# Adam Smith's Interval Valued Approach to Probability: Why was It Overlooked by Academics for 239 Years?

32 Pages Posted: 24 May 2015 Last revised: 30 May 2015

Date Written: May 23, 2015

### Abstract

Adam Smith was the first economist, mathematician, or historian to present an explicit, detailed, interval valued approach to the estimation of probabilities. His first application occurred in the Wealth of Nations on pp.106-109 and involved a comparison-contrast between two occupational choice options - the uncertain/ambiguous lawyer option and the risky shoemaker option. This choice-option problem is identical to the first Ellsberg urn example between two urns presented by Ellsberg in 1961. Ellsberg’s first urn has 50 red and 50 black balls in it. The second urn has 100 red and black balls in it, but the decision maker does not know the number of balls of each color in that second urn. The general result was that decision makers were uncertainty/ambiguity averse. The majority would choose to bet on balls drawn from the risky urn. A smaller percentage of decision makers preferred uncertainty/ambiguity to known risks.

This type of analysis was presented by Smith in another two option problem involving a choice between the domestic trade and the foreign trade, given equal or nearly equal returns. This was his second example in the Wealth of Nations of an Ellsberg type analysis.

Further analysis by Smith demonstrated that it was not possible to make precise or exact estimates of the probabilities of outcomes involving losses of shipping at sea, losses from fire, or the probability of being captured during wartime activities.

In general, this lack of precision or exactness mean that the expected value and/or expected utility (subjective expected utility) rules can’t be applied in the real world of decision making because there is missing evidence that does not allow the decision maker to use point probabilities.

Smith also recognized the nonlinear, sub proportional nature of the tradeoff between the outcome and the probability in the Expected Value (Expected Utility) model, as opposed to the standard assumption made by neoclassical economists that the tradeoff is linear and proportional.

These unique and seminal contributions by Smith have not been discussed anywhere in the Literature by any economist, philosopher, historian or mathematician in any article, note, monogram, periodical, or book since the Wealth of Nations was published in 1776, except for Brady (2012, 2015). There are two possible reasons for this. The first is the dominance among economists of Jeremy Bentham’s act utilitarian approach to decision making that claims that all decision makers are able to precisely calculate option outcomes and probabilities (called “uncertainties” by Bentham). The second is the undue influence on economists of papers and books published by Viner, Schumpeter, Rashid, and Rothbard. These contributions argued that Smith had not made a single original, unique or seminal contribution to political economy or economics, be it micro or macro. These claims are examined and found to be generally erroneous. Rashid and Rothbard also claimed that there was overwhelming evidence that Smith was a plagiarizer. In fact, exactly the opposite is the case. Smith was a two eyed, intellectual giant in the land of the blind, utilitarian, political economists of the 18th century whose contributions to probability, decision theory, and the uncertainty versus risk distinction were not equaled until J M Keynes published his A Treatise on Probability in 1921.

This author can find no mention/analysis/evaluation of any of the above points made by any Adam Smith scholar or any scholar associated with the Adam Smith Institute in England. There is no mention/analysis/evaluation of any of the material or topics covered in this article by any published article in any volume of The Adam Smith Review.

**Keywords:** interval estimate, imprecise, uncertainty, Ellsberg paradox

**JEL Classification:** B10, B12, B20, B22

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