Violations of normative theories of utility maximising behaviour appear to suggest that behavioural factors influence attitudes to uncertain choices. In effect, decision-makers appear to draw categorical distinctions between payoffs which are certain, possible, or highly
improbable. The concept of a continuous utility function able to differentiate objectively between payoffs across all levels of risk is therefore open to question. The most obvious inconsistencies appear to lie at the extremes; either modelling potentially high, risky payoffs against certain alternatives or when considering potential outcomes with vanishingly small likelihoods.
In practice, decision-makers appear not to evaluate risky choices in isolation but apply some context with regard to consequences. The most obvious component of the latter appears to be related to changes in wealth states. Therefore, many individuals are prepared to play objectively unfair gambles in the form of lotteries offering extremely low probabilities of a high return as that high return is likely to be transformative in terms of wealth state while refusing to accept even small risks with relatively insignificant payoffs (Baillon & Bleichrodt, 2012). Conversely, already wealthy individuals may place greater importance upon
preserving that wealth state, leading to a subjective over-discounting of risk. In such circumstance, risk preferences can be explained by the application of subjective functions
w(p) which weight the probability p of risky events. The willingness to bet on a particular event E can then be defined by; w(p(E)), which may differ from p(E).
Elements of this become clear if we re-examine the basic proposition advanced by Allais in which decision-makers are presented with the following paired choices;
A: receive £1,000,000 with certainty. B: receive £5,000,000 with probability .1,
receive £1,000,000 with probability .89, receive nothing with probability .01. and
C: receive £1,000,000 with probability .11, receive nothing with probability .89. D: receive £5,000,000 with probability .10,
receive nothing with probability .90.
A preference for option D over option C is, of course, consistent with probability weighted expected payoffs assumed by expected utility theory; the expected payoff associated with option C being £110,000 while that of D is £500,000. This is, however, reversed in the case of preferring A to B; the expected monetary payoff from B is £1.39m (£500,000 + £890,000), while A guarantees a risk free return of £1m.
Loss aversion and the certainty effect, overweighting certain outcomes relative to those which are considered only to be objectively probable (Kahneman & Tversky, 1979), have been suggested as explanations for axiom violations. Bell (1982) has extended this to include
regret. Thus, when comparing risky options with those with certain outcomes, decision- makers consider the disappointment they might feel should the actual outcome from the risky choice fall short of probabilistic expectations. Both loss aversion and anticipated regret as a consequence of actual outcomes are thus assumed to apply a higher discount rate to risky prospects notwithstanding knowledge of objective probabilities. Loss aversion can also appear to align with regret in the form of opportunity cost. Thus, given the propositions presented by Allais, while there is no possibility of an actual loss being incurred, there is a clear potential opportunity cost in relation to the alternative foregone. Consequently,
accepting the gamble represented by prospect B involves the opportunity cost associated with rejecting the certain receipt of £1m represented by prospect A. If the certain return of £1m is deemed by the decision-maker to be sufficiently transformative in terms of wealth states, the basis exists for the subjective preference for A over B notwithstanding the mathematically higher expected payoff associated with the latter. An issue remains in relation to how such behaviour can be generalised in model form.
One generalised framework for overcoming the restrictions of standard additive approaches has been proposed using a Choquet Integral to accommodate multi-criteria decision-making (Chateauneuf, Eichberger and Grant 2002). The model involves applying a non-extreme-outcome (neo-additive) function to model optimistic and pessimistic attitudes towards uncertainty, overweighting best and worst outcomes, while also applying weights to interactions between decision-criteria. The aggregate function (Choquet Integral) is defined with respect to a set function acting on all possible combinations of a set of criteria. As indicated, weights (defined as fuzzy measures) are applied across explicit decision criteria and also interactions between combinations of decision criteria. The latter provides significant flexibility when distinguishing between preferences.
Price Cleanliness Service
Hotel A 0.7 0.6 0.9
Hotel B 0.6 0.7 0.9
Hotel C 0.6 0.7 0.4
Hotel D 0.7 0.6 0.4
A practical example of the process is provided by Li, Law, Vu, Huy quan and Rong, (2013), explained briefly as follows: a traveller selecting a hotel has identifies a number of criteria which will be used to arrive at a preference ordering between the various options. Assume that three criteria are deemed the most important: price; cleanliness and service. On that basis, four possible hotels are identified and the traveller has assigned utility values against each of the three criteria for each hotel. This results in the following;
Table 4. Hotel Choices With Utility Scores Assigned to Decision-Criteria
It might be concluded straight away that; A > D and B > C. In order to make a final decision, the pairs A v B and C v D must still be evaluated. In order to achieve full rankings, preferences for combined criteria might be considered. For example, assuming that service levels are high (A and B above), price may then be considered the most important secondary criterion. On that basis, Hotel A would be ranked higher than Hotel B. However, in the case where service is relatively poor (C and D), cleanliness may be considered more important than price. Applying these filters, we would then find the following preferences; A > B > C > D. Standard additive weighting models would not explain this outcome. For example, if wp,wc and ws are taken to represent unequal weights applied to each of the three criteria, the
preference for A > B implies wp > wc. However, the order C > D holds only when wp < wc.
This therefore fails to explain the observed preference of the decision-maker.
The failure of the simple additive model stems from the fact that it assumes that the criteria are mutually independent. If, however, we allow utility-deriving interactions between
Criteria Groups Weight v() 0 v ({price}) 0.4 v ({cleanliness}) 0.3 v ({service}) 0.75 v ({price, cleanliness}) 0.2 v ({price, service}) 0.9 v ({cleanliness, service}) 0.6
v ({price, cleanliness, service}) 1
Hotel A Hotel B Hotel C Hotel D
C( 0.73 0.7 0.59 0.58
the criteria, a solution can be found. Given utilities associated with a set of N criteria, N = {x1, x2, x3}, a fuzzy measure, v, represents weightings across all criteria; v({1}), v({2}) and
v({3}) together with all interactions between the criteria; v({1, 2}), v({1, 3}), v({2, 3}) and v({1, 2, 3}). The discrete Choquet Integral (CI) for each fuzzy measure, v, is then given by;
Cv(X) = ∑ 𝑥(𝑖) [v({ j xj ≥ xi }) - v({ j xj ≥ xi+1 })] Equation 8
where x(1), x(2),….. x(n) is an ordered (non-decreasing) permutation of the input x.
Assuming the following with regard to fuzzy measures;
Table 5. Fuzzy measures across all choice criteria permutations
Applying the weights above to the formula in Equation 8 gives; Table 6. CI-derived combined weight utilities for each option
On this basis, the preference ranks: A > B > C > D is derived in accordance with observed preferences.
Multi-criteria-decision models of the type described above are potentially promising approaches when clear decision criteria can be derived or inferred. In particular, they offer
solutions to more basic additive models which can fail to explain preferences across choices with multiple utility-inducing attributes. By allowing for the expression of general
preferences within the context of ambiguity, the models relax Savage’s Sure-Thing Principle. By allowing greater probability weights to be applied to the best and least favourable options, CI-derived models enable potentially problematic behaviours at the extremes of normal utility functions to be more easily accommodated. Similarly, a preference for certainty with
minimum payoff over higher-potential payoff risky options can also be accommodated. Application of the models empirically is, however, a potentially complex task as, in many cases, attributes and criteria can be extensive. Nevertheless, attempts have been made to apply the approach to practical decision tasks (Büyüközkan, Feyzioğlu, & Göçer, 2018).
2.2. Discussion
The current chapter discussed significant advances to expected utility theory, describing some key approaches to modelling uncertainty and ambiguity. Challenges to axiomatic approaches to behaviour were considered along with further developments of more complex models with elements relating in part to process as well as outcome. In that regard, multi- criteria-decision models were found to offer a flexible framework from which inferences about preferences relating to complex structures of criteria and their interdependencies could be derived. Such models offer significant advantages over and above simple additive
probabilistic models and provide a basis for resolving some of the more challenging paradoxes which had called into question key assumptions of axiomatic behaviour.
changes in wealth states might play when decision-makers evaluate uncertain outcomes. The concept of changes in wealth, as opposed to its quantum, is present in the early literature, although it remained relatively underdeveloped for some time in terms of a robust framework. The concept itself implies that decision-makers may attach importance to reference points, evaluating preferences in relation to those reference points and more general goals. This has significance in terms of how utility might be expressed. The next chapter takes up this theme, covering one of the most significant advances in modelling behaviour in the form of Prospect Theory.