European Journal of Economics, Finance and Administrative Sciences ISSN 1450-2887 Issue 15 (2009) © EuroJournals, Inc. 2009 http://www.eurojournals.com/EJEFAS.htm
A Meta Analysis of the Empirical Evidence on Expected Utility Theory
Muhammad Zafar Yaqub Center for Business Studies, University of Vienna, Brünner Strasse 72 A-1210, Vienna, Austria Tel: +43 1 4277 37945; Fax: +43 1 4277 38174 E-mail: mzyaqub7@yahoo.co.uk Gökhan Saz Center for Business Studies, University of Vienna, Brünner Strasse 72 A-1210, Vienna, Austria E-mail: g_saz@hotmail.com Dildar Hussain Center for Business Studies, University of Vienna, Brünner Strasse 72 A-1210, Vienna, Austria E-mail: hussaindildar@yahoo.com Abstract The authors have conducted an exhaustive literature survey of the experimental studies on Expected-utility Theory (EUT) in order to account for the violations of its axiomatic foundations. Beyond identifying and classifying the (behavioral) bias phenomena according to the axiom(s)-violated, the possible background conditionals for such violations are reported in this paper.
Keywords: EU Theory, EU axioms, Behavioral biases, Preference reversal
1. Introduction
Partly owing to the testability feature, the scientific theories have to constantly prove their truthfulness in the face of critical testing from their rivals (Sekaran, 1984). As a natural consequence of this process, the theories under examination either get corroborated (in case of finding the supporting evidence), or these have to undergo some revision(s) in case credible contradictory evidence emerges. Being no exception, Expected utility theory (EUT), too, had been subjected to such critical examinations ever since it had been propounded (by Daniel Bernoulli and Gabriel Cramer) in the 18th century. A number of experimental studies have been conducted to testify the truthfulness of its axiomatic foundation. Even though results from some of these experiments supported the baseline axioms, a fairly large number have reported instances where EU axioms have been violated leading to the discovery of a variety of (behavioral) bias phenomenon. This paper addresses a threefold agenda: 1) to take an account of the EU axioms’ violations, 2) to identify the behavioral biases, and 3) to explain the background conditionals of these violations and/or biases, as revealed by the previous studies. We start with presenting a brief note on the evolution of EU Theory in section 1. Section 2
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discusses the methodology used to find relevant literature as well as the relevant facts of the matter from this literature. Section 3, presents our findings where we discuss the EUT axioms’ violations along with the nature, origin and background conditionals of the behavioral biases. The final section concludes the whole discussion. A couple of tables have been presented to show a consolidated picture of the phenomenon of interest.
2. Expected Utility Theory
In the 17th century, the efforts to describe rational behavior in mathematical terms ran into problems with the discovery of St. Petersburg paradox. Daniel Bernoulli and Gabriel Cramer were the first to solve and/or explain the St. Petersburg Paradox in 18th century. The solution to this paradox revealed that the value of a gamble is not, in general, equal to its expected value. Individuals place subjective values (utilities) on monetary outcomes and the value of a gamble is the expectations of these utilities (Bernoulli, 1738). Since Bernoulli’s statement (or theory) was based on a cardinal (interval) utility scale, it failed to attract much attention of economists (until the 1950’s) who were more keen in developing ordinal theories. John vonNeumann (a German-Hungarian mathematician) and Oskar Morgenstern (an Austrian economist) were the first scientists who formally proved that EUT was a viable theory for rational decision making, and that it can be derived from certain axioms. They propounded that any "normal" preference relation over a finite set of states can be written as an expected utility. That is why it is also called vonNeumann-Morgenstern utility. The theory became important as; 1) it was developed shortly after the Hicks-Allen “ordinal revolution” of the 1930's; 2) it revived the idea of cardinal utility in economic theory Although the economic and simple tractability of EUT cannot be criticized, a crucial question is whether it provides a sufficiently accurate representation of actual choice behavior. Even though most decision theorists would regard the establishment of the expected utility (EU) paradigm as being the most important development in modeling behavior under risk and uncertainty, yet it was not long before other researchers were drawing attention to a number of apparently systematic patterns of behavior which seemed to be dissonant with one or more of the basic EU axioms. Allais (1953), Ellsberg (1961), Lichtenstein and Slovic (1971) and Kahneman and Tversky (1979) are some notable examples. The debate still continues even after so many decades have passed. In rest of this paper, we will present results from various experimental studies where the authenticity of the EU axioms have been subjected to the critical testing. Before, we elaborate on these experiments, we discuss the methodology of this inquiry in the following section.
3. Methodology
In order to account for the EUT axioms’ violations as reported in the empirical literature, a sample of 69 articles has been selected using an adapted version of the approach used by David and Han (2004) in their assessment of the TCE empirical literature. David and Han identified a representative sample of studies that empirically tested the core tenets of Williamson’s TCE by using the following procedure; 1. Search for published journal articles only. 2. Search the ABI/Inform and EconLit databases. 3. Ensure substantive relevance by requiring that selected articles contain at least one primary keyword in their title or abstract. 4. Eliminate substantively irrelevant articles by requiring that selected articles also contain at least one of 12 additional keywords in their title or abstract.
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5. Ensure empirical content by requiring that selected articles also contain at least one of the following seven ‘methodological’ keywords in their title or abstract: data, empirical, test, statistical, finding*, result* or evidence. 6. Eliminate substantively irrelevant articles by only selecting articles that appear in journals in which multiple articles appear. 7. Ensure substantive and empirical relevance by reading all remaining abstracts for substantive context (i.e., discussion of the core tenets of the theory) and empirical content (i.e., mention of statistical analysis). 8. Further ensure substantive and empirical relevance by reading all remaining articles in their entirety for substantive context (i.e., article tests the core tenets of the theory) and adequate empirical content (i.e., article presents results of statistical tests). 9. Consolidate results from ABI/Inform and Econ-Lit and eliminate duplicate articles. While obtaining the present sample, the above-mentioned procedure has been used with the following adaptations; First, this study restricts the search by including articles published in ‘scholarly’/peer-reviewed journals only (criterion 1). Referring to the work by Light and Pillemer (1984), David and Han (2004) argued that by restricting their search to journal articles (as opposed to book chapters or unpublished works, for example), they enhanced quality control. Given this logic, it is felt that by further restricting the search to the articles published in ‘scholarly’ journals, the quality of the articles returned would be further ensured for because of the rigorous peer- review process to which these articles are subjected prior to their publication. Second, in place of EcoLit, ScienceDirect has been used for its wider adoption in the contemporary research works. Third, the substantive keywords used to identify articles via criterion 3 have necessarily been adapted to make the search relevant to the EU Theory. Whereas David and Han (2004) initially selected articles that contained either transaction* or cost* in their abstract or title, the present study initially selects articles that contain either "Decision making", "Decision theory", EU, "Expected Utility" or "Utility theory" in their abstract or title. Fourth, David and Han (2004) identified 12 additional keywords that articles must have in their title or abstract in order to be considered relevant to TCE (criterion 4). In present study, the following additional keywords have been used to further identify articles that are substantively relevant to the EU Theory in step 4: "Allais Paradox", Axiom*, Bias*, "Preference reversal", Violat*. Fifth, the methodological keyword test, has been used together with 2 more key words i.e. Empirical*, and Experiment*. This was done in order to account for the empirical evidence stemming from the experimental inquires only. Sixth, in determining substantive relevance via criteria (7) and (8), articles were retained only if they revealed some violation of certain axioms of EU theory in their abstracts. The search process mentioned in above was still augmented with snow ball method which resulted in finding (even though a small number) useful articles not listed in any of the two databases. Full texts of the articles selected after the aforementioned filtration process were studied in order to find the relevant facts of matter which are being reported in the next section.
4. Results and Discussion
4.1. Sample Profile Following the process outlined in the previous section, we selected 69 articles. As Table 1 shows majority of these articles pertain to what Muller terms as the “Behavioural Economics Era”. (www.muellerscience.com). It may also be interesting to note that majority of the authors opted for Journal of Risk and Uncertainty and Management Science for publishing their results.
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Table 1:
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Frequency Distribution of the Research Papers
Decade-wise Era 1950s 1960s 1970s 1980s 1990s 2000s w.r.t. Muller’s Classification Era Description 1954-1969 The rise of rational decision theory & operations Research 1970-1989 Studies of Biases & Heuristics 1990-date Behavioural Economics w.r.t. The Scholarly Journals Rank Journals 1 Journal of Risk and Uncertainty 1 Management Science 2 Organizational Behaviour and Human Decision Processes 2 American Economic Review 2 Economic Journal 3 Journal of Experimental Psychology 4 Econometrica 4 Journal of Economic Behaviour & Organization 4 Journal of Mathematical Psychology 4 Journal of Risk Insurance 4 Economic Letters 4 Psychology Review 5 Others F 01 01 10 17 28 12 F 02 27 40 F 7 7 5 5 5 4 3 3 3 3 3 3 18 % 01.44 01.44 14.49 24.63 40.58 17.39 % 02.89 39.13 57.97 % 10.14 10.14 07.24 07.24 07.24 05.79 04.34 04.34 04.34 04.34 04.34 04.34 26.08
4.2. Axioms’ Violations and the Bias Phenomena We accounted for the violations of five axioms of EUT: Independence, Betweenness, Transitivity, Monotonicity, and Reduction. Table 2 presents a summary of our findings concerning such violations as highlighted by the empirical evidence reported in scholarly journals.
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Table 2:
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EUT Axioms’ Violations & the Bias Phenomenon
Key Contributors Allais (1953), Bleichrodt et al., (2001),, Bleichrodt and Pinto (2000), Burke et. al. (1996), Carlin (1990), Cohen and Jaffray (1988), Conlisk (1989), Cubitt et. al., (1998), Fan (2002), Finkelshtain and Feinerman (1997), Grether and Plott (1979), Harrison (1994), Hershey and Schoemaker (1980, 1985), Kahneman and Tversky (1974, 1979, 1981, 1986, 1992), Lichtenstein and Slovic (1971), Lindman (1971), Laughhunn et al. (1981), Macdonald and Wall (1989), McCord and de Neufville (1986), Oliver (2002, 2003), Payne et al. (1980, 81), Reilly (1982), Schoemaker, (2000), Schoemaker and Kunreuther (1979), Slovic et al. (1977), Thaler et al (1997), Tversky et. al. (1990), Wu and Gonzalez (1996). Battalio et al. (1990), Bernasconi (1994), Blavatskyy (2005, 2007), Camerer (1989), Camerer and Ho (1994), Chew and Waller (1986), Conlisk (1987), Coombs and Huang (1976), Evans et. al., (1991), Gigliotti and Sopher (1993, 2004), MacDonald et. al. (1991), Prelec (1990) Birnbaum et. al. (1999), Birnbaun and Gutierrez (2007), Birnbaum and Schmidt (2008), Bradbury and Nelson (1974), Humphrey, (2001), Loomes and Taylor (1992), Loomes et. al. (1989, 1991), Mellers and Biagini (1994), Shafir (1994), Starmer and Sugden, (1998), Tversky, (1969) Birnbaum, (1992), Birnbaum and Sutton (1992), Birnbaum and Thompson (1996), Birnbaum et. al. (1992), Mellers et. al. (1992, 1995), Weber et. al. (1992) Bar-Hillel (1973), Bernasconi (1994), Birnbaum (2004, 2007), Carlin (1992), Friedman (2005),Kahneman and Tversky (1979), Segal (1988), Behavioural Biases Common-consequence effect Common-ratios effect Reference-point effect Response-mode bias Probability-weighting bias Loss-aversion Framing Effect Anchoring and adjustment Context effects Preference reversal Fanning-out/in effect Context effect Boundary effect Ordering effect Framing effect Event-splitting bias Regret aversion Ordering effect Dimension interaction Framing effect Probability-weighting bias Ordering effect Splitting/coalescing effect. Anchoring and Adjustment Certainty effect
Axiom Violated Independence (Including consistency)
Betweenness (A weaker form of Independence)
Transitivity
Monotonicity
Reduction
Background Conditionals for Axiomatic Violations
Independence Size (low vs. high) and the nature (hypothetical vs. real) of monetary outcomes, limited ability to process information, probability transformations/distortions, structure of experiment, aspiration levels, (high/low) anchors, probability thresholds, limited sensitivity to low probability events, frequency of outcome evaluations by the decision makers, procedure and description invariance, nonlinear evaluation of probabilities, greater sensitivity to losses than to gains, over /underweighting certain dimensions or specific information, use of heuristics for simplification Nonlinearities in the indifference curves, random errors, lottery’s location in probability triangle, folded ordering, quasi-concavity and quasi-convexity of preferences Contrast effect of comparison within dimension, Novelty, strength and direction of preference,, use of a comparative rather than absolute approach in evaluation, similarity among attributes Distribution of cash value offered for comparison in gambles, probabilities of loss or zero outcome, decision makers’ view points, judgements about the attractiveness of gains/losses Overestimation of the expected value of compound lotteries in comparison to their reduced forms, coalescing/splitting of branches, higher probability of winning in earlier stages, difference in CEs for the compound and the reduced lotteries
Betweenness Transitivity Monotonicity Reduction
4.2.1. Independence Axiom The independence axiom holds that the preference function must be represented as an expectation with respect to the given distribution of a fixed utility function defined over the set of possible outcomes (i.e. ultimate wealth levels). In other words, the preference function is constrained to be a linear function over the set of distribution functions, or, as commonly phrased, "linear in the probabilities” (Machina, 1982). Therefore a decision maker’s preference over two lotteries should be independent of any combination with an inclusion of a third lottery. According to Starmer (2000), the independence axiom places strong restrictions on the form of preferences and gives the standard EU theory most of
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its empirical content. Therefore we can say without losing generality that the independence axiom is the most crucial in the set of all EUT axioms and as it will turn out in the paragraphs below that most of the systematic deviations from the predicted outcomes stem from the violations of this axiom. 4.2.1.1. Allais Paradox/Common Consequence effect Common-consequences effect is the generalization of Allais Paradox which reveals that EUT violations by the decision maker result from decision reversals. In the experiments manifesting this phenomenon, the decision maker chooses between a certain outcomes for sure and a lottery in first stage of the decision problem. Then, he is asked to make a decision between two risky outcomes. The outcomes clearly violate EUT and show a systematic bias. It was first empirically discovered by Allais (1953) and later generalized by Tversky and Kahneman (1979, 1981) in their historical works in behavioral economics. After the initial findings of Allais, many researchers followed up with studies in which new experimental designs were employed to; 1) replicate these findings; 2) reveal relevant properties of the EUT axioms’ violation. Burke et al. (1996) and Harrison (1994) tested if violations of EUT in an Allais paradox environment are sensitive to monetary incentives. They found that the violations were significantly reduced when the lotteries were real rather than hypothetical. Fan (2002) found that imposing hypothetical or real cash rewards induce a change in the structure of Allais Paradox (resulting in the near disappearance of the paradox) when rewards of the lotteries were small compared to the original setting of the Allais experimental design. In a different design involving non-monetary outcomes, Oliver (2002b) found that majority of the experimental subject(s) demonstrated strict and systematic violations of independence axiom, with many in the direction predicted by Allais. However, after an experimental inquiry in the loss domain, Macdonald and Wall (1989) attributed the inconsistencies in their experimental data to the violations of independence axiom and found that such violations were insensitive to the level of monetary incentives involved. Despite the vast confirmatory literature, there is still some evidence which falsifies the truthfulness of independence axiom’s violation in Allais Paradox. An intriguing experimental design had been dissipated by Carlin (1990) where he reframed lotteries without a reference to decimals or fractions. Doing that resulted in a (statistically) significant reduction in violations of the independence axiom. His study had primarily intended to test robustness of the Allais paradox by changing the frame of lotteries in a seemingly trivial fashion. Reframing also proved to be prolific in reducing such violations in an experimental study by Conlisk (1989), who examined three variants of the Allais example. Some evidence in the literature also suggests that the apparent EUT violations in experiments might be due to the factors other than these axiomatic violations. One such demonstration came from Finkelshtain and Feinerman (1997) who concluded that violations of EUT under the Allais Paradox were mainly due to certain choice errors (e.g. misinterpretations of the questions, bad instructions of the questioners etc.) rather than systematic violations of the EUT axioms. Similar findings have also been documented by Blavatskyy (2005). 4.2.1.2. Common ratio Effect The common ratio effect is very akin to the common consequence and/or the Allais paradox. Decision makers tend to switch their choices from one prospect to the other when the probability is lowered proportionally across the lotteries, which is inconsistent with EUT. In their study Cubitt et al., (1998) focused on the common ratio effect in the context of dynamic choice experiments using real financial incentives. Their findings, however, are inconsistent with several existing theories which purport to explain the common ratio effect and instead propound a new effect/principle called timing independence.
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4.2.1.3. Preference Reversal The preference reversal effect corresponds to the empirical findings that people tend to change their actual decision when confronted with different contexts. For example, in winning a small prize with high probability vs. a high prize with small probability, people tend to choose the small prize with high probability. But when asked for buying the lottery for a certain monetary amount they reverse their decision and choose the high prize with small probability. The first studies on the preference reversal effect were published by Lichtenstein and Slovic (1971) and Lindman (1971). Lichtenstein and Slovic (1971) conducted three experiments in which undergraduate males chose their preferred bet from pairs of bets and later bid for each bet separately. In each pair, one bet had a higher probability of winning (p bet); the other offered more to win ($ bet). Bidding method (selling vs. buying) and payoff method (real-play vs. hourly wage) were varied. Results showed that when the p-bet was chosen, the $- bet often received a higher bid leading to the preference reversal phenomenon. A similar study was conducted by Lindman (1971) who performed five decision-making experiments and found that subjects, in choosing among gambles, tended to prefer gambles with high probabilities of winning. The same subjects, while naming selling prices, preferred gambles with small probabilities of winning larger amounts. In two follow-up studies by Payne et al. (1980), three different experiments were conducted by varying the relationship of pairs of gambles to an assumed reference point by adding or subtracting a constant amount from all outcomes. Such translations of outcomes resulted in the reversal of choice within pairs of gambles. Later on, Payne et al. (1981) extended their previous work by analyzing risky choice behavior in two different experiments. The results showed convincing evidence for the validity of their earlier explanation. The culmination of several competing explanations of the preference reversal effect led Grether and Plott (1979) to an exposition on various experiments on the preference reversal effect up to 1979 and to propose additional (alternative) explanatory frameworks. The main aim of their study, however, has been to incorporate and/or control for the theoretical causations in their experimental design and see if the preference reversal phenomenon existed in situations where economic theory is generally applied, which was eventually shown to be affirmative. Owing to the sensitivity of preference reversal effect to the experimental setting, Reilly (1982) incorporated some alterations in Grether and Plott's (1979) previous experiments and found the level of preference reversals to be a function of the particular experiment's structuring. 4.2.1.4. Procedure and Description Invariance After the initial unsuccessful quest for a satisfactory explanation of the preference reversal effect, the scientific community turned to a different approach mainly driven by results supporting the sensitivity of this phenomenon to the experimental designs/conditions. They found the preference reversals to be a consequence of the procedure invariance (elicitation effects) and/or the description invariance (framing effects) instead of any violation of the independence axiom within the EUT framework. In their experimental study employing different utility elicitation methods, Hershey and Schoemaker (1985) found that the probability equivalent methods and certainty equivalent method (although theoretically assumed to be equal) yield inconsistent results. They concluded that the main reason of this divergence lies within the different decision framing of the subjects. Instead of a certainty equivalent method, McCord and de Neufville (1986) used a lottery equivalent method to elicit the preferences in their experiments by eliminating certain amounts in the lottery. Their simplified method, however, reduced context dependencies as well dependency of the utility function on the probability involved. A similar finding has been reported by Cohen and Jaffray (1988) who developed a multiplicative form of the utility function and tested if the certainty effect or the probability distortions cause violations of EUT. They concluded the former to be most relevant in explaining the inconsistencies whereas discarded the later on a (poor) evidential-base. The emergence of evidence for the procedure and description invariance finds its climax in the classical paper of Tversky et al. (1990) who reveal that preference reversals cannot be adequately
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explained by violations of the independence, reduction, or transitivity axiom and propound that the main cause of preference reversal is the procedure invariance. In their concluding remarks they describe the complex and rigid predicament of contemporary decision theorists by stating: “the failures of description invariance (framing effects) and procedure invariance (elicitation effects) pose a greater problem for rational choice models than the failure of specific axioms, such as independence or transitivity, and they demand descriptive models of much greater complexity; …it does not seem possible to construct a theory of choice that is both normatively acceptable and descriptively adequate.” (p.215). 4.2.1.5. Response Mode Bias The concept of response-mode bias is very akin to the procedure invariance. As a variation of independence, consistency is defined as the property that the preference order should be independent of the method used to elicit it. Preference reversals between different methods violate the consistency principle. Hershey and Schoemaker (1985) found in their experiments that PE (probability equivalent) elicitations lead to systematically different utilities than CE (Certainty equivalent) elicitations. In the health domain, several authors have shown that different procedures to elicit utilities lead to systematically different results (Bleichrodt et al., 2001; Oliver, 2003a). Bleichrodt et al. (2001) controlled for these biases and proposed new corrected formulas based on prospect theory to evaluate answers to PE and CE measurements and showed that these new formulas were able to resolve the systematic discrepancy between PE and CE utilities. They argued that decision makers’ preferences are affected by probability weighting and loss aversion. These biases are quite explicitly modeled by Prospect theory (Kahneman and Tversky, 1979) and Cumulative Prospect theory (Tversky and Kahneman, 1992), some of the main contenders for a complete descriptive theory of decision under risk. 4.2.1.6. Probability Weighting Bias The probability weighting bias reveals that people tend to behave nonlinear in probabilities. Empirical studies have shown that the most common pattern for the probability weighting function is inverse Sshaped; overweighting small probabilities and underweighting large probabilities (Bleichrodt and Pinto, 2000; Gonzalez and Wu, 1999; Tversky and Fox, 1995; Tversky and Kahneman, 1992). Macdonald and Wall (1989) attributed consistency violations in their experimental findings to the nonlinearities in the probabilities, problem representation, context effects and the related expected utility violating effects. Kahneman and Tversky (1992) incorporated the previous findings on the inverse s-shaped probability weighting function along with the tenets of rank dependent utility into a new theory called cumulative prospect theory (CPT) which provides an even better description of choice under risk and uncertainty compared to the one given by prospect theory. 4.2.1.7. Loss aversion Loss aversion refers to the tendency in people to strongly prefer avoiding losses than acquiring gains. A number of experimental studies have questioned this pervasive assumption. In their study Slovic et al., (1977) observed that people refuse to protect themselves against losses whose probability is below some threshold. They postulate a utility function that is convex over losses which clearly contradicts the EUT postulations. In reference to the findings of Slovic et al. (1977), Schoemaker and Kunreuther (1979) conducted further experiments on insurance and demonstrated that the main reason for utility violations was people's limited abilities to process information. Hershey and Schoemaker (1980) in their experiments found that less than 40 percent of their subjects would pay $100 to protect against a 0.01 chance of losing $10,000 (a fair insurance). They found that when low probability and very high loss lotteries are examined, discrepancies were quite apparent in risk taking attitudes. Finally, empirical studies on managers’ and investors’ choice behavior show that business managers are risk-seeking when presented with gambles having outcomes below target return levels and that for investors loss
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aversion is the combination of a greater sensitivity to losses than to gains and a tendency to evaluate outcomes frequently (Laughhunn et al., 1980; Thaler et al., 1997). 4.2.1.8. Framing Effect Kahneman and Tversky (1981, 1986) were the pioneers in theorizing this concept according to which the decision makers systematically violate EU hypotheses due to the different psychological framing of the contexts. People choose different reference points when asked different questions. They tend to focus on the probability when directly asked to choose one of the lotteries. When asked for paying, they tend to focus on the monetary outcome. Thus if the same problem is presented in a different manner, the choice of people may not be the same. In their empirical study Schoemaker and Kunreuther (1979) discovered substantial influence of contextual effects on decision making. They demonstrated that the main reason for EUT violations is people's limited abilities to process information. A number of related experimental studies like Hershey and Schoemaker (1985) and Schoemaker and Russo (2001) attributed EU violations to the decision framing and quite convincingly revealed that choice and valuation tasks invoke different mental processes which in turn generate different orderings of the given pair(s) of prospects. 4.2.1.9. Reference point effect Decision makers base their evaluation of alternatives on certain reference point(s). They assess departures from this reference point as gains or losses. The gains and losses relative to a reference point have been referred as the “reference point effect” by Kahneman and Tversky (1979, 1981). Although EU hypothesis suggests that alternatives should be evaluated with respect to their effects on final wealth levels, it is (cognitively) easier for the decision makers to assess options in terms of gains and/or losses relative to some reference point (Schoemaker, 2000). They use certain heuristics as the simplifying method for optimal decision making. That is why Grether and Plott’s (1979) view of the decision maker is: “the one who is continually searching for systematic procedures that will produce quick and reasonably satisfactory decisions”. Finally, Payne et al. (1980) reveal that reference point effects (e.g. target return) are variations on the concept of the aspiration level. 4.2.1.10. Anchoring and Adjustment Lichtenstein and Slovic (1971) and Tversky and Kahneman (1974) were the first to highlight this concept according to which individuals normally over weight certain dimensions or specific information during the decision making process. They normally start with an “anchor” (a reference point) and then make adjustments to reach the final decision. Bleichrodt (2001) has revealed ‘anchoring and insufficient adjustment’ to be a convincing explanation for the observed violations of EU hypothesis. 4.2.3. Betweenness Axiom The betweenness (a weaker form of independence) axiom holds that a lottery offering outcomes A and B have an attractiveness level intermediate to those of A and B. The pioneering empirical work on this axiom was published by Coombs and Huang (1976) who reported two experiments on portfolio theory where significant violations of the betweenness condition were observed. Similar results were reported by Chew and Waller (1986) and Battalio et al. (1990) who found that around 30% of their experimental subjects systematically violated the betweenness axiom. In a subsequent study, Camerer (1989) reported empirical evidence on the fanning out and fanning in of indifference curves and the quasiconcavity and quasiconvexity of preferences, indicating that the subjects in their experiments systematically violated the betweenness axiom in a non-random fashion. Gigliotti and Sopher (1993) attributed the violations of betweenness axiom to the nonlinearities in the indifference curves. Interestingly, empirical experiments on the betweennes axiom are not restricted to the human choice domain only. MacDonald et. al. (1991) conducted an experiment with rats and found systematic
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violations of the betweenness axiom ultimately implying that indifference curves are convex rather than linear in the probabilities. In a recapitulatory empirical paper Camerer and Ho (1994), however, assert that the violations of betweenness axiom are not as frequent as the violations of independence. In their summary on different empirical studies on the violations of betweenness they depict that Coombs and Huang (1976), Chew and Waller (1986), Conlisk (1987), Camerer (1989), Prelec (1990), Gigliotti and Sopher (1993), Battalio et al. (1990), and Bernasconi (1994) all found significant violations of this axiom in their experimental ventures. Their concluding remarks on the causes of such violations point the attention to non-linearity in probabilities and framing effects, ultimately showing a limited descriptive value of EU hypothesis under risky choices. Similar results on the context dependency of such violations have been reported by Evans et al., (1991) in their financial-choice experiments. However, after his reexamination of several experimental studies Blavatskyy (2006, 2007) postulates (by introducing a stochastic version of the EUT) that violations of the betweenness can be the result of random errors in choice under risk. 4.2.4. Transitivity Axiom Transitivity is defined as: If A is preferred to B, and B is preferred to C, then A must be preferred to C. Transitivity therefore holds that all “indirect" links which can be established via some third element, are also explicitly contained in that relation. A number of experimental studies (e.g. Birnbaum et. al., 1999; Bradbury and Nelson (1974; Humphrey, 2001; Loomes et. al., 1989, 1991; Loomes and Taylor, 1992; Starmer and Sugden, 1998; Tversky, 1969) have reported violations of the transitivity axiom. Referring to a similar study by Smedslund in 60s, Bradbury and Nelson (1974) (in a noneconomic context) found that children and adults frequently showed non-random intransitive responses and thus violated transitivity systematically. They ascribed the inconsistencies in child preferences to the psychological concept of novelty and provided further evidence that the violations declined with age but remained systematic. Looms et al., (1989) observed the preference reversal phenomenon in an experimental design in which individuals made straight choices rather than being required to report reservation prices. In each case, majority of the transitivity violations were in the direction corresponding with the classic preference reversal phenomenon indicating that the explanation of the preference reversal by violations of transitivity looms larger than by information processing effects alone. After their initial findings Looms et al., (1991) extended their experimental work through a design that was insensitive to the violations created by different value elicitation methods, incentive systems and diverging outcomes between valuation and choice tasks. These factors were all believed to create the preference reversal effect, as revealed in the previous literature. However, the results showed that none of these factors including random errors were valid explanations of the preference reversals in their experiments with the exception of regret theory thus concluding that the descriptive validity of the transitivity axiom is questionable. Later on, Starmer and Sugden (1998) designed four experiments to replicate the findings from Loomes et al. (1989, 1991). The results showed that the main reasons for transitivity violations lie in the realm of decision making heuristics and framing effects. In a subsequent empirical study, Humphrey (2001) developed a new experimental design by controlling and testing for the relative contributions of event-splitting effects and framing effects. He successfully replicated previous findings on systematic transitivity violations consistent with regret theory, but found that (contrary to the existing literature) event splitting effects had a limited explanatory power for transitivity violations. Similar to the findings of Starmer and Sugden (1998), he propounded decision making heuristics and framing effects as the main reasons for inconsistent behavior. Most of the experimental designs testing transitivity up to the beginning of 1990s were mainly concerned with decisions that only involved gains. One of the first studies that analyzed the loss domain is that of Loomes and Taylor (1992) who after conducting a series of experiments revealed that transitivity is not a robust and indispensable principle of choice in the domain of losses. They observed significant asymmetries in the patterns of nontransitive cycles all of which were in the direction
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predicted by regret theory. They concluded that regret aversion is capable of predicting a broad range of systematic violations of transitivity. One of the problems in studying transitivity violations is about deciding whether these are “true/systematic” violations or just “random errors/unsystematic” violations. Scholars have distinguished between systematic violations of transitivity from momentary fluctuations of judgment and have extended the deterministic structure of the common transitivity axiom into the probabilistic domain by forming the concept of weak stochastic transitivity. Weak stochastic transitivity is defined as: if P(A,B) ≥ 1/2 and P(B,C) ≥ 1/2 then P(A,C) ≥ 1/2, where P(A,B) is the probability of choosing A over B (see Birnbaum et al., 1999). One of the earlier proponents of this concept Tversky (1969) reported systematic and predictable violations of weak stochastic transitivity in two separate experiments, one dealing with gambles and the other with college applicant decisions. In their empirical work on evidence against rank dependent utility theories Birnbaum et al. (1999) tested if any connection between stochastic dominance and transitivity exists and found that violations of stochastic dominance can produce but are not explained by violations of transitivity. Strong stochastic transitivity is defined in accordance to weak stochastic transitivity (P(A,B) ≥ 1/2 and P(B,C) ≥ 1/2 then P(A,C) ≥ 1/2) as the consequence that P(A,C) should exceed the larger of P(A,B) and P(B,C). The research on the transitivity of preferences is not restricted to the human behavior alone. Shafir (1994) tested a proximate-level hypothesis about decision making in honey bees showing that three out of fifteen bees violated weak and strong stochastic transitivity in making decisions between different flowers. They propounded that the bees use a comparative rather than absolute approach in evaluation, which explains these intransitivities. Two recent studies on transitivity, however, show mitigating results. Birnbaum and Gutierrez (2007) conducted two new experiments in which hundreds of participants made choices among the same gambles as studied by Tversky (1969) and concluded that very few people repeated intransitive patterns. Similarly, Birnbaum and Schmidt (2008) found no evidence for systematic intransitivities in a recent study. They concluded that the absence of violations is consistent with theories that assume transitivity. 4.2.5. Monotonicity Axiom There are two types of monotonicity i.e. 1) outcomes, and 2) probabilities. The first holds: If two alternatives are otherwise identical, but one gamble has one outcome that is preferred to the corresponding outcome of the other gamble, then the gamble with the better outcome should be preferred. Similarly, if two gambles are identical, except one has a greater probability of a preferred outcome and lower probabilities of less preferred outcomes, the one with the higher probability of the better outcome should be preferred. The concept of stochastic dominance combines these two types of monotonicity. The dominance principle states that one should prefer the option with consequences that are at least as good as those of other options for any state of the world. Although outcome monotonicity seems a very reasonable axiom for the rational decision maker, a number of experiments have found situations in which human judgments appeared to violate this principle systematically (Birnbaum, 1992; Birnbaum et. al., 1992; Birnbaum and Sutton, 1992; Mellers et. al., 1992). In the experiments conducted by Birnbaum et al., (1992), subjects judged the values of lotteries from three points of view: the highest price a buyer should pay, the lowest price a seller should accept, and the “fair" price. Data showed violations of branch independence and monotonicity (dominance). The rank order of judgments changed as a function of decision makers’ point of view. Birnbaum (1992) found that violations of monotonicity persisted even when gambles are ordered by choice based certainty equivalents (based on comparisons between gambles and sure amounts of money). He also found that the inferred certainty equivalent (the value of cash that it is preferred half the time to the gamble) depends on the distribution of cash values offered for comparison against the gambles, which further ads to the difficulty of comparing certainty equivalents when the set of comparison values for the gambles being compared is different. Later, Birnbaum and Thompson (1996) found violations of monotonicity with choices between gambles and a fixed set of
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amounts of money that would be received for certain. In their experiments, a set of relations had been defined on choice proportions as: $A\succ _{c} B$ (Gamble A is preferred to B on relation c) if and only if the proportion of choices favouring amount c over Gamble A is less than the proportion choosing c over B. Results revealed systematic violations of monotonicity when c is greater than the minimum outcome of the superior gamble, but not when c is less than this value. Configural-weight theory explains the effect of viewpoint, the violations of branch independence, and the violations of monotonicity using a single scale of utility that is independent of the lottery and the point of view. Monotonicity violations occur because the worst outcome of zero has a much lower weight than a worst outcome that is a small positive amount (for the same low probability). Birnbaum and Sutton (1992) revealed systematic violations of dominance that can be explained by assuming that the configural weight of zero (when it is the lower value) has smaller weight at low probabilities than nonzero outcomes. Weber et al., (1992) observed that people judged both the attractiveness and risk of lotteries to win or lose money. They found that risk judgments were more sensitive to the probability of losses and zero outcomes compared to attractiveness judgments, which were more sensitive to the probability of gains. Mellers et al., (1995) refuted the view that configural weighting is caused by a shift in strategy that would apply only to two-outcome gambles. They observed that people violate the dominance principle even with three-outcome gambles with financial incentives. 4.2.6. Reduction Axiom The Reduction of Compound Lotteries axiom holds that a decision maker should be indifferent between two lotteries with the same probability of winning and the same prize for winning. However, empirical results have not always supported this (e.g. Bar-Hillel, 1973; Kahneman and Tversky, 1979; Friedman, 2005 etc.). The violations of this axiom are often found in studies where subjects are required to choose between lotteries (e.g. Kahneman and Tversky 1979, Bernasconi 1994). These studies have shown that decision makers tend to overestimate the expected value of compound lotteries compared to their reduced forms. The violation is more frequent and intense in compound lotteries with a higher probability of winning occurring in the earlier stages. In her experiment, Bar-Hillel (1973) found that people tend to overestimate the probability of conjunctive events and underestimate the probability of disjunctive events. These biases can be explained as the effects of anchoring: i.e., the stated probability of the elementary event provides a natural starting point from which insufficient adjustment is made to arrive at the final answer. Segal (1988) revealed that the preference reversal phenomenon can be traced back to violations of the reduction axiom. He suggested a decision mechanism for the preference reversals lotteries which is transitive and satisfies the independence axiom, but not the reduction axiom. Friedman (2005) during his experiment observed that subjects systematically preferred compound lotteries to their reduced forms. He found a (statistically) significant difference between the certainty equivalents (CEs) derived for compound lotteries, and those derived for their reduced forms, with the CEs for the compound form being higher. Carlin (1992) reveals that violations of expected utility theory associated with the Allais paradox and common ratio effect are sensitive to the reduction process. In his inquiry, Birnbaum (2004) also concluded coalescing (reduction) to be the main factor behind the EUT violations by stating: “the exception in this study is that a variable has been identified that not only undoes the Allais paradox, it significantly reverses it. That variable is the splitting or coalescing of branches, which appears to give the best explanation of common consequence paradoxes.” (p.105). Later on, Birnbaum (2007) finds further evidence that coalescing (reduction) is crucial for the EUT inconsistencies. He refutes cumulative prospect theory (CPT) as an appropriate explanatory framework for “Allais paradoxes”.
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5. Conclusion
In this paper, we have taken an account of the violations of EUT axioms reported in scholarly journals. We selected our sample of research papers (reporting such violations) using an adapted version of David and Han procedure used by them in order to identify a representative sample of studies that empirically tested the core tenets of Williamson’s Transaction Cost Economics (TCE). A total of 69 articles have been included in the investigation majority of which have been published during the “Behavioural Economics Era (1990 to date) in the Journal of Risk and Uncertainty, and Management Science. Majority of the experimental studies have reported a violation of the independence axiom which has been attributed to a host of behavioural biases like common-consequences effect, common ratio effect, preference reversals, framing effects, reference point effects, anchoring and adjustment bias, probability-weighting bias, response-mode bias and the loss-aversion. The background conditionals for such violations include size (low vs. high) and the nature (hypothetical vs. real) of monetary outcomes, decision makers’ limited ability to process information, probability transformations/distortions, structure of experiment, aspiration levels, (high/low) anchors, probability thresholds, limited sensitivity to low probability events, frequency of outcome evaluations by the decision makers, procedure and description invariance, nonlinear evaluation of probabilities, greater sensitivity to losses than gains, over/under weighting certain dimensions or specific information, and the use of heuristics for simplification. Betweenness (a weaker form of independence) is the second most violated axiom. These violations have been attributed to the fanning-out/in effects, context effects, boundary effects, ordering effects, and the framing effect. The notable factors behind these violations include the lottery’s location in probability triangle, nonlinearities in the indifference curves, random errors, quasiconcavity and quasiconvexity of preferences, and the folded ordering. Transitivity violations (the third in that order) have been attributed to the event-splitting bias, regret-aversion, dimension-interaction, ordering effect, and the framing effects. They salient background conditionals of these violations include contrast effect of comparison within dimension, novelty, strength and direction of preference, use of a comparative rather than absolute approach in evaluation, and the similarity among attributes. The violations of monotonicity axiom have been attributed to the probability weighting bias, and the ordering effects. The key factors behind these violations include the distribution of cash value offered for comparison in gambles, probabilities of loss or zero outcome, decision makers’ view points, and judgements about the attractiveness of gains/losses. Finally, the violations of reduction axiom have been attributed to the splitting/coalescing effects, anchoring and adjustment effect, and the certainty effect. Overestimation of the expected value of compound lotteries in comparison to their reduced forms, coalescing/splitting of branches, higher probability of winning occurring in earlier stages, and the difference in CEs for the compound and the reduced forms of lotteries have been revealed as the most important background conditionals. Despite these widely-reported violations of its axiomatic foundations, we still believe that EUT is and will continue to act as a crucial reference framework for the emerging explanations of decision behaviour especially under risk and uncertainty.
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European Journal of Economics, Finance And Administrative Sciences - Issue 15 (2009) Allais, M., 1953. “Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’école Américaine”. Econometrica, 21, pp. 503-46. Bar-Hillel, M., 1973. “On the subjective probability of compounded events”, Organizational Behavior and Human Performance, 9, pp. 396-406. Battalio, R., J. Kagel, and K. Jiranyakul, 1990. “Testing between alternative models of choice under uncertainty: Some initial results”, Journal of Risk and Uncertainty, 3, pp. 25–50. Bernasconi, M., 1994, “Nonlinear preference and two-stage lotteries: theories and evidence”, Economic Journal, 104, pp. 54–70. Bernoulli, D., 1738, originally published in 1738, translated by Dr. Lousie Sommer (January 1954), “Exposition on a new theory on the measurement of risk”, Econometrica, 22(1), pp.2236 Birnbaum, M. H., 1992. “Violations of monotonicity and contextual effects in choice-based certainty equivalents”, Psychological Science, 3, pp. 310-14. Birnbaum, M. H., and S. E. Sutton, 1992. “Scale convergence and utility measurement”, Organizational Behavior and Human Decision Processes, 52, pp. 183-215. Birnbaum, M. H., G. Coffey, B. A. Mellers, and R. Weiss, 1992. “Utility measurement: Configural-weight theory and the judge's point of view”, Journal of Experimental Psychology: Human Perception and Performance, 18, pp. 331-46. Birnbaum, M. and L. Thompson, 1996. “Violations of monotonicity in choices between gambles and certain cash” The American Journal of Psychology, 109(4), pp. 501-23. Birnbaum, M. H., J. N. Patton, and M. K. Lott, 1999, “Evidence against rank-dependent utility theories: Violations of cumulative independence, interval independence, stochastic dominance, and transitivity”, Organizational Behavior and Human Decision Processes, 77, pp. 44-83. Birnbaum, M.H., 2004. “Causes of Allais common consequence paradoxes: An experimental dissection”, Journal of Mathematical Psychology, 48 (2), pp. 87-106 Birnbaum, M.H., 2007. “Tests of branch splitting and branch-splitting independence in Allais paradoxes with positive and mixed consequences”, Organizational Behavior and Human Decision Processes, 102 (2), pp. 154-73. Birnbaum, M. and R. Gutierrez, 2007. “Testing for intransitivity of preferences predicted by a lexicographic semi-order”, Organizational Behavior and Human Decision Processes, 104(1), pp. 96-112. Birnbaum, M. and U. Schmidt, 2008. “An experimental investigation of violations of transitivity in choice under uncertainty”, Journal of Risk and Uncertainty, 37, pp. 77–91. Blavatskyy, P. (2005). “Back to the St. Petersburg Paradox?” Management Science, 51 (4), 677-78. Blavatskyy, P., 2006. “Violations of betweenness or random errors?”, Economic Letters, 91(1), pp. 34-38. Blavatskyy, P., 2007. “Stochastic expected utility theory”, Journal of Risk and Uncertainty, 34(3), pp. 259-87. Bleichrodt, H., and J. L. Pinto, 2000. “A parameter-free elicitation of the probability weighting function in medical decision analysis”, Management Science, 46, pp. 1485-96. Bleichrodt, H., 2001. “Probability weighting in choice under risk: An empirical test”, Journal of Risk and Uncertainty, 23, pp. 185-98. Bleichrodt, H. J. L. Pinto, and P. P. Wakker, 2001. “Using descriptive findings of prospect theory to improve prescriptive applications of expected utility”, Management Science, 47(11), pp. 1498–1514. Bradbury, H. and T. M. Nelson, 1974. “Transitivity and the patterns of children's preferences”, Developmental Psychology, 10(1), pp. 55-64.
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