Why Baltimore was correct to kick an extra point for an eight-point lead against Washington

With 4:47 remaining in their game vs Washington, the Baltimore Ravens scored a touchdown that increased their lead from 21-20 to 28-20.  You probably know what happened next: the Redskins drove down the field, overcoming the loss of RG3 along the way, with Kirk Cousins hitting Pierre Garcon for a touchdown with 32 seconds remaining and then successfully executing a QB draw for the tying two-point conversion; Washington then went on to win the game in overtime.  In the aftermath of the game, some analysts, such as Grantland's Bill Barnwell and espn.com's Gregg Easterbrook (aka TMQ) suggested that Baltimore ought to have attempted a two-point conversion after their touchdown, which if successful would have given them a nearly insurmountable nine-point lead.  They both suggest that the two-point try offered great upside with little risk, pointing out that even if it failed, Washington would still almost certainly only end up tying the game with a touchdown.

But closer examination of the issue reveals that this is a rare case where Barnwell and Easterbrook (normally very sharp football thinkers) are wrong, and the conventional wisdom manifested in John Harbaugh's decision is right.  In fact, Barnwell and Easterbrook have fallen prey to the very same error that underlies many of the legitimately foolish strands of football's conventional wisdom: defining the issue in terms of vaguely-specified risk/reward criteria, instead of a rigorous analysis of the win probabilities involved.  The math involved in this particular case is actually too simple to be of much inherent interest, but it still serves as a valuable reminder of how important it is to think through these questions in quantitative terms - we see that even top-caliber analysts can be led astray when they rely entirely on their intuition. 

Here, the key point is that by kicking an extra point, Baltimore made it considerably more difficult for Washington to tie the game with a touchdown, as they now needed to convert a two-point try, rather than simply kicking an extra point.  The fact that Washington would have faced a much easier road to a tied game following a failed two-point attempt by Baltimore may not trigger any visceral perception of risk, but it still endows the extra point with a fair amount of upside in relative to a failed two-point try.  In order to estimate the success probability at which Baltimore's two-point try would have been a break-even proposition, we need to weigh this upside against the greater reward offered by a successful two-point conversion, which would result in a nine-point lead. 

We will use P(B) to denote the probability of Baltimore successfully executing a two-point conversion, P(W) to refer to the probability of Washington executing a successful two-point conversion.  (For simplicity, we assume that probability of extra point success is 100% - this variable largely cancels out on both sides of the equation, and the logic of the conclusion can easily be restated in order to allow this assumption to be relaxed.)  Now, restricting ourselves to the space of potential outcomes where Washington outscores Baltimore by exactly one touchdown over the remainder of the game (since these are the only feasible outcomes where Baltimore's post-TD decision is directly reflected in the final result) we see that Baltimore faces the following dilemma:

Option 1: Kick an extra point.  Now with probability P(W), the game goes to overtime, and with probability 1-P(W) Baltimore wins in regulation.
Option 2: Attempt a two-point conversion.  Now with probability P(B), Baltimore wins the game in regulation, and with probability 1-P(B), the game goes to overtime.

So to first approximation, Baltimore's two-point try breaks even if P(B) = 1-P(W); i.e. if their probability of success is as high as Washington's probability of failure.  While we cannot say for certain that this hurdle was not reached in the game, we note that it requires that at least one team to have a two-point success probability of over 50%* - in which case they ought to be adopting the two-point conversion as their general post-touchdown strategy.  This conclusion can be summarized by pointing out that statistically, a team is better off needing to defend a two-point conversion than they are needing to make one of their own, unless one or both teams would be successful more than half the time. 

And in fact, the two-point conversion attempt has yet another factor in its disfavor.  As indicated above, the game situations where Washington outscores Baltimore by exactly one touchdown over the remainder of the game are the only ones in which Baltimore's post-TD decision has direct bearing on the final score.  But the probability of the various rest-of-game outcomes is not an entirely exogenous factor to their decision; in particular, a successful two-point conversion, by clarifying Washington's scoring requirements for the remainder of the game, would make it slightly more likely that Washington would outscore them by one TD plus another score of some variety, thereby winning the game outright themselves.  This is still a very low-probability event - scoring twice in the final five minutes is always a tall order - but it still means that a slight downward adjustment to the equity of the two-point try is in order.  Therefore, even if the nominal break-even point of P(B) = 1-P(W) were reached, we conclude that the two-point attempt would still be an incorrect decision. 

* Here is we can bring extra-point probability back into the mix: the actual threshold is not 50%, but P(XP)/2.  But the general consideration remains the same: if a team's two-point probability were in excess of this value, they ought to attempt a two-point conversion after virtually every touchdown.

A Response to Stephen Metcalf's Critique of Libertarianism, Part 2

This is a continuation of my response to Stephen Metcalf's slate.com article on libertarianism. In Part 1, we marveled at Mr Metcalf's powers of imagination, as at a stroke, he replaced the rich and variegated history of libertarian thought with a caricature entirely of his own invention. In this new version, the movement featured nothing but a bunch of cranks and apologists for the upper class, with the sole exception of a temporary mental lapse by a Harvard professor, which manifested itself in a semi-popular book in defense of libertarianism. Further, we saw that despite having essentially rigged the game so that he only had one man to beat, Mr Metcalf nonetheless managed to whiff badly on his shot at the open goal, as his discussion of ASU only manages to establish his utter failure to understand the nature of the claims made by its author.

Of course, these tortuous attempts to engage with the output of other writers could be forgiven if Mr Metcalf were to provide us with some truly interesting thoughts of his own on the merits of libertarian ideas. But when he finally turns his efforts in this direction in the final section of the article, we find that the results are, if anything, even less impressive than the ill-conceived Nozick-related drivel with which he has been regaling us throughout the preceding pages. In the first two paragraphs of this final section, Mr Metcalf provides two distinct attacks on libertarianism:

1) An implication that libertarianism rests on semantic sleight of hand: "The ploy is to take libertarianism as Orwell meant it and confuse it with libertarianism as Hayek meant it; to take a faith in the individual as an irreducible unit of moral worth, and turn it into a weapon in favor of predation." Here Mr Metcalf is once again replacing the actuality of the matter with an altered version more congenial to his own purposes - the "ploy" is entirely of his own making. His criticism has doubtless caused great consternation among the libertarian occupants of his imaginary parallel universe, who start from an axiom that some vague notion of individual liberty is inviolable, and only then proceed to work out the definition of what liberty actually is. But, of course, libertarians here on Earth do not do this. They argue, through a variety of methods, for a very specific non-interference principle as an overriding consideration in the construction of an ideal government; since this implies a state of affairs closely related to the accepted meaning of the word "liberty," this word has become associated with their ideas. But it is the precise content of the ideas themselves that must be addressed; having failed to do so, Mr Metcalf's suggestion that Hayek's form of liberty constitutes "predation" is nothing but a flagrant piece of question-begging.

2) A claim that critics of libertarianism are inherently "bullied," "manipulated," and "hectored into silence" by the discourse of their opponents. Unfortunately, his invocation of these terms reveals that Mr Metcalf simply does not understand the logical structure of the libertarian position - that in contrast to other political philosophies, it ascribes non-trivial content to the question of what a government is properly allowed to do, a question that is logically prior to the question of what it should do in a given situation. Mr Metcalf is correct, if not particularly felicitous in his mode of expression, when he says: "The non-trivial question is: What risks...must be removed from the oasis [??] and placed in the framework [??] ... if liberty is to be kept a substantive reality, and not a vacuous formality?" But he has absolutely no justification to suggest, as he does in the following sentence, that libertarians, simply by virtue of their firm negative answer, "seek to prevent a reasoned discussion [of this question] from ever taking place." Libertarians merely direct their "reasoned discussion" toward a subsidiary question that Mr Metcalf apparently has no use for: the prior question of the proper limits (if any) to be placed on government power at the outset. If one accepts the premise that government ought to do virtually anything that is dictated by the democratic process, then the libertarian position would indeed seem unjustly dogmatic. But it does not take extensive familiarity with libertarianism to realize that it is this very premise that they reject. Whether the approach adopted by libertarian writers is primarily consequentialist (as in Hayek's Road To Serfdom) or deontological (as in Nozick's ASU) they are in every case more than sufficiently clear as to the nature of their contention: that the entity of government ought to have its power carefully circumscribed at the very point when it comes into existence, before the question of its application to any particular issue even arises. That their opponents are compelled to answer this question, instead of skipping directly to the question that they would apparently prefer to answer, is due to the inherent logical relationship of the two questions - not to any unsavory tactics on the part of libertarians.

This latter mode of attack is worthy of further examination, both because, fallacious though it is, it is appallingly common in everyday discourse on the subject, and because elucidating the nature of the mistake helps to reveal Mr Metcalf's fundamental failure to properly understand libertarian ideas. Indeed, it is this error that also underlies the haughtily dismissive tone with which our author, in the article's opening paragraph, addresses the notion that libertarians are "above" the political spectrum. In fact, this description is particularly apt, as with their focus on the prior question, true libertarians have little in common with any mainstream position, all of which give the green light to virtually any government activity, at least in principle. This explains libertarians' hesitation to participate in conventional political discussion; like our author's Amtraker, I have personally found that there is very little to be gained by engaging in a discussion that presumes the truth of the very proposition that I maintain to be false.

But by far the most insidious application of this mistaken thought process is the attribution of "hate," "greed," or some similarly distasteful sentiment to the libertarian position. It is doubtless true that there have been cases where people already possessed of such emotions have gravitated to libertarianism as the political philosophy with the greatest degree of superficial similarity to their chosen mode of interaction. But it should go without saying that this has absolutely no bearing on libertarian ideas in general - the fact that a given conclusion can be reached via a flawed mode of reasoning tells us absolutely nothing about whether the conclusion itself is true or false. To reiterate: libertarians direct their focus at a level that logically precedes the point at which interpersonal sentiment of any type would begin to play a role in generating the content of political belief. For instance, they favor the eradication of active government intervention on behalf of the unfortunate not because they are insensitive to the personal tragedies faced by such people (many libertarians - myself included - attempt through voluntarily-undertaken activity to ameliorate these sad states of affairs) but because according to their interpretation, the intellectual evidence suggests strongly that such government activity never ought to have been on the table to begin with; to remove it now is merely to correct a long-standing error that ultimately possesses nothing less than the potential to bring our free and prosperous civilization to ruin, no matter how laudably humane its consequences appear to be in the short run. Those who suggest that libertarians necessarily possess insufficiently enlightened feeling for their fellow human beings commit an error analogous to that made by Mr Metcalf in his attack on Nozick: they impose onto the discussion their own view of one of the component questions (in this case, by presuming that entire discussion of normative political propositions consists of the simple question of what it is that we "want" government to do), and evaluate the disagreement offered by their opponents through this prism.

In order to demonstrate that libertarianism ultimately provides worthwhile answers to the questions of political philosophy, it is of course necessary to validate the arguments that have been made in support of such an inherently limited conception of government, against the challenges offered by opponents. A complete rehash of this ongoing debate would take us well beyond the scope of this review; fortunately, it suffices for our current purposes to point out that beyond his failed attack on the Wilt Chamberlain example from ASU, Mr Metcalf's five pages of shrill attacks on libertarianism contain precisely one sentence that engages with the libertarian position at a fundamental level (and you'd better sit down here, lest the demonstration of logical power that you are about to witness blow you away):

"When I think with my own brain and look with my own eyes [??], it's obvious to me that some combination of civil rights, democratic institutions, educational capital, social trust, consumer choice, and economic opportunity make me free."

Well then!

Now, if libertarianism were truly the intellectual horror-show that our author, ad nauseam, suggests it to be, it should not be in the least bit difficult to confront its distinguishing premise head-on. What are we to make of Mr Metcalf's failure to do so? As far as I can see, we are left with two possibilities: 1) either Mr Metcalf is not aware that this is the question that needs to be answered (which is nothing less than an indication that he does not know what libertarianism is), or 2) he is simply unable to provide a satisfactory answer to the question, and therefore has deliberately attempted in this article to reach his desired conclusion through smoke and mirrors - a never-ending stream of irrelevant, ill-informed, and/or fallacious arguments. It is not intended as a compliment to Mr Metcalf that I am quite uncertain which of these possibilities is more likely; regardless, the inescapable conclusion is that his article, purporting as it does to evaluate as a "scam" an idea to whose defining premise he has offered no rebuttal, is a complete failure.

But we are not done, as Mr Metcalf has obligingly saved his crowning moment of idiocy for the final paragraph:

"Large-scale, speculative risk, undertaken by already grossly overcompensated bankers, is now officially part of the framework, in the form of too-big-to-fail guarantees made, implicitly and explicitly, by the Federal Reserve."

I will go ahead and quote this again, in case your eyes disbelieve what they read the first time around:

"Large-scale, speculative risk, undertaken by already grossly overcompensated bankers, is now officially part of the framework, in the form of too-big-to-fail guarantees made, implicitly and explicitly, by the Federal Reserve."

Surely benefit of the doubt can no longer be maintained by even the most generous of readers. The truth has been laid bare for all to see: our author has simply been presumptuous enough to write an article claiming to offer a comprehensive evaluation of the merits of an idea, without going to the very great trouble of ascertaining what that idea actually is. The distinction between the pure capitalism advocated by libertarians and the crony capitalism that underlies the phenomenon of overcompensated bankers, the concept of Too Big To Fail, and even the very existence of the Federal Reserve, should not be obscure to anyone with even a rudimentary familiarity with the current political landscape - if nothing else, a sizable clue is provided by the fact that the one prominent American politician to maintain an explicit affinity for libertarian ideas, Ron Paul, has written a best-selling book called End The Fed. That Mr Metcalf initially attempted to discredit libertarianism by associating it with the interests of the wealthy was reprehensible enough; that he is now essentially defining it as such - by including as examples of the negative consequences of libertarianism the very phenomena that libertarians (more than any other political faction) have consistently and vocally opposed - is simply beyond words. "Sad" and "repugnant" are merely a start at forming a proper evaluation, both of this disgraceful piece of trash journalism, and of the fact that it is representative of what passes for meaningful political discourse in today's intellectual climate.

A Response to Stephen Metcalf's critique of libertarianism

Edited 6/27/11, to better explain the issue of Nozick's conception of justice.

One of the indispensable attributes for any libertarian in the early 21st-century is a very thick skin. We must constantly hear our ideas misrepresented, twisted, summarily dismissed via idiotic pop economic reasoning, summarily dismissed via even-more-idiotic Keynesian economic reasoning, and/or generally placed on a par with some combination of devil-worship and geocentrism. In most cases, if we are to maintain both our sanity and good relations with our fellow human beings, we must rise above such episodes with little more than a knowing smile. However, slate.com recently published an article by Stephen Metcalf that is so egregiously awful that it cannot be passed over in silence. It is titled "The Liberty Scam - Why even Robert Nozick, the philosophical father of libertarianism, gave up on the movement he inspired." The author claims that libertarianism, which, as we all know, is little more than an excuse for rich capitalist people to continue their never-ending quest to exploit the poor, gained a potentially troublesome veneer of intellectual respectability when Harvard philosophy professor Robert Nozick published Anarchy, State, and Utopia - a book-length defense of libertarian ideas - in 1974. But not to worry: our hero has identified a fatal flaw in Nozick's reasoning; moreover, he informs us that Nozick himself abandoned libertarianism later in his career. And while libertarian ideas continue to exert a regrettable influence on real-world events (such as when the Federal Reserve bails out Too Big To Fail banks) we can rest assured that with the movement's erstwhile champion safely disposed of, the rest of the civilized world will soon come around to the author's final conclusion that a political organization based on unfettered freedom of individual action (and its counterpart, unmitigated individual bearing of risk) is "not only sad, but repugnant."

How does a libertarian respond to this? What has our fearsome adversary gotten wrong in his bold assault? In a word: everything. Even sparing any extended response to the absurd motive-mongering association of libertarian ideas with the interests of the wealthy - which ought to be an outright embarrassment to intellectually-mature readers of any political persuasion - we are still spoiled for choice when it comes to points of rebuttal.

First, Mr Metcalf badly overestimates Nozick's stature within the libertarian movement. It is doubtless true that the popularly-accessible style of Anarchy, State, and Utopia has been responsible for a more widespread familiarity with libertarian ideas among the general populace. But while serious libertarians remain grateful for this high-profile effort on their behalf, (and for the not-infrequent moments of genuine philosophical brilliance contained in the book), this is mixed with a tinge of embarrassment at the shallow treatment that Nozick gives to the ultimate foundations of the libertarian edifice in comparison with writers such as Jan Narveson, in The Libertarian Idea, or Roger Pilon, whose obscure but brilliant essay "Ordering Rights Consistently: Or What We Do and Do Not Have Rights To", published in The Georgia Law Review of 1979, may be the most convincing purely philosophical treatment of all. But by far the greatest shortcoming in the author's implied hierarchy of libertarian ideas is the short shrift given to the so-called Austrian economists - in particular, the triumvirate of Mises, Hayek, and Rothbard. To be fair, the author does grant a passing reference to the first two: these economists, perhaps the two greatest thinkers in the history of the Austrian tradition, are included by Mr Metcalf among the "nutters and shills" of post-WWII libertarians, not on the basis of any direct evaluation of their writings, but because the solid, if unspectacular, roster of professorial appointments they held in their careers (which, of course, is the ultimate standard by which a thinker's ideas are to be judged) was apparently tainted by the influence of (gasp!) corporate sponsorship; and because the infallible John Maynard Keynes once deigned to write a critical note in the margin of his copy of Hayek's Prices and Production, thereby settling the matter once and for all. Meanwhile, Mises's American protege Murray Rothbard, who was known in his lifetime as "Mr Libertarian", who was directly responsible for sparking Nozick's interest in libertarian ideas (as Nozick states in the preface of ASU) and who explored these ideas at much greater depth than Nozick (cf. The Ethics of Liberty), is never even mentioned. Once one is aware of the awe-inspiring depth with which these thinkers treat libertarian themes, it becomes clear that at an attempt to discredit libertarianism via an attack on Nozick is like trying to refute the theory of relativity by quibbling with a Bill Nye presentation.

It may be objected at this point that our author's primary concern is the status of libertarian ideas within the general zeitgeist of American thought, and that the timing of the publication of Nozick's book fits so closely with the increase in prominence of these ideas that it must have played a causal role. While this may be correct to some degree, it is both irrelevant to any serious critical evaluation of libertarian thought (which should clearly focus on the best writings from that tradition, not merely the most popular) and grossly overstated - Mr Metcalf suggests, with no apparent trace of irony, that the publication of a pop philosophy account of libertarianism somehow resulted in the award of Nobel Prizes in economics to Hayek in 1974 (after all, thanks to Keynes's marginal scribblings, we know that he couldn't have won it on his own merits) and Milton Friedman in 1975. In fact, a much more plausible explanation for the sudden academic success enjoyed by free-marketeers at this time is the dismal economic performance of the 1970s. It became apparent to a great many people with a modicum of economic literacy (a subset that our author has clearly not bothered to join, judging by his rather confused references to Keynes throughout the article, among many other errors) that the interventionist Keynesian orthodoxy of the time was inadequate, leaving Milton Friedman's monetarism as the dominant tradition in macroeconomic thought. The 1970s also featured the incipient resurgence of the Austrian school, which has continued an accelerating expansion to this day; they can now boast of faculty appointments at several prominent American universities, with no corporate sponsorship involved.

Second, the claim that Nozick "abandoned" his earlier libertarianism is simply incorrect. Here is what Nozick himself had to say, in an interview shortly before his death in 2002: "What I was really saying in The Examined Life was that I was no longer as hardcore a libertarian as I had been before. But the rumors of my deviation (or apostasy!) from libertarianism were much exaggerated." As we can see here, it is true that Nozick did indeed soften the stance set forward in ASU. Had Mr Metcalf seen fit to quote these modifications in their proper context, and subject them to substantive discussion, they could conceivably have provided interesting material with which to criticize the libertarian position. Instead, he crudely distorts this modulation of Nozick's thought in order to use it as a would-be point of fact in support of his simplistic hatchet-job on the libertarian movement. The loss that results accrues entirely to his own credibility.

Perhaps Mr Metcalf thought it unnecessary to explore Nozick's deviation from ASU in any meaningful manner because he was so confident in his own criticism of Nozick's earlier work. But this attack (which, like many critical responses to ASU, focuses on the famous "Wilt Chamberlain example") is thoroughly defective. In his book, Nozick introduces Wilt Chamberlain in order to illustrate the point that any chosen end-state conception of justice (D1) is liable to be upset - virtually instantaneously - by voluntary transactions to the mutual benefit of all parties, such as when a large number of people pay to watch Wilt Chamberlain play basketball. The resulting end-state D2 is highly different from D1 (Wilt Chamberlain has become much wealthier) but surely, no matter how great one's attachment to the D1 pattern, there can be nothing unjust about D2, if it was reached in this manner. All of this is dutifully summarized by Mr Metcalf. Then he goes on the attack: the Wilt Chamberlain example is used as a justification of capitalism...but what relevance can it have in this context when it makes no reference to "real" capitalist activity? He suggests that by focusing on the similarities between Chamberlain profiting from his talent and a capitalist profiting from his nefarious activities, while ignoring the potential morally-relevant differences, Nozick is surreptitiously smuggling exploitative conduct into his vision of justice, hidden behind an example that is carefully chosen not to offend the moral sensibilities of his readers. (In one of the article's particularly cringe-worthy nadirs, Mr Metcalf even brings up the spectacularly irrelevant factor of Chamberlain's race.)

Mr Metcalf has erred here by imposing onto the discussion of justice the very element that Nozick seeks to call into question. In a sense, he misrepresents Nozick's thought experiment as being of the form: "You say D1 is OK, by end-state principles. But surely D2 (with a rich Wilt Chamberlain) is OK by end-state principles too! Doesn't he deserve his money?" In fact, by hypothesis, D2 is supposed to be (and remain) unacceptable according to the chosen end-state principles that generate D1, which is stipulated to be of a sufficiently egalitarian bent that no one should be as wealthy as Chamberlain becomes under D2. Nozick's point is not that D2 is just because Chamberlain "deserves" his money (in an end-state sense) in a manner that a capitalist might not (although it should be noted that beyond allusions to the general horror of "capitalist" activity so oddly de rigeur among modern intellectuals, Mr Metcalf provides no substantive argument for such a moral difference.) Nozick's claim is that since D2 was reached by taking unobjectionable deviations from an unobjectionable starting point, but violates D1 in the process, that D(x) will not work as a criterion for justice - that no matter how strongly our intuitions (or viscera) point to end-states as the barometers of justice, such an account is likely to prove untenable when subjected to rigorous analysis, and we may need to conclude that justice is determined, first and foremost, by the process through which an end state came into being. (Nozick provides his own solution in accordance with this framework, which he calls the Entitlement Theory of Justice - this, along with the direct attacks on Rawls, are probably the most rewarding material to be found in the book.) It is true that Nozick chose a particularly uncontroversial example for the transition from D1 to D2, but this is because the existence of a single such acceptable transition is sufficient to establish the difficulty faced in maintaining a patterned conception of justice. It appears that Mr Metcalf believes that the legitimacy of this particular D1-D2 transition is supposed to imply the acceptability of other (capitalist) D1-D2 transitions. But this is incorrect: in an extreme case, we could even stipulate that no other transition would be allowed, and Nozick's point would still be made, provided that the information of the resulting D2 end-state were not contained in the original formulation of D1. The Wilt Chamberlain example does indeed function as a defense of capitalism, but only indirectly - it seeks to undermine the end-state-derived conception of justice that is invoked whenever income inequality, per se, is used as evidence against the moral legitimacy of capitalism. The subtlety of this thought process is entirely lost on Mr Metcalf.

That Mr Metcalf has badly misunderstood Nozick's thought would be obvious from the outset to any reader familiar with ASU, and would already suffice to render his continued discourse on the subject unworthy of further attention (anyone who was wise enough to skip over the next few paragraphs would have spared themselves the particularly galling experience of reading the author's fatuous and impertinent suggestion - offered more than once - that Nozick's analysis confuses capital with human capital. I shudder to think how many intellectually-chic Slate readers are eagerly parroting this worthless bromide in conversation even as I write.) However, he manages to compound his error still further in his pre-emptive counterattack on what he envisions as a possible libertarian rejoinder. He suggests that a libertarian might obviate the distinction that he has drawn between Chamberlain and a proper capitalist (which we have already seen to be thoroughly irrelevant) through an economic argument showing that there is no difference between the market process by which Chamberlain's services are ultimately priced, and the analogous framework through which capitalists reap their ill-gotten gains - that in a competitive market, as he puts it, "rewards commensurate [sic] perfectly with utility." That such a statement (once it is converted into proper English) is in fact true - at least to close approximation - is suggested by a long-established body of economic theory. Has our intrepid author prepared a well-chosen piece of recent scholarship with which to launch his assault on this venerable proposition? Bah, such a tedious approach is for the plebeians! Our author has seen right to heart of the matter: he notes that this analysis, invoking a state of affairs where unseen economic processes bring about an optimal result, possesses utopian characteristics - a "free-market paradise," as he puts it. To which he offers, as his sole point of response: "Maybe; and maybe in a socialist paradise, no one will catch the common cold." Let us draw the curtain of charity over the rest of the scene.

Thus far, we have seen that with his article's main thesis, Mr Metcalf has managed to achieve the rarely-seen Triple Crown of rhetorical failure: his central (indeed, titular) fact - that Nozick "abandoned" the libertarian movement - is simply wrong; his implication that this fact, were it true, would deal a significant blow to the libertarian edifice rests on a risibly shallow understanding of the history of libertarian ideas; and finally, his attempt to reinforce his anti-Nozick conclusion via his own original analysis is based on a complete misunderstanding of the latter's argument. At this point, having conferred such ignominy upon our author's effort, we might be expected to ease up on our critical tone, offer a generous remark by way of a conciliatory conclusion, and move on to other pursuits. In fact, we are just warming up, as the errors exposed to this point - grievous though they are - pale in comparison with the author's truly stunning ignorance of what libertarianism even is. This will be explored in Part 2 of this review.

This is an extract from a letter I wrote to espn.com's TMQ, in which I defended Mike Tomlin's decision to attempt a two-point conversion late in their Wild Card playoff game against the Jacksonville Jaguars following the '07 season. The Steelers had scored just a touchdown, reducing their deficit to five point, with 10:25 remaining in the game. A holding penalty was called on the two-point attempt, moving the line of scrimmage back to the 12-yard line. At this point, many commentators (including TMQ) apparently believe that Tomlin erred in repeating the two-point try, rather than kicking an extra point. An exception was footballcommentary.com, whose calculations suggest that the break-even probability of success on the two-point try was 0.17 - a relatively low threshold. Unfortunately, the process used by footballcommentary.com is rather opaque, using a dynamic programming model based on league-average probabilities of possible game events. As a result, I have found in the course of my discussions of this issue that casual fans do not hesitate to dismiss footballcommentary.com's conclusions out of hand, whenever they violate conventional intuition on the subject, as was certainly the case here.

In order to vindicate footballcommentary.com's two-point conversion chart - and also to show how the break-even point can be modified based on our knowledge of the specific teams in question - I will attempt to work through the issue in a more step-by-step manner. This will undoubtedly sacrifice some degree of accuracy in comparison with footballcommentary.com, but should serve to validate their conclusions by showing how they can be approximated in a thoroughly transparent manner. While I will refer only to the particular case faced by the
Steelers in this game, the general intuition can be applied to many late-game situations where a two-point try must be considered.

Our approach will be to partition all possible outcomes of the game by conditioning over the possible future scores by our opponents after the possible two-point attempt, and then evaluating how the two-point conversion attempt holds up in comparison with an extra point in each case, with each case weighted by its estimated likelihood of occurring in the game. This would ordinarily be a very tedious (and very imprecise) process; in this case, however, we can safely assume that if the opponent scores twice more, our win probability is so small that it can safely be disregarded. (This may seem a controversial assumption, but I believe it is justified: even if the opponent scores two field goals, this requires that we score twice more as well - and their scoring possessions presumably consume a large amount of the time remaining in the game. So while not 0, the probability of winning if the opponent scores twice more following the two-point try is of an order of magnitude that its effect on the final conclusion will be negligible.) This leaves us only needing to consider three cases:

a) The opponent does not score for the rest of the game. Then the
two-point conversion attempt is clearly superior to the extra point,
as if successful, it allows us to tie the game with a field goal. So
it is superior from a win probability standpoint by a margin of P(2) x
P(FG) x 0.5.

b) The opponent scores a field goal later in the game. This is more complicated, as the extra point after this touchdown would allow us to tie the game with a subsequent TD + XP, while a two-point attempt, depending on its success or failure, would either leave us needing a TD + 2-pt in order to tie, or allow us to take the lead after a TD + XP. So the win probability value of the extra point, holding other things equal, is P(TD) x 0.5 (I'm assuming throughout this discussion that all extra points succeed, as this assumption works unequivocally against the case I am attempting to make in favor of the two-point try), and the win probability value of the 2-pt attempt is [P(2) x P(TD)] + [(1-P(2)) x P(TD) x P(2) x 0.5]. So the net value of the two-point conversion attempt rather than the extra point is the second expression minus the first.

c) The opponent scores a touchdown later in the game, and kicks an extra point (regardless of our two-point decision and its success/failure, they would have no reason to attempt a two-point
conversion of their own at this point.) Then it makes virtually no difference whether an extra point or two-point was attempted after this touchdown, as one (and only one) two-point conversion will be required in order to allow a field goal to tie the game, and it can be equally well attempted after the next touchdown.

So in case (a), the two-point try is unequivocally superior, and in case (c), the two-point and XP attempts are of equal value. This leaves only case (b) in which the XP could come out ahead. However, we can easily see that if we use a conservative approximation of 0.4 for P(2) in
these expressions, the net value of the two-point conversion is still positive, as it simplifies to 0.52 x P(TD) - 0.5 x P(TD). In fact, for any value of P(2) greater than 0.38, going for two strategically dominates the extra point, as it provides equal or greater win probability regardless of what the opponent does for the rest of the game. Thus, it is undoubtedly the right call in a normal situation.

Things become more interesting when one assumes that P(2) is significantly less than 0.38, as was presumably the case when Pittsburgh attempted their conversion from the 12-yard line. Then one needs to estimate values for P(TD) and P(FG), as well as for P(Opp. No Score) and P(Opp. FG), which gives the respective weight to be assigned to cases (a) and (b). Plugging in league-average values gives a break-even point for P(2) (i.e. where the gain from case (a)
exactly offsets the loss from case (b)) of 0.156 - quite close to the footballcommentary.com value of 0.17. However, one can pursue the question still further by using team-specific and game-specific values for the various probabilities. For example, using FootballOutsiders' Drive Stats, with a simple average taken between the values for Pittsburgh Offense and Jacksonville
Defense, or vice versa, gives a more appropriate value for the break-even point than using league average values does. (There may well be an even more accurate method of estimating these team-specific probabilities, but this suffices as a first approximation.) By this method, since the Pittsburgh defense was generally so good at preventing the other team from scoring, the break-even point for this particular P(2) is actually reduced all the way to 0.12. Surely this threshold is low enough that Tomlin's decision was justified. [In a discussion of Bill Belichick's decision to attempt a 4th and 13 conversion rather than kick a FG in SB XLII, footballcommentary.com uses a success probability of 25% in their analysis - the probability here is lower, since a defensive penalty would not result in an outright success, but certainly cannot be pegged below the 12% threshold.]

A simplified version of the "two-point conversion equation" for this game situation (trailing by 5 following the TD, and where the amount of time remaining is such that we can safely disregard situations where the opponent scores two or more times) is the following:

P(2) x P(Opp. No Score) x P(FG) x 0.5 - [P(Opp. FG) x P(TD) x [0.5 - (1.4 x P(2))] = 0.

This establishes the break-even probability for P(2). Analogous equations could be constructed for other game situations, and empirical testing of the degree of inaccuracy introduced by the assumptions could refine the conclusion still further. Regardless, the primary purpose of this discussion was to show that footballcommentary.com's chart of break-even probabilities makes good logical sense, and by obtaining a close correspondence with its result in one of the most apparently counter-intutive cases, I think that it succeeds in this endeavor.

Thoughts on NFL Overtime

It is apparent to many observers of the NFL that the current overtime system (coin toss to determine which team kicks off, first points thereafter decide the game) is flawed. In a large number of cases, the team who receives the opening kick scores on this first possession, ending the game before the other team has had a chance at an offensive possession. Overall, teams winning the toss won 60% of all OT games between 2000 and 2007 (http://www.advancednflstats.com/2008/10/how-important-is-coin-flip-in-ot.html) and 59% of all OT games from 1994-2003 (http://www.footballcommentary.com/otauctions.htm). This means that a team will gain or lose 10% of win probability depending on a completely random event, clearly an unpalatable state of affairs. But it's worth examining the issue in more detail, as it is not as easy a case to resolve decisively as it may appear at first, and it gives rise to some interesting considerations along the way.

1) Proponents of the current system often attempt to entirely dismiss the primary objection - that the game can be decided without both teams having possession - by saying that the losing team in this scenario should have "played some defense." This is line of reasoning is obviously inadequate. It rests on the fallacy - still all too common in sports discussions - of treating game events as arising deterministically, rather than probabilistically, from the actions of the players. The fact that a team allows a score on a given possession does not necessarily imply that their performance was in any way sub-par; it is perfectly possible to perform at an average level - or even above this - and still suffer a reversal due to the random breaks of the game. (Note that these are not random in the strict quantum mechanical sense - but are determined by margins that are too fine to be precisely correlated with physical and/or mental actions of the players.) A given level of performance implies a probabilistic distribution of game outcomes, and vice versa. In the absence of further information, the end result of one possession is much too small a sample to infer anything about how well either team played. In general, NFL teams score on around 30% of their possessions. This does not mean that teams "played some defense" 70% of the time, and failed to do so on the other 30%. A good defensive performance on a possession may reduce the average probability of a score to 25% or 20%, but it does not reduce it to anywhere near 0% - that's just not how the world works. So even if the defensive team does all that can be asked of them on the first overtime possession, they have still been exposed to a substantial risk of losing immediately, while their opponents have faced only the comparatively trivial risk of losing on a turnover retured for a score.

2) Opponents of the current system often introduce a red herring into the discussion by calling it "unfair." As those who support the system tend to gleefully point out in response, this is not correct - both teams have an equal shot at the advantage resulting from the coin toss, so no fairness criteria are violated. A more precise way to state the objection is that the result of the game is highly influenced by an exogenous factor. After all, a system that determined the result of a game that was tied at the end regulation entirely by a coin toss (winner of the toss wins the game, no OT play at all) would be fair to both teams - but it would also be pretty stupid. One of the most important goals in constructing rules for a game should be to ensure that the greatest predictor of success is performance across a set of related skills. Some element of physical randomness is unavoidable (see #1 above) and even desirable. But introducing external random factors - such as coin tosses - into situations where they have a large impact on the outcome unnecessarily weakens the correspondence between performance and result.* If one were to graph the win probability of both teams over the course of the game, one of the largest single-event changes would result from the OT coin toss - it would rank alongside the game's biggest plays, and those in the "clutchiest" of situations, as one of the most significant events in determining a winner. When considered in these terms, the objection to the coin-toss/sudden death system should be obvious.

3) Even with the validity of the objection established, commonly-suggested alternatives often introduce flaws of their own. The most obvious possibility - stipulating an equal number of possessions for each team - risks making the game last substantially longer, exposing the players to greater risk of injury and (silly as this consideration sounds) interfering with TV scheduling. It also increases the possibility of the game ending in a tie, which many NFL fans seem to find an insufferable proposition. On the other hand, resorting to alternating possessions from the 25-yard line, as in NCAA football, offends football purists who deplore such gimmickry.

4) As a result, it may seem that we are trapped in a "de gustabus non disputandum est" situation, where every possible solution has its own strengths and weaknesses, and it is a matter of taste whether one prefers the current system or one of the alternatives. However, we could circumvent this problem if we were able to state the major objectives of an optimal overtime system, and present solutions that dominate the current system by performing equal or better in all categories. It seems to me that the main criteria are:

1) Minimize the probability of a tie
2) Minimize the amount of time it takes to reach a result
3) Minimize the effect of exogenous factors on the determination of the result
4) Maximize the similarity between OT and regulation play (i.e. avoid gimmicks)
5) Maximize "excitement" (the subjectivity of this criterion means that it can probably serve as a catch-all for all other specific criteria that one might think of)

Clearly, the current system does pretty well at 1, 2, 4 and 5. Solutions involving equal possessions or a fixed-time OT period improve on 3, but at the cost of a decline in 1 and 2. So whether one prefers these alternatives to the current system depends on how much weight one assigns to 1-3 on the list above. This makes it hard for supporters of the respective solutions to make any progress in persuading the other.

But there seem to be good candidates for solutions that dominate across all five criteria:

1) Simply extend the fourth quarter. When time runs out in the fourth quarter with the game still tied, add 15 minutes to the clock, and continue play as before, with the next points deciding the game. This still gives one team the advantage of having the first opportunity to score, but the difference here is that there is no exogenous factor - the advantage is built-in to the game state at the end of the fourth quarter, such that both teams have ample time with which to factor it into their plans earlier in the game. Not only is a coin flip not responsible for a sudden spike in win probability - there is no spike in probability at all. (To confirm this, consider that the last moments of regulation are de facto sudden death, as there will be no time left for a score in response. So if a team keeps possession across the nominal end of the fourth quarter and into overtime, their win probability will not change, except to increase as their distance to field goal range decreases.)

So this possibility obviously does at least as well as the current system in categories 1, 2, and 3. In my opinion, it gets a particular added boost from its performance in category 5. Under the current system, a team driving down the field with little time left, trailing by 3 or 7 points, will often play for a tie, taking their 50% chance of winning in overtime minus whatever chance there is that their opponent scores in regulation. Under the extended fourth quarter overtime scenario, teams in this situation would know that the opponent would in effect be guaranteed to win the overtime coin toss, giving them greater incentive to play for an outright win in normal time. Teams trailing by 3 with a minute remaining might forego a 30-yard field goal on 4th and 2, and play for a touchdown instead, and teams who score a late touchdown when trailing by 7 may choose to attempt a two-point conversion. The decisions made by coaches in such situations are often excessively conservative, failing to maximize win probability (as well as being relatively boring.) It is therefore a beneficial secondary consequence of this overtime system that it would give them a nudge in the right direction. Besides being more exciting and representing less frustratingly bad decision-making, this incentive to play for a win also feeds back into the second criterion, making it more likely that the game will end sooner, without any extra time played at all.

Several people have objected that this system would take away the two-minute drill at the end of a game, as teams now know that with the ability to continue a possession beyond the end of the quarter, they now have no reason to hurry. But this only applies to situations where the game is tied - teams trailing by anywhere from one to eight points would still have to run their two-minute offense. And when the game is tied, the two-minute offense only has to reach the opposing team's 30-yard line in order to have a decent shot at a winning field goal. Is this really the most exciting thing in the world? (Perhaps Patriots fans have reason to be a little biased on this question, recalling SB XXXVI). In any case, this consideration is also counterbalanced by the fact that it avoids a situation where a team is pinned deep in their own territory late in the game decides to run out the clock and play for overtime rather than attempt to do anything with the final possession of regulation (what John Madden thought that the Patriots should have done.)

2) The key feature of the extended fourth quarter system is that both teams know which team will have the first chance to score in the sudden death period, before this period starts. This allows them to spread the win probability deficit out throughout many game events throughout regulation, and allows them to increase their chance of overcoming it entirely through the decisions they make. Therefore, overtime systems that make use of this principle allow us to keep the sudden death feature (which cuts down on time and on ties) without the ugly spike in win probability that occurs when one team suddenly finds out that they are the first to have their heads on the chop block (no pun.) There are other ways to accomplish this: giving the first overtime possession to the home/road team, performing the OT coin flip at half-time, and having the opening coin flip count as the OT coin flip are the most obvious. In both cases, the team who will be at a sudden-death disadvantage may have slight incentive throughout the game (or the second half in the case of the half-time flip) to unbalance the score by attempting two-point conversions in situations where they otherwise would not. This could be kind of cool, though I still prefer to give the incentive to unbalance the game to the team who trails last. Retaining the coin flip, either at the outset of the game or at halftime, allows the overtime period to begin with a kickoff, which might be construed as superior performance in the fourth criterion, and takes care of the two-minute drill problem (though it allows teams in some cases to sit on the ball and force overtime, as in the current system.) Personally, the extended fourth quarter remains my favorite of these possibilities.

3) Another solution entirely is to keep the sudden death element of overtime, but to determine possession by bidding on field position. For a full discussion of this, see http://www.footballcommentary.com/otauctions.htm. It could be maintained that this is subpar according to the fourth criterion, as it involves a coaching decision the like of which never appears at any other point in a football game. But it fulfills the basic requirement of dominating across the first three categories without resorting to anything excessively foolish, and therefore would be a better choice than the current system.

Why do people overrate the punt?

My last post tried to show that the criticism of Belichick's fourth-down decision is largely misguided - that a very realistic value of the probability of the Colts scoring from a short field causes the decision to break even, even if all other assumptions are made so as to favor the punt. If I am correct, this would fit the general trend in NFL practice and commentary: most coaches choose to punt far more often than a probabalistic analysis indicates that they ought to, and most pundits compound the situation by criticizing coaches on the rare occasions where they correctly forego a punt (but almost never vice versa.) So perhaps we should follow up by looking at the meta-question: why are people so eager to believe in the punt? After all, if people are going to go around spouting irritating nonsense about sports, we can at least use this as a natural sociological experiment to identify the sorts of conceptual errors to which human beings are particularly prone. Here is a letter I sent to Boston Metro on Monday morning following the Colts game, in which I offer a possible answer for this in the case of punting:

--------------------------------

The mad scramble by fans and pundits alike to condemn Belichick's 4th-and-2 gamble on Sunday (believe it or not, the numbers suggest that he probably made the right decision) is evidence of a more general issue worthy of discussion: the punt is currently the most overrated strategy in American sports. Contrary to what the "Trust your defense!" crowd seems to believe, punting does not send the ball across a magical Rubicon River that ensures that the other team will fail to score. It adds some distance - generally around 40 yards - to how far they will have to go, making it somewhat less likely that they will succeed. This can make it a valuable tactical option in certain game scenarios. But in a great many cases, it is merely a foolish waste of an opportunity to do something productive with an offensive possession, whether scoring points, or, in this case, running out the clock without the other team having a chance to touch the ball. The fact that NFL teams punt far more often than they ought to has been mathematically demonstrated several times: the ground-breaking analysis was done by David Romer of UC Berkley in a paper called Do Firms Maximize? Evidence From Professional Football, and espn commentator Gregg Easterbrook (aka TMQ) routinely exposes the folly of the automatic fourth-down punt in his columns.

So why is it so firmly ensconced in conventional wisdom that punting (in all but a tiny minority of fourth down situations) represents "playing the percentages," when in fact the opposite is often true? The answer seems to lie in a phenomenon known by economists as "risk aversion" - we are hard-wired to prefer the certainty of some fixed amount of most things (food, money, etc) rather than an uncertain outcome where we will either end up with a great deal, or none at all. This is true even if the weighted average of the uncertain outcome is higher than the certain one: when signing a ten-year contract, most of us would choose a guaranteed income of $50,000 rather than a possibility of either $5000 or $100,000, to be decided by a coin flip. (Risk aversion is one of the major reasons why people purchase insurance.) This is a sensible response to uncertainty when making economic decisions: personal well-being, not money, is the ultimate currency here, and for most people, being rich is not that much better than being comfortable, while poverty can be ruinous. But risk aversion is not rational when the uncertain quantity is win probability in sports, as this probability itself is the ultimate standard by which the decision is to be evaluated. (As such, it is not subject to decreasing marginal utility - a strictly linear relationship is maintained at all values.) Here, ten times out of ten, we should prefer a decision that will leave us with either a coin-flip 5% or 100% of winning to one that will guarantee us a 50% chance of winning.* While the specific numbers differ widely from case to case, this general template underlies the various possibilities arising from a fourth down decision. A punt will guarantee a relatively negligible impact on win probability - we don't score, and we lower the probability that the other team scores. Nothing much has changed. An attempted fourth-down conversion, on the other hand, will have a much greater impact on win probability in one direction or the other: either we keep the ball, with a good chance to put points on the board, or we give the other team the ball with good field position. We instinctively balk at this latter scenario, treating a coach who chooses it as though he were a portfolio manager exposing his client's 401k to ruin in pursuit of a big payoff. In fact, we should be celebrating coaches like Belichick who are bold enough to look past the irrationality of the conventional wisdom, and make decisions that put their team in the best position to win.

---------------------------

I made a few changes here to what was actually submitted before. In the actual letter, I included a footnote that tried to tie the analogy together by comparing the win probabilities following the various fourth down outcomes with those of the salary possibilities under the analogous choice under uncertainty. This seemed to cause more confusion than it was worth (people seemed to think that I was actually suggesting that these numbers corresponded to the specific win probabilities in this case) so I removed it here.

Comments

on anything I say here (or anything in general, for that matter) can be e-mailed to dajepson at gmail dot com.