Editing Cardinal utility (section)

== History ==
In 1738, [[Daniel Bernoulli]] was the first to theorize about the marginal value of money. He assumed that the value of an additional amount is inversely proportional to the pecuniary possessions which a person already owns. Since Bernoulli tacitly assumed that an interpersonal measure for the utility reaction of different persons can be discovered, he was then inadvertently using an early conception of cardinality.<ref>{{cite journal |last1=Kauder |first1=Emil |title=Genesis of the Marginal Utility Theory: From Aristotle to the End of the Eighteenth Century |journal=Economic Journal |date=1953 |volume=63 |issue=251 |page=648 |jstor=2226451 |doi=10.2307/2226451}}</ref>

Bernoulli's imaginary [[logarithmic scale|logarithmic]] utility function and Gabriel Cramer's {{math|''U''&nbsp;{{=}}&nbsp;''W''<sup>1/2</sup>}} function were conceived at the time not for a theory of demand but to solve the [[St. Petersburg paradox|St.&nbsp;Petersburg's game]]. Bernoulli assumed that "a poor man generally obtains more utility than a rich man from an equal gain"<ref>{{cite journal |last1=Samuelson |first1=Paul |title=St. Petersburg Paradoxes: Defanged, Dissected, and Historically Described |journal=Journal of Economic Literature |date=1977 |volume=15 |issue=1 |page=38 |jstor=2722712}}</ref> an approach that is more profound than the simple mathematical expectation of money as it involves a law of ''moral expectation''.

Early theorists of [[utility]] considered that it had physically quantifiable attributes. They thought that utility behaved like the magnitudes of distance or time, in which the simple use of a ruler or stopwatch resulted in a distinguishable measure. "Utils" was the name actually given to the units in a utility scale.

In the [[Victorian era]] many aspects of life were succumbing to quantification.<ref>{{cite book |last=Bernstein |first=Peter |date=1996 |title=Against the Gods. The Remarkable Story of Risk |location=New York |publisher=John Wiley and Sons |page=191 |isbn=978-0-4711-2104-6 |url=https://books.google.com/books?id=G5TKA6B0pyEC}}</ref> The theory of utility soon began to be applied to moral-philosophy discussions. The essential idea in [[utilitarianism]] is to judge people's decisions by looking at their change in utils and measure whether they are better off. The main forerunner of the utilitarian principles since the end of the 18th century was [[Jeremy Bentham]], who believed that utility could be measured by some complex introspective examination and that it should guide the design of social policies and laws. For Bentham a scale of pleasure has as a unit of intensity "the degree of intensity possessed by that pleasure which is the faintest of any that can be distinguished to be pleasure";<ref name="Stigler, George 1950">{{cite journal |last=Stigler |first=George |date=August 1950 |url=http://www.ppge.ufrgs.br/GIACOMO/arquivos/eco02277/stigler-1950.pdf |title=The Development of Utility Theory. I |journal=Journal of Political Economy |volume=58 |issue=4 |pages=307–327 |jstor=1828885 |doi=10.1086/256962 |s2cid=153732595 |access-date=2013-03-06 |archive-url=https://web.archive.org/web/20130908090842/http://www.ppge.ufrgs.br/GIACOMO/arquivos/eco02277/stigler-1950.pdf |archive-date=2013-09-08 |url-status=dead }}</ref> he also stated that as these pleasures increase in intensity, higher and higher numbers could represent them.<ref name="Stigler, George 1950"/> In the 18th and 19th centuries utility's measurability received plenty of attention from European schools of political economy, most notably through the work of [[Marginalism|marginalists]] (e.g., [[William Jevons|William Stanley Jevons]],<ref>{{cite journal |last1=Jevons |first1=William Stanley |title=Brief Account of a General Mathematical Theory of Political Economy |journal=Journal of the Royal Statistical Society |date=1862 |volume=29 |pages=282–287 }}</ref> [[Léon Walras]], [[Alfred Marshall]]). However, neither of them offered solid arguments to support the assumption of measurability. In Jevon's case he added to the later editions of his work a note on the difficulty of estimating utility with accuracy.<ref name="Stigler, George 1950"/> Walras, too, struggled for many years before he could even attempt to formalize the assumption of measurability.<ref>{{cite journal |last1=Jaffé |first1=William |title=The Walras-Poincaré Correspondence on the Cardinal Measurability of Utility |journal=Canadian Journal of Economics |date=1977 |volume=10 |issue=2 |pages=300–307 |doi=10.2307/134447 |jstor=134447}}</ref> Marshall was ambiguous about the measurability of hedonism because he adhered to its psychological-hedonistic properties but he also argued that it was "unrealistical" to do so.<ref>{{cite journal |last1=Martinoia |first1=Rozenn |title=That Which is Desired, Which Pleases, and Which Satisfies: Utility According to Alfred Marshall |journal=Journal of the History of Economic Thought |date=2003 |volume=25 |issue=3 |page=350 |url=http://cas.umkc.edu/econ/economics/faculty/Lee/courses/502/reading/mad2.pdf |access-date=21 May 2015 |doi=10.1080/1042771032000114764 |s2cid=31350151 }}</ref>

Supporters of cardinal utility theory in the 19th century suggested that market prices reflect utility, although they did not say much about their compatibility (i.e., prices being objective while utility is subjective). Accurately measuring subjective [[pleasure]] (or [[pain]]) seemed awkward, as the thinkers of the time were surely aware. They renamed utility in imaginative ways such as ''subjective wealth'', ''overall happiness'', ''moral worth'', ''psychic satisfaction'', or {{lang|fr|ophélimité}}. During the second half of the 19th century many studies related to this fictional magnitude&mdash;utility&mdash;were conducted, but the conclusion was always the same: it proved impossible to definitively say whether a good is worth 50, 75, or 125 utils to a person, or to two different people. Moreover, the mere dependence of utility on notions of [[hedonism]] led academic circles to be skeptical of this theory.<ref>{{cite journal |last=Stigler |first=George |date=October 1950 |title=The Development of Utility Theory. II |journal=Journal of Political Economy |volume=58 |issue=5 |pages=373–396 |jstor=1825710 |doi=10.1086/256980 |s2cid=222450704 }}</ref>

[[Francis Edgeworth]] was also aware of the need to ground the theory of utility into the real world. He discussed the quantitative estimates that a person can make of his own pleasure or the pleasure of others, borrowing methods developed in psychology to study hedonic measurement: [[psychophysics]]. This field of psychology was built on work by [[Ernst H. Weber]], but around the time of World War I, psychologists grew discouraged of it.<ref name="Collander, David 2007">{{cite journal |last1=Colander |first1=David |date=Spring 2007 |title=Retrospectives: Edgeworth's Hedonimeter and the Quest to Measure Utility |journal=Journal of Economic Perspectives |volume=21 |issue=2 |pages=215–226 |jstor=30033725 |doi=10.1257/jep.21.2.215|doi-access=free }}</ref><ref>{{cite magazine |last=McCloskey |first=Deirdre N. |title=Happyism |url=https://newrepublic.com/article/103952/happyism-deirdre-mccloskey-economics-happiness |magazine=New Republic |date=June 7, 2012 |access-date=11 March 2013}}</ref>

In the late 19th century, [[Carl Menger]] and his followers from the [[Austrian school|Austrian school of economics]] undertook the first successful departure from measurable utility, in the clever form of a theory of ranked uses. Despite abandoning the thought of quantifiable utility (i.e. psychological satisfaction mapped into the set of real numbers) Menger managed to establish a body of hypothesis about decision-making, resting solely on a few axioms of ranked preferences over the possible uses of goods and services. His numerical examples are "illustrative of ordinal, not cardinal, relationships".<ref>{{cite journal |last1=Stigler |first1=George |title=The Economics of Carl Menger |journal=Journal of Political Economy |date=April 1937 |volume=45 |issue=2 |page=240 |jstor=1824519 |doi=10.1086/255042 |s2cid=154936520 }}</ref>

However, there are other interpretations of Carl Menger's work. Ivan Moscati and J. Huston McCulloch argue that Menger was a classical cardinalist, as his numerical examples are not merely illustrative but represent explicit arithmetic proportions of value between economic goods.<ref>{{cite journal |last1=Moscati |first1=Ivan |title=Were Jevons, Menger and Walras Really Cardinalists? On the Notion of Measurement in Utility Theory, Psychology, Mathematics and Other Disciplines, 1870-1910 |journal=History of Political Economy |date=2013 |volume=45 |issue=3 |pages=373-414 |doi=10.1215/00182702-2334758}}</ref><ref>{{cite journal |last1=McCulloch |first1=J. Huston |title=The Austrian Theory of the Marginal Use and of Ordinal Marginal Utility |journal=Zeitschrift für Nationalökonomie|date= April 1977 |url=https://www.asc.ohio-state.edu/mcculloch.2/papers/AustrianOrdinalMU.pdf |publisher=Ohio State University |accessdate=2024-09-26}}</ref> Arithmetic proportions, sums, and multiplications are inherently cardinal and do not exist in an ordinal paradigm. Menger also explicitly states the following: ''"Only the satisfaction of our needs has direct and immediate significance to us. In each concrete instance, this significance is measured by the importance of the various satisfactions for our lives and well-being. We next attribute the '''exact quantitative magnitude''' of this importance to the specific goods on which we are conscious of being directly dependent for the satisfactions in question"''<ref>{{cite book |last=Menger |first=Carl |title=Principles of Economics |translator=James Dingwall and Bert F. Hoselitz |publisher=Ludwig von Mises Institute |year=2007 |url=https://cdn.mises.org/principles_of_economics.pdf |page=152 |isbn=978-1-933550-12-1}}</ref>

Around the turn of the 19th century [[neoclassical school of economics|neoclassical economists]] started to embrace alternative ways to deal with the measurability issue. By 1900, [[Vilfredo Pareto|Pareto]] was hesitant about accurately measuring pleasure or pain because he thought that such a self-reported subjective magnitude lacked scientific validity. He wanted to find an alternative way to treat utility that did not rely on erratic perceptions of the senses.<ref name="teaching.ust.hk">{{cite journal |last1=Lewin |first1=Shira B. |date=September 1996 |title=Economics and Psychology: Lessons for our Own Day from the Early Twentieth Century |url=http://teaching.ust.hk/~mark329y/EconPsy/Economics%20and%20Psychology%20-%20Lessons%20for%20Our%20Own%20Day%20From%20the%20Early%20Twentieth%20Century.pdf |journal=Journal of Economic Literature |volume=34 |issue=3 |pages=1293–1323 |jstor=2729503 |url-status=dead |archive-url=https://web.archive.org/web/20101011071242/http://teaching.ust.hk/~mark329y/EconPsy/Economics%20and%20Psychology%20-%20Lessons%20for%20Our%20Own%20Day%20From%20the%20Early%20Twentieth%20Century.pdf |archive-date=2010-10-11 }}</ref> Pareto's main contribution to ordinal utility was to assume that higher indifference curves have greater utility, but how much greater does not need to be specified to obtain the result of increasing marginal rates of substitution.

The works and manuals of Vilfredo Pareto, Francis Edgeworth, [[Irving Fischer]], and [[Eugene Slutsky]] departed from cardinal utility and served as pivots for others to continue the trend on ordinality. According to Viner,<ref>{{cite journal |last1=Viner |first1=Jacob |date=August 1925 |title=The Utility Concept in Value Theory and its Critics |journal=Journal of Political Economy |volume=33 |issue=4 |pages=369–387 |doi=10.1086/253690 |jstor=1822522 |s2cid=153762259 }}</ref> these economic thinkers came up with a theory that explained the negative slopes of demand curves. Their method avoided the measurability of utility by constructing some abstract [[Indifference curve#Map and properties|indifference curve map]].

During the first three decades of the 20th century, economists from Italy and Russia became familiar with the Paretian idea that utility does not need to be cardinal. According to Schultz,<ref>{{cite journal |last1=Schultz |first1=Henry |title=The Italian School of Mathematical Economics |journal=Journal of Political Economy |date=February 1931 |volume=39 |issue=1 |page=77 |jstor=1821749 |doi=10.1086/254172 |s2cid=154718427 }}</ref> by 1931 the idea of ordinal utility was not yet embraced by American economists. The breakthrough occurred when a theory of [[ordinal utility]] was put together by [[John Hicks]] and [[R. G. D. Allen|Roy Allen]] in 1934.<ref>{{cite journal |last1=Hicks |first1=John |last2=Allen |first2=Roy |date=February 1934 |title=A Reconsideration of the Theory of Value |journal=Economica |volume=1 |issue=1 |pages=52–76 |doi=10.2307/2548574 |jstor=2548574}}</ref> In fact pages 54–55 from this paper contain the first use ever of the term "cardinal utility".<ref name=Mos>{{cite web |last=Moscati |first=Ivan |title=How Cardinal Utility Entered Economic Analysis During the Ordinal Revolution |url=http://eco.uninsubria.it/dipeco/quaderni/files/QF2012_05.pdf |work=Working Paper |publisher=Universita Dell'Insubria Facolta di Economia |date=2012 |access-date=9 February 2013 |archive-url=https://web.archive.org/web/20140714181142/http://eco.uninsubria.it/dipeco/quaderni/files/QF2012_05.pdf |archive-date=14 July 2014 |url-status=dead }}</ref> The first treatment of a class of utility functions preserved by affine transformations, though, was made in 1934 by Oskar Lange.<ref>{{cite journal |last1=Lange |first1=Oskar |title=The Determinateness of the Utility Function |journal=Review of Economic Studies |date=1934 |volume=1 |issue=3 |pages=218–225 |doi=10.2307/2967485 |jstor=2967485}}</ref>

In 1944 Frank Knight argued extensively for cardinal utility. In the decade of 1960 Parducci studied human judgements of magnitudes and suggested a range-frequency theory.<ref>{{cite thesis |last=Kornienko |first=Tatiana |date=April 2013 |title=Nature's Measuring Tape: A Cognitive Basis for Cardinal Utility |publisher=University of Edinburgh |page=3 |url=http://www.gtcenter.org/Archive/2013/Conf/Kornienko1637.pdf}}</ref> Since the late 20th century economists are having a renewed interest in the measurement issues of [[Happiness economics|happiness]].<ref>{{cite journal |last1=Kahneman |first1=Daniel |last2=Wakker |first2=Peter |last3=Sarin |first3=Rakesh |year=1997 |title=Back to Bentham? Explorations of Experienced Utility? |journal=Quarterly Journal of Economics |volume=112 |issue=2 |pages=375–405 |doi=10.1162/003355397555235|url=https://repub.eur.nl/pub/23011/QJE_1997_112_375.pdf }}</ref><ref>{{cite book |editor-last1=Kahneman |editor-first1=Daniel |editor-first2=Ed |editor-last2=Diener |editor-first3=Norbert |editor-last3=Schwarz |date=1999 |title=Well-Being: Foundations of Hedonic Psychology |location=New York |publisher=Rusell Sage Foundation |isbn=978-1-6104-4325-8 |url=https://books.google.com/books?id=-wIXAwAAQBAJ}}</ref> This field has been developing methods, surveys and indices to measure happiness.

Several properties of cardinal utility functions can be derived using tools from [[measure theory]] and [[set theory]].

=== Measurability ===
A utility function is considered to be measurable, if the strength of preference or intensity of liking of a good or service is determined with precision by the use of some objective criteria. For example, suppose that eating an apple gives to a person exactly half the pleasure of that of eating an orange. This would be a measurable utility if and only if the test employed for its direct measurement is based on an objective criterion that could let any external observer repeat the results accurately.<ref>{{cite journal |last1=Bernadelli |first1=H. |date=May 1938 |title=The End of the Marginal Utility Theory? |journal=Economica |volume=5 |issue=18 |page=196 |doi=10.2307/2549021 |jstor=2549021}}</ref> One hypothetical way to achieve this would be by the use of a [[hedonometer]], which was the instrument suggested by Edgeworth to be capable of registering the height of pleasure experienced by people, diverging according to a law of errors.<ref name="Collander, David 2007"/>

Before the 1930s, the measurability of utility functions was erroneously labeled as cardinality by economists. A different meaning of cardinality was used by economists who followed the formulation of Hicks-Allen, where two cardinal utility functions are considered the same if they preserve [[Preference (economics)|preference]] orderings uniquely up to positive [[affine transformation]]s.<ref name="Ellsberg, Daniel 1954">{{cite journal |last=Ellsberg |first=Daniel |year=1954 |title=Classic and Current Notions of 'Measurable Utility' |journal=Economic Journal |volume=64 |issue=255 |pages=528–556 |doi=10.2307/2227744 |jstor=2227744}}</ref><ref>{{cite journal |last1=Strotz |first1=Robert |year=1953 |title=Cardinal Utility |journal=American Economic Review |volume=43 |issue=2 |pages=384–397}}</ref> Around the end of the 1940s, some economists even rushed to argue that von Neumann–Morgenstern axiomatization of expected utility had resurrected measurability.<ref name="teaching.ust.hk"/> 

The confusion between cardinality and measurability was not to be solved until the works of [[Armen Alchian]],<ref name="www2.uah.es">{{cite journal |last1=Alchian |first1=Armen A. |date=March 1953 |title=The Meaning of Utility Measurement |url=http://www2.uah.es/econ/MicroDoct/Alchian-Utility%20Measurement_1953.pdf |journal=American Economic Review |volume=43 |issue=1 |pages=26–50 |jstor=1810289 |access-date=2010-03-21 |archive-date=2012-03-21 |archive-url=https://web.archive.org/web/20120321224736/http://www2.uah.es/econ/MicroDoct/Alchian-Utility%20Measurement_1953.pdf |url-status=dead }}</ref> William Baumol,<ref>{{cite journal |last1=Baumol |first1=William |date=December 1958 |title=The Cardinal Utility Which is Ordinal |journal=Economic Journal |volume=68 |issue=272 |pages=665–672 |doi=10.2307/2227278 |jstor=2227278}}</ref> and John Chipman.<ref name="Chipman, John 1960">{{cite journal |last1=Chipman |first1=John |date=April 1960 |title=The Foundations of Utility |journal=Econometrica |volume=28 |issue=2 |pages=215–216 |doi=10.2307/1907717 |jstor=1907717}}</ref> The title of Baumol's paper, "The cardinal utility which is ordinal", expressed well the semantic mess of the literature at the time.

It is helpful to consider the same problem as it appears in the construction of [[level of measurement|scales of measurement]] in the natural sciences.<ref>{{cite journal |last1=Allen |first1=Roy |date=February 1935 |title=A Note on the Determinateness of the Utility Function |journal=Review of Economic Studies |volume=2 |issue=2 |pages=155–158 |doi=10.2307/2967563 |jstor=2967563}}</ref> In the case of [[temperature]] there are two ''degrees of freedom'' for its measurement{{snd}} the choice of unit and the zero. Different temperature scales map its intensity in different ways. In the [[celsius scale]] the zero is chosen to be the point where water freezes, and likewise, in cardinal utility theory one would be tempted to think that the choice of zero would correspond to a good or service that brings exactly 0 utils. However this is not necessarily true. The mathematical index remains cardinal, even if the zero gets moved arbitrarily to another point, or if the choice of scale is changed, or if both the scale and the zero are changed. Every measurable entity maps into a cardinal function but not every cardinal function is the result of the mapping of a measurable entity. The point of this example was used to prove that (as with temperature) it is still possible to predict something about the combination of two values of some utility function, even if the utils get transformed into entirely different numbers, as long as it remains a linear transformation.

Von Neumann and Morgenstern stated that the question of measurability of physical quantities was dynamic. For instance, temperature was originally a number only up to any monotone transformation, but the development of the ideal-gas-thermometry led to transformations in which the absolute zero and absolute unit were missing. Subsequent developments of thermodynamics even fixed the absolute zero so that the transformation system in thermodynamics consists only of the multiplication by constants. According to Von Neumann and Morgenstern (1944, p.&nbsp;23), "For utility the situation seems to be of a similar nature [to temperature]".

The following quote from Alchian served to clarify once and for all{{citation needed|date=August 2017}} the real nature of utility functions:
{{quote
|text=Can we assign a set of numbers (measures) to the various entities and predict that the entity with the largest assigned number (measure) will be chosen? If so, we could christen this measure "utility" and then assert that choices are made so as to maximize utility. It is an easy step to the statement that "you are maximizing your utility", which says no more than that your choice is predictable according to the size of some assigned numbers. For analytical convenience it is customary to postulate that an individual seeks to maximize something subject to some constraints. The thing {{snd}} or numerical measure of the "thing"{{snd}} which he seeks to maximize is called "utility". Whether or not utility is of some kind glow or warmth, or happiness, is here irrelevant; all that counts is that we can assign numbers to entities or conditions which a person can strive to realize. Then we say the individual seeks to maximize some function of those numbers. Unfortunately, the term "utility" has by now acquired so many connotations, that it is difficult to realize that for present purposes utility has no more meaning than this.
|author=[[Armen Alchian]]
|source=The meaning of utility measurement<ref name="www2.uah.es"/>}}

=== Order of preference ===
{{Details|Preference (economics)}}

In 1955 [[Patrick Suppes]] and Muriel Winet solved the issue of the representability of preferences by a cardinal utility function and derived the set of axioms and primitive characteristics required for this utility index to work.<ref>{{cite journal |last1=Suppes |first1=Patrick |first2=Muriel |last2=Winet |date=April 1955 |url=http://suppes-corpus.stanford.edu/article.html?id=11 |title=An Axiomatization of Utility Based on the Notion of Utility Differences |journal=Management Science |volume=1 |issue=3/4 |pages=259–270 |jstor=2627164 |doi=10.1287/mnsc.1.3-4.259 |access-date=2010-06-10 |archive-url=https://web.archive.org/web/20100721014253/http://suppes-corpus.stanford.edu/article.html?id=11 |archive-date=2010-07-21 |url-status=dead |url-access=subscription }}</ref>

Suppose an agent is asked to rank his preferences of {{math|''A''}} relative to {{math|''B''}} and his preferences of {{math|''B''}} relative to {{math|''C''}}. If he finds that he can state, for example, that his degree of preference of {{math|''A''}} to {{math|''B''}} exceeds his degree of preference of {{math|''B''}} to {{math|''C''}}, we could summarize this information by any triplet of numbers satisfying the two inequalities: {{math|''U<sub>A</sub>'' > ''U<sub>B</sub>'' > ''U<sub>C</sub>''}} and {{math|''U<sub>A</sub>'' − ''U<sub>B</sub>'' > ''U<sub>B</sub>'' − ''U<sub>C</sub>''}}.

If {{mvar|A}} and {{mvar|B}} were sums of money, the agent could vary the sum of money represented by {{mvar|B}} until he could tell us that he found his degree of preference of {{mvar|A}} over the revised amount {{math|''B''′}} equal to his degree of preference of {{math|''B''′}} over {{mvar|C}}. If he finds such a {{math|''B''′}}, then the results of this last operation would be expressed by any triplet of numbers satisfying the relationships {{math|''U<sub>A</sub>'' > ''U''<sub>''B''′</sub> > ''U<sub>C</sub>'' }} and {{math|''U<sub>A</sub>'' − ''U''<sub>''B''′</sub> {{=}} ''U''<sub>''B''′</sub> − ''U<sub>C</sub>''}}. Any two triplets obeying these relationships must be related by a linear transformation; they represent utility indices differing only by scale and origin. In this case, "cardinality" means nothing more being able to give consistent answers to these particular questions. This experiment does not require measurability of utility. [[Itzhak Gilboa]] gives a sound explanation of why measurability can never be attained solely by [[introspection]]:
{{quote
|text=It might have happened to you that you were carrying a pile of papers, or clothes, and didn't notice that you dropped a few. The decrease in the total weight you were carrying was probably not large enough for you to notice. Two objects may be too close in terms of weight for us to notice the difference between them. This problem is common to perception in all our senses. If I ask whether two rods are of the same length or not, there are differences that will be too small for you to notice. The same would apply to your perception of sound (volume, pitch), light, temperature, and so forth...
|author=Itzhak Gilboa
|source=Theory of decision under uncertainty<ref>{{cite book |last=Gilboa |first=Itzhak |date=2009 |url=http://www.econ.hit-u.ac.jp/~kmkj/uncertainty/Gilboa_Lecture_Notes.pdf |title=Theory of Decision under Uncertainty |publisher=Cambridge University Press |isbn=978-1-1077-8251-8 |access-date=2010-03-30 |archive-url=https://web.archive.org/web/20180219002606/http://www.econ.hit-u.ac.jp/~kmkj/uncertainty/Gilboa_Lecture_Notes.pdf |archive-date=2018-02-19 |url-status=dead }}</ref>}}

According to this view, those situations where a person just cannot tell the difference between {{mvar|A}} and {{mvar|B}} will lead to indifference not because of a consistency of preferences, but because of a misperception of the senses. Moreover, human senses adapt to a given level of stimulation and then register changes from that baseline.<ref>{{cite book |last=Poundstone |first=William |date=2010 |title=Priceless: The Myth of Fair Value (and How to Take Advantage of It) |location=New York |publisher=Hill and Wang |page=39 |isbn=978-1-4299-4393-2 |url=https://books.google.com/books?id=xWjTZhw1MiUC&pg=PA39}}</ref>