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The fact that mathematics nurtures not only the development of new technologies but academic fields from A to Z, deserves to be better known to "ordinary" people, and doing this for Mathematics and Statistics Awareness Month with regard to some small aspects of economics is my goal...
To what extent is mathematics unreasonably effective in getting insights into subjects outside of mathematics? Many scientists are familiar with the paper by Eugene Wigner (1902-1995), the Nobel Prize-winning (1963) physicist entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences."
His paper includes this quotation:
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
In honor of Mathematics and Statistics Awareness Month I raise the issue of both the possible unreasonable effectiveness and unreasonable ineffectiveness of mathematics for subjects which might seek to draw insights from mathematics. While certainly mathematicians don't need subjects outside of mathematics to practice their subject--internal issues can more than keep mathematicians busy--when it comes to convincing "mathematics skeptics" that mathematics deserves their attention, it helps to argue that mathematics affects their lives in many ways. The fact that mathematics nurtures not only the development of new technologies but academic fields from A to Z, deserves to be better known to "ordinary" people, and doing this for Mathematics and Statistics Awareness Month with regard to some small aspects of economics is my goal.
Not everyone in the natural sciences would agree with Wigner but to what extent does Wigner's point apply to fields outside the natural sciences? Here I would like to argue that mathematics has indeed been reasonably effective in economics. Certainly, not everyone would agree that mathematics is useful in economics and some argue it muddies the waters so much to use mathematics that perhaps this route hurts economics.
When mathematics is used outside of mathematics itself, the perspective adopted in recent times is that what is being done is constructing a model (simplification) for the situation one wants to examine using mathematics. Here are some examples of situations one might be trying to understand in economics:
a. How to successfully allocate some resources, say, graduate school housing to new graduate students?
b. Is understanding the consequences of a new technology being brought to market analogous to the insights into the market for expensive women's dresses and men's suits?
c. How can we smooth out the operation of the economy so as to avoid bubbles and recessions?
I am not prepared to offer up a book-length treatment of how mathematics has come to the assistance of those trying to give insights into economics, though this would be what would be required to do justice to mathematics' contributions. I will settle for an extremely skeletal account of some history of economics, with emphasis on some contributors who have brought mathematics or mathematically related contributions to the field.
It is not easy to begin to chart the development of economics as a field of thought or the starting point for where mathematics began to be seen as a way to get insights that were important. Various economic "facts" have come to be called "laws" even if the use of the term, suggesting that something is always true, is not actually valid.
So let me begin with the work of Jean Baptiste Say (1767-1832).
Today, it is common to hear about "Say's Law," the intuitive idea being that when something is produced (supply) it creates its own demand. In hindsight it is hard to know what Say meant in saying what today is called Say's Law because Say stated no "law" as such. Among other issues related to supply and demand is the notion of a "glut," the idea that some items might be produced in such a great supply that they could not be "cleared" by the market. With hindsight we realize that supply and demand issues are complicated and that money and price are related to the way that "markets" function. Economics has moved beyond where items change hands through barter on a one person to another person basis. However, the way that Say's work is looked at, in particular the reaction that John Maynard Keynes (1883-1946) had to "Say's Law," represents the attempt to understand what Say might have been trying to get at in more "precise" or quantitative terms.
Today if one takes a course in economics in high school or college, one is likely to see a diagram such as the one in Figure 3. Such diagrams are designed to get ideas across rather than being necessarily representative of "real" supply-demand curves. The way that mathematics is used in the world involves mathematical modeling: the simplification of complicated situations so that one gets a simplified form of the situation to investigate. Here the mathematics involved is analytical geometry (and sometimes calculus) developed by Descartes and Fermat in the 17th century. Also, ideas related to curves and functions emerged from Newton and Liebniz's work on calculus.
Say's Law tries to use words to capture an economic reality. However, having conceptualized about supply and demand, Figure 3 is designed to help understand important intuitive ideas about supply and demand. The diagram shows a supply curve (in blue) and two demand curves shown in red. The vertical axis shows prices and the horizontal axis shows the quantity of some good delivered. Each point on the blue curve represents the amount of the good delivered to market for the price involved. Where do such graphs/curves come from? Does the fact that the blue curves (supply) rise as one moves to the right (more quantity) make sense? Does it make sense that the demand curves go down as one moves to the right (that is, as quantity increases)? What kind of good are we talking about? Perhaps supply and demand issues for milk behave differently than for sports cars? What happened to the costs of bringing this particular good to market? What about revenue (the amount of money a supplier makes that takes account of the income one makes and the cost to get sales)? Is it really a fact that given a price, we can actually predict the quantity of some good that is delivered to a market?
There are chicken-and-egg questions lurking here. As the price goes up, the amount of goods supplied presumably will rise. Figure 3 shows curves in blue and red but many economics texts use straight lines for the supply and demand curves. While this stretches realism further it does mean that one can write down the equations of the supply-demand curves. When using lines, their slope is the key to their qualitative behavior. Negative-slope lines go down to the right and positive-slope curves go up to the right. For the more general case in Figure 3 it is the positive or negative slope of a tangent line to the curve at a point on a "well-behaved" supply or demand curve that governs whether the curve is going up or down as one moves to the right. Presumably it is not realistic to have supply-demand curves with horizontal or vertical stretches, but what thinking about the mathematics does is to encourage thinking about the economic interpretation of what is mathematically possible. While it is not realistic that these supply-demand curves be straight lines, this stronger modeling assumption allows one to simplify the analysis of what is going on.
Not all goods that are supplied or demanded in an economy are alike; economists have used mathematical ideas to try to understand the behavior of different kinds of goods. One example of the way mathematics can try to give insight into microeconomics is to understand the shape or nature of the supply and demand curves such as those in Figure 3. Graphs of mathematical functions can be either convex (a line segment joining any two points the graph lies above--or on--the graph) or concave (a line segment joining any two points on the graph lies below--or on--the graph). The three graphs in Figure 3 are convex. Would it ever be the case that a supply curve or demand curve be concave? If so, what kind of goods would have this property and if it is impossible, what about supply and demand makes this happen?
Calculus is a tool for understanding how functions and curves change, enabling one to study how fast/slowly some variable is changing with respect to another variable. Calculus for example can be used to acquire an understanding of the economic concept of price elasticity, a term which can be used with respect to demand or supply. Thus, a good or service can be said to show price elasticity or price inelasticity of demand. Intuitively, price elasticity of a good (or service) G measures the response in the amount of the good (service) demanded when the price of the good changes. In the spirit of mathematical modeling, the simplification is made that nothing else about the complex issues of demand changes except the price and one wants to "predict" what will happen to the demand for this good G after the "sudden" price change. One common aspect of the discussion of what will happen to a particular good is "substitutability."
If overnight the price of orange juice doubles, how the demand will change depends to some extent on the fact that people may like orange juice for breakfast but they are "willing" to have apple, cherry, or pomegranate juice instead, providing the price of these substitutes are "competitive." At the same price, people will make selections for what juice they prefer as their breakfast drink but they might not actually buy that preferred drink if they have other drinks that are approximately equal in appeal that can be purchased for less money. However, other goods are such that there are fewer substitutes than example about orange juice. If one must drive one's car to work (no buses, trains, car services, etc. available as alternatives) and the price of gasoline quadruples overnight, there is little choice but to pay the higher price as unpleasant as the thought is. You might make compensations in other spending to deal with the reality of greater expenditures for gas because your income did not quadruple overnight. On the other hand, and there typically are at least several other hands, perhaps one might negotiate with one's employer to work from home for as long as possible until the price of gas returns to a more "reasonable" level. Or one might rush out and buy an electric car immediately!
Mathematical notions allow us to express the qualitative issues alluded to above in more "exact" terms. Price elasticity of demand gives the percent change in the amount of a good or service demanded as a consequence of a one-percent change change in price. Actually getting data for "elasticity of demand," no matter how carefully defined, may not be an easy task. Goods may vary in elasticity at different seasons--the world of consumer behavior is very complicated.
Certainly, different goods seem to show different degrees of demand elasticity but in many cases it would seem that the "elasticity" of a good or service is not something every person would agree about. One way that price may change suddenly is to go to a different part of the country or to a different country. Cheesecake might have a particular price in the area in which you live but have a noticeably higher price in another country. And yet, one might be willing to buy the cheesecake abroad at triple the price while one would not buy cheesecake at your local bakery if the price tripled. (In my family this involves what I like to call "vacation rules.")
Parallel to the ideas about price elasticity of demand are ideas about price elasticity of supply. Now one is interested in what happens if there is a sudden increase in the price of a good or service, will more or less of this good now be produced (supplied)? One can carry out a similar discussion to the one above "on the supply" side.
Kenneth Arrow (1921-2017) is probably best known for his work that "created" the field of social choice. His famous theorem, which in essence says that when a group is trying to use individual preference ordinal rankings to get a preference ranking for the group, when the rankings involve three or more choices, there is no perfect method of accomplishing this goal, perfect in the sense of obeying a small list of appealing fairness axioms.
However, Kenneth Arrow, a winner of the Nobel Memorial Prize for Economics (1972) made many very important contributions to mathematical economics. His "complete works," published many years before his death (and thus missing some things) span six volumes.
In particular Arrow together with Gérard Debreu (1921-2004) made an important theoretical development related to prices, supply, and demand. The mathematical model developed by Arrow–Debreu argues that under certain economic/behavioral assumptions there have to exist a set of prices such that "supplies" will equal "demands" for all of the commodities produced in the market involved.
The Debreu-Arrow "equilibrium" model theorem depends for its proof on the use of a sophisticated mathematical apparatus, in particular the use of fixed point theorems. Some economists have argued that such results don't help provide true insight into economics because their assumptions are not satisfied by real world markets/economies and by the people who participate in such economies. Yet attempts to understand the world which have unrealistic assumptions do offer the possibility of making better assumptions closer to what one sees in the "real world," thereby offering the chance that progress in mathematical tools will make it possible to solve problems in the future that cannot be solved now. New applied mathematics grows from theoretical mathematics regularly.
Above we used the term money. In the past (but sometimes still today) goods got exchanged without the use of money. A farmer might provide a part of his crop to someone who provided him/her with tools. Such exchanges are referred to as barter. The development of paper money is one of the dramatic developments by human societies. In the past, the value of money was "secured" by precious metal (usually gold) but recently many currencies don't guarantee that you can "turn in" and receive gold or silver in exchange. In fact, there are now cryptocurriences that are exchanged on markets that have no government backing at all. Mathematics had been used to understand the way that changes in the "money supply" affect what goes on in a country's or the world's economic system.
In modern times many economists talk about being influenced by the economist John Maynard Keynes (1883-1946). Modern acrimonious discussions of economics often center on "Keynesian" approaches to government activity in the economy. Some of this plays out in Keynes's views of Say's Law.
While many know Keynes's name only as an economist, he was also an extremely talented mathematician. In particular, he was the 12th highest scorer for 1905 (12th wrangler) on the famous examination known as the Mathematical Tripos at Cambridge University. The person with the highest score was known as the senior wrangler. Many senior wranglers have name recognition for having gone on to successful careers in mathematics and/or the sciences. Thus, the list of senior wranglers included: Edward Waring, George Airy, John Herschel, Arthur Cayley, James Inman, George Gabriel Stokes, Isaac Todhunter, Morris Pell, Lord Rayleigh, Arthur Eddington, J. E. Littlewood, Jayant Narlikar, Frank Ramsey, Harold Scott MacDonald Coxeter (my own academic grandfather), Jacob Bronowski, Lee Hsien Loong, Kevin Buzzard, Christopher Budd, Ben Green, and John Polkinghorne. The Mathematical Tripos was known to be a very difficult examination to perform well on, but students who did often got high recognition for their accomplishment. This encouraged some students to devote great amounts of time to doing well on this examination, even if it meant they neglected going to their "classes." It was common for strong students to work with special coaches to help them do better on the Tripos. The distinguished number theorist G.H. Hardy (1877-1947) was very unhappy with the way that the Mathematical Tripos distorted the mathematical experience for students at Cambridge and worked hard to have it downgraded in importance.
He felt performing well on the Tripos had become an end in itself rather than a means to an end. Hardy himself had been a 4th wrangler in 1898. (Many today see some of the international comparison exams related to mathematics as distorting what goes on in individual countries because good performance on these examinations has become an end in itself for some countries rather than having the mathematics education system of the country serve the needs of the students in the country.)
Keynes is particularly known for his work in probability theory. He authored the book A Treatise on Probability in 1921, while he was still at Cambridge. His views on probability were complex and not fully in accord with what came before. His work set off a reaction by Frank Ramsey (1903-1930), and many other philosophers and mathematicians were inspired to take a closer look at how probability theory could be used in the "real world."
Historically, probabilities were often interpreted as "stabilized" relative frequencies. This approach may make sense for situations where a biased or fair coin is being tossed or a fair die is being rolled. But for the kinds of issues that often drew Keynes's attention and that of others concerned with the making of decisions in an economic context, this approach could not be defended. Many decisions were made in environments that had never been seen before and would not be seen again. This led to probability being thought of in terms of degrees of belief, and gave birth to what is often described as subjective or personal probability. Leonard Savage (1917-1971), whose doctorate degree in mathematics at Michigan was entitled "The Application of Vectorial Methods to the Study of Distance Spaces," wrote about this approach in his important book of 1954 about probability theory.
In particular, the idea of utilitarianism discussed below sometimes involves decision making based on the notion of "expected utility." To do these calculations requires assumptions about what probabilities are and how human beings arrive at them and act on their understanding of probability. Utilities and expected utilities also play a big part in mathematical game theory which is an important part of economics and mathematical economics.
Many people who contributed to economics did so in ways that are also viewed as contributions to philosophy.
When a government runs an economic system and individuals take actions within this system (or in situations where there is "no government"), there are questions about how the individual should act and/or the government should act. One point of view involves what has come to be called utilitarianism. This term refers to the ethical notion that individuals and governments should maximize "utility," which in this context refers to well being. The pioneers (though there are other "ancestors" for the ideas involved) of this way of thinking were Jeremy Bentham and John Stuart Mill.
However, what does "utility" mean and how can we measure it? From a more recent perspective, utility is sometimes described as the pleasure, satisfaction, or value of some action or having possession of some item. One gets "utility" from owning a house, from getting a college education, from proving a new theorem, etc. Bentham put it this way: "it is the greatest happiness of the greatest number that is the measure of right and wrong" Mill tried to flesh out the meaning of some of what Bentham did and to chart ideas about moral/ethical behavior. Thus, for example, he argued against slavery and for women's rights at a time when slavery was widespread, including legal in the United States. He also wrote about women's rights at a time when women could not vote in the US or Great Britain, and the rights of women within their marriages not to be "subservient" to their husbands. With regard to utilitarianism he had more nuanced views than Bentham in that he explicitly discussed that different things or actions/possessions might provide different amounts of utility. On the subject of happiness, certainly a notion that means different things to different people, he wrote: "the ultimate end, for the sake of which all other things are desirable (whether we are considering our own good or that of other people) is an existence as free as possible from pain and as rich as possible in enjoyments."
Sometimes utilitarianism is described as a "policy" which promotes the greatest "good" or happiness for the largest number of people. Later I will look at some of the issues about how to measure "good" or "happiness" and whether one can sum the individual happiness of people in a group to get the "happiness" of the group. However, it is noteworthy that simple examples show it is possible to give great "happiness" to a very small group of people in a big group, making it look like the happiness of the group is great when in fact the average (mean) happiness for the group is not truly large. One billionaire living in a small town may make it look as if the mean wealth of the people in the community is high when in fact the wealth of a single person is distorting the impression of the wealth of the group. Something similar can happen for "utility," and one thing that might go into measuring the "utility" of a group is the wealth of its members.
It is surprisingly recent that the American philosopher John Bordley Rawls (1921-2002) looked at issues related to calling into question whether the greatest good for the largest number of people was the "proper" optimization measure for policy.
Vastly oversimplifying Rawls, instead of arguing for the greatest good for the greatest number of people he argued for a maximin principle--acting in a way that made the outcome for the worst well-off person to be as favorable (best) as possible. While not directly mathematical, Rawls's arguments raise the question of measuring the "justice" or "utility outcome" for individuals and groups. Over and over again in the history of using mathematical ideas in economics (or philosophy) one sees discussions of the subtle issues of measuring things that are at least as complex to measure as lengths, weights, and time. In such environments over and over again questions about preference and intensity of preference are involved. John is better off than Mary, right? But by a little bit or a lot? Does "trickle down" happen? If most people are not very well off but there is a group of people who are very well off, over time will things get better or will things "get worse" in terms of economic well-being? It is not surprising that Kenneth Arrow, with his mathematical and philosophical bent, reacted with keen interest to Rawls's work.
Many aspects of utilitarianism have attracted attention. How does one measure how much pleasure or happiness accrues to a specific person from an action or the result of obtaining a "good?" Often this is done by using some kind of "scale" to measure, say, your pain from having a vaccination via injection. Do you use the scale 0 to 9 (0 low, 9 high), 0 to 99 (0 low, 99 high), or a verbal scale such as the one below:
worst pain I have ever felt,
a little painful, or
not painful at all ?
Vaccinations are given relatively rarely, but what if one uses the numerical scales above to rank how much pleasure one gets from eating, say, the last banana he/she ate versus the last strawberry eaten (which as it happens was eaten a few hours earlier for lunch)? Would a person use the same numbers on the scale 0 to 99 the evening after having had the fruits for lunch versus doing the comparison a week after he/she had eaten the fruits? Suppose you and your best friend, who had the same vaccination, use the same number on the scale from 0 to 9 as a pain scale to describe your pain. Does this mean you both "experienced" equal amounts of pain or might there be differences between people's experiencing pain that have little to do with the numbers they use to rate pain? If one has a group of people who use the same scale for rating the utility of something, is it "legal" to sum these utilities to get a "group" opinion or value? Does it make sense for public decision-making to do standard statistics on these numbers? Some experts about these matters think that "interpersonal" comparisons of utility are meaningless and others say that it is acceptable and even perhaps important to do such calculations.
What insights and ideas does mathematics have to offer here? By adopting an "axiomatic" approach, mathematics can investigate what rules different "types" of utilities might obey. Mathematics makes precise the difference between ordinal measurement and cardinal measurement. Suppose a person rates a particular brand A of coffee as having utility of 120 for an 8 oz. cup. This number 120 will be said to measure 120 "utils" for the experience. A util in this context is similar to using the unit of yards or meters to measure lengths. Now the same person tests 8 oz. of herbal tea and decides to assign it 80 utils. As a "control" for the "pleasure" of these drinks the person is given an 8 oz. cup of water and gives it 40 utils. Note that it might be reasonable in understanding utility of different drinks (or fruits) to ask what represents the zero mark. Thus, above we assigned water, perhaps a rather "neutral" drink, a utility above 40. What drink would the person have assigned as having 0? Also, what arithmetic can be done with the numbers above that express "pleasure" amounts for drinks? Can we say that coffee is better than tea by the same amount that tea is better than water? Under the assumptions made about cardinal utilities this conclusion is warranted (but one might be nervous that the person really should have given 119 utils to the coffee). Can one conclude that a cup of tea is 2/3 as pleasurable as a cup of coffee? Again, one might ask the question where the "zero" of the utilities that the person is working with lies. Is zero utility for a drink one where it is "worthless" to have it? Thus, if the "utility function" allows one to "transform" the utilities with a different zero, what is the meaning of the numbers one is working with? If we "reposition" the zero for the numbers above so that it is -20, then coffee now has 140 utils and tea now has 100 utils while water has 60 utils. Before the repositioning of the zero point coffee had 3 times as many utils as water but in the "new" approach coffee is only twice as "pleasurable." So what does this mean in terms of what insight we have into the pleasures of the person who provided us with these numbers?
Part of the reason that cardinal utilities have gone "out of fashion" is alluded to with some of the discussion above. When one assigns coffee a utility of 120, tea a utility of 80 and water a utility of 40, presumably one is saying that one prefers coffee to tea to water. If one is offered additional drink choices, say apple juice, presumably the person can in a consistent way add this choice to the "ranking" of the these 4 drinks. One way this might be denoted is:
Note these rankings of drinks is very similar to the issue of rankings by individuals of candidates running in an election. It is not common to assign "utilities" to the candidates in a cardinal sense but there are definitely ballots of a cardinal type which have been studied by mathematicians, economists, political scientists, and other scholars. As regards producing ordinal preferences for, say drinks, while this is presumably a lot "easier" than producing the more "detailed" information needed for cardinal utilities, it is still not without issues regarding whether people can really carry out this kind of task. For example, for 4 drinks there are 6 comparisons that must be made between pairs of drinks (for example, coffee vs. tea; coffee vs. apple juice; apple juice vs. water, etc.). While many people can carry out saying which drink in each pair is preferred (we assume ties are not allowed, but in reality many people prefer some drinks equally), often when they look at the information given by the preference pairs they see something that bothers them. For example they might say coffee is preferred to tea, tea is preferred to apple juice, apple juice is preferred to water but water is preferred to coffee. When this happens mathematicians say that the "relation" of preference is not "transitive." Many relationships mathematicians study obey "transitivity." Thus, in Euclidean geometry if line l is parallel to line m, and line m is parallel to line n, it must be true that line l is parallel to line n. Many people feel that preference relationships should obey transitivity, so when they do not it makes people uncomfortable with "theory" that is built on generating such relationships.
If one takes economics in high school and/or tries to follow articles about economics that appear regularly in such papers as The New York Times or The Wall Street Journal, one needs working knowledge of a wide array of terms.
The Council for Economics Education has compiled a list of key concepts for high school classes in America that deal with economics. Of course, these are also key topics for economics in college as well.
List of Economic Topics (Council for Economics Education):
Decision Making and Cost-Benefit Analysis
Division of Labor and Specialization
Trade, Exchange and Interdependence
Budget Deficits and Public Debt
Employment and Unemployment
Monetary Policy and the Federal Reserve
Real vs. Nominal
Competition and Market Structures
Elasticity of Demand
Government Failures/Public-Choice Analysis
Markets and Prices
Price Ceilings and Floors
Roles of Government
Balance of Trade and Balance of Payments
Barriers to Trade
Benefits of Trade/Comparative Advantage
Foreign Currency Markets/Exchange Rates
Personal Finance Economics
Risk and Return
Saving and Investing
The mathematical point of view takes terms like these and tries to make sure that when one sees these terms/phrases one knows what the content and issue involved is. I will not try to take on how mathematics has clariifed the economics related to each of these various terms. But mathematics provides a service to both economics and mathematics. Using mathematics we can elucidate subtle distinctions between what sometimes seem to be the same thing and can generate attempts to get even better ways to conceptualize about things we see both in the real world and the mathematical world.
In lower grades of school one receives instruction from one teacher who integrates the learning one does across boundaries of subjects such as physics, mathematics, English, history, and social and behavioral science. Not until later in one's education does one "change classes" and go for more differentiated learning of the topics above, but typically one does not have a course in economics. One learns about economics in high school but relatively rarely is economics the title of a class one takes there. Paying attention to how mathematics interacts with what one sees related to economics in K-12 education might help students better understand what mathematics--and its nature--is.
Enjoy exploring the way mathematics and economics interact for Mathematics and Statistics Awareness Month!
Arrow, K. and G. Debreu, G., Existence of an equilibrium for a competitive economy Econometrica: Journal of the Econometric Society, (1954) 265-290.
Arrow, K., Collected Papers of Kenneth J. Arrow: Applied Economics, Six Volumes, Harvard University Press, 1985.
Arrow, K., and A. Sen, K. Suzumura, eds.), Handbook of Social Choice and Welfare (Vol. 2), Elsevier, 2010.
Aumann, R. and S. Hart, (eds.), Handbook of Game Theory with Economic Applications, Volumes 1, 2, 3, North Holland, New York. (Note: A 4th volume edited by P. Young and S. Zamir appeared in 2015.)
Barbera, S. and P. Hammond, C. Seidel (eds.), Handbook of Utility Theory, Volume 1, Springer, 1998.
Barbera, S. and P. Hammond, C. Seidel (eds.), Handbook of Utility Theory, Volume 2, Kluwer, Dordrecht, 2004.
Debreu G.,Theory of value: An axiomatic analysis of economic equilibrium. Yale University Press, New Haven, 1987.
Eatwell, J. and M. Milgate, P. Newman, (eds.,) Game Theory (The New Palgrave), Norton, NY, 1989.
Fishburn, P., Utility theory, Management science, 14 (1968) 335-378.
Fishburn, P., Nonlinear preference and utility theory, Baltimore: Johns Hopkins University Press, Baltimore, 1988.
Freeman S., (ed.), The Cambridge Companion to Rawls, Cambridge University Press, Cambridge, 2003.
Gilboa, I., Theory of Decision under Uncertainty, Cambridge University Press,
Kagel, J. and A. Roth, (eds.), The Handbook of Experimental Economics, Volume 2, Princeton University Press, Princeton, 2016.
Keynes, J., A Treatise on Probability. The Collected Writings of John Maynard Keynes, Vol. VIII, Royal Economic Society, 1973.
Kuhn, H., (ed.), Classics in Game Theory, Princeton U. Press, Princeton, 1997.
Kreps, D., Notes on the Theory of Choice, West-view Press, Boulder, 1988.
Morrow, J., Game theory for political scientists, Princeton University Press, Princeton, 1994.
Quiggin, J., Generalized expected utility theory: The rank-dependent model,. Springer Science & Business Media, New York, 2012.
Ramsey, F. Mr. Keynes on probability, The British Journal for the Philosophy of Science. 40 (1989) 219-222. (Reprint of something Ramsey originally published in 1922.)
Rawls J., A theory of justice, Harvard University Press, Cambridge, 1971.
Rawls, J., Some reasons for the maximin criterion, The American Economic Review 64 (1974): 141-146.
Rawls, J., Justice as fairness: A restatement, Harvard University Press, 2001.
Runde, J., Keynes after Ramsey: in defense of ’A treatise on probability’, Stud. Hist. Philos. Sci., 25 (1994), 97-121.
Savage, L., Foundations of Statistics, Wiley, New York, 1954.
Stigler, G., The development of utility theory, I. Journal of Political Economy, 58 (1950) 307-327.
Swalm, R., Utility theory-insights into risk taking. Harvard Business Review, 44 (1966) 123-136.
Theocharis, R., Early developments in mathematical economics, Macmillan, New York, 1961.
Tversky, A., Utility theory and additivity analysis of risky choices, Journal of experimental psychology, 75( (1967) 27.
Varian, H., Gerard Debreu's contributions to economics, Scandinavian Journal of Economics, 86 (1984) 4-14.
Velupillai, K., The unreasonable ineffectiveness of mathematics in economics, Cambridge Journal of Economics, 29 (2005).849-872.
Weintraub, E., (ed.), Toward A History of Game Theory, Duke U. Press, Durham, 1992.
Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials.
The AMS encourages your comments, and hopes you will join the discussions. We review comments before they're posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.
The AMS encourages your comments, and hopes you will join the discussions. We review comments before they're posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.