What Expected Utility Means
When a decision maker is faced with several acts, he will choose the act with the highest expected value’ (Machina, 2003). Expected utility is ‘the sum of the utility that would be achieved in each outcome multiplied by the probability that the outcome will actually occur’ Eaton, Eaton and Allen (1995). Economists attribute the introduction of the concept expected utility to Daniel Bernoulli.
John von Neumann and Oskar Morgenstern made a land mark use of the expected utility theory. They introduced the consideration of probabilities and lotteries in utility. They believed that these would make additive comparison of expected utility unambiguous. According to Machina (1987), rather than using expected value x̄ =∑(xi)pi gambles are evaluated on the basis of expected utility ū=∑U(xi)pi. The primary motivation for introducing expected utility instead of the expected value of outcomes is to explain the attitudes towards risks (Thomas, 2008). The expected utility theorem of von Neumann and Morgenstern has the implication that if a decision maker can choose between the lotteries, he/she has a utility function which we can add and multiply by real numbers. The interpretation of this is that the ‘utility of an arbitrary lottery can be calculated as linear combination of utility of its parts’ (Wikipedia, 2009). An implication of expected utility is that as property of linearity in the payoffs is dropped, linearity in probability is retained by the preference function.
Axioms of Expected Utility Theory
Four axioms were offered by the von Neumann and Morgenstern (vNM) Expected Utility Theory (EUT). These axioms give the assurance that preferences over gambles and lotteries can be represented through simple utility functional form. The axioms are better explained with the aid of the Marschack-Machina Triangle (M-M Triangle). The implication of linearity in probability is that indifference curves in the M-M Triangle are parallel to the probability simplex.
1. Completeness axiom: For two lotteries X and Y, either X ≻Y or Y ≻X. This guarantees that preferences are defined over the whole set of possible lotteries ( Gaudeul 2009).
2. Transitivity axiom: For any X, Y, Z, if X≻ Y, and Y≻ Z, then, X≻ Z. This assures that in preferences, there are no cycles.
3. Continuity/convexity/Archimedean axiom: For any Z≻ Y ≻X, there is a unique, α, 0 ≤ α ≤ 1 such that αZ + (1 – α)X ∼ Y. With the uniqueness of α, the axiom is saying that this guarantees that indifference curves are continuous.
4. Substitutability or Independence axiom: For any X, Y, Z such that X ≻ Y, then for any α ∈ (0,1), (1- α)X + αZ ≻ (1- α)Y + αZ. This axiom guarantees that in the M-M triangle, indifference curves are parallel straight lines.
The axioms are reasonable, but ‘not necessarily prescriptive or necessarily backed up or drawn from experimental evidence.’ Gaudeul (2009). Truly one can see that the actions of decision makes do not necessarily fit into the Expected Utility Theory and this implies that some of the axioms are not ‘respected’. ‘Experimental studies have shown that the key behavioral assumption of expected utility theory, the so-called "independence axiom," tends to be systematically violated in practice.’ (Machina 1982).
Rabin (2000) posits that ‘expected utility theory gives absurd results under the calibrations they perform.’ He is of the view that ‘a low level of risk aversion with respect to small gambles leads to a high, and absurd, level of risk aversion with respect to large gambles’ (Safra and Segal (2006)). These positions are true.
Experimental Evidences
Currently, there are experimental evidences from where one can deduce that individual preferences are not linear in probabilities; instead, they reflect systematic departure from this property. The Allais paradox (1953) is a strong evidence of this violation of linearity in probabilities and through the common ratio effect and the common consequence effect, the paradox has systematically contradicted the independence axiom of expected utility. The argument of Allais is that for choice under risk to be properly analysed, a separate specification of attitudes towards uncertainty and a cardinal specification of utility as a function of wealth under certainty are required. The deviation from linearity in probabilities led to the generalisation of the expected utility model. Researchers have developed non-linear functional forms which are appropriate for individual preference function. Many of the non functional forms are ‘flexible enough to exhibit the properties of stochastic dominance preference, risk aversion/risk preference and fanning out’. (Machina 1987)
Transitivity axiom of expected utility is also violated by the ‘preference reversal phenomenon’ (the P-bet, $-bet) in which among a number of pairs of bets, the decision maker will choose a bet and he would be requested to value each of the bets by stating the certainty equivalents. Psychologists Sarah Linchtenstein and Paul Slovic (1971) capture this as:
*$-bet: outcomes (Y, y) with probabilities (q, (1-q)), versus
*P-bet: outcomes (X, x) with probabilities (p, (1-p)).
Holt (1986) and Kerni and Safra (1987) are of the views that this phenomenon will truly reveal preference but the concern is how and whether experiments will address the issue. From psychologists positions, choice and valuation are processes that are distinct and possibly conditional upon different influences.
Framing effect has been discovered to be disturbing phenomenon. It is believed that when equivalent choice problems are probabilistically ‘framed’, the will result in systematic difference in choice. Duplex gamble and reference point are the two framing effects problems that involve identical distribution over final wealth of the decision maker. Loomes, Starmer and Sugden (1991) identify the violation of the transitivity axiom because agents, using techniques that are less sophisticated than expected utility theory in evaluating lotteries will go for lotteries with the higher probability if the payoffs are close. As a reflection of subjective probability manipulations, it is evident that when individuals are asked to estimate or revise probabilities for themselves, they will make mistakes or they may not even do it at all. Machina (1987)
Conclusions
Apart from the developments of non-linear expected utility function, the contradictions that are glaring in the expected utility theory have caused the evolution of many other alternatives. Though Rabin (2000) recommends that ‘it is time for economists to recognise that expected utility is an ex-hypothesis, a strong position that that the expected utility is still relevant in addressing important economic and social issues.
Bibliography
Machina, M. (1982)’Expected Utility’ Analysis without Independent Axiom. Econometrica (50), March 1982 pp 277-323.
Allais, M. And Hagen, O., (1979) Expected Utility Hypothesis and the Allais Paradox. Dordrecht: D. Reidel. 1979.
Lichtenstein, S. And Slovic, P. (1971) Reversal of Preferences between Bids and Choices in Gambling Decisions. Journal of Experimental Psychology. 89 (1), July, pp 46-55.
Gaudel, A. (2009) A (Micro) Course inMicroeconomic Theory for MSc Students. Munich Personal RePec Archive, Paper 15388 May 2009.
Machina, M. (1987) ‘Choice Under Uncertainty: Problems Solved and Unsolved’. The Journal of Economic Perspectives, 1 (1) pp 121-154
Rabin, M. (2000), ‘Risk Aversion and Expected-Utility Theory’. Econometrica 68 (5) pp 1281 – 1292.
Loomes, G., Starmer, C., and Sugden, R. (1991) ‘Observing Violation of Transitivity by Experimental Methods’, Econometrica 59 (2), 425-439.
Eaton, B., Eaton, D., Allen, D. (1995), Microeconomics. Ontario: Prentice Hall, Canada.
Holt, C., ‘Preference Reversals and the Independence Axiom’, America Economic Review, June 1986, 76, pp 508-515.
Machina, R. (1998) ‘Expected Utility Hypothesis’, The New Palgrave: A Dictionary of Economics. Palgrave Publishers, pp 232-238
Wikipedia (2009), ‘Expected Utility’, Wikipedia Encyclopaedia.
Thomas, G. (2008), ‘Expected Utility Theory’, International Encyclopaedia of the Social Sciences
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