The correct formula of 1979 prospect theory for multiple outcomes

Whereas original prospect theory was introduced over 40 years ago, its formula for multi-outcome prospects has never yet been published, resulting in many misunderstandings. This note provides that formula.


Introduction
When Kahneman and Tversky (1979) introduced (original) prospect theory, they formally restricted their theory to lotteries with at most two nonzero outcomes. The extension to more outcomes is straightforward, as they wrote, but they only stated it in words and did not provide the actual formulas. Up to today, over 40 years after, and 20 years after the shared prize in memory of Nobel, the formulas of original prospect theory have never yet been written in public. While understood by specialists, the formulas nevertheless continue to cause numerous misunderstandings. 1 This note provides the formulas, hoping to put an end to the confusions.

The formulas
I will refer to the 1979 version of prospect theory as original prospect theory (OPT). The current version (Tversky & Kahneman, 1992), called prospect theory here, 2 will not be discussed in this note.
By (p 1 :x 1 , …, p n :x n ) we denote a prospect (lottery) assigning probability p j [ 0 to outcome x j [R, j = 1, …, n. All outcomes are different and the variable n can be any natural number. Deviating from Kahneman and Tversky (1979), we always write the outcome 0, so that always p 1 ? _ ? p n = 1. By v: R ! R we denote the value function. It is nondecreasing with v(0) = 0. p: [0,1] ? [0,1] is the weighting function. It is nondecreasing with p(0) = 0 and p(1) = 1.
Outcome 0 is the reference point, positive outcomes a [ 0 are gains, and negative outcomes a \ 0 are losses. We first consider gain-prospects with x 1 [ _ [ x n C 0. Thus, x n is the smallest outcome with positive probability, which may be 0. The OPT value of the gain-prospect (p 1 :x 1 , …, p n : The minimum outcome x n , called riskless by Kahneman and Tversky, plays a special role. It is received with certainty-maybe even more will be received. Its value (''utility'') is, therefore, not weighted. For the other outcomes, their extra value, which is not certain but risky, is weighted.
, the formula is reflected. Here x 1 is the ''best'' loss with positive probability, and x 1 may be 0. The OPT value of the lossprospect (p 1 :x 1 , …, p n : The minimum loss, x 1 (possibly 0), is what one surely loses-maybe more. As with gains, the outcome closest to the reference point is perceived as quasi-certain. For the other outcomes, their extra value loss, which is not certain, is weighted. For mixed prospects, where both a gain and a loss occur with positive probability, there is no perception of a certain threshold and the OPT value of (p 1 :x 1 , …, p n :x n ) then is It is important to notice that this equation is a natural mathematical combination of Eqs. 1, 2-and it is the only one so. To see this point, split a mixed prospect up into a gain-part (all losses replaced by 0) and a loss-part (all gains replaced by 0). Then Eq. 3 results as the sum of the OPT values of the gain-and loss-part. Note that both the gain-and loss-part then involve outcome 0 as quasi-certainty-whose weight is immaterial because v(0) = 0. Thus, Eq. 3 is indeed the only possible combination of Eqs. 1, 2. The three formulas on the three different subparts of the domain naturally fit together as one well-defined functional on the domain of all simple (finite-valued) prospects.
Another point of note is that OPT of 1979 and the current prospect theory (Tversky & Kahneman, 1992) agree for prospects with no more than two outcomes (n B 2). 3 The two theories only diverge for three or more outcomes. Thus, Eqs. 1, 2 already contained part of the idea of rank dependence, central to current prospect theory.
There is no clear extension of OPT to infinite-valued prospects. Rieger and Wang (2008) considered plausible candidates but revealed several difficulties. For instance, extensions may depend on p only through p 0 ð0Þ and may depend much on the particular discrete approximations chosen.
A theory different than OPT, called separable prospect theory results if Eq. 3 is also used for nonmixed prospects, rather than Eqs. 1, 2. It was used in the very first experimental study of risk attitudes (Preston & Baratta, 1948) and was widely used in the psychological literature until the 1980s (Edwards, 1962). It lost popularity when it was discovered that it violates stochastic dominance (Fishburn, 1978). The violations are not only normatively undesirable, but also descriptively. They do not fit with the few violations that have been found empirically and that always crucially depend on misleading framings. Further, they become extreme, and the formulas become absurd, if there are many outcomes. The following lottery illustrates this point: 0:01 : 1 þ 1 Â 10 À5 ; . . .; 0:01 : 1 þ 99 Â 10 À5 ; 0:01 : 0 À Á : Its certainty equivalent, with the parametric estimates of Tversky and Kahneman (1992), is 6.9, which exceeds the maximal outcome of the lottery more than sixfold. This does not make any sense, neither theoretically nor empirically. Yet, separable prospect theory has continued to be used sometimes, for instance by Camerer and Ho (1994) who do carefully distinguish it from OPT in their endnote 16. Unfortunately, other papers often confused separable prospect theory with OPT as referenced in §1, also for lotteries with only two outcomes that are both nonzero, even though Kahneman and Tversky (1979 Eq. 2) explicitly wrote a deviating formula for these lotteries. Finally, OPT and separable prospect theory do agree if outcome 0 has a positive probability.
3 Background Kahneman and Tversky (1979) had much impact. Merigó et al. (2016) listed it as the most cited paper in economics. Kahneman and Tversky (1979) only gave a verbal statement of the general Eqs. 1, 2. We repeat their text here, where the additions between square brackets serve to extend to plurality of non-minimum outcomes.
… prospects are segregated into two components: (i) the riskless component, i.e., the minimum gain or loss … which is certain to be obtained or paid; (ii) the risky component, i.e., the additional gain[s] or loss[es] … which is[are] actually at stake…. That is, the value of a strictly positive or strictly negative prospect equals the value of the riskless component plus the value-difference between the outcomes, multiplied by the weight associated with the more extreme outcome [s]. The essential feature … is that a decision weight is applied to the value difference … which represents the risky component of the prospect, but not to … the riskless component. (Kahneman & Tversky, 1979 p. 276).
That Eqs. 1, 2 are correct also appears from Kahneman and Tversky (1975 p. 18), a first working paper version of their 1979 paper. They did explicitly write the formula of PT for multiple outcomes there. 4 They also emphasized (their p. 12) that the riskless outcome should not be weighted, as in our Eqs. 1, 2 and as opposed to separable prospect theory where it is weighted. In other papers, whenever relevant, they pointed this out (Tversky & Kahneman, 1981 Footnote 5;Tversky & Kahneman, 1986 Footnote 2). Kahneman and Tversky (1979) only described the formula of their original prospect theory (OPT) for multiple outcomes in words. This has led to many misunderstandings, and dozens of papers confused separable prospect theory with thedifferent-OPT. Amazingly, I am not aware of even one paper that wrote the correct formula of OPT for multiple outcomes. This paper has provided it.

Conclusion
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