Abstract
How should we decide? And how do we decide? These are the two central questions of Decision Theory: in the prescriptive (rational) approach we ask how rational decisions should be made, and in the descriptive (behavioral) approach we model the actual decisions made by individuals. Whereas the study of rational decisions is classical, behavioral theories have been introduced only in the late 1970s, and the presentation of some very recent results in this area will be the main topic for us. In later chapters we will see that both approaches can sometimes be used hand in hand, for instance, market anomalies can be explained by a descriptive, behavioral approach, and these anomalies can then be exploited by hedge fund strategies which are based on rational decision criteria.
As soon as questions of will or decision or reason or
choice of action arise, human science is at a loss.
Noam Chomsky
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Notes
- 1.
We usually allow all real numbers as outcomes. This does not mean that all of these outcomes have to be possible. In particular, we can also handle situations where only finitely many outcomes are possible within this framework. For details see the background information on probability measures in Appendix A.4.
- 2.
It is possible to extend this definition from finite lotteries to general situations: state dominance holds then if the payoff in lottery A is almost nowhere lower than the payoff of lottery B and it is strictly higher with positive probability. See the appendix for the measure theoretic foundations to this statement.
- 3.
Often this concept is called first order stochastic dominance, see [Gol04] for more on this subject.
- 4.
- 5.
EUT is sometimes called Subjective Expected Utility Theory to stress cases where the probabilities are subjective estimates rather than objective quantities. This is frequently abbreviated by SEU or SEUT.
- 6.
Sometimes this property is called “strictly risk-averse” . “Risk-averse” then also allows for indifference between a lottery and its expected value. The same remark applies to risk-seeking behavior, compare Definition 2.9.
- 7.
If the lotteries are given as probability measures, then the notation coincides with the usual algebraic manipulations of probability measures.
- 8.
Sometimes this is also called “Archimedian Axiom”.
- 9.
In order to make this concept work for non-discrete lotteries, one needs to take a slightly more complicated approach. We give this general definition in Appendix A.6.
- 10.
This approach has recently found a revival in the works of Kahneman and others, compare [KDS99].
- 11.
We use these standard names, although they are not coherent with the use of the similar term “risk-averse” where a strict inequality occurs.
- 12.
The function u 2 is, by the way, an unorthodox suggestion to resolve Allais’ Paradox which we will meet in the next section.
- 13.
This example might remind the reader of Example 2.32 that demonstrated how Mean-Variance Theory can lead to violations of the Independence Axiom.
- 14.
It is historically interesting to notice, that a certain variant of the key ideas of Kahneman and Tversky have already been found 250 years earlier in the discussion on the St. Petersburg paradox: Nicolas Bernoulli had the idea to resolve the paradox by assuming that people underweight very small probabilities, whereas Gabriel Cramer, yet another Swiss mathematician, tried to resolve the paradox with an idea that resembles the value function of Prospect Theory.
- 15.
The definition of CPT can be generalized if we use different weighting function w − and w + for negative resp. positive outcomes. To keep things simple, we assume that \(w_{-} = w_{+} = w\).
- 16.
This is not the case in the original formulation of CPT when applying the weighting function on cumulative probabilities in losses and de-cumulative probabilities in gains.
- 17.
If α < β, however, even a value of \(\lambda <1\) can lead to loss aversion. – In fact, when measuring \(\lambda\) on experimental data one often gets values substantially smaller than 2.
- 18.
Here we consider again only the form defined in this book. In the original formulation we would need to write down two integrals for negative and positive outcomes and invert the direction of integration on the latter one. Compare the remark after Definition 2.37.
- 19.
This characterization is mathematically quite involved. Brave readers might want to look into the original paper [Wak93].
- 20.
Unless the weighting function w satisfies the symmetry property \(w(1 - p) = 1 - w(p)\).
- 21.
Compare Theorem 2.25 (ii).
- 22.
The indicator function is of course not continuous, but one can work around this problem by approximating the indicator function by continuous functions – a quite useful little trick that works here.
- 23.
This definition is not related to the “Continuity Axiom” of von Neumann and Morgenstern (Axiom 2.18), even though the (unfortunate) name of the axiom suggests this.
- 24.
For a definition of Dirac masses, see Appendix A.4.
- 25.
Sometimes there are attempts in the literature to use both words for slightly different concepts, but so far there seems to be no commonly accepted definition, hence we take them as synonyms and will usually use the word “uncertainty”.
- 26.
We have seen that CPT models such decisions quite well, and that the rational decisions modeled by EUT are not too far away from CPT.
- 27.
This framing seems at least to be used frequently enough to produce proverbs like “A stitch in time saves nine” and “Never put off till tomorrow what you can do today”.
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Hens, T., Rieger, M.O. (2016). Decision Theory. In: Financial Economics. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49688-6_2
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