Risk and choice: A research saga

Christian Gollier James K. Hammitt Nicolas Treich JEL Classification D 0 ) Center for Risk Analysis, Harvard University , 718 Huntington Ave., Boston, MA 02115, USA The economic theory of decision making under risk has seen remarkable advances over the 50 years since Pratt's (1964) characterization of risk aversion under expected utility. We review developments in three key areas to which Louis Eeckhoudt has made significant contributions: (1) increases in risk and risk taking; (2) self-protection and risk aversion; and (3) higher (and lower) order derivatives of utility. For each, we identify seminal papers, puzzles, and recent developments. The saga of research on these topics reveals that important contributions were made long ago and yet significant gains in understanding continue to be made. Recent advances often have roots in early results and researchers can profit by examining the old as well as the new papers. For as long as Homo sapiens have existed, they have made decisions without being certain of the consequences. Arguably the first normatively compelling theory of decision making under risk was developed in response to the St. Petersburg paradox, which was proposed just 300 years ago in a letter from Nicolas Bernoulli to Pierre Raymond de Montmort dated September 9, 1713. The St. Petersburg paradox posited a wager with infinite expected value, for which no individual seems prepared to pay more than a modest amount. In its best-known form, the gambler is offered a monetary prize of 2n1 monetary units if a series of coin tosses produces its first heads on the nth toss. Several solutions to the paradox were offered in the early 18th century (Seidl 2013). These include neglecting outcomes whose probability falls below some threshold or prizes larger than can realistically be paid, concepts which later reappear in the guise of acceptable risk (e.g., a mortality risk smaller than one in a million; Starr 1969; Kelley 1991) and worst-case analysis (Sunstein 2009). What has been accepted as the standard solution was - proposed by Daniel Bernoulli (Nicolas cousin) in 1738. He proposed that the value of a prize to an individual was an increasing but strictly concave function of the prize (specifically, a logarithmic function). If the degree of concavity is sufficient, the expected utility of the gamble is finite and the certainty equivalent may be quite modest. For example, an individual whose utility function is logarithmic with an initial wealth of 10 monetary units would not be ready to pay more than 2.88 monetary units to play the St. Petersburg game. The three centuries following the introduction of the St. Petersburg paradox have provided a wealth of development in theories and applications of decision making under risk, most of it occurring in the lifetime of Louis Eeckhoudt (a correlation that reflects partial causation). To celebrate his many contributions and continue the development of theories of decision making under risk, the Toulouse School of Economics hosted Risk and choice: A conference in honor of Louis Eeckhoudt on July 1213, 2012. This special issue of the Journal of Risk and Uncertainty includes selected papers from that conference. Expected utility theory (EUT) was proposed and axiomatized by von Neumann and Morgenstern (1944, the year of Louis birth). A critical limitation of EUT is that it assumes the probabilities of the alternative consequences are known. While this is acceptable for classical games of chance such as tossing dice and spinning roulette wheels, it is inappropriate for betting on horse races and other sporting contests, and more significantly for financial, insurance, business, health, and public policy decisions. It is often written that Frank Knight (1921) first drew attention to this distinction, calling the first case (where probabilities are objective) one of risk and the second one of uncertainty. For cases of Knightian uncertainty, EUT was generalized to subjective expected utility by Savage (1954). More recently, EUT has been further generalized to nonlinearity in probabilities (Quiggin 1982; Tversky and Kahneman 1992) and ambiguity (Gilboa and Schmeidler 1989; Klibanoff et al. 2005). Bernstein (1996) in his remarkable story of risk provides an entertaining summary of these three centuries of research on risk. We focus on the last 50 years, developed mostly under EUT. A key and simple property under EUT is that risk aversion is equivalent to the concavity of the utility function. Another important property is the equivalence between the statistical concepts of first-order and second-order stochastic dominance and, respectively, the positivity of the first and negativity of the second derivative of the utility function. Many other properties associated with the derivatives of the utility function have been derived under EUT, and we present some of them below. EUT is not unanimously accepted as a normative model of how people ought to choose and there is much evidence of violation, especially in laboratory experiments. Nevertheless, it remains the most widely used theory under risk and uncertainty. This is especially true outside the field of pure decision theory, for instance in public economics, macroeconomics, game theory and information economics. It is a good place here to recall that some of the most prominent economists have contributed to the research on risk. To illustrate, let us cite some theoretical works produced by recipients of the Nobel Prize in Economic Science. The list includes: Milton Friedman and Harry Markowitz on the convex-concave utility function explaining both insurance and gambling choices (Friedman and Savage 1948; Markowitz 1952), James Tobin on the preference for liquidity under risk aversion (Tobin 1958), Paul Samuelson on the benefit of diversification under risk aversion (Samuelson 1963, 1967), Kenneth Arrow on the measure of risk aversion and analysis of risk taking (Arrow 1965), Gary Becker on the substitutability of self-insurance and self-protection with market insurance (Ehrlich and Becker 1972), Joseph Stiglitz (Rothschild and Stiglitz 1970) and more recently Robert Aumann (Aumann and Serrano 2008) on measures of risk, Maurice Allais (Allais 1953) and Daniel Kahneman (Kahneman and Tversky 1979) for their criticism of the EUT paradigm, and the list could go on. A starting point to summarize the development and influence of research under EUT in the last 50 years is the seminal paper by Pratt (1964), Risk aversion in the small and in the large, published in Econometrica. This paper introduces an index of risk aversion in the small, namely using a second order approximation of the risk premium, and then obtains a result about comparative risk aversion across agents in the large, namely for all risks. Pratts paper has more than 5,000 citations on GoogleScholar at the time of this writing. But it has in reality many more citations, as the paper became so famous that it is oft (...truncated)