Proposing A Normative Basis For the S-Shaped

Value Function

Malcolm E. Fabiyi

Why the S-Shaped Function? Kahneman & Tversky’s 1979 Prospect Theory

indicate that a S-Shaped value function best captures utility

S-shaped value function characterized as• Concave above reference point• Convex below it• Steeper in losses than in gains• Defined on deviations from the reference point

S-shaped value function confirmed in the field e.g.,• Galanter & Pliner, 1979; • Fishburn & Kochenburger, 1979; • Collins et al, 1991

Why is the Value Function S-Shaped?

S-Shaped value function very pervasive, yet no comprehensive explanation exists• Friedman (1989) suggested emergence of S-Shape when

agents maximize expected sensitivity at actual choice opportunities Approach requires prior knowledge of opportunity

distributions Inconsistent with reality of choice decision making

• Robson (2002) argues that S-Shaped utility function emerges from evolutionary considerations Requires framing as principal agent problem Nature is the principal

Prospect Theory is Descriptive Most robust characterization of choice that matches choice

reality Theory remains largely descriptive, no overarching normative

theory exists Fundamental questions

• Why is the utility curve S-Shaped?• Why is the curve steeper in losses than in gains• What is the normative basis for the S-Shaped utility profile?• What factors determine the shape of the curve and how can they

be influenced?• Can the curve be established a priori?• Is the curve valid across temporal dimensions?• What is the significance of the reference point, how is it to be

determined, and can it be established a priori?

Proposing a Normative Basis

We propose an evolutionary rationale• Pervasive emergence suggests S-Shaped value function

intrinsic to human nature• Therefore must have evolutionary basis for its selection

Rust (2005) has applied an evolutionary approach• S-Shaped profile emerges from evolutionary scenario

where behavioral tendencies select according to a hazards model

• The survival function differs from Prospect theory value function as it has finite limits

• Valuable because it demonstrates that evolutionary approach can lead to S-Shaped value function

Laying out the Theory Evolutionary goal for average decision agent is survival, i.e., the

perpetual representation of genes• Survival is comprised of two components: Gene (G) and Self (S) survival• Evolutionary goal is joint assurance of Gene and Self Survival• Goes beyond mere reproductive success• Having progeny does not assure survival

Progenitor must provide shelter, food, clothing, protection For genes to be transferred, progeny must access quality mates

Possess intra-generational competitiveness – health, good earnings Progenitor is necessary to assure this happens

Every behavior manifested by a decision agent impacts the capacity for survival

All behavior manifested by decision agents has three (3) impacts• Fitness enhancing• Fitness neutral• Fitness reducing

Impact of Behavior on Fitness

Primary assumption is that survival (fitness) set comprises two components – Gene and Self Survival

All behavior impacts both components in 3 ways • Enhancement = +

• Neutral = 0

• Diminishment = -

There are nine (9) possible fitness outcomes from any behavior

These are depicted in Table 1

Outcome states

G(+) G(0) G(-)

S(+) G(+)S(+) G(0)S(+) G(-)S(+)S(0) G(+)S(0) G(0)S(0) G(-)S(0)S(-) G(+)S(-) G(0)S(-) G(-)S(-)

Rules for Analysis Real number weights can be assigned to the members of the

fitness set The valence of the members of the fitness set carry nominal

values given as +1 for Enhancement, -1 for Diminishment and 0 for Neutral (Zero) impact

The magnitudes of the members of the fitness set are given as G, and S respectively for the Gene and Self survival members

A relative magnitude , of the weights of the members of the fitness set is defined, and is given as: = G/S

The relative magnitude of the weights of the members of the fitness set is specified within the bound - ≤ ≤ .

• The maximization of fitness (assurance of perpetual presence of genes in gene pool) requires that >1.

• This is so because gene permanence is possible only if gene survival is assured.

Analysis

We define the fitness set as; F = G() + S() & represent the valence of the fitness set, and

can take the values +1, -1, and 0

• Since = G/S; F = S () + S()

• F = S(+ ) It is possible to develop a Table of values for

the fitness set• Normalized with respect to G(+)S(+); at which

F = S(+ )

Fitness set

Goal state Fitness value Normalized fitness value

G(+)S(+) 1S 11

1

SS

G(+)S(0) S 11

SS

G(+)S(-) 1S 1

111

SS

G(0)S(+) S 11

1 SS

G(0)S(0) 0 010 S

G(0)S(-) S 11

1

SS

G(-)S(+) 1S 1

11

1

SS

G(-)S(0) S 11

SS

G(-)S(-) 1 S 11

1

SS

Translating Fitness Values to Utility Values The utility or value of a stimulus is equivalent to the

magnitude of its impact on survival The motive or goal states correspond to utility states

since they are the currency by which the fitness impacts of stimuli are measured.

The ultimate goal of the individual is the attainment of the G(+)S(+) state

Evolutionary imperatives require the avoidance of the G(-) and, or S(-) states

x is defined as the subjective value of the stimulus (prospects) evaluated by the Decision Agent (DA)

Transform goal:

0

0:

xUF

xUF

Transform Rules Since the maximum attainable goal state G(+)S(+) corresponds

to +1, the corresponding utility is +1. Whereas the transformation of fitness values to utilities on the

positive dimension is intuitive, establishing how to conceive of the transformation from fitness value to utilities in the negative domain is not as obvious

We assume that fitness maximization constraints require that no fitness utility is derivable from any goal state that compromises the total survival set i.e. when F < 0 then U = -1

The transform F_ is minimized not at G(-)S(-), but at any of the

G(-), S(-) states that returns the first negative fitness value. Based on these considerations, we can specify that:

0G(0)S(-));G(-)S( G(-)S(0); G(-)S(-);1

0)()(1)(

min

max

xMaxu

xSGuxu

Determination of the Nature of the Transform Function A Transform Function U(x) is required for converting fitness

to utility It is required that U(x) is continuous, differentiable and

bounded within the domain U(x):(-1,1). • Specifying this constraint necessarily limits the choice of

functional types suitable for the transformation. We will adopt the use of the hyperbolic Tan (Tanh) as a

Transform Function that satisfies the necessary conditions. Two fitting variables + and - are defined We therefore have that U(x) = Tanh(x) and we reformulate

the problem thus:

0

0)(

xxTanh

xxTanhxu

Applying the Transform relations (I)

Utility function for = 0.25, where + and - have the values 20 and 5

0.25

G-S States x uG(0)S(+) 1.0 1.00G(0)S(0) 0.8 u+G(+)S(0) 0.6 u+G(-)S(+) 0.2 u+G(-)S(0) 0.0 0.00G(-)S(-) -0.2 -1.00G(0)S(-) -0.6 -1.00G(+)S(-) -0.8 -1.00G(0)S(-) -1.0 -1.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Applying the Transform relations (II)

Utility function for = 1.0, where + and - have the value 8.3

1G-S States x u

G(+)S(+) 1 1G(+)S(0) 0.5 u+G(0)S(+) 0.5 u+G(+)S(-) 0 0G(0)S(0) 0 0G(-)S(+) 0 0G(0)S(-) -0.5 -1G(-)S(0) -0.5 -1G(-)S(-) -1 -1 -1.500

-1.000

-0.500

0.000

0.500

1.000

1.500

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Applying the Transform relations (III)

Utility function for = 3.0, where + and - have the values 4 and 12

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

3.0

G-S States x uG(+)S(+) 1.0 1.00G(+)S(0) 0.8 u+G(+)S(-) 0.5 u+G(0)S(+) 0.3 u+G(0)S(0) 0.0 0.00G(0)S(-) -0.3 -1.00G(-)S(+) -0.5 -1.00G(-)S(0) -0.8 -1.00G(-)S(-) -1.0 -1.00

Scenario Analysis

What happens when < 0: Non S-shaped Utility = 0 : Stable S-shaped value function

Conclusions

Evolutionary considerations can explain S-Shaped value function

Consistent with notion of value function being intrinsic (cardinal) to the individual

Future work• Is the S-Shaped value function valid in all

stimulus contexts?

• Do more appropriate Transform functions exist?