Extensions of the Psychophysical Basis of Rational Choice

1. Extensions of the Psychophysical Basis of Rational Choice Malcolm E. Fabiyi

2. Overview • The Expected Utility Theory (EUT) has come under pressure due to its inability to accommodate several observations of the actual behavioral decisions made by agents e.g., Allais Paradox. • Several Non-Expected Utility Theories have been proposed • Kahneman & Tversky’s (1979) prospect theory has been successful in explaining many anomalies • It is based on psychophysics • Focused on the psychophysical transformation of value (utility) and probabilities • In this work, we; • Apply new interpretations for the psychophysical transformations of lotteries, • Develop novel formulations for risk mediated choice • Propose implications for inter-temporal choice • By applying the methods derived from this approach, we are able to match all the modal responses to a series of questions posed by Kahneman and Tversky (1979; 1984). • Implications of the theory for inter-temporal choice are also discussed.

3. Prospect Theory • Kahneman and Tversky’s (1979, 1992) Prospect theory is the most seriously considered alternative to Expected Utility Theory (Starmer, 2000). • Has been successfully applied to such confounds as • The equity premium puzzle (Epstein & Zin, 1990), • Option trader behavior (Fox et al, 1996) • The low incidence of tax evasion (Bernasconi, 1998). • Despite its obvious strengths, there have been recent challenges to its viability as a replacement for Expected Utility Theory • None of the parameterizations obtained for prospect theory can simultaneously accommodate gambling on unlikely gains and Allais type behavior (Nielson & Stowe, 2002). • Blavatskyy (2004) and Rieger & Wang, (2004) have recently suggested that some conventional parameterizations of CPT do not accommodate the St Petersburg paradox.

4. The Underpinnings of Prospect Theory • Prospect theory is derived fundamentally from the view that • Choice is mediated by cognitive and psychophysical determinants • Involves the application of psychophysical analytical aids to the transformation of money and to probability (Kahneman & Tversky, 1984) for the evaluation of preferences. • In order to extend the predictive validity of psychophysical approaches to decision making, we believe it is necessary to extend the range of variables within the decision set to which psychophysical transformations are applied. • In this work, we extend the psychophysical approach to lotteries in several important ways. • We accommodate the influence of the order of presentation of the lotteries on perceptions of relative magnitude; and • We apply psychophysics for the evaluation of subjective time in inter-temporal choice situations.

5. Psychophysical Basis for Decision Making • The transformation of stimuli into cognitively relevant percepts follows power or logarithmic transform laws (Fechner,1860/1966; Stevens, 1961/1962). • Kahneman & Tversky (1979, 1984) have successfully applied power law relations for the psychophysical transformation of money and probabilities associated with lotteries. • There is a Time Order Error (TOE) that is associated with the subjective perception of sequential stimuli • The TOE error of estimations holds across almost every stimulus dimension ever tested (Fechner, 1860; Hellström, 1985). • There are two broad classes of theories that are applied to explaining TOE. The first class of theories suggests that observers judge a second stimulus, not relative to the first one, but relative to its memory trace (Helson, 1947, 1964; Pratt, 1933). • An alternative theory – the quantitative adaptation-weighting model of (Hellström, 1977, 1979, 1985) argues that different weights are associated with the corresponding sensation magnitudes of the stimuli, and that these weights will depend on the relative attention paid to the stimuli. • Both theoretical viewpoints have been successful in explaining variant observations of TOE (Tse et al, 2004). • While the existence of TOE is established in psychophysics, the concept of the TOE is yet to be systematically incorporated into the choice theoretic metric.

6. The Need to Update Prospect Theory • Choice tasks involve sequential presentations of lotteries • Given that lotteries are presented, at least cognitively, in sequential fashion, there is a need to account for TOE in preference tasks involving prospects. • In this work, we adopt the view that all subsequent lotteries are scaled relative to the first cognitively relevant lottery, i.e., the first lottery that is perceived by the decision agent. • The sensation magnitude of this initial stimulus is then used to scale all subsequent stimuli. • Organisms transform objective stimuli or physical parameters into subjective percepts using psychophysical laws (Birnbaum, 1994; Fechner, 1860/1966; Stevens, 1961; 1962). • We assume that when a stimulus (or prospect) is presented to the decision agent, it is transformed into a cognitively relevant equivalent using Stevens’ psychophysical power law function • Fechner’s logarithmic function (Fechner, 1860/1966) provides an alternate formulation for the psychophysical transform function. • However it appears that Stevens’ power law more accurately matches the experimental evidence (Stevens, 1961)

7. 4 Classes of Stimuli • Discrete physical variables which are directed at the five major senses of sense, touch, taste, smell, sight e.g., sound, light • Discrete physical variables which are not directed at any of the senses, and which have a universal modulus e.g. time, which is delineated on the basis of a universal circadian periodicity - for the human organism e.g., in Inter-temporal choice tasks • Discrete physical variables which are not directed at any of the five senses and which do not have a universal modulus e.g., monetary prospects • Statistical inference variables (or probabilities) which serve to modify the magnitude of the exponent of the power law

8. The Stevens Power Law Relation & Other Relevant Factors • According to the Stevens power law relation, given a stimulus E, the magnitude of the percept is given as: • Where k is a constant related to the modulus or intrinsic reference state for the individual, Es is the magnitude of the percept, E is the magnitude of the stimulus and  is the power law exponent. • A cognitively relevant stimulus is fundamentally defined as any influence that evinces a response from the object to which it is directed. • These responses can be physical or mental and psychological; externally manifested or internally enacted. • In economics, the stimuli of interest would generally comprise prospects or lotteries, inference weights (or probabilities) and time.

9. The Relevant Psychophysical Transformations • Transformation of Probabilities • Already applied in prospect theory • Modified power law transform utilized in PT • Logarithmic alternatives exist e.g., Prelec, 1998 • Transformation of Lotteries • Psychophysics currently applied in PT • TOE not acknowledged • Transformation of Time • Psychophysics not currently applied to time in PT • Circadian basis for timing not acknowledged

10. The Psychophysical Transformation of Probabilities • Probabilities are known to be subject to psychophysical laws (Tversky & Kahneman, 1974). • The processes by which decision agents handle probabilities is characterized by the use of heuristics and biases (Tversky & Kahneman, 1974; Lichtenstein et al., 1977). • Decision agents consistently (but systematically) mishandle and misinterpret probabilities (Kahneman & Tversky, 1982) – suggesting subjective transformations are occurring • The subjective transformations of probability have been well characterized (Kahneman & Tversky, 1979; Prelec, 1998, 2000). • We adopt the power function probability weighting transform relation of Tversky & Kahneman (1992) given as:

11. The Psychophysical Transformation of lotteries • Economic stimuli often involve a lottery L, which we will assume to be transformed into percept (subjective) equivalents using the Stevens power law transform to give: • Where is a constant corresponding to the wealth modulus and is equal to the inverse of the reference wealth state for the decision agent, while is the power law exponent.

12. The Psychophysical Transformation of Time • Einstein’s theory of relativity established a rationale for the subjective observance of time • Provides a basis for the consideration of time as a stimulus – in much the same way as other stimuli such as light and sound are considered (Einstein, 1905/1923). • Time is deemed to be subject to the Stevens power law. • We can therefore speak of a subjective time given by the relation: • Where is subjective time, is objective time in days, t is the exponent, and is a constant. • corresponds to the inverse of the time modulus. Since we have suggested that this modulus is defined in terms of circadian periods (i.e., 1 day), hence when t is defined in days • We assume that the relevant cognitive time period is one for which the relevant period cycles are daily (or circadian) ones.

13. Support for a Circadian View of Time • Time and its measurement in years, months, hours, minutes and seconds is on an evolutionary scale a recent occurrence. • In mammals, the suprachiasmatic nucleus (SCN) in the hypothalamus has been identified as the site controlling circadian behavioral rhythmicity (Hastings & Maywood, 2000). • There is evidence that the circadian rhythmicity has a controlling influence on the overall behavior of organisms (Wagner-Smith & Kay, 2000). • The adaptational utility of the circadian rhythm or any event timer for organisms is now better understood • Evolutionary imperatives require such pervasive mechanisms to have adaptational utility i.e., aid survival • Examples of functional areas in which intrinsic event timers could aid survival include communication - through its requirement of signal transmission via sequence duration (Michelsen et al, 1985; Kyriacou et al, 1992); and navigation (Seeley, 1995).

14. Integration of the Various Transform Relations • We will call the exponent of the power law function the fitness quotient & further assume the fitness quotient to be an intrinsic (although variable) internal property of organisms that mediates the psychophysical transformation of all non-sensate, primarily psychological stimuli. Thus • We assume that lies within the bounds . . It appears intuitive that for normal functioning, should strictly lie within the bounds . • Where a probability is associated with a lottery i.e. for cases of risk mediated choice, then the subjective wealth transform must incorporate the risk measure – usually a probability or inference value. • We opine that the adaptational utility of risk must have been to serve as a forced update factor for the subjective wealth evaluations. • To this effect risk functions primarily to moderate the effective strength of the fitness value, modifying it to permit a modulation of the expected values. • We assume that the mediation of the fitness value by risk follows a linear multiplicative function; i.e., where is the risk mediated fitness quotient, is the nominal fitness quotient, and is the subjective probability. • Inter-temporal choice is mediated by exponential discounting, and subjective time is in turn mediated by a two-part variable , which comprises a constant discount rate, r; and the consequence, . We define consequence as the ratio of the nominal values for the lottery compared to the reference wealth i.e., . Thus we have that , where is a constant. • The subjective value for any prospect is therefore given as:

15. Time Order Errors & Scaling • When stimuli are presented to a decision agent, they are aurally, spatially or temporally spaced. • Ultimately all sequential displacements of stimuli reduce cognitively to temporal displacements. • We have proposed earlier that in order to accommodate TOE in decision analysis, we will adopt the approach that the first cognitively relevant stimulus, typically the first stimulus that is registered cognitively is used to scale all subsequent stimuli. • It would appear that this should be a wholly reasonable way in which to deal with the obvious fact that real world prospects do not all occur at the same time, and within the same context. All decisions are at a minimum, discrimination between temporally sequential binary items. • Each prospect is scaled and assigned a scaling value, i.

16. Applying a Scaling Procedure • A complete prospect set will generally include a lottery, L; probabilities, P; and for the case of inter-temporal prospects, a time delay, t • These quantities give: • We adopt a scaling procedure such that the normalization of the first cognitively relevant lottery acquires the normalized value of 1. • For lotteries A and B, The ratios of interest are: • Generally, we have that: • Where is the prospect under evaluation, while is the first cognitively acknowledged prospect

17. Analysis - I • The subjective lottery transform corresponds to a fitness value • The fitness value can be written as: • Given as the multiplication of the consequence of a prospect by its scaled trace • The impact of the consequence on utility can be significant given that, as the resource horizon extends, the consequence diminishes • Generally, given options , the choice will be preferred if

18. Analysis - II • A 2-part value function can be defined given as: • The function is defined as: • At t = 0, we obtain:

19. Predictive validity • Predictive validity of model tested using a set of 22 questions posed by Kahneman & Tversky (1979; 1984) • All of the modal responses returned by the respondents are anticipated by our method • The approach is robust in its predictive abilities.

20. Interesting Implications I – Why people choose ($0,$0) over ($92,$8) • Recall that • Why do people choose ($0,$0) over ($92,$8)? • Subjects often select the irrational option ($0, $0) over ($92, $8) when they are made the offer by an anonymous subject who has been tasked with splitting $100 between the two parties (Rabin, 2002). • The agent who has rejected the offer of $8 goes into the choice task with a prior probability of fairness, P • This probability is used to estimate the agent’s share of the initial $100 • An agent with an absolute belief in fairness expresses P = 1. • For instance, an agent with fairness probability Pwould evaluate their share of the prospect according as , • while the offer Ls is transformed accordingly as • Clearly if the share offered is such that, then rational considerations dictate that the decision agent reject the offer since for such conditions. • The rejection behavior is favored unambiguously where P is high.

21. Interesting Implications II – Why poor people play the lottery • Recall that • Lower income individuals spend higher proportion of income on lotteries (Clotfeller, 2000)* • Lotteries are attractive when is large • or is Large • is low • Low income earners • People who have a small relevant financial period e.g., people paid in short financial periods i.e., weekly or daily wage earners *Data compiled from National Opinion Research Organization Surveys, 1999.

22. Interesting Implications III - Does a Subjective Time Really Exist? • Roelofsema (1994) reported that implicit discount rates differed depending on whether respondents matched on amount or time. • When subjects were asked how much they would demand in compensation to allow a purchased bike be delivered 9 months late, they were ready to accept a median compensation of 250 florin. • When the question was posed to another group in terms of how long they would be willing to delay delivery in exchange for 250 florins, the mean response was 3 weeks (about 21 days). • The inference drawn from this evidence was that the method by which discount rates are elicited can influence the results obtained (Frederick et al, 2002; Roelofsema, 1994). • Another possibility exists – subjective time. Given that hence both times are equivalent if • Possibility that fitness quotient can be elicited by comparing subjective and objective time values for a given preference value

23. Future work • What is the form of the appropriate probability transformation function? • Is the discount rate in inter-temporal choice an intrinsic species constant parameter, or is it mediated by individual difference? • How quickly are the changes in fitness value incorporated into the choice model, and how is the change in fitness value mediated? • How are non-objectively specified lotteries to be evaluated? • Are the axioms of Expected Utility Theory to be defined in nominal or real (hence subjective terms); what implications does this paradigm hold for the axioms of EUT? • What is the correct inter-temporal discount rate model? • Although we have not concerned ourselves with inter-temporal choice, an exponential discount factor is assumed • What is the appropriate psychophysical transformation function? • While we have adopted the use of the Stevens power law function in this work, the suitability of the Weber-Fechner logarithmic law is not precluded for classes of preferences.