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

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

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:

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:

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:

Applying the Transform relations (I)  Utility function for  = 0.25, where  + and  - have the values 20 and 5

Applying the Transform relations (II)  Utility function for  = 1.0, where  + and  - have the value 8.3

Applying the Transform relations (III)  Utility function for  = 3.0, where  + and  - have the values 4 and 12

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?