Mises Daily Articles
What Mises Can Teach the Quants
Scott Patterson, reporter for the Wall Street Journal and author of The Quants (2010), tells the story of
traders and financial engineers who used brain-twisting math and superpowered computers to pluck billions in fleeting dollars out of the market. Instead of looking at individual companies and their performance, management and competitors, they use math formulas to make bets on which stocks were going up or down. By the early 2000s, such tech-savvy investors had come to dominate Wall Street, helped by theoretical breakthroughs in the application of mathematics to financial markets, advances that had earned their discoverers several shelves of Nobel Prizes.
The fall of the quants in 2007 was even more precipitous than their rise. Patterson tells of the experience of one outfit in particular, Morgan Stanley's Process Driven Trading, during the week of August 6.
PDT, one of the most secretive quant funds around, was now a global powerhouse, with offices in London and Tokyo and about $6 billion in assets (the amount could change daily depending on how much money Morgan funneled its way). It was a well-oiled machine that did little but print money, day after day.
That week, however, PDT wouldn't print money — it would destroy it like an industrial shredder.
The unusual behavior of stocks that PDT tracked had begun sometime in mid-July and had gotten worse in the first days of August. The previous Friday, about half a dozen of the biggest gainers on the Nasdaq were stocks that PDT had sold short, expecting them to decline, and several of the biggest losers were stocks PDT had bought, expecting them to rise. It was Bizarro World for quants. Up was down, down was up. The models were operating in reverse.
The market moves PDT and other quant funds started to see early that week defied logic. The fine-tuned models, the bell curves and random walks, the calibrated correlations — all the math and science that had propelled the quants to the pinnacle of Wall Street — couldn't capture what was happening.
The quants were among the first of many victims of the bursting of that decade's economic bubble. Patterson would have us believe that they were also among the chief causes of the financial crisis that ensued. But the bubble and its bursting were made virtually inevitable by the prior actions of the Federal Reserve, as anyone who adheres to the Austrian business-cycle theory will tell you. However, exactly who is involved the most in the inflating of the bubble, and exactly who suffers the most when it bursts, depends on the actions of the market participants in question. And undoubtedly the quants were foremost among their own gravediggers.
The ultimate test of market strategies should always be profit and loss, and therefore it is worse than useless to regulate such investment strategies as President Obama has tried to do. But if Ludwig von Mises were alive today, he probably would not have been in the least surprised that the quants' methods ultimately bore such bitter fruit. Their models "couldn't capture what was happening" for two fundamental reasons:
They did not integrate an accurate understanding the business cycle.
They treated a matter of case probability as if it were a matter of class probability.
Mises's distinction between these two kinds of probability is the cornerstone of his theory of uncertainty, which in turn hinges on his theory of "the specific understanding." In this article, I will endeavor to explain both of these little-understood areas of Mises's thought.
Class Probability and Case Probability
According to Mises, there are two approaches that the human mind can take concerning incomplete knowledge of real affairs: class probability and case probability.
The most important thing to note about this distinction is that class probability has to do with frequency, and case probability does not. Another important thing to note is that class probability has to do with causality and nature, while case probability has to do with teleology and human choice.
Mises defines the two kinds of probability in Human Action.
Class probability means: We know or assume to know, with regard to the problem concerned, everything about the behavior of a whole class of events or phenomena; but about the actual singular events or phenomena we know nothing but that they are elements of this class.
Case probability means: We know, with regard to a particular event, some of the factors which determine its outcome; but there are other determining factors about which we know nothing.
Here are some examples Mises gives of each type.
|Class Probability||Case Probability|
|Gambling (Lottery Tickets, Roulette, etc.)||Betting (Sports Wagers, Political Bets, etc.)|
|Engineering Margins and Medical Prognoses||Historical Scholarship|
|Business-Inventory Vicissitudes||Entrepreneurial Speculation|
To take the example of a lottery, a prospective ticket buyer may know everything about the class "Tickets for the MegaMillions Lottery Ending May 31." He may know exactly how many tickets there are, and exactly how many will be winning tickets. He would thus have complete knowledge concerning the "winning frequency" about the class in question. However, as he looks at four tickets hanging on the wall behind the drugstore counter, he would know absolutely nothing about the cases in question (the particular tickets he is considering) except that they are members of the MegaMillions-Tickets class. He has relevant (and, in this situation, complete) class knowledge, but no relevant case knowledge. Placing a stake according to class probability is Mises's definition of a gamble.
Now, the state of affairs would be very different if one were placing a stake on human action. For example, let us say an ancient Roman named Quintus is placing a wager on whether Caesar will lead his army across the River Rubicon toward Rome, an act that will commit the Roman commander to war against the forces of Pompey and the Roman Senate, thus plunging Rome into a civil war.
Unlike the situation with the lottery ticket, Quintus has no relevant class knowledge concerning Caesar. Quintus can classify Caesar according to any number of class distinctions (or "real types," as discussed further below) his mind can conceive of: "Roman consul," "Roman of patrician rank," "human born on 13 July." But any survey he makes of members of such classes would be useless to him for predicting his choice. That is because, as Mises says,
the distinctive mark of what we call the human sphere or history or, better, the realm of human action is the absence of … a universally prevailing regularity. Under identical conditions stones always react to the same stimuli in the same way; we can learn something about these regular patterns of reacting, and we can make use of this knowledge in directing our actions toward definite goals. Our classification of natural objects and our assigning names to these classes is an outcome of this cognition. A stone is a thing that reacts in a definite way. Men react to the same stimuli in different ways, and the same man at different instants of time may react in ways different from his previous or later conduct. It is impossible to group men into classes whose members always react in the same way.
So when we are dealing with uncertainty concerning any event determined by human choice, we are
grappling with an individual, unique, and nonrepeatable case. The case is characterized by its unique merits, it is a class by itself. All the marks which make it permissible to subsume it under any class are irrelevant for the problem in question.
Therefore Quintus's wager on whether Caesar will cross the Rubicon is not a matter of class probability.
Now, while the prospective lottery ticket buyer has no case knowledge about particular tickets, is it possible to have case knowledge about the particular man Caesar? Is it possible to have insight into the individual, unique, and nonrepeatable cases we call human beings?
Case Probability, the Specific Understanding, and Thymology
Yes, case knowledge is undeniably possible. In our daily lives, we all have experience in achieving psychological insight into the emotions, motivations, ideas, judgments of value, and volitions of unique individuals. Mises calls such knowledge "thymology," and he calls the mental process that results in such knowledge "the specific understanding" (or, in German, "verstehen").
Because it is possible to achieve psychological insight into the emotions, motivations, ideas, judgments of value, and volitions of the individual named Caesar, Quintus can have case knowledge: That is, as Mises defined it, knowledge of, with regard to a particular event (the crossing or noncrossing of the Rubicon), some of the factors which determine its outcome. Placing a bet based on case probability is Mises's definition of a bet.
The specific understanding, as well as the thymological knowledge which it produces,
is applied by everybody in daily intercourse with all his fellows. It is a technique employed in all interhuman relations. It is practiced by children in the nursery and kindergarten, by businessmen in trade, by politicians and statesmen in affairs of state. All are eager to get information about other people's valuations and plans and to appraise them correctly.
Thymology is the insight that people refer to when they speak of a biographer's insight into the "psychology" of his subject or an advertiser's insight into the "psychology" of the potential consumers of his product. However, to avoid confusion, Mises used the term "thymology" instead of "psychology" because the latter term had already, by his time, come to be associated with naturalistic disciplines like psychopathology and neuropathology, which deal with physiological states and outward behaviors, but not the "inner life" of ideas and values.
It must be understood that the specific understanding is not some kind of magical inspiration. In producing thymological knowledge, the specific understanding does depend on experience — just not on controlled experiments or quantitative data.
Thymology is on the one hand an offshoot of introspection and on the other a precipitate of historical experience. It is what everybody learns from intercourse with his fellows. It is what a man knows about the way in which people value different conditions, about their wishes and desires and their plans to realize these wishes and desires.
For example, for Quintus to understand anything about Caesar, or anybody else, requires some introspection. Our direct experience of our own emotions, ideas, values, and volitions is the root of our understanding of what it means to have emotions, ideas, values, and volitions in the first place. Furthermore, through introspection, we can empathize with, and thus better understand, the mental states of other individuals. But introspection by itself is not enough. Understanding also requires experience. This experience can be personal conversation with Caesar himself, or it can be hearing from others about the character and actions of Caesar.
Maybe Quintus judges based on these sources that Caesar has longed to be a dictator, either out of lust for power or a desire to reconstitute the republic. And maybe he judges that if Caesar does not cross the Rubicon, and instead relinquishes control of his army, that he will never be able to enter Rome again without being prosecuted. And perhaps Quintus judges that Caesar himself is fully aware of that. These factors may contribute toward the likelihood of Caesar crossing the Rubicon.
But then there may be counterfactors. Quintus may expect Caesar to abhor the prospect of the bloodletting among his fellow Romans that would ensue in a civil war, and to be concerned particularly for his loved ones.
There may be dozens of such factors that Quintus may think will weigh on Caesar's mind in making the decision. Quintus must try to estimate as best as possible, using his own judgments concerning the importance of each factor for Caesar, which factors will ultimately hold sway.
As illustrated above, the specific understanding has two tasks in constructing thymological knowledge for use in forecasting human choice:
It must establish the factors (goals, judgments of value, ideas, etc) that may influence the choice.
It must establish the relative importance of each factor.
But of course this is all necessarily educated guesswork. Quintus may be wrong about what factors exist, or (and this is the trickiest issue) about the importance of each factor, or both.
Understanding is always based on incomplete knowledge. We may believe we know the motives of the acting men, the ends they are aiming at, and the means they plan to apply for the attainment of these ends. We have a definite opinion with regard to the effects to be expected from the operation of these factors. But this knowledge is defective. We cannot exclude beforehand the possibility that we have erred in the appraisal of their influence or have failed to take into consideration some factors whose interference we did not foresee at all, or not in a correct way.
For this reason, the operation of the specific understanding always deals with probabilities, and not certainties.
Thymology … can never predict in the way the natural sciences can. It can never know in advance with what weight the various factors will be operative in a definite future event.
Because choice factors and their relative importance are unquantifiable, the probability dealt with by the specific understanding (case probability) is also necessarily unquantifiable, and therefore it can have nothing to do with quantitative frequency.
The term "probability" in philosophical discourse used to include such nonquantitative uncertainties, but, according to the quantophrenetic trend of modern thought, the term now often is given an exclusively mathematical connotation. Due to this, Mises also suggested referring to "case probability" as a "likelihood" instead, if that helps avoid confusion.
In dealing with case probability, people may speak of one human event having a "greater" or "lesser" likelihood of occurring than another human event. But just as there is no way of measuring how much "greater" is one's valuation of apples over oranges, there is no way of even crudely measuring such "differences" in likelihood.
Also, in dealing with uncertainty in human affairs, people may verbally assign quantitative probabilities to their judgments of likelihood. But this does not mean class probability has any bearing on human affairs. Mises gives the example of someone, on the eve of the 1944 US presidential election saying that they estimate the chances of Franklin Roosevelt winning to be 9 to 1.
This is a proposition about the expected outcome couched in arithmetical terms. It certainly does not mean that, out of ten cases of the same type, nine are favorable for Roosevelt and one unfavorable. It cannot have any reference to class probability. But what else can it mean?
It is a metaphorical expression. Most of the metaphors used in daily speech imaginatively identify an abstract object with another object that can be apprehended directly by the senses. This is not a necessary feature of metaphorical language, but merely a consequence of the fact that the concrete is as a rule more familiar to us than the abstract. As metaphors aim at an explanation of something that is less well known by comparing it with something better known, they consist for the most part in identifying something abstract with a better-known concrete. The specific mark of our case is that it is an attempt to elucidate a complicated state of affairs by resorting to an analogy borrowed from a branch of higher mathematics, the calculus of probability. As it happens, this mathematical discipline is more popular than the analysis of the epistemological nature of understanding.
There is no use in applying the yardstick of logic to a critique of metaphorical language. Analogies and metaphors are always defective and logically unsatisfactory. It is usual to search for the underlying tertium comparationis. But even this is not permissible with regard to the metaphor we are dealing with, for the comparison is based on a conception that is in itself faulty in the very frame of the calculus of probability, namely the gambler's fallacy. In asserting that Roosevelt's chances are 9:1, the idea is that Roosevelt is in regard to the impending election in the position of a man who owns 90 percent of all tickets of a lottery in regard to the first prize. It is implied that this ratio 9:1 tells us something substantial about the outcome of the unique case in which we are interested. There is no need to repeat that this is a mistaken idea.
There are other sources besides personal experience with, and hearsay about, Caesar that Quintus could draw from to try to forecast his choice. Quintus could also draw from his experience with human nature, Roman civilization, patrician culture, or, "the military man."
Now, this may seem to run contrary to what was said above about the irrelevance of class concepts with regard to understanding and predicting human action. However, such terms are not "class concepts," but instead they are what Mises called "ideal types."
Mises distinguished the ideal types used in contemplating human action from the class concepts used in contemplating natural phenomena.
The natural sciences classify the things of the external world according to their reaction to stimuli. Since copper is something that reacts in a definite way, the name copper is denied to a thing that reacts in a different way. In establishing the fact that a thing is copper, we make a forecast about its future behavior. What is copper cannot be iron or oxygen.
Individual humans, on the other hand, do not react in definite, fixed ways to given stimuli, and so men, ideas, customs, institutions, and artifacts cannot be classified according to stimulus reaction. Instead, they can only be grouped according to human meaning.
Some human things can be grouped according to human meaning into rigidly definable class concepts. Mises calls such class concepts in human affairs "real types."
In acting-in their daily routine, as well as in technology and therapeutics, and also in history-people employ "real types," that is, class concepts distinguishing people or institutions according to neatly definable traits. Such classification can be based on concepts of praxeology and economics, of jurisprudence, of technology, and of the natural sciences. It may refer to Italians, for example, either as the inhabitants of a definite area, or as people endowed with a special legal characteristic, viz., Italian nationality, or as a definite linguistic group. This kind of classification is independent of specific understanding. It points toward something that is common to all members of the class.
But as discussed above, class concepts constructed concerning human affairs (real types) are useless with regard to dealing with uncertainty in human affairs.
Ideal types are distinct from real types in that
The characteristic mark of an "ideal type," on the other hand, is that it implies some proposition concerning valuing and acting. If an ideal type refers to people, it implies that in some respect these men are valuing and acting in a uniform or similar way. When it refers to institutions, it implies that these institutions are products of uniform or similar ways of valuing and acting or that they influence valuing and acting in a uniform or similar way.
Ways in which people concretely value and act are often noticeably similar, but their marks of similarity are not such that they can be abstracted in order to construct neatly definable class concepts (real types). The specific understanding, however, can perceive such resemblances and make judgments concerning their importance. Based on such perceptions and judgments, the specific understanding can recognize affinity between various instances in human affairs, and construct an enumeration of connected traits based on this "meaning affinity." Such a construction is an "ideal type."
An ideal type is not a class concept, because its description does not indicate the marks whose presence definitely and unambiguously determines class membership. An ideal type cannot be defined: it must be characterized by an enumeration of those features whose presence by and large decides whether in a concrete instance we are or are not faced with a specimen belonging to the ideal type in question. It is peculiar to the ideal type that not all its characteristics need to be present in any one example. Whether or not the absence of some characteristics prevents the inclusion of a concrete specimen in the ideal type in question depends on a relevance judgment by understanding. The ideal type itself is an outcome of an understanding of the motives, ideas, and aims of the acting individuals and of the means they apply.
For example Quintus may consider Caesar to possibly be a "demagogue/striver" — an ideal type constructed in his mind characterized by these traits:
- Love of power
- Love of prestige
- High self-esteem
- Ability to sway the masses (rank-and-file soldiers as well as city mobs)
Quintus may have constructed this ideal type in his mind when reading about Pisistratus, an ancient Greek who never ceased his dramatic machinations until he was tyrant of Athens. He may see much of the character of Pisistratus and other Greek tyrants in the character of Caesar. Again, "demagogue/striver" is not a class concept with a rigid definition. Quintus may even consider some of the enumerated traits to be wholly inapplicable to Caesar. For example, Quintus may consider Caesar not to be unscrupulous, unlike Pisistratus. But he may consider Caesar to fit the type well enough for it to be applicable to him.
Also, Quintus may not be able to judge from personal experience or hearsay whether some of the enumerated traits apply to Caesar or not: for example, whether Caesar is personally daring. But he may infer from how Caesar fits the type in other regards that he likely fits the type in that regard as well.
What thymology achieves is the elaboration of a catalogue of human traits. It can moreover establish the fact that certain traits appeared in the past as a rule in connection with certain other traits.
This may seem like a prejudicial simplification on the part of Quintus, and it is. But such simplifications are pragmatic necessities in our daily task of carving meaning out of the bewildering profusion of data in human affairs.
In referring to ideal types the historian of the past as well as the historian of the future, i.e., acting man, must never forget that there is a fundamental difference between the reactions of the objects of the natural sciences and those of men. Ideal types are expedients to simplify the treatment of the puzzling multiplicity and variety of human affairs. In employing them one must always be aware of the deficiencies of any kind of simplification.
According to Mises, ideal types, unlike class concepts (including real types), can be used to forecast the future in human affairs. How can this be? Well, the use of an ideal type may help us infer a trait that we could not discover through personal experience or hearsay. For example, as discussed above, the use of the ideal type "demagogue/striver" may lead Quintus to conclude that Caesar was a daring person. And considering Caesar to be a daring person might lead Quintus to expect the former to cross the Rubicon.
It must always be noted that ideal types are highly tentative tools for the highly tentative sciences of history and human forecasting, and that the usefulness of an ideal type depends on the insight and judgment used in formulating and applying it.
The service a definite ideal type renders to the acting man in his endeavors to anticipate future events and to the historian in his analysis of the past is dependent on the specific understanding that led to its construction. To question the usefulness of an ideal type for explaining a definite problem, one must criticize the mode of understanding involved.
More on Class Probability and Case Probability
It is not only bet-placers who deal with case probability. Everybody does in dealing with his fellow man. For example, Pompey himself was most likely vitally concerned with whether Caesar would cross the Rubicon or not, and thus would have had to use his imperfect specific understanding to try to glean incomplete thymological knowledge from introspection and experience about Caesar's ideas and aims in order to prepare as best he could.
And it is not only forecasters that deal with case probability. Historians do too. A historian may be virtually certain, based on documentary evidence, that Caesar did indeed cross the Rubicon. But to glean why he crossed the Rubicon, the historian must use his specific understanding to try to glean from introspection and experience (including experience with the documentary evidence) thymological knowledge about Caesar's desires and beliefs in order to judge what motivated him. Of course, such conclusions can only ever be probabilities, and never certainties. And, as with anticipations of future human action, such probabilities can have nothing to do with quantitative frequency.
Economics is immensely concerned with one kind of forecaster of human action in particular: the entrepreneur. The entrepreneur tries to forecast the patterns in which future consumers will buy and abstain from buying (consumer demand). Based on these uncertain forecasts, he seeks to buy and sell on the market so as to attain profits and avoid losses.
The only source from which an entrepreneur's profits stem is his ability to anticipate better than other people the future demand of the consumers.
This kind of speculation too is a matter of case probability. The entrepreneur must use his specific understanding to glean thymological knowledge about the desires and values of consumers. Again, such conclusions can only ever be probabilities, and never certainties. And, again, such probabilities can have nothing to do with quantitative frequency.
However, not all incomplete knowledge in business affairs is a matter of case probability. Mises characterizes the vicissitudes of inventory management as a matter of class probability.
Every businessman includes in his normal cost accounting the compensation for losses which regularly occur in the conduct of affairs. "Regularly" means in this context: The amount of these losses is known as far as the whole class of the various items is concerned. The fruit dealer may know, for instance, that one of every fifty apples will rot in this stock; but he does not know to which individual apple this will happen. He deals with such losses as with any other item in the bill of costs.
Uncertainty in the realm of human action (case probability) is, by definition, in the province of the entrepreneurial function. Uncertainty in the realm of natural events (class probability) is not.
One must not confuse entrepreneurial profit and loss with other factors affecting the entrepreneur's proceeds.
The fact that the bursting of bottles reduces the output of champagne does not affect entrepreneurial profit and loss. It is merely one of the factors determining the cost of production and the price of champagne.
Accidents affecting the process of production, the means of production, or the products while they are still in the hands of the entrepreneur are an item in the bill of production costs. Experience, which conveys to the businessman all other technological knowledge, provides him also with information about the average reduction in the quantity of physical output which such accidents are likely to bring about. By opening contingency reserves, he converts their effects into regular costs of production. With regard to contingencies the expected incidence of which is too rare and too irregular to be dealt with in this way by individual firms of normal size, concerted action on the part of sufficiently large groups of firms take care of the matter. The individual firms cooperate under the principle of insurance against damage caused by fire, flood, or other similar contingencies. Then an insurance premium is substituted for an appropriation to a contingency reserve. At any rate, the risks incurred by accidents do not introduce uncertainty into the conduct of the technological processes. If an entrepreneur neglects to deal with them duly, he gives proof of his technical insufficiency. The losses thus incurred are to be debited to bad techniques applied, not to his entrepreneurial function.
Implicit in the above are these conclusions:
Insofar as the businessman is dealing with forecasts of consumer choices (which are human events, and therefore are a matter of case probability) he is an entrepreneur, and those parts of his net returns that are dependent on the success of such forecasts are his profit/loss.
Insofar as the businessman is dealing with forecasts of natural events (which are a matter of class probability), he is a technician (a kind of laborer), and those parts of his net returns that are dependent on the success of such forecasts are wages.
The businessmen confronting class probability with regard to rotting apples and bursting bottles brings up another important point: class probability does not require the certainty and precision concerning frequency that you find in the case of a lottery or a dice throw. The apple businessman above may not know with certainty that exactly one of every fifty apples will certainly rot. But because the rotting of apples is a process of nature, in which universal regularity does prevail, class concepts can provide relevant knowledge concerning frequency, even if technical knowledge of the natural phenomenon in question is not yet sufficient for perfect knowledge concerning frequency to be had. For this reason, the margins of error of the engineer and the prognoses of the physician are also matters of class probability, as are even the rough judgments laymen make every day: for example, a fellow judging the chance of rain, and whether he should bring his umbrella, based on nothing but eyeballing the sky. Class probability forecasts concerning natural events,
are based either on statistical information or simply on the rough estimate of the frequency derived from nonstatistical experience.
As Donald Rumsfeld might put it, class probability is not only a matter of "known unknowns," but also a matter of "unknown unknowns."
Of course, human actions and natural phenomena both occur in the same universe, and affect each other. So acting men often have to deal with both class probability and case probability concerning the same choice, as Mises points out:
The outcome of horse racing depends both on human action — on the part of the owner of the horse, the trainer, and the jockey — and on nonhuman factors — the qualities of the horse. Most of those risking money on the turf are simply gamblers. But the experts believe they know something by understanding the people involved; as far as this factor influences their decision they are betters.
And, again, gambling is a matter of class probability and betting is a matter of case probability.
Furthermore, as discussed above, business forecasting can involve both class probability (as in the case of forecasting inventory vicissitudes) and case probability (as in the case of forecasting consumer demand). And sometimes a single business decision may depend on both kinds of forecasts. For example, in deciding how much to invest in the rice market, an investor may try to forecast the natural events that affect the rice market (future rainfall, etc.) as well as human factors (consumer tastes, institutional factors such as tariffs, etc.).
Nonetheless, in all such cases, the natural factors can only be dealt with successfully by the human mind as matters of class probability. And the human factors can only be dealt with successfully as matters of case probability. The ultimate choice may be a simple "do" or "do not," but the deliberation leading up to the choice may be a composite of different modes of reasoning.
Class probability and case probability are similar in that they both deal with incomplete knowledge. In all other regards, they are worlds apart, and thus require different approaches ("methodological dualism"). Until mainstream scholars take such differences seriously, and give due regard to the distinct methodological challenges of the sciences of human action, they will always be at best, stymied, or at worst, lost.
Businessmen too would benefit by paying due regard to such distinctions. By dealing only in class "patterns" and ignoring the specific characteristics of
the "individual companies and their performance, management and competitors" within those classes,
the consumers those companies served,
and the institutional factors which constrained those companies,
the Wall Street quants completely abjured case probability, the specific understanding, and thymology. And entrepreneurship is at bottom all about the application of these tools, even if the entrepreneur himself is wholly ignorant of the distinct nature of the cognitive tools he is using, or of the Misesian terms for them.
But even now the enthusiasm for heavily class-probability-reliant approaches seems little abated among many investment strategists. For some, the lesson of the financial crisis is the need to prepare for low-frequency "black swan" market catastrophes with statistical "tail-risk hedging." Again, the proof in the market pudding is in the eating of profit and loss, so the SEC has no business sticking its nose in the matter. But buyer beware of class-probability gamblers posing as case-probability entrepreneurs.
 By Mises's time, the concept of verstehen already had a long tradition in German epistemology.
 Inappropriately relying on statistics and mathematical results for their own sake.
 According to Herodotus (The Histories, Book 1), Pisistratus even went so far as ride into Athens in a golden chariot, accompanied by an unusually tall woman impersonating the goddess Athena, so that the Athenians would believe she favored him.
 Ibid. "Ideal types are constructed and employed on the basis of a definite mode of understanding the course of events, whether in order to forecast the future or to analyze the past."