the perspective function that addresses this latter computational challenge. This explicit representation can be used to generate (tighter) algebraic reformulations for a variety of different problems containing disjunctive convex sets, and we report computational results using this representation for several nonlinear disjunctive problems.

Descartes’ “Meditations on First Philosophy”: First Meditation

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Introduction

I plan on examining Rene Descartes’ “Meditations on First Philosophy” in a series of posts; interestingly, the subtitle of Descartes’ work is “In which the existence of God and the immortality of the soul are demonstrated”. The treatise was first published in 1641 in Latin as a series of six Meditations, and subsequently translated into French. However, I will be using John Cottingham’s English translation, available online here, in what follows.

In this post, I’ll focus on his First Meditation, subtitled “What Can Be Called Into Doubt”. I plan on discussing the text in detail, clarifying concepts and critically examining some of Descartes’ arguments.

As an aside, Descartes is widely considered to be the father of modern Western philosophy, and this particular work continues to exert much influence in contemporary philosophical discussions. Specifically, within the area of epistemology, his “method of doubt” is an approach that continues to inform skeptical approaches to knowledge, while in the area of metaphysics, his doctrine of “Cartesian dualism” (which essentially states that material and immaterial things, like bodies and souls, respectively, are distinct and separate) continues to frame discussions around the soul, consciousness, and other aspects of the philosophy of mind (among other things).

First Meditation – What can be called into doubt

Descartes begins this work by remarking that:

“Some years ago I was struck by the large number of falsehoods that I had accepted as true in my childhood, and by the highly doubtful nature of the whole edifice that I had subsequently based on them. I realized that it was necessary, once in the course of my life, to demolish everything completely and start again right from the foundations if I wanted to establish anything at all in the sciences that was stable and likely to last”.

Descartes’ first sentence is revealing. Descartes was a foundationalist, and as such, believed that it was essential to secure a foundation of beliefs that were certain upon which his remaining beliefs could rest. In other words, he believed that knowledge could only be built on (foundational) beliefs that were apodictically true (i.e. certain); otherwise, those remaining beliefs could not be taken to be (necessarily) true since they would rest on shaky or uncertain foundations. The approach that Descartes uses to establish these secure foundations is through a process of dialectic: he begins by positing a hypothetical set of beliefs that could potentially be certain, then proceeds to show, through counter-examples and thought experiments, that the posited beliefs could be doubted (and therefore, could not serve as beliefs that were certain). He then updates those doubtful beliefs to new beliefs that are potentially more certain by informing himself of the reasons for rejecting the old beliefs, only to show in similar fashion that those new beliefs could be doubted as well via new counter-examples and thought experiments. This process goes on until, it is hoped, he arrives at “an Archimedean point” – a belief that could not be doubted and that could serve as a foundation for his edifice of knowledge. It is critical to appreciate that throughout this process, Descartes adheres to the principle of “withholding assent” to beliefs that he could doubt; as such, if he could doubt them, then these beliefs were not certain, and therefore should not serve as his foundations. Indeed, he explicitly states this in the following paragraph:

“But to accomplish this, it will not be necessary for me to show that all my opinions are false, which is something I could perhaps never manage. Reason now leads me to think that I should hold back my assent from opinions which are not completely certain and indubitable just as carefully as I do from those which are patently false. So, for the purpose of rejecting all my opinions, it will be enough if I find in each of them at least some reason for doubt”.

The argument from the senses

Descartes begins his exercise by identifying the senses as his most widely used method of obtaining (what he believes up to this point are) true beliefs, only to quickly dismiss it as inadequate:

“Whatever I have up till now accepted as most true I have acquired either from the senses or through the senses. But from time to time I have found that the senses deceive, and it is prudent never to trust completely those who have deceived us even once”.

The formal argument for the dismissal of the senses

P1: The senses deceive us sometimes

P2: It is prudent never to always trust those (things) that have sometimes deceived us

C: It is prudent never to always trust our senses

Descartes then proceeds dialectically. He acknowledges that you cannot always trust your senses, but surely, he thinks, this does not imply that you could never trust your senses:

“Yet although the senses occasionally deceive us with respect to objects which are very small or in the distance, there are many other beliefs about which doubt is quite impossible, even though they are derived from the senses – for example, that I am here, sitting by the fire, wearing a winter dressing-gown, holding this piece of paper in my hands, and so on. Again, how could it be denied that these hands or this whole body are mine?”

In other words, Descartes is saying that just because some beliefs obtained through the senses could be doubted (regarding objects that are very small or in the distance) does not imply that all beliefs derived from the senses could be doubted (e.g. that he is here, sitting by the fire etc). As an example, imagine walking in the desert and seeing what appears to be an oasis in the distance, only to realize, as you got closer, that this was merely a mirage; as such, you would have been deceived by your senses. However, would you then necessarily conclude, on that basis, that your senses were deceiving you when you looked at your hands? Would you dismiss the notion that you had hands merely because you used your senses to arrive at that conclusion? If you are convinced by this argument, then you believe the following updated conclusion to be true:

C’: Although it is prudent never to always trust our senses (because they can sometimes deceive), it is foolish to always dismiss our senses (because some beliefs obtained through our senses cannot be doubted)

The argument from insanity

Descartes’ next move is to counter the updated conclusion C’ (that some beliefs obtained from our senses cannot be doubted) by invoking the argument from insanity:

“Unless perhaps I were to liken myself to madmen, whose brains are so damaged by the persistent vapours of melancholia that they firmly maintain they are kings when they are paupers, or say they are dressed in purple when they are naked, or that their heads are made of earthenware, or that they are pumpkins, or made of glass”.

Here Descartes is essentially arguing that even supposed transparent truths such as “he is here, sitting by the fire, that his hands are his etc” could be doubted if he were insane. Indeed, if we think of people afflicted with schizophrenia, they often see things that we would consider delusions but that they firmly believe are “real”. As such, if Descartes was truly insane, then his beliefs that he is sitting here, by the fire etc, could be seen to be the delusions of a mad-man, thus casting doubt on his claim that transparent truths such as these (and that are obtained through the senses) could not be doubted.

The formal argument from insanity

P1: It is possible that I am insane

P2: If I am insane, then under all circumstances, I have reason to doubt that I can distinguish through my senses between what is “real” and what is not

C: Under all circumstances, I have reason to doubt that I can distinguish through my senses between what is “real” and what is not

What’s interesting about the argument from insanity is that for the argument to not be vacuous (I explain what I mean by that term below), P2 seems to presume that a meaningful distinction between what is “real” vs. what is not can be acquired through the senses if we were not insane; as such, if we were insane, that meaningful distinction could no longer be acquired though the senses. So P2 seems to be presuming the truth of P2’:

P2’: If I am not insane, then under at least some circumstances, I can distinguish through my senses between what is “real” and what is not

I say that it presumes the truth of P2’ because if it didn’t, then whether I was insane or not would be irrelevant to whether my senses could be used to distinguish between what is “real” vs. not. Indeed, the argument from insanity would become vacuous since in both cases, my senses could not be used to determine the difference between what was real and what was not (in at least some circumstances). Note that if P2’ was false, this wouldn’t falsify Descartes’ argument from insanity (i.e. P2 could still be true if P2’ was false); it would simply render it vacuous, or meaningless, as an argument. Our reaction would be yes, but so what (in the sense that we could not use our senses to distinguish between what was real vs. not even if we were not insane). As such, in order for this argument to have any force, we must pre-suppose that P2’ is true. But is P2’ true?

Descartes doesn’t address the above point. He does, however, dismiss the argument from insanity when he states:

“But such people are insane and I would be thought equally mad if I took anything from them as a model for myself”

I’m not quite sure what Descartes means by this. Is he invoking public ridicule as his reason for dismissing the argument? Alternatively, is he dismissing the argument because it would take seriously the notion that he was not rational (by presuming he was insane), and therefore put in jeopardy his entire philosophical exercise?

The dreaming argument

Regardless of his reasons for dismissing the insanity argument, Descartes continues with his philosophical endeavor by presuming that he is not insane (i.e. he rejects P1 above). He then makes yet another argument to cast doubt on the senses being purveyors of certainty – the dreaming argument.

The formal dreaming argument:

P1: There are no sure ways of distinguishing between dream experience vs. waking experience (so I could be dreaming right now while I experience something)

P2: Unless I know that I am not dreaming, then under all circumstances, I have reason to doubt that I can distinguish between what is “real” and what is not

C: Under all circumstances, I have reason to doubt that I can distinguish between what is “real” and what is not

Rejection of P1?

Could we not attack P1? Can we really say that there is no way to distinguish between dream experience and waking experience? Doesn’t the mere positing of the categories of “dream experience” vs. “waking experience” imply that we have, experientially-speaking, a conception of their difference (for amongst other things, why label two experiences that are the same differently)? Perhaps. But haven’t we all experienced “waking up in our dreams”, only to realize later when we “really wake up” that we were dreaming that we were awake? If so, then it must be acknowledged that the “dreaming experience” is experientially different in some sense to the “waking-up-in-our-dream-experience”, and yet, we were still dreaming in both. Now presumably, the “waking-up-in-our-dream-experience” is different than the “waking experience”. But could it not be possible that the “waking experience” was really just a second-order “waking-up-in-our-dream-experience” (i.e. an experience where we begin by dreaming that we are awake, only to realize that that experience was merely us dreaming that we were awake, only for us to be really dreaming all along). Indeed, could it not be possible that there really isn’t a “waking experience” at all, but only a series of higher-order “waking-up-in-our-dream-experiences” every time we think we are “really awake”? In other words, can we ever be sure that we “really wake up” at all? As such, the fact that we may have, experientially speaking, a conception of the differences between these experiences doesn’t negate the possibility that we could still be dreaming in all of these experiences. As a side note, I think something like this happens in the movie “Inception”, but it’s been a while since I watched it, and I remember it being so convoluted at the time that I can’t be sure at this point.

On a different note, it’s worth thinking about whether this argument pre-supposes the meaningfulness of an objective conception of reality, given that it seems to require a distinction between what is “really” happening vs. what we are experiencing. Indeed, when it is argued that the “waking experience” could really be a higher-order “waking-up-in-our-dream-experience”, and therefore that we are still “really” dreaming, the word “really” implies some non-subjective reality that actually exists objectively (i.e. the objective reality that we are dreaming). If we were to reject the conception of objective reality, then could we not argue against this line of thinking, since the idea of what was “really happening” would become nonsensical (i.e. reality would just be what we experienced)? Perhaps that is too high a price to pay, and I won’t pursue this any further, although I may return to the Dreaming Argument in a later post.

Rejection of P2: appeal to the Painter’s Analogy

At any rate, Descartes seems to attack P2. He acknowledges, for the sake of argument, that he is dreaming, and that

“[…] these particulars – that my eyes are open, that I am moving my head and stretching out my hands – are not true”.

Nevertheless, he claims that:

“[…] it must surely be admitted that the visions which come in sleep are like paintings, which must have been fashioned in the likeness of things that are real, and hence that at least these general kinds of things – eyes, head, hands and the body as a whole – are things which are not imaginary but are real and exist. For even when painters try to create sirens and satyrs with the most extraordinary bodies, they cannot give them natures which are new in all respects; they simply jumble up the limbs of different animals.”

Descartes is arguing by analogy here. Indeed, he analogizes paintings with dreams to make the point that although the paintings, in some sense, are not “real”, their inspiration or the different elements of the paintings are based on something “real”. As such, he would argue, by analogy, that although our dreams are not “real” in some sense, the elements of our dreams are based on something real, for where else would those elements have come from? He then ups the ante even further, by strengthening the requirements around the kind-of-elements that could be deemed to be real when he states:

“Or if perhaps they manage to think up something so new that nothing remotely similar has ever been seen before – something which is therefore completely fictitious and unreal – at least the colours used in the composition must be real. By similar reasoning, although these general kinds of things – eyes, head, hands and so on – could be imaginary, it must at least be admitted that certain other even simpler and more universal things are real. These are as it were the real colours from which we form all the images of things, whether true or false, that occur in our thought”.

Descartes seems to be making the argument that those things that are more “primary” or “general” in nature cannot be doubted to be real (such as the colours of the painting), whereas other less general things (such as eyes, heads etc) can be doubted to be real. What things does Descartes consider to be “primary”?  He answers:

“[…] the shape of extended things; the quantity, or size and number of these things; the place in which they may exist, the time through which they may endure, and so on”.

This leads Descartes to the tentative conclusion that although disciplines such as physics, medicine and other subjects that depend on the study of composite things could be doubted, truths deriving from more “general” subjects such as arithmetic and geometry could not be doubted, because:

“[…] whether I am awake or asleep, two and three added together are five, and a square has no more than four sides. It seems impossible that such transparent truths should incur any suspicion of being false”.

The Omnipotent God argument

Given Descartes’ religious background, the idea of an omnipotent God pervades his thinking. As such, Descartes questions how we would know that this omnipotent God had not brought about the appearance of an external world as well as the concepts of shape, size, and the more general things presumed to exist in the previous section – and presumably everything that could be believed – when in reality, none of these things “truly existed”?:

“How do I know that he has not brought it about that there is no earth, no sky, no extended thing, no shape, no size, no place, while at the same time ensuring that all these things appear to me to exist just as they do now?”

The formal Omnipotent God Argument

P1: An omnipotent being (God) exists

P2: An omnipotent being could have brought about the appearance of everything, when in reality, nothing truly exists

C1: I have reason to doubt that anything truly exists

Attempt at rejecting C1 by rejecting P2: the appeal to God’s Goodness

Descartes proceeds to reject C1 by arguing that P2 could not be true since God is supremely good, and therefore would not have created him in a way where he was always deceived about everything since that would be inconsistent with his goodness. The argument is as follows:

P3: God’s nature is supremely good

P4: A supremely good nature is inconsistent with me being created in such a way that I am totally deceived (i.e. deceived about everything)

C2: God’s supremely good nature is inconsistent with me being created in such a way that I am totally deceived

As soon as Descartes makes the above argument, he moves to counter that by showing that it would not, after all, be inconsistent with God’s supremely good nature for him to be deceived about everything (i.e. he rejects P4), given that he is sometimes deceived:

“But perhaps God would not have allowed me to be deceived in this way, since he is said to be supremely good. But if it were inconsistent with his goodness to have created me such that I am deceived all the time, it would seem equally foreign to his goodness to allow me to be deceived even occasionally; yet this last assertion cannot be made”.

P5: If God’s supremely good nature is inconsistent with me being totally deceived, then God’s supremely good nature is also inconsistent with me being sometimes deceived

P6: I cannot deny that I am sometimes deceived

C3 (P3+P6): Therefore, given that God is supremely good, it cannot be inconsistent with God’s supremely good nature that I am sometimes deceived

C4 (C3+P5): Therefore, it is not inconsistent with God’s supremely good nature for me to be totally deceived

Conclusion C4 follows by assuming the truth of C3 (which follows from P3 and P6) and P5. Indeed, P5 has the form of “if A then B”, where A is “God’s supremely good nature is inconsistent with me being totally deceived”, and B is “God’s supremely good nature is also inconsistent with me being sometimes deceived”. Given C3, then B must be false, which implies that A must be false as well (otherwise, A must be true, which implies that B must be true, which is a contradiction; in formal logic, this kind of argument is called Modus Tollens: not B, therefore not A). In fact P5 can be re-phrased as P5’:

P5’: If it is not inconsistent with God’s supremely good nature for me to be sometimes deceived, then it is not inconsistent with God’s supremely good nature for me to be totally deceived

Again, C3 and P5’ would lead to C4, as above.

Note that P5 (or P5’) is necessary to arrive to C4. But is there any necessary logical reason for P5 (or P5’) to be true? In other words, if God’s supremely good nature is inconsistent with me being totally deceived, does that necessarily imply that his nature is also inconsistent with me being deceived only sometimes? Isn’t it possible to imagine an omnipotent supremely good God who also has a sense of humor and who deceives his creations sometimes for fun, but who is still compassionate enough, given his Goodness, to not always deceive them? Alternatively, couldn’t this omnipotent supremely good God deceive his creations only sometimes in order to “protect them” or because occasional deception is in their interests, but refuse to completely deceive them because that would be harmful? Consider the analogy of “white lies”: don’t we sometimes engage in “white lying” because telling the truth at that moment is more costly than beneficial; and yet, we still insist that always lying is harmful? As such, one could argue that P5 (or P5’) need not be true, and therefore reject that premise. If so, then we would re-affirm C2 (that God’s supremely good nature would be inconsistent with him totally deceiving us), and by extension, be successful in rejecting P2.

Attempt at rejecting C1 by rejecting P1

Be that as it may, Descartes believes that that the rejection of P2 is not tenable, and considers next the rejection of P1:

“Perhaps there may be some who would prefer to deny the existence of so powerful a God rather than believe that everything else is uncertain”

Having shown that the existence of an omnipotent God, despite his supremely good-nature, could lead to being totally deceived (assuming one buys into P5), he considers the possibility that some would rather deny the existence of this God (in order to avoid this scenario of being totally deceived); for presumably, if this God did not exist, then there is at least a possibility of finding something that cannot be doubted.

Although the rejection of P1 would clearly falsify C1 in the Omnipotent God argument, Descartes claims that the rejection of P1 would affirm C1 (that we could doubt everything) through a different argument. Apparently, the rejection of an omnipotent God would ablate the possibility of ever guaranteeing that he is “well-enough put together” to avoid being in constant error:

“The less powerful they make my original cause, the more likely it is that I am so imperfect as to be deceived all the time”.

His “original cause” here is God. Descartes seems to implicitly connect his creator’s omnipotence to his own imperfections, and imperfections to deception. He seems to be saying that the less powerful his creator, the more likely he is of being imperfect, and therefore the more susceptible to deceit he is. The argument is as follows:

P7: The less omnipotent God is, the more imperfect I am

P8: The more imperfect I am, the more susceptible to deceit I am

P9: An omnipotent being (God) does not exist (i.e. not P1)

C5: I am susceptible to deceit (and the less omnipotent God is, the more susceptible to deceit I am)

Descartes’ argument above is intended to show that even if P1 is rejected, C1 must still necessarily be the case. As such, Descartes arrives at a dilemma, given that whether an omnipotent God exists or not, he must admit that all of his former beliefs are in doubt. However, is this argument convincing? The above argument implies that if God was completely impotent (this would presumably be the case if God didn’t exist at all), then we would be completely imperfect such that we would be totally susceptible to deceit. And yet, many people do not believe in God, and although none of them would claim to be perfect, none of them would also claim to be completely imperfect (and neither would we claim that they were completely imperfect). As such, though they may fall prey to deceit sometimes as a result of some imperfections, surely they are not always being deceived. For who would have the power to do that, if an omnipotent being such as God did not exist?

Conclusion

By the end of the First Meditation, Descartes is convinced that he has reason to doubt everything. However, being convinced of something and remembering to adhere to that something in practice are two different things. As such, in order for Descartes not to fall prey to habit and lapse back into beliefs he has shown he could doubt, he creates an ingenious mechanism – the malicious Demon – to remind himself of the conclusion that he has reason to doubt everything. This demon is malicious in the sense that he is constantly trying to deceive Descartes, and thus, forces him to withhold assent to any of his beliefs lest he fall prey to demonic deceit.

The formal Malicious Demon argument

P1: I have beliefs

P2: A malicious demon could be systematically making me believe only things that are false

C: All my beliefs could therefore be false (therefore, I have reason to doubt all my beliefs)

The malicious demon will make a further appearance in the Second Meditation as Descartes’ foil. There, Descartes claims to finally find his “Archimedean point” – I think, therefore I am (although not stated in this exact way) – that not even a malicious demon could cause him to doubt. I plan on exploring the Second Meditation, and Descartes’s Archimedean point, in a subsequent post.

Clinton vs. Sanders Primaries: Update 5-17

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Clinton vs. Sanders (Nigel Paray for CNN)

Actual Results:

Since our last update, Sanders gained some ground against Clinton as a result of wins in Indiana, West Virginia and Oregon (he lost Guam and Kentucky). The pledged delegate count is now 1771 vs 1499 in favor of Clinton. As such, he now trails Clinton by 272 pledged delegates. Clinton now holds a 54.2% to 45.8% advantage vs. Sanders in total pledged delegate %. Furthermore, Sanders now needs 67.5% of the remaining 781 pledged delegates to win (vs. 32.7% for Clinton). This means that if Sanders wins 67.5% of the remaining pledged delegates, he would win the battle for pledged delegates. Put differently, Sanders would need to win, on average, every contest until the end of the primaries by 34.8% to win the pledged delegate race (of course, if he wins some primaries by less than that, or straight-up loses future primaries, he would need to win other primaries by more than 34.8%). In our “Analysis Going Forward” section (see below), we present some scenarios under which Sanders could still win the pledged delegate nomination; however, keep in mind that all these scenarios are highly unlikely.

Clinton vs Sanders Graph1 5-17-16

Fig 1. The graph shows (off the left-hand axis) the number of pledged delegates won by Clinton and Sanders by state and cumulatively (Clinton in blue and Sanders in red); the remaining pledged delegates in the race (in gray); and the % of those remaining delegates needed to win for each candidate (off the right-hand axis).

A few notes worth mentioning:

  1. We’ve reached the 80%+ mark in the race. There are only 781 pledged delegates (out of 4051) in the remaining races.
  2. Sanders added an additional delegate in Maine once final numbers came in, as well as an additional delegate in Mississippi. On the other hand, he lost the 2 additional delegates he had won at the NV county convention in early April (relative to the district caucuses back in February) once the final state convention was held this past weekend. Although the state convention was marred with controversy, the end result is that the final NV delegate count reflects the original delegate count per by the district caucuses (20-15 in favor of Clinton).

Analysis of Actuals vs. Projections:

The model did OK in its projections of the primaries between April 26th and May 17th. Cumulatively, Clinton won 108 delegates vs. Sanders’ 127 delegates. The model predicted 117 for Clinton vs. 118 for Sanders. As previously discussed in our last update, we expected Sanders to win West Virginia and Oregon (which he did), and for him to be competitive in Indiana and Kentucky (he won the former by 5%, and lost the latter by less than 1%). Guam was a crap-shoot given lack of demographic data (Clinton won by a 60-40% margin). Overall, the model had projected a lead of 458 pledged delegates for Clinton at this point, for a 57.0% vs. 43.0% Clinton lead, so the model is still over-projecting in favor of Clinton (by +2.8%). That overshoot can be largely attributed to Sanders exceeding expectations during his caucus run in March. Final model projections are +590 pledged delegates and a 57.3% vs. 42.7% win for Clinton. Projections were developed after Super Tuesday on March 1st and have not changed since then. See here.

Clinton vs Sanders Graph2 5-17-16

Fig 2. The graph shows (off the left-hand axis) the delegates won or lost by Clinton after each primary (blue bars mean Clinton won the state; red means Sanders won); the size of the bars reflect the difference in delegates won or lost for each state. The bars are staggered in “water-fall” fashion to reflect Clinton’s actual total delegate lead, which is compared against model projections of Clinton’s delegate lead (black dots). The graph also tracks (off the right-hand axis) actual delegate % won for Clinton (blue line) vs. Sanders (red line), and compares against model projected % for Clinton (blue dots) vs. Sanders (red dots)

Clinton vs Sanders Table 5-17-16

Table 1. The table tracks actual pledged delegates won by Clinton and Sanders vs. model projected delegates, and calculates the delta between the two

Analysis Going Forward:

Going forward, the largest remaining chunk of delegates up for grabs occurs on June 7th, but before that, the Virgin Islands and Puerto Rico will hold their caucus and primary, respectively. Given the paucity of good demographic data for both the VI and PR, it’s hard to make any reasonable prediction, although the expectation is that Clinton will at least win PR (she won in a landslide against Obama in 2008). Aside from California, which we will discuss later, Sanders is expected to win Montana, North Dakota and South Dakota, and possibly big, given the favorable demographics and format (Montana is an open primary, North Dakota is an open caucus, and South Dakota is a semi-closed primary). On the other hand, Sanders is expected to lose New Jersey, New Mexico and the District of Columbia, all states whose demographics favor Clinton. This leaves California, which has 475 of the remaining 781 delegates, and which I believe will be a relatively close contest that Sanders is likely to lose, although not by the large margin projected by the model (the semi-closed format still allows independents to vote in the primary if they register by the appropriate date). Of course, given how far behind Sanders is in the delegate count, he not only must win California, but he needs to win it big. There are different scenarios that can still lead to a Sanders victory, but all are highly improbable:

  1. If the demographic projections per the model are accurate, Clinton is expected to win 173 vs. Sanders’ 133 delegates in remaining races ex-California, for a 56.5% vs. 43.5% margin. That would mean that Sanders would need to win California 82.9% to 17.1% (which would give him 394 delegates to Clinton ‘s 81 delegates) in order to win the pledged delegate race.
  2. If Sanders exceeds his demographic expectations and splits the remaining delegates ex-California 50%-50% with Clinton, he would need to win California by a margin of 78.7% to 21.3%.
  3. If Sanders far exceeds his demographic expectations and wins the remaining ex-California delegates by a margin of 67.5% to 32.5%, he would still need to win California by that same margin of 67.5% to 32.5%.

Of course, the alternative to winning the pledged delegate vote is to flip super-delegates at the convention, as Clinton is unlikely to get to the 2383 pledged delegates she needs to outright win the nomination, but that’s an even more unlikely scenario given that Clinton would have won the majority of the pledged delegates, the majority of the popular vote, and she is the party establishment (I also don’t believe the argument that Sanders does better than Clinton against Trump per the latest polls will sway the super-delegates, nor should it; polling 6 months out of an election has very little predictive power). Therefore, I continue to maintain that realistically, unless Clinton gets indicted between now and the convention, the best Sanders can hope for is to influence the Clinton agenda at the convention (a worthy endeavor) by staying in the race and trying to get as many delegates as he can. Next update after the primaries on June 7th.

How the model “works”:

The model regressed delegates won by Clinton vs. Sanders for primaries on March 1st and before against the “racial makeup” of those states. The resulting regression coefficients are then used to project future primaries based on the “racial makeup” of those future states.

Clinton vs. Sanders Primaries: Update 4-26

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Clinton vs. Sanders (Nigel Paray for CNN)

Actual Results:

Sanders continues to lose ground to Clinton as a result of some big losses in PA and MD. He now trails Clinton by 291 pledged delegates. Clinton now holds a 54.8% to 45.2% advantage vs. Sanders in total pledged delegate %. Furthermore, Sanders now needs 64.4% of remaining pledged delegates to win (vs. 35.7% for Clinton). This means that if Sanders wins 64.4% of the remaining pledged delegates, he would win the battle for pledged delegates (and by extension, the popular vote). Put differently, Sanders would need to win, on average, every contest until the end of the primaries by 28.7% to win the pledged delegate race (of course, if he wins some primaries by less than that, or straight-up loses future primaries, he would need to win other primaries by more than 28.7%).

Clinton vs Sanders Graph1 4-26-16

Fig 1. The graph shows (off the left-hand axis) the number of pledged delegates won by Clinton and Sanders by state and cumulatively (Clinton in blue and Sanders in red); the remaining pledged delegates in the race (in gray); and the % of those remaining delegates needed to win for each candidate (off the right-hand axis).

A few notes worth mentioning:

  1. We’ve reached the 3/4 way mark in the race. There are 1016 pledged delegates remaining (out of 4051) in the remaining races.
  2. Sanders won an additional 3 delegates at the CO state convention. If we add the 2 delegates he won at the NV state convention and the 1 delegate change in the GA primary, that’s 6 additional delegates that Sanders has won (and Clinton has lost) in primaries pre-March 1. Given that these were technically additional races independent of the district races, I’ve created a new “pre-March 1” bucket where I’ve lumped those changes together (and penalized my predictions as a result of that).
  3. Clinton gained 1 delegate in the IL race once final numbers came in.

Analysis of Actuals vs. Projections:

The model continued to do well in its projections of the April 26th primaries. Cumulatively, Clinton won 218 delegates vs. Sanders’ 166 delegates. The model predicted 212 for Clinton vs. 172 for Sanders. As previously discussed, we expected Clinton to slightly exceed her demographic projections because of the closed nature of the primaries, which is what occurred (with the exception of Rhode Island, which was a semi-closed primary; as such, registered independents could vote, and Sanders slightly exceeded his demographic projections). Still, overall, the model had projected a lead of 459 pledged delegates for Clinton at this point, for a 57.6% vs. 42.4% Clinton lead, so the model is still over-projecting in favor of Clinton (by +2.8%). That overshoot can be largely attributed to Sanders exceeding expectations during his caucus run over the past month. Final model projections are +590 pledged delegates and a 57.3% vs. 42.7% win for Clinton. Note that we have undone the retroactive changes in delegates before March 1st (GA and NV) from the last update (in addition to adding CO), thus penalizing our projections (relative to last update) to reflect, fairly I think,  changes that occurred as a result of “new” races in pre-March 1 primaries. As such, we go back to the original projected delegate delta of 590 in favor of Clinton. Projections were developed after Super Tuesday on March 1st. See here.

Clinton vs Sanders Graph2 4-26-16

Fig 2. The graph shows (off the left-hand axis) the delegates won or lost by Clinton after each primary (blue bars mean Clinton won the state; red means Sanders won); the size of the bars reflect the difference in delegates won or lost for each state. The bars are staggered in “water-fall” fashion to reflect Clinton’s actual total delegate lead, which is compared against model projections of Clinton’s delegate lead (black dots). The graph also tracks (off the right-hand axis) actual delegate % won for Clinton (blue line) vs. Sanders (red line), and compares against model projected % for Clinton (blue dots) vs. Sanders (red dots)

Clinton vs Sanders Table 4-26-16

Table 1. The table tracks actual pledged delegates won by Clinton and Sanders vs. model projected delegates, and calculates the delta between the two

Analysis Going Forward:

The expectation going forward (in primaries/caucuses pre-June 7th) is for Sanders to win West Virginia and Oregon, and to be competitive in Indiana (open primary) and Kentucky. Guam, the Virgin Islands and Puerto Rico are all caucuses for which we have poor demographic data, so it’s a bit of a crap-shoot, although I would expect Clinton to at least win PR. Of course, Sanders is so far behind at this point that only a California miracle on June 7th can save his campaign. Indeed, assuming he splits all remaining delegates 50-50 with Clinton with the exception of California (and that’s being generous given the Clinton-favorable NJ with its 126 delegates, the largest remaining non-California state), he would need to win California 85%+ vs. 15% to win the pledged delegate vote (assuming proportional allocation, he would actually need a 90-10 win, but given allocation rules, he would get all delegates if he crosses the 85% mark; this implies that, depending on how much better than 50-50 he does, he may still need 85%+ in CA to win the pledged delegate race). The alternative to winning the pledged delegate vote is to flip super-delegates at the convention (I don’t expect Clinton to get to the 2383 pledged delegates she needs to outright win the nomination), but that’s an even more unlikely scenario. Realistically, unless Clinton gets indicted between now and the convention, the best Sanders can hope for is to influence the Clinton agenda at the convention (a worthy endeavor) by staying in the race and trying to get as many delegates as he can. Next update after the Oregon primary on May 17th.

How the model “works”:

The model regressed delegates won by Clinton vs. Sanders for primaries on March 1st and before against the “racial makeup” of those states. The resulting regression coefficients are then used to project future primaries based on the “racial makeup” of those future states.

Clinton vs. Sanders Primaries: Update 4-19

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Clinton vs. Sanders (Nigel Paray for CNN)

Actual Results:

As a result of his big New York loss, Sanders has taken a hit since my last update. He now trails Clinton by 243 pledged delegates. Clinton now holds a 54.6% to 45.4% advantage vs. Sanders in total pledged delegate %. Furthermore, Sanders now needs 58.7% of remaining pledged delegates to win (vs. 41.4% for Clinton). This means that if Sanders wins 58.7% of the remaining pledged delegates, he would win the battle for pledged delegates (and by extension, the popular vote).

Clinton vs Sanders Graph1 4-19-16 v2

Fig 1. The graph shows (off the left-hand axis) the number of pledged delegates won by Clinton and Sanders by state and cumulatively (Clinton in blue and Sanders in red); the remaining pledged delegates in the race (in gray); and the % of those remaining delegates needed to win for each candidate (off the right-hand axis).

A few notes worth mentioning:

  1. There were some small changes in pledged delegates won in previous primaries that were retroactively adjusted as a result of final voting numbers coming in. Clinton lost 1 pledged delegate in Georgia, 2 in Arizona, but gained 1 in Hawaii, for a net loss of 2 delegates relative to our last update.
  2. Sanders won the Nevada County Convention on April 2. The 12 pledged delegates that were up for contention were won 7 vs. 5 for Sanders despite the fact that Clinton had presumably won those delegates 7 vs.5 during the Precinct caucuses. Apparently, Clinton’s delegates didn’t show up. The remaining 23 delegates were not in contention. As a result of that, Sanders gained 2 delegates (and Clinton lost 2). To be clear, nothing is officially “won” until the Nevada Democratic Convention on May 14-15. Until then, it’s important to remember that all pledged delegate calculations are estimates only.
  3. Although Sanders won the popular vote in Wyoming 56% to 44%, both he and Clinton ended up with 7 pledged delegates apiece. This is due to rounding. Recall that delegates are allocated proportionally on a district by district level according to their popular vote in those districts (in addition to the PLEOs and delegates at large for the state). Sometimes, a candidate can be unlucky, in the sense that rounding on a district by district level doesn’t end up reflecting overall popular vote numbers (keep in mind that this has happened to Clinton before as well).

Analysis of Actuals vs. Projections:

The model did really well in its projections of Wisconsin, Wyoming and New York. Cumulatively, Clinton won 184 delegates vs. Sanders’ 163 delegates. The model predicted exactly those cumulative numbers for a delta of 0. Still, overall, the model had projected a lead of 413 pledged delegates for Clinton at this point, for a 57.8% vs. 42.2% Clinton lead, so the model is still clearly over-projecting in favor of Clinton (by +3.2%). That overshoot can be largely attributed to Sanders exceeding expectations during his caucus run over the past month. Final model projections are +584 pledged delegates and a 57.2% vs. 42.8% win for Clinton. As a result of the retroactive changes in delegates before March 1st (Georgia and Nevada), the projected delegate delta went down from 590 to 584. Projections were developed right after Super Tuesday on March 1st. See here.

Clinton vs Sanders Graph2 v2 4-19-16

Fig 2. The graph shows (off the left-hand axis) the delegates won or lost by Clinton after each primary (blue bars mean Clinton won the state; red means Sanders won); the size of the bars reflect the difference in delegates won or lost for each state. The bars are staggered in “water-fall” fashion to reflect Clinton’s actual total delegate lead, which is compared against model projections of Clinton’s delegate lead (black dots). The graph also tracks (off the right-hand axis) actual delegate % won for Clinton (blue line) vs. Sanders (red line), and compares against model projected % for Clinton (blue dots) vs. Sanders (red dots)

Clinton vs Sanders Table v2 4-19-16

Table 1. The table tracks actual pledged delegates won by Clinton and Sanders vs. model projected delegates, and calculates the delta between the two

Analysis Going Forward:

The expectation is for Sanders to continue losing ground in the next 5 states that vote on the 26th of April. I expect Connecticut and Rhode Island to be close, and Maryland and Delaware to be big victories for Clinton. Pennsylvania is an interesting one, in the sense that I expect it to be closer than what most polls are predicting, but I still expect Clinton to win it. All 5 states are primaries, and with the exception of Rhode Island, all of them are closed primaries (Rhode Island is semi-closed). As such, I expect Clinton to slightly outperform her demographic projections. Next update after the 26th of April round of primaries.

How the model “works”:

The model regressed delegates won by Clinton vs. Sanders for primaries on March 1st and before against the “racial makeup” of those states. The resulting regression coefficients are then used to project future primaries based on the “racial makeup” of those future states.

Bernie Sanders’ Record on Palestine

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My article  “Bernie Sanders’ Record on Palestine” has been published at Mondoweiss.

“Bernie Sanders is clearly more progressive on the Palestinian issue than any other major candidate for the Presidency including Hillary Clinton. Still, Nicolas Sawaya says a review of his record on key issues in support of the Palestinian struggle for freedom and justice falls well short”. – Mondoweiss

Clinton vs. Sanders Primaries: Update 3-27

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Clinton vs. Sanders (Nigel Paray for CNN)

Actual Results:

Sanders did very well since my last update. He has cut into Clinton’s pledged delegate lead by almost 100 delegates, whittling her advantage from 326 to 230. Clinton now holds a 55.0% to 45.0% advantage vs. Sanders in total pledged delegate %. Furthermore, Sanders has reduced his % of pledged delegates remaining needed to win the pledged delegate vote, from 58.0% on our last update to 56.6% today. This means that if Sanders wins 56.6% of the remaining pledged delegates, he would win the battle for pledged delegates (and by extension, the popular vote).

Clinton vs Sanders Graph2 3-27-16

Fig 1. The graph shows (off the left-hand axis) the number of pledged delegates won by Clinton and Sanders by state and cumulatively (Clinton in blue and Sanders in red); the remaining pledged delegates in the race (in gray); and the % of those remaining delegates needed to win for each candidate (off the right-hand axis).

Analysis of Actuals vs. Projections:

The model projected a lead of 397 pledged delegates for Clinton at this point, for a 58.6% vs. 41.4% Clinton lead, so the model is clearly over-projecting in favor of Clinton (by +3.6%). Final model projections are +590 pledged delegates and a 57.3% vs. 42.7% win for Clinton. Projections were developed right after Super Tuesday on March 1st and will not be changed. See here.

Clinton vs Sanders Graph1 3-27-16

Fig 2. The graph shows (off the left-hand axis) the delegates won or lost by Clinton after each primary (blue bars mean Clinton won the state; red means Sanders won); the size of the bars reflect the difference in delegates won or lost for each state. The bars are staggered in “water-fall” fashion to reflect Clinton’s actual total delegate lead, which is compared against model projections of Clinton’s delegate lead (black dots). The graph also tracks (off the right-hand axis) actual delegate % won for Clinton (blue line) vs. Sanders (red line), and compares against model projected % for Clinton (blue dots) vs. Sanders (red dots)

Clinton vs Sanders Table 3-27-16

Table 1. The table tracks actual pledged delegates won by Clinton and Sanders vs. model projected delegates, and calculates the delta between the two

The model projections under-performed relative to Sanders’ actual win totals. This can be explained because of 2 reasons:

1) Given that the model works based on a regression of “racial demographics”, it’s important to have good demographic data. The misses on Democrats Abroad, Alaska and Hawaii can be attributed to no demographic and poor demographic data, respectively (we had no demographic data for Democrats Abroad, and Alaska and Hawaii don’t fit our simple White/Black/Hispanic bucketing very well). As such, these misses are not big surprises.

2) A more interesting phenomena is Sanders’ performance in caucus states. Although the model predicted that Sanders would win Idaho, Utah and Washington (and Kansas previously), clearly, he does much better than expected based on projections using simple demographics. I would argue that this is because the caucus format rewards Sanders’ base much more so than Clinton’s.

The so-called “enthusiasm gap” manifests itself in these formats, where people are expected to caucus for significant chunks of the day, and Sanders certainly has very dedicated supporters. As such, when the caucus format is over-layed on top of demographics, this exacerbates the difference in victory in favor of Sanders. This can be contrasted to the primary format, which is less “demanding” of voters and where Clinton does much better than Sanders, and where demographic projections seem right in line with results.

One could also add an additional variable around whether the primary/caucus is open or closed (closed implies only registered Democrats can vote, whereas open implies anyone can register and vote; there are formats in-between as well), where Clinton seems to do better than expected in the closed format given her large lead with registered Democrats.

As such, in retrospect, adding secondary variables around “primaries vs. caucus format” and perhaps “open vs. closed” in the model to complement the primary predictive demographic variables would have likely enhanced results.

Analysis Going Forward:

The bad news for Sanders is that are only 4 remaining caucus states left (Wyoming, Guam, Puerto Rico, Virgin Islands), and they are all relatively small and closed caucuses. Going forward, Wyoming is a closed caucus, Wisconsin is an open primary, and the next states through April 26 are all closed primaries (Rhode Island is semi-closed), including the big states of New York and Pennsylvania. As such, I expect Clinton to slightly outperform her demographic projections. Despite Sanders’ strong performance over the past couple of weeks, I still strongly believe that Clinton will beat Sanders, and I suspect when it’s all said and done, she will do so within ~3% of initial model projections. More to come. Next update after New York.

How the model “works”:

The model regressed delegates won by Clinton vs. Sanders for primaries on March 1st and before against the “racial makeup” of those states. The resulting regression coefficients are then used to project future primaries based on the “racial makeup” of those future states.

Clinton vs. Sanders Primaries: Update 3-18

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Clinton vs. Sanders (Nigel Paray for CNN)

Projections were developed right after Super Tuesday on March 1st and will not be changed. See here.

Right after Super Tuesday, Clinton led Sanders by 199 pledged delegates and had won 59.7% of the pledged delegates (vs. 40.3% for Sanders). Since then, Clinton has widened her lead to 326 pledged delegates and leads Sanders 58.1% vs. 41.9% in total pledged delegate %. The model projected a lead of 389 pledged delegates for Clinton at this point, for a 59.6% vs. 40.4% Clinton lead, so the model is slightly over-projecting in favor of Clinton (by +1.5%). Final model projections are +590 pledged delegates and a 57.3% vs. 42.7% win for Clinton. See table and graph below:

Clinton vs Sanders Graph 3-18-16

Fig 1. The graph shows (off the left-hand axis) the delegates won or lost by Clinton after each primary (blue bars mean Clinton won the state; red means Sanders won); the size of the bars reflect the difference in delegates won or lost for each state. The bars are staggered in “water-fall” fashion to reflect Clinton’s actual total delegate lead, which is compared against model projections of Clinton’s delegate lead (black dots). The graph also tracks (off the right-hand axis) actual delegate % won for Clinton (blue line) vs. Sanders (red line), and compares against model projected % for Clinton (blue dots) vs. Sanders (red dots)

Clinton vs Sanders Table 3-18-16

Table 1. The table tracks actual pledged delegates won by Clinton and Sanders vs. model projected delegates, and calculates the delta between the two

How the model “works”:

The model regressed delegates won by Clinton vs. Sanders for primaries on March 1st and before against the “racial makeup” of those states. The resulting regression coefficients are then used to project future primaries based on the “racial makeup” of those future states.

Clinton vs. Sanders Primaries: Update 3-14

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Clinton vs. Sanders (Nigel Paray for CNN)

For fun, I plan on updating the Clinton vs. Sanders nomination battle after each major set of primaries and tracking actual pledged delegate counts vs. projected delegate counts based on the model I developed right after Super Tuesday (March 1st). See previous post here.

Projections were developed right after March 1st and will not be changed but I will update the actual count to compare against model projections.

Right after Super Tuesday, Clinton led Sanders by 199 delegates and had won 59.7% of the pledged delegates (vs. 40.3% for Sanders). Since then, Clinton has widened her lead to 223 pledged delegates and leads Sanders 58.4% vs. 41.6% in total pledged delegate %. The model projected a lead of 249 delegates for Clinton at this point, for a 59.4% vs. 40.6% Clinton lead, so the model is slightly over-projecting in favor of Clinton. Final model projections are +590 delegates and a 57.3% vs. 42.7% win for Clinton. See table and graph below:

Clinton vs Sanders Graph 3-14-16

Fig 1. The graph shows (off the left-hand axis) the delegates won or lost by Clinton after each primary (blue bars mean Clinton won the state; red means Sanders won); the size of the bars reflect the difference in delegates won or lost for each state. The bars are staggered in “water-fall” fashion to reflect Clinton’s actual total delegate lead, which is compared against model projections of Clinton’s delegate lead (black dots). The graph also tracks (off the right-hand axis) actual delegate % won for Clinton (blue line) vs. Sanders (red line), and compares against model projected % for Clinton (blue dots) vs. Sanders (red dots)

Clinton vs Sanders Table 3-14-16

Table 1. The table tracks actual pledged delegates won by Clinton and Sanders vs. model projected delegates, and calculates the delta between the two

How the model “works”:

The model regressed delegates won by Clinton vs. Sanders for primaries on March 1st and before against the “racial makeup” of those states. The resulting regression coefficients were then used to project future primaries based on the “racial makeup” of those future states.

Why Bernie Sanders won’t win: Demographics

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Clinton vs. Sanders (Nigel Paray for CNN)

Much has been made about Bernie Sanders’ poor performance with minorities so far in the democratic primaries (see here and here). Indeed, an analysis by ABC news of exit polls of all democratic primaries so far reveals that only 15% of black voters and 36% of Hispanics have voted for Sanders (vs. 83% and 63%, respectively, for Clinton); by contrast, Sanders has picked up 48% of the white vote (vs. 50% for Clinton).

ABC Analysis of Democratic Exit Polls

Sanders supporters acknowledge this fact, but counter this by pointing to a more favorable landscape going forward in remaining states, especially post March 22nd. Although this point is true (as we’ll see below), the question remains: is it favorable enough for Sanders to win the nomination. As such, we’re interested in answering the following questions:

  • Does the “racial makeup” of a state (i.e. White / Black / Hispanic demographic split) have any value in predicting the pledged delegate vote?
  • If it does, what do Sanders and Clinton’s delegate projections look like for the remaining races?
  • What would it take for Sanders to win the pledged delegates race?

In order to answer the first question, we regress pledged delegates won by Clinton vs. Sanders for primaries on March 1st and before against the “racial makeup” of those states and check the quality of the fit. The purpose of the regression is to find a set of optimal coefficients that, when multiplied by the “racial demographics” of each state and then by the number of delegates available for that state, result in a set of calculated pledged delegates that match as closely as possible the actual delegates won by each candidate for states that have already voted. The results of the regression are listed in Table 1:

Table 1 Clinton vs Sanders

Table 1. Actual pledged delegates won by Clinton vs. Sanders compared against calculated pledged delegates from regression model

The quality of the fit can intuitively be appreciated by comparing the results of the regression (columns labeled “Projected Delegates”) vs. number of delegates actually won (columns labeled “Delegates Won”) on a state by state level. The closer the regression results are to the actuals (i.e. the closer the “Projected Delegates Delta” columns are to 0), the better the fit. Even though the regression is not perfect (in the sense that the delta between our regression results and the actuals is not 0 for each state), the deviations are relatively minor. Mathematically, this intuition can be assessed more formally by checking the R2 coefficient of the regression (there are other ways as well). The closer this coefficient is to 1, the better the regression is. Given that our R2 coefficient is 0.99 for Clinton and 0.965 for Sanders (the adjusted R2 for Clinton is 0.986 and 0.952 for Sanders), we can conclude that this a good regression (see here and here for a good introduction to regression). This implies that there is a strong correlation between candidate preference and the “racial make-up” of a state, at least based on the states that voted on March 1st and before. Although this strong correlation doesn’t necessarily imply anything causal on its own (in the sense that this could be a spurious correlation), there seems to be a lot of independent evidence that this is indeed a meaningful correlation.

And what of the regression coefficients that yielded the above results? It will come as no surprise that the coefficients reflect what we expected, which is to say that minorities play a very important part of Clinton’s success, while the White vote explains most of Sanders’ success.  For completeness, we list the coefficients here:

Regression coefficients Clinton vs Sanders

In order to answer the second question, we multiply the regression coefficients by the “racial makeup” of future states and then by the pledged delegates available for those future states to project future primaries. Our total projected delegates for these future states is presented in table 2:

Table 2 Clinton vs Sanders

Table 2. Projected pledged delegates from regression model

If we look at the projections based on our regression method, we see that Sanders performs better than Clinton in states with a large fraction of White voters, while Clinton does better in more diverse states. Overall, we expect that Clinton will win 2321 delegates (57.3% of total pledged delegates) vs. Sanders’ 1730 delegates (42.7%). As such, we can see that Sanders loses the pledged delegates vote handily.

The third and final question can now be asked: what would it take for Sanders to win? We choose to answer this question by asking a proxy question: what would the candidate preference by White / Black / Hispanic voters have to be going forward for Sanders to win the nomination?

There are multiple ways to do this, and we run three different scenarios. For all scenarios, we assume that the future popular vote won is proportional to the number of pledged delegates won; this is a reasonable assumption, on average, as the Democratic primaries proportionally allocate their delegates once a 15% threshold of the vote is met:

  • Assuming minorities continue voting as they have in states that have already voted, Sanders would have to win about 70% of the White vote going forward. Given that Sanders has only managed to win 48% of the White vote so far, expecting him to win an additional 20%+ going forward seems implausible.
  • Assuming Sanders continues to capture the same amount of the White vote as he has, he would need to capture 70% of the Black and Hispanic vote to win the nomination. This seems even more implausible given his current percentages with minorities.
  • Assuming Sanders captures 60% of the White vote (a +12% increase vs. today), he would need to capture 50% of the Black and Hispanic vote (a +36% and +14% increase vs. today, respectively) to win the nomination. As such, even with such a drastic increase in the White vote captured, Sanders would somehow have to triple his percentage with Black voters, and significantly increase his support with Hispanics. Again, this simply does not seem realistic.

Note that throughout this analysis, we have ignored super-delegates (who favor Clinton). One of the interesting conclusions resulting from this exercise is that talk of super-delegates in this race is superfluous: unless Clinton gets forced out of the race (because she gets indicted, imprisoned, or some other far-fetched scenario) Sanders is extremely unlikely to win the popular vote, because of demographics.

Some notes:

“Racial makeup” in red in Tables 1 and 2 come from 2016 democratic primary exit polls; in black from 2008 democratic primary exit polls (note that this helps Sanders as minorities, which Clinton wins handily, have increased their percentages over the past 8 years); in blue for Florida from the latest 2016 democratic primary poll available; in green from the “racial make-up” of the state as a whole (no better data was available) based on, for the most part, the 2010 census; in gray for Democrats Abroad based on an assumption that neither candidate is favored given lack of information.