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Double rank dependent expected utility (DRDEU) model in individual judgments
Yenalyev Maxim Mihailovich, Petro Mohyla Black Sea State Universityy postgraduate studenty the Faculty of Economics E-mail: [email protected]
Double rank dependent expected utility (DRDEU) model in individual judgments
Abstract: This study aims to present a new model of decision making in risk that uses double weighting (for probabilities and potential outcomes) process (DRDEU model). In article is shown that the new rank dependent model can explain empirical data better than other coalescing models. It’s shown that Allais paradox and Ellsberg paradox can be explained within this model’s frameworks too.
Key words: utility theory, decision making, paradox, Ellsberg paradox.
Introduction
The recent investigations have shown that most famous and popular models in utility theory and decision making theory contradict the new experimental data [1-4]. These models (Rank and Sign Dependent Expected Utility, Cumulative Prospect Theory, Prospect Theory (without editing phase)) are belonging to one class of models — models that
satisfy coalescing principle (coalescing principle models, CP models).
In last researches were shown several new paradoxes that can’t be predict or even explain in framework of CP-models. These paradoxes may be classified as follow:
—Violations offirst order stochastic dominance (FSD)
—Violation of Branch Independence (BI)
—Violation of Upper and Lower Cumulative Independence (LCI and UCI)
However, as Birnbaum showed, models that satisfy event-splitting effect can predict and explain these violations. Such inconsistency between CP-models in utility theory and descriptive relevance of event splitting models allows some researchers to make an assumption that coalescing principle is generally inconsistent. However, there is no enough theoretical explanation arising of event splitting effect. And it’s the most important reason for development a new models that would explain new paradoxes in risky choice.
Description of the model
The first stage in development of the new models in decision making is defining basic assumptions
coalescing principle, event-splitting effect, Allais
or axioms in axiomatic framework. The key improvement for DRDEU model is a new approach to individual’s process of decision making in risk or uncertainty. In many models this process can be evaluated as a function:
U (G ) = fu (xt )W (pt)
i=l
Where G — is evaluated alternative, x{ e {X} — is a set of the possible outcomes this alternative and p{ e {P} — a set of probabilities of these outcomes, U (■) — is utility function of outcomes and W (■) — is a probability weighting function.
Central in DRDEU model is an assumption about key role of differences between possible outcomes and its influence on gamble evaluation in all. Such methodology was suggested by A. Michalos [10, 11] in context of evaluating life satisfaction. The key point of this methodology is measuring of gap between what one has and what one wants. This approach is widely used in subjective wellbeing [13] and quality of life researches [7].
From this point of view individual who evaluates probability gamble (or lottery) with nonnegative outcomes may think that the smallest outcome is a something like guaranteed minimum or a new reference point for evaluation. In this case other possibility results may lead to increase his minimum result on the difference between second smaller result (SSR) and “guaranteed” result (GR) and on difference between third smaller result (TSR) and SSR and so on. Assumption that individual evaluates utilities of such differences is important for further investigation.
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Section 7. Economic theory
Probabilities these outcomes (GR, SSR-GR, TSR-SSR) are transformed in manner: probability of the GR is equal one, probability of the increasing GR on the difference SSR-GR is equal 1 - w (pGR), probability of the increasing SSR on the difference TSR-GR is equal 1 - w (pGR + pSSR-GR) where w() is increasing function that generate inverse S-shape curve on interval [0, 1] and w (0) = 0, w (l) = 1.
Ifwe range possible outcomes in increasing order xj <x2 < < xn, where x1 is the lowest potential
result (GR) from all n probabilities wins, we can write a general rule for evaluating some n-branches lottery G with nonnegative outcomes:
U (G ) = fü (x, - x,. _,)(1 - w (Jp ))
i=1 j=1
Where U (x0) = 0, ^p. = 0.
j=1
Explanation of Allais and Ellsberg paradoxes in DRDEU model
DRDEU model can predict Allais paradox which was firstly revealed by Maurice Allais. In [8] Allais paradox was shown as switching the median individual choice in two pairs of lotteries:
J A = {2500,0.33; 2400,0.66; 0,0.01} vsB = {2400,1} (1) (C = {2500,0.33; 0,0.67} D = {2400,0.34; 0,0.66}
As we can see, in the second pair of lotteries common branch (branch is probability-consequence pair in lottery) in the first pair {2400, 0.66} (this record denotes possible outcome in 2400$ with probability 66%) turned into {0, 0.66}, note that lottery В in linearity probabilities framework may be presented as {2400,0.34; 2400,0.66}.
The data show [8] that 82% of the participants chose lottery В in first pair and 83% of them chose lottery C in the second pair. That is an evidence of inconsistency of independence axiom.
Allais paradox demonstrated inconsistency of independence axiom and therefore it initiated process rejecting expected utility theory (EUT ) by von Neumann — Morgenstern and Subjective Expected Utility (SEU) by Lenard Savage.
Also Maurice Allais’ research initiated the emergence of new alternative class models of Nonexpected utility theories, which has received intensive development in the next few decades in [6, 8, 9, 12, 14] and others.
DRDEU models can explain and predict Allais paradox. System of inequities (1) may be shown in DRDEU as a new system:
jU (2400 )(1 - w (0.01)) + U (100 )(l - w (0.67)) < U (2400) [u (2500)(l - w (0.67 ))> U (2400)l - w (0.66))
Or after some transformations as:
U (100 )(1 - w (0.67 ))< U (2400)w (0.0l) (2)
[U (2500 )(l - w ( 0.67 )) > U (2400)(l - w (0.66 ))
In standard conditions [14] where U (x) = x088,
w
(P ) = -
0,724p0
- the system (2) may
be cal-
0.724p061 + (1 - p)0 culated as:
J 27,205 < 39,658 [462,207 > 452,34
That satisfy the conditions of Allais paradox. Ellsberg paradox [5] show us individual uncertainty aversion in risky alternatives. In one urn variation, Ellsberg paradox can be described as a hypothetical experiment with urn that contains 30 red balls and 60 other balls that are either yellow or black.
On the first stage individual has to make a choice between lottery A (if individual drains red ball he receives 100$ and otherwise nothing) and lottery B (he receiving 100$ if he drains black ball and otherwise nothing). As far as precise number of black
2
known it denotes p2 , 0 < P2 < 3 . On the second
stage if he drains red or yellow ball he receives 100$ and otherwise nothing (lottery C) and he receives 100$ in case black or yellow ball otherwise nothing (lottery D). The probability of drawing black ball — 2 2
p3, lies between 0 and —, moreover p2 + p3 = —. Median choices are shown in system (33
A ^ B & 1u (100) >- p2U (100)
C < D &
3+p
Л
(3)
U (100) U (100)
3
The first stage choice within DRDEU framework describes as follow:
( 2 V
U (0 ) + u (100 - 0) +u (100 - 0)
1 - w
\ 3
> U (0) +
o\ 0 1
1 - w +
13 90 JJ
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Double rank dependent expected utility (DRDEU) model in individual judgments
Where i denotes quantity of black balls and i e [1, 59 ]. Within Laplacian approach to uncertainty, we can rewrite function
r 1 60 - i Л .90 - L.
= 1 - w (-
1 - w
as:
v 3 + 90 J 90
t 1 ^ ( 90 - i ^
1----> w -------
59 ы ^ 90 J
And for the first pair of lotteries, after some algebra’s manipulations, individual’s choice can be presented as:
1 ^ f 90 - i} f 2 ^ ,
(4)
-Yw 591=
> w
3 j
V 90 j
2 1 59 f 90 - i ^
It’s easy to show that — = —^ let’s write inequity (4) as: 3 59 i_1
^90-i\ 4
1 59
--fw
59 ы
90
> w
J
1 59
1f
59(
V 90 у ( 90 - i Y
then
90
JJ
For the second pair of lotteries such manipulations can be written as follow:
U (0) + U (100 - 0)
+U (100 - 0)
1 - w
1 - w
(1 ^
V 3 J
'j_ Л
90 j
<
и (0)
+
In such case within Laplacian approach to uncertainty and U (0 ) = 0 relation between second pair of lotteries is described as:
1 49 ( i Л (1 ^ .
— > w - (5)
.90 j
-Yw 591=
1
as:
As far as — = —Y
3 591=1
1 49 ( i 4
3 j
1 59
—Yw
59t!
f j_ Л
90 y
90
> w
then (5) can be written
1 49 f i Y
■Y 59 t!
V 90 yy
In such case resolving of Ellsberg paradox can be equivalent to the existence non-empty set of solutions to the system of inequalities:
1 49 ( 90 - i}
fw
59 It
1 59
—fw
59Tt
90
90 j
> w
> w
1 f ( 90 - i Y
59 ti
1 59
- f
59 Tt
90
JJ
59 f i W
V 90 JJ
This condition is equivalent to the existence su ch
1 2
function w(-) that is convex in points — and —.
3 3
This is generally in line with experimental data about the properties of a probability weight function [8,14].
In such case DRDEU model can predict violation of independence axiom (Allais paradox) and uncertainty aversion (Ellsberg paradox).
Violation of FSD in DRDEU model
Accordance to M. Birnbaums recipe violation of FSD, it needs to hold at least one condition:
— Violate of coalescing principle
— Violate of monotonicity
— Violate of transitivity
The DRDEU model satisfies coalescing in manner:
G = {, p; x,q; y,1 - p - q}~ G =
= {x, p + q; y ,1 - p - q}
Accordance with DRDEU model, individual will be evaluating the utility of G as: U (G) = U (y) + U (x - y )[1 - w (1 - p - q)] +
+ U (x - x )[1 - w (1 - p)].
That is equivalent next one view:
U (G ) = U (y) + U (x - y )[1 - w (1 - p - q)]
From the other hand, utility of gambling G is evaluated within framework of DRDEU, as: U (£) = U (y ) + U (x - y )[1 - w (1 - p - q)].
In such case we can posit the identity of two lotteries: DRDEU (G )~ DRDEU (g).
Also DRDEU model satisfies transitivity, in sense performing the sequence U(A) > U(B) and U (B)> U(C) means U (A) > U(C).
But in special cases DRDEU can violate monotonicity principle. For example in the three branches lotteries increasing of middle branch can decrease total utility of gamble. Let’s examine next two lotteries: A = {x,p;y,p2;z,p3} and B = {x,p;y,p2;z,p3}. Where x < y < z, x < y < z and у < у . In such case DRDEU (A ) = U (x ) + U (y - x }[o - w (-)] +
+ U (z - y )[0 - w(pl + -2 )]and DRDEU (B ) = U (x ) + + U(y -x)[l-w(p,)] + U(z-y ) [1 -w(p, + p2)]. According with monotonicity principle is must hold that DRDEU (A ) < DRDEU (B ). But let’s make an assumption that DRDEU (A)> DRDEU (B) . Hence U (y - x )[l - w (p y + U (z - У )[l - w (pi+ p2)]> >U(y -x)[l-w(pi)p U(z-ypl-w(pi + P2)]
■U (y - x ) - w( p1 )] + Up - y) For at least some pl and p2 it is equal to:
U (y - x) + U (z - y)> U (y - x) + U (z - y )(6)
Let’s imagine outcome у as linear combination of x and z, such that y = ax + (1 — a)z . Then у
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Section 7. Economic theory
may be write as y = ßx + (1 — ß)z. Where 0 < a < 1 and 0 < В < 1. In such case we can re-write inequality (6) as follow:
U (ax + (1 -a)z - x) + U (z -ax + (1 -a)z) >
> U(ßx + (1 - ß)z -x) + U(z -ßx + (1 - ß)z) Note that by virtue of the fact that у > у then ß>a . After transformation it can be shown as:
U ((1 -a)(z - x)) + U (a(z - x)) > U ((1 - ß)(z - x) + + U(ß(z -x)).Note that U((1 -a)(z -x}) isequal to U (1 -a)U (z - x).
Or after simplification: U(1 — a) + U(a) >
> U(1 — ß) + U(ß). Let’s denote vector {l — a'),a} as a and {(l-ß),ß} as b . Then, accordance with Karamata’s inequality if coordinates of vectors a and b are arranged in descending order and b ma-jorizate a ( b У a ) that means for instance (in case
1 <a<ß):
2
fa> 1 -a
1b> i-b° (7)
a<ß
1 -a + a = 1-ß+ß
^b у a ^U(a)>U(b)
The last part of expression (7) is held for some concave function U(•) . Therefore there are some conditions that allow violating monotonicity principle, at least for some probability distributions. Note that such conditions may arise more often when ß ^ 1 or ß ^ 0, in other words when median outcome is closely to the lowest or the highest possible results.
C onclusions
The descriptive value of CP-models in utility theory is being criticized by some researches by virtue of their inconsistency in explaining the range new paradoxes. The most important of them is violation of FSD. The existing coalescing models can’t predict or explain this behavior pattern in risky choice. But if coalescing model violates monotonicity principle it can predict violate of FSD and potential some other violations (e. g. upper and lower cumulative independence). Suggested model that called DRDEU in some circumstances can demonstrate violation from monotonicity principle. Thereby this model can predict violations of First-order stochastic dominance, which can appear in situation when middle branch in three branches lottery is closely to lower or higher branch.
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Double rank dependent expected utility (DRDEU) model in individual judgments
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