Risk and Uncertainty: A Fallacy of Large Numbers (pdf, 1963) Oct 10 2017 13:43 languageMoneyScience
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By Paul A. Samuelson
Experience shows that while a single event may have a probability spread, a large repetition of independent single events gives a greater approach toward certainty. This corresponds to the mathematically provable Law of Large Numbers of James Bernoulli. This valid property of lnrge numbers is often given an invalid interpretation. Thus people say and insurance company reduces its its risk by increasing the number of ships it insures. Or they refuse to accept a mathematically favorable bet, but agree to a large enough repetition of such bets: e.g. believing it is almost a sure thing that there will be a million heads when two million symmetric coins are tossed even though it is highly uncertain there will be one head out of two coins tossed. The correct relationship (that an insurer reduces total risk by subdividing) is pointed out and a strong theorem is proved: that a person whose utility schedule prevents him from ever taking a specific favorable bet when offered only once can never rationally take a large sequence of such fair bets, if expectad utility is maximized. The intransitivity of alternative decision criteria-such as selecting out of any two situations that one which will more probably leave you better off is also demonstrated.
The Paper “Risk and Uncertainty: A Fallacy of Large Numbers” by Paul A. Samuelson, was published in Scientia in April-May, 1963. It was later reprinted in the Collected Scientific Papers of Paul A. Samuelson, Volume 1, pp- 153-8, MIT Press, 1966 _ It had a very distinguished influence on the ideas of risk and portfolio for investment applications. The paper first got the attention of Pratt, Zeckhauser, and other mathematical economists and thereby spawned several related papers. Unfortunately it apparently did not reach the one group most concerned with property and casualty insurance. We hope that the current republication will rectify this.