Abstract: Decision-making implies formation of conceptual models, conscious or not, in order to prioritize available choices. Rational ordering by decision utility value does not distinguish, however, within ranges of choices perceived as nearly optimal. In such situations I postulate that counteracting behavioral tendencies towards variance and conceptual continuity and simplification lead to wave characteristics in modeling and decision-making probabilities. This paper considers behavioral wave interference phenomena, in which disparate parallel applications or choices evolve from an original core value or priority. Specific examples include evolution of conceptual paradigms in societies and effects on decision-making induced by verbalization of priorities.

Decision-making is traditionally assumed to be optimizing in theoretical modeling. That is, a consumer or investor always chooses $2 values over $1 values in transactions with fixed prices. Prominent economic theories assuming decision-making optimization include expected utility optimization theory and portfolio and efficient market pricing theories, which have attained worldwide popular acceptance and application in financial markets. These theories assume rational optimization of quantifiable expected monetary value and/or perceived utility (Savage, 1954)

More recently, behavioral finance theorists have explored the role of irrational decision-making in human economic behavior (Shiller, 1997; Thaler, 1992). Decisions in their models are made based on systematic, i.e., deterministic, behavioral tendencies and perceptions (Day, 1997; Wang, 1998) rather than on quantitative optimization of expected monetary gain or other utility. In anchoring phenomena (Kahneman and Tversky, 1974), for example, decision-makers repeat choices made earlier by themselves or by others, even when aware that earlier choices, e.g., of regularly purchased consumer brand, were partially arbitrary or random and not necessarily optimal. Magical thinking (Skinner, 1948) occurs when decisions arise from irrational perceptions of optimization, e.g., choosing lucky or favorite items, numbers and rituals.

Intrinsic randomness in choices and actions (Grobstein, 1994) has been mostly neglected in theoretical modeling until very recently (McKelvey and Palfrey, 1995, 1998). Behavioral forces driving choices often contain a significant random component, however. Many consumer and other mundane choices in life are made not by consistent optimization of a utility function, but with significant dependence on arbitrary taste and/or mood at the time of decision. Such choices exhibit limited impact on perceived utility return, resulting from limited experience and discipline applicable, due to cost (time and effort) in obtaining information, and/or simply due to desire for variety (Cox, 1969; Grobstein, 1994).

In another paper (Wang, preprint) I have derived a theory of sub-optimal decision-making behavior mathematically analogous to quantum mechanical particle theories (Feynman and Hibbs, 1965; Landau and Lifshitz, 1977). Quantum theories postulate that particle motions are explicable only in terms of probabilistic wavefunction amplitudes. The exact functional distribution for a particle depends on the surrounding environment as well as on parameters of the particle. As particle energy, momentum, and mass increase, quantum mechanical behavior approaches the classical behavior of particles under Newtonian laws.

A wavefunction theory of decision-making provides a framework for unification of apparently disparate behavioral theories through the dual principles of conceptual simplification and variability. In general, all choices occur probabilistically, following frequency distributions dependent on choices available, resources allocated to decision, and decision-maker characteristics. Rational expected utility optimization theories, irrational behavioral anomalies (Shiller, 1997; Thaler, 1992), and random variance (Grobstein, 1994; McKelvey and Palfrey, 1995, 1998) describe behavior in different limiting regimes. When choices are clearly prioritized by utility/value, individuals and groups seek deterministically maximum expected utility in choices of actions and preferences. Consumers will almost always prefer to buy the same or very similar item at 30% sale discount to buying at regular price. This regime is analogous to that of classical potential theory in physics, stating that particles tend to move closer due to gravity.

When optimal choices are not obvious and essential, preferences for conceptual simplification can dominate decision-making. Arbitrarily broken symmetries in selecting among similarly optimal choices are manifested in apparently irrational behavioral anomalies. Consumers and investors in dynamic market environments arbitrarily “anchor” estimates of market values with reference to prices earlier observed or obtained, waiting to “break even” regardless of likely changes in underlying values with time. In magical thinking arbitrary “lucky” numbers, clothing or rituals are identified as optimizing outcome value. This regime is analogous to that of classical theories of motion, in which particles possess inertia, tending to maintain position in the absence of strong forces.

An important identifying characteristic of areas of mundane behavior is regular repetition of the same action, weekly, daily or with other frequency. The same decision, e.g., exactly what to eat, wear, and read, is repeated over a range of similarly optimal choices. “Classical” rational optimization then provides minimal predictive power, for non-life-altering choices between different cereals or color of ties and socks on a given day. This regime is analogous to that of quantum theory, in which observations can only reveal a probability distribution of results. Within this analogy, wave properties can arise from countering decision-maker tendencies towards variation and continuity. In seeking to improve understanding of such properties of mundane and other behavior, this paper identifies and examines a phenomenon of behavioral wave interference, the evolution of disparate parallel applications of an original core priority or concept.

Evidence for wave characteristics in decision- and conceptual model-making behavior will be most convincing from systems displaying continuous evolution of behavior between “classical,” such that utility optimization dominates, and “wave-like,” where the opposite is true. In the analogous phenomenon of wave interference in physical systems, wave propagation generates variations in wave intensity at distances from an isolated origin of intensity. This paper identifies and analyzes several familiar or readily visualized cognitive and behavioral systems as behavioral interference phenomena. Section I of this paper introduces a Hamiltonian formalism for decision-making and conceptual modeling behavior and derives a unified formalism for modeling of deterministic and probabilistic behavior. In Section II I describe the phenomenon of physical wave interference and outline the analogy to behavioral wave interference. In Section III analogies are identified and defined in detail between physical wave interference and four examples of behavioral interference in decision and conceptual model making: consumer choices subsequent to stating reasons for choices, evolution of scientific paradigms, reserve definition in insurance accounting, and sect formation in Christianity.

I. Decisions Within Ranges of Near-Optimal Choices

A. Hamiltonian Formalism for Decision and Conceptual Model Making

An individual situated in an environment in stable equilibrium (excluding, e.g., artificial environments such as prisons) will seek to maintain a spectrum of available choices for most activities and items under consideration in the course of daily life. Even in the most primitive societies, food is prepared in different ways and at different times. Clothing and shelter vary to suit each individual’s taste and requirements. Daily and seasonal rituals and routines are generally undertaken with at least minimal flexibility in scheduling and details, albeit avoiding extreme variations. Rituals and holidays departing from regular daily routine are necessary features of societal structure (Cox, 1969).

For a given choice i of activity or item to be made by an individual, assume that all activities and/or items available can be identified by various parameters. In Figure 1(a) different options concurrently available for a choice i are “located” at various points x along the horizontal x-axis. The single variable x may thus represent location of and/or relative magnitudes of resources applied towards available items and/or pursuits (Goetzmann and Spiegel, 1997; Briley, Morris and Simonson, 2000). As a specific example, choices for one meal might be classified as multi-dimensional variables x, parametrized in terms of ingredients, preparation style, amounts, etc.

For choice i, the aggregate net amount of resources necessary to undertake option x defines a “net cost” function U(x). In the example of choosing food, U(x) equals cooking labor and costs minus net physical nourishment for given choice x. When x represents choices requiring identical or no material resource expenditure, U(x) may represent intangible quantities such as motivation, experience, and training necessary to pursue the choice specified by x, net of psychological reward. In games and laboratory environments U(x) may represent perceived expectation of reward for correct or optimal decisions. Finally, the concept of net utility return can be extended to conceptual modeling, by defining in terms of total range or number of facts or observations logically explained, or inverse of number of anomalies unexplained, given choice of theoretical assumption x.

Figure 1. (a) Requisite resources U(x) as a function (dotted line) of choice of activities or items (parametrized by horizontal axis coordinate x). The minimum value of U(x) is Umin at x = 0 [u = U(x) – Umin]. (b) 2 separate choice functions x(t) plotted versus time t (vertical axis), one continuous and deterministic (curved path, left) and one discontinuous and probabilistic (solid vertical segments connected in sequence by dotted segments, right). (c) Short-term measured frequency distributions P(x) for the deterministic (smooth curve, left) and probabilistic (discrete solid bars, right) choice functions in (b). U(x) and Umin from (a) are replotted for comparison (different scale from P(x)). Long-term probabilistic frequency distribution P(x) may be a continuous function. (click graph to expand)

It is assumed that available options are such that x can be defined “naturally,” so that U(x) is a smooth function of x. In this case, when U(x) is near an optimal minimum Umin, U(x) can be visualized as a wide and shallow “bowl” of height u resting at Umin [Fig. (1a)]. If a decision-maker has finite resources u allocable towards choice i, then he/she can only afford options x such that u exceeds U(x), “filling” the bowl U(x) up to height u. In this paper I assume that U(x) does not vary within relevant time scales and that variations in x do not alter available resources u; given x can occur more than once.

Even if U(x) is time-invariant, I have earlier noted that choices x(t) change with time t. [Fig. 1(b)] Macroeconomic modeling may generate continuous and deterministic average trends and parameters x(t), implicitly assuming reduction of stochastic variance in large systems, i.e., the central limit theorem or “law of large numbers.” The dependence of such trends on U(x) is analogous to the influence of potential forces in “classical” Hamiltonian theories of physical motion. At the level of the individual or small group, however, the relative impact of random unforeseeable events and variability of behavior, or free will (Grobstein, 1994), causes future outcomes and actions x(t) to evolve with significant variance, discontinuously, and with less direct dependence on environment U(x). [PROBABILISTIC curve in Fig. 1(b)] After a brief duration of pursuing steady habits and lifestyle, consumers and even serious investors may try “impulse buying” just for “a change of pace.” Fractional percentage interest rate changes may predictably alter corporate and national production and investment activity, but have indiscernible effects on financial habits of one individual or family. As discussed in the Introduction, analogous empirical limitations on accuracy of particle measurement and subsequent research historically necessitated fundamental revisions to classical theories of particle motion, yielding the modern theory of quantum mechanics.

B. Probabilistic Decision Functions Within Hamiltonian Formalism

Self-consistency at both the level of the individual and of large populations is attained within a unified model of decision-making constructed in terms of non-negative probability distributions P(x, t) for decisions (McKelvey and Palfrey, 1995, 1998), where

P(x, t) dx = 1, (1)

rather than in terms of deterministic dynamics. As discussed in preceding subsection A., x(t) may in general vary discontinuously, with long-term frequency determined by P(x, t). The curve P(x) to right in Fig. 1(c) labeled PROBABILISTIC shows an example frequency distribution for discontinuous x(t) observed over a finite time interval. For large heterogeneous populations, the central limit theorem allows x(t) to approach deterministic behavior with expectation values satisfying smooth Hamiltonian dynamics implied by U(x). [DETERMINISTIC P(x) in Fig. 1(c), left] P(x) is intuitively expected to exhibit continuity for all systems in general. That is, probabilities of two choices should approach one value as two choices are made more similar. Evidence in support of this appears in the success of mass production, commodities and generic products, which succeed to the extent that consumer demand is the same for similar products.

As discussed in the Introduction, decision-making does not depend exclusively upon rational optimization of utility value when many near-optimal choices are available (Grobstein, 1994). It instead exhibits discernible and reproducible, if sometimes irrational and non-optimizing, patterns and/or probability distributions. Review of literature on various anomalous behavioral phenomena (Shiller, 1997; Thaler, 1992) suggests an underlying central theme of simplification and “self-filtering” of perception, or non-equilibrium belief formation (James, 1890; Lévi-Strauss, 1966; Costa-Gomes and Zauner, to be published). “Anchoring” (Kahneman and Tversky, 1974), cognitive dissonance (Festinger, 1957) and overconfidence, for example, all involve irrational adherence of thought or opinion to initial assumptions. Cognitive dissonance is specifically identified as denial of later contradicting information after initial formation of a belief. Within probabilistic systems such adherence results in arbitrary peaks in P(x), corresponding to the inertial resistance postulated to exist in continuous, deterministic systems. In other experiments individuals and entities evaluate and respond to a spectrum of choices or items in terms of a few arbitrarily defined “mental compartments.” Magical and quasi-magical thinking (Shafir and Tversky, 1992; Skinner, 1948) involve arbitrary association of causality between initial actions and events, resulting in adherence to “false” optimal choices.

The analogous phenomenon of “broken symmetry” (Anderson, 1984) occurs in physical systems in nature, such as particle spins and magnetic materials which preserve fixed magnetic alignment even after elimination of external electromagnetic influence. Broken symmetry occurs because interaction energies are optimized when directional alignments are preserved over large areas, countering entropic tendencies towards randomness. In behavioral systems, in complete analogy, when no clear process exists for optimal decision, effort and uncertainty are minimized by mental simplification, “self-filtering” to form maxima in P(x), countering fears of sub-optimal choices.

Mental simplification will be greatest when resources such as information and experience are most limited. Conversely, prior analysis and decisions will be most complete for available choices x when resources are sizeable, e.g., for macroeconomic and corporate entities. As an example, novice investors might initially limit their choices to index funds such as the Dow Jones Industrial and NASDAQ representing broad economic sectors. [Figure 2(a)] As investors gain experience, wealth, and confidence, i.e., decision resources u, they expand investment choices among more specialized economic sectors and individual business entities. A length Dx can be defined corresponding to characteristic distance between points x of maximal P(x, t). From the above discussion, Dx corresponds to degree of conceptual generalization and simplification and will vary inversely with u. In the example of investors, Dx might equal % of total market capitalization represented by each of his/her index fund choices.

Figure 2. Probability distribution P(x) of investment selections by novice investor. Investor analysis is constrained by risk and reward requirements; subsequent selection is made from universe of stocks. Formation of “mental compartments” results in selections with maximal P(x) from a few discrete broad categories of width Dx as % of total stock market capitalization. (b) Behavioral interference pattern in life insurance reserve valuation measures originating from core objective of evaluating insurance business profitability. Concurrent insurance industry focus on different business components of profitability leads to conceptual interference and definition of separate reserves. (click graph to expand)

Irrational “self-filtering” of perception and multiple maxima in P(x, t) are often direct consequences of prior external influence on or restriction of perception (James, 1890; Costa-Gomes and Zauner, to be published), manifested in experiments as hinting or suggestive format of questioning (Shiller, 1997; Briley, Morris and Simonson, 2000; Fitzsimons and Shiv, 2001). With initial restriction of perspective, P(x, t) still ultimately depends on optimality characteristics, “absolute” attractiveness, of available choices. An initial external influence may be manifested, however, via subsequent constructive and destructive “behavioral interference” in the decision-making process, enhancing or reducing the preference for various choices, generating several maxima and minima in P(x, t). In the next two sections I attempt to define behavioral interference in decision-making via analogies to wave interference in physics and examples in real life.

II. Behavioral Interference as a Wave Phenomenon

Light wave interference occurs when coherent light waves emit from a small physical aperture, propagate in the absence of intermediate obstacles, and illuminate distant surfaces in alternating concentric light and dark regions silhouetting the original aperture shape. This interference occurs due to superposition of light waves originating from different points of the aperture, alternately enhancing and “canceling out” each other at different angles. [Fig. 2(b)] The angular width Dq of one light region depends on aperture width w and wavelength L as

Dq ~ L / w. (2)

L generally increases for different types of waves as wave energy density decreases.

The proportional dependence between Dq and L is intuitively clear, since longer wavelength requires larger distances for wave phenomena to develop. An inverse dependence between Dq and w arises because waves emerging from the aperture are more uniform as w decreases. Wave interference effects then only appear at larger angles Dq, as interference between waves originating at various points of the aperture becomes significant [Fig. 2(b)].

In the formalism of Section I, restrictive conditions on a decision correspond to a restricted range of choices with low U(x). In behavioral interference this region of acceptable choices or concepts is analogous to a narrow physical aperture admitting light. wx represents “size” of this region in decision or conceptual parameter-x space. Evolving probability P(x, t) of future choices subsequent to satisfying initial restrictive conditions corresponds to the propagation of light from an aperture. P(x, t) observed for specific future decision points x corresponds to the illumination pattern observed at a distance from the aperture. In the investment example, wx might equal a small fraction of all basic asset categories deemed desirable for investment by restrictive criteria. P(x, t) then equals the actual asset allocation ultimately made from available securities.

In behavioral interference, P(x, t) subsequent to an initial “successful” choice exhibits multiple maxima analogous to the light and dark interference patterns described in the preceding paragraph. Dx, analogous to Dq in Eq. (2), expresses the degree of conceptual simplification or “compartmentalization” in choices from a spectrum of choices subsequent to an initial restricting concept or value. Smaller u implies reduced resources available for evaluation of optimal concepts or choices at conceptual “distances.” Less refinement in decision-making then occurs, corresponding to larger L. This is analogous to the energy dependence of physical waves on L. Reduced resources u and therefore larger L clearly implies greater simplification necessary in decisions and broader Dq in P(x, t), as in Eq. (2). Abundant applicable resources u and small L in contrast allows optimally rational choices and actions, corresponding to a “classically” straight-line light beam projection without wave patterns (infinitesimal Dx). Large conceptual restriction “width” wx implies a broadly applicable initial core concept or set of decision-making criteria. Extension to new decisions or situations is then easier. More acceptable and optimal choices implies narrower maxima in P(x, t) and smaller Dx.

III. Examples of Behavioral Interference

Recent research on cultural mechanisms underlying consumer decision-making (Briley, Morris and Simonson, 2000) gives striking evidence for the concept of behavioral interference. The authors themselves propose an interpretation of cultural values as a lens which may or may not be applied to shape behavioral disposition for any specific decision. Cultural lens and behavioral interference are similar metaphors, both invoking perception altered by interactions within a localized intermediate region.

The main body of their paper presents experimental data and analysis supporting the postulate that cultural influences on individual decision-makers are manifested through specific applications of reasons and values. Absent prior questioning, subjects from different cultures exhibited similar “conservative” tendencies towards choices compromising quality and low cost. Probabilities of choices P(x) were significantly altered, however, when subjects first specified their reasons prior to making their choices. Measures of individual characteristics of disposition correlated poorly with choices made, confirming that specification of reasons had a specific influence on decision-making apart from dispositional traits.

The studies by Briley, Morris and Simonson (2000) necessarily involved simplified laboratory situations. Only three choices were made available for each decision under study, limiting the applicability of a continuous model. The results can nevertheless be interpreted in terms of behavioral interference as follows. Absent prior questioning regarding reasons for choices, consumer choices of one product are analogous to an light beam propagating undeflected and therefore without dependence on l. Asking specific reasons for consumer decision-making is analogous to passing light through a restricted aperture. Decision-making subsequent to providing reasons is analogous to light emanating from the aperture. Probability distribution P(x) of reasons given and choices made among available items is analogous to the resulting light intensity pattern projected onto a distant surface parameterized by x, the degree of preference for product quality versus low cost.

In studies of decision-making sans questioning, distributions of choices centered on the compromise choice identically (40 to 50%) for groups from various cultures. The compromise choice consistently predominated regardless of actual items presented. This confirmed that items represented price/quality trade-offs of approximately equal utility value U(x), analogous to smooth surfaces well-centered beneath light projections. The dominance of compromise choices presumably reflects universal instinctive risk-avoidance at unfamiliar choice extremes (Kahnemann and Tversky, 1979). This would be analogous to the natural tendency of light beams to maintain initial direction in the absence of interference effects, fading at extreme angles from the original direction.

In the study by Briley et al., requiring reasons for decision-maker choices established awareness of restrictive priorities on decision-making. Decision-makers were forced to consider the positions of choices relative to each other, analogous to light passing through a narrow aperture. In light wave interference waves emerge from all points of an aperture [as in Fig. 2(b)] and recombine constructively along the path emerging straight ahead, at a zero angle. At slightly different angles, waves from the two ends of the aperture cancel out with each other, resulting in patterns of alternating brightness at different angles. The appearance of light interference patterns corresponds to the appearance of significant variations in P(x) in behavioral interference. For decision-makers under study by Briley et al., this arises from conscious evaluation of two product qualities available only with trade-offs. Conflicting priorities for optimization of each quality can result in increased P(x) for extreme choices.

Poor correlation between individual disposition and actual choices made in the study by Briley et al. further supports interpretation of data in terms of behavioral interference. Consumer decision-makers under questioning seek to maintain conceptual self-consistency. Independent of disposition, each reintegrates the original trade-off, quality versus low price, following different priorities with different resulting choices x. An analogous property of light wave interference is that pattern intensity regions are not identifiable with different points of the narrow aperture originating the light waves. Pattern regions arise simultaneously from each light wave passing through and interacting with the aperture.

The behavioral interference model allows quantitative interpretation and parameterization of decision-making behavior of consumers from different cultures. A major finding of Briley et al. (2000) is that questioning of American and East Asian consumers resulted in reduced and increased selection of compromise options, respectively. As discussed above, controlled questions and available choices correspond to constant wx. Differences in interference effects in P(x) can therefore be attributed solely to differences in l in Eq. (2). Increased selection of compromise options by East Asians implies narrower maxima in P(x), therefore smaller l. This was argued in Section II to correlate with greater resources u allocated to decision-making. Larger l characteristic of American decision-makers implies greater conceptual generalization in decision-making. This is consistent with conclusions of Briley et al. (2000) that American subjects tended to invoke absolute principles when giving reasons for making choices. Japanese, in contrast, have been observed to review attributes carefully for individual choices, allocating greater decision-making resources u (Myers and Simonson, 1992). A final observation in support of the above analysis is that East Asian choices P(x) systematically changed less upon questioning of reasons (3 6% versus 9 33% change by European-Americans). Greater u allocated by East Asians should in fact result in more “classical,” optimized decision-making, not subject to interference. In the analogous light-aperture system, high-energy low-l light waves exhibit weakest interference effects, minimally different from intensities emerging unrestricted through large apertures.

In the above example of behavioral interference in consumer decision-making, data supports a model of individual decision-making shaped by cultural lenses or restrictions triggered for specific decisions. As discussed in Subsection I.A., analogy to wavefunction theory generally implies that “average” choices by large populations should be more “classically” predictable and optimal. This concept is in some sense a crucial premise favoring open market over bureaucratic command economies. Relative benefits are often not easily and/or quickly valued, however, within ranges of abstract conceptual choices, e.g., theologies and scientific models. Such choices are often not readily parameterized for automatic use as with commodity prices, but must be communicated between individuals. Behavioral interference characteristic of individual decision-making may then be observed. This is analogous to independent formation of light interference patterns by different color components of initially white light on oil surfaces. The remaining examples below identify behavioral interference in evolution of collective conceptual model formation by intellectual and other communities.

A familiar example is evolution of scientific paradigms, or “thinking within the box” (Kuhn, 1996). After initial success of an intellectual and/or cultural core conceptual theme, e.g., Galilean invariance in physics or mass production in economics, the same concept is applied with imperfect extrapolation throughout an expanded regime. Interference consists of additional “dark region” caveats and assumptions that must be invoked, affirming the original concept only with interspersed anomalies. Perhaps the best-known example of scientific paradigm evolution is that of Ptolemaic geocentric theory. Classical Greek and medieval European astronomers sought to incorporate all astronomical data within models of circular orbits around the Earth in this theory. As data was obtained with increasing precision, modeling inconsistencies and complexities required introduction of increasingly complex and unwieldy modifications to the theory, e.g., including orbits within orbits and exceptional non-circular paths. The development of heliocentric and eventually relativistic models ultimately allowed successful unified modeling of all observed stellar and planetary motions based on a few axiomatic (Newton’s and ultimately Einstein’s) equations.

Geocentric theory represents a narrow conceptual restriction in astronomical theory. Advancements and error levels in measurements correspond to resource level u and wavelength l in Eq. (2). With the earliest measurement accuracy, modeling reveals few anomalies and geocentric theory is appealing in its simplicity. In terms of Eq. (2), l and ∆ are so large that interference is not visible within a diffuse conceptual projection. As inaccuracy and l decrease, modifications to geocentric orbits necessary in modeling newer data correspond to dark anomalous regions in developing conceptual interference patterns. The eventual generalization to orbit centers other than the Earth represents conceptual aperture enlargement, allowing consistent extrapolation to improved data and elimination of conceptual interference. As an aside, modern advances in air and ocean navigational tools and techniques correspond to l and ∆ reduced such that restriction to geocentric maps and globes induces no conceptual interference for modern pilots.

The historical development of life insurance accounting in the United States provides a second example of conceptual interference. Life insurance risk is conceptually unique as a business liability, involving contractual events highly uncertain in both time and magnitude, and therefore difficult to evaluate although of substantial social utility (Black and Skipper, 1994). In consequence, accounting principles have developed over decades defining numerous different life insurance liability components, each subject to different oversight and calculation. [Fig. 2(b)] Life insurance companies in the United States must calculate separate reserves for guaranteed interest and mortality contracts, premium deficiency, asset default risk (AVR), crediting guarantees (IGR), and separate accounts (NY State Reg. 128, SSA). Asset categories must be reserved for deferred acquisition costs (DAC), for first year expenses (CRVM), for interest rate fluctuation risk (IMR), as well as other conventional business depreciation. Statutory company surplus moreover must exceed calculated minimum values (RBC) to cover financial risks over a range of probabilities.

The initial core concept in this example is viability of managing contractual insurance risk, through receipt and investment of policyholder premiums over time exceeding contractual payments and administrative expenses. Insurance businesses can be characterized in a conceptual space parameterized by various accounting quantities, e.g., insurance in-force, rate of sales, premium rates, contract durations, crediting rates, etc., both in total and by type. Viability occurs in a restricted optimal region in this space yielding positive expected profit u over time, with discounting for risky business parameter configurations (e.g., speculative investments). The historical development of a variety of separate and distinct calculated reserves [Fig. 2(b)] represents a conceptual spectrum of modern business accounting valuation measures evolved from the core concept of viability. Relative size of different reserves x provides a measure of P(x, t) as the relative importance of valuation measure x in evaluating insurance business viability.

A traditional pairing between investment assets and policy reserves represented a natural formation of mental compartments in evaluating insurance activities, between simple contractual obligations to receive and to pay out monies, not distant from the original concept of business viability. In recent decades, however, many business tools and practices, swaps and other derivatives, synthetic GIC contracts, and reinsurance, are no longer easily classified within traditional accounting reserves (Donahue, 2001). Reserve categories have consequentially proliferated and increased in complexity in response to the modern expanded continuum of business practices. Resulting interference patterns appear in the arbitrary distinctions based on restrictive perceptions of insurance business.

Consistent with this interpretation, increased computing power in recent years has allowed innovations in business valuation based on techniques such as cash flow testing and stochastic scenario and sensitivity analysis. These techniques project business operations and results with unsimplified realistic assumptions, directly addressing the original core theme of long-term business viability. Increased computing power corresponds to larger u and smaller l. Consistent with Eq. (2), this allows more accurate projection and evaluation in terms of the original core concept of viability, reducing interference effects and returning to analysis of “the bottom line.”

The historical development of Christianity provides the final example of behavioral interference. The initial core concept here is proper lifestyle, defined as worship of Jesus Christ as divine Messiah and intercessionary for humanity, according to His own teachings. P(x, t) equals the percentage of population at time t espousing religious interpretation or denomination x defined within a space parameterized by different philosophical and theological themes and values. The ascetic lifestyle of early Christians (at time 0) represents a restricted region of low requisite resources U(x, 0) isolated within the diversity of sophisticated and hedonistic lifestyles in the early Roman Empire (Kiefer, 1971). The corresponding high surplus u allocable by early Christians fueled the focused evangelical drive with which Christianity has since propagated throughout the world (Chidester, 2000). Propagating the concept of the life of Christ sacrificed for all humanity encouraged expansion of Christianity both to new populations as well as to previously neglected social classes such as slaves, throughout and eventually beyond the Roman Empire. This social expansion was facilitated by and concurrent with evolution of a broad underlying intellectual framework (Kiefer, 1971; Küng, 1976), built upon the written New Testament, subsequent theological writings and thought, and adaptations from other religions, most notably Judaism. To summarize, the historical evolution of Christianity over two millennia has extended broadly in many dimensions since its early restrictive heritage of asceticism and persecution by the state.

Richly detailed cultural and intellectual structure, corresponding to small Dx, characterizes modern Christianity. A myriad different denominations have developed, Catholic, Protestant, Greek and Russian Orthodox, each with its own emphasis on distinct essential rituals and beliefs, corresponding to different peaks in P(x, t). This diversity originates in part from secular social and geopolitical forces, corresponding to non-uniform U(x, t) and interactions among various demographic groups. But to significant extent, diversity has resulted from purely intellectual developments in Christian theology (Kiefer, 1971). Many denominations are distinguished primarily by different self-consistent interpretations of the fundamental nature of Jesus Christ, as human or divine, and of God, as Unity or Trinity, mystical or rational. Associated behavioral interference is even a familiar theme in literature, e.g., in resolving the paradox of an omnipotent God allowing injustice and evil:

Human beings, in their generous endeavor to construct a hypothesis that shall not degrade a First Cause, have always hesitated to conceive a dominant power of lower moral quality than their own; and, even while they sit down and weep by the waters of Babylon, invent excuses for the oppression which prompts their tears. (Hardy, Thomas, The Return of the Native (1955), p. 434.)

In terms of Eq. (2), small Dx and diverse denominations characterizing P(x, t) results from large u (small l) of the early Christians propagating their all-inclusive theology (large wx). Modern Christian missionary efforts in underdeveloped nations provide a supplementary case study, wherein available local resources u are low (large l). Consistent with larger Dx expected from Eq. (2), Christian missionary theology has conformed more to broad core themes of faith in God and dignity of life, without significant formation of new theological schisms.

IV. Conclusions

References throughout this paper confirm that behavioral interference, the imperfect cognitive extension of an initial core concept, is a familiar phenomenon in social sciences dealing with cognitive and behavioral anomalies. I have attempted to establish formal analogy between behavioral and physical wave interference through analysis of four example applications. Oscillatory wave-like characteristics in conceptual model and decision-making probabilities arise under restrictions of limited resources for decision-making, due to simultaneous behavioral preferences for variance and conceptual continuity and simplification. In a more complete extension of the analogy between behavioral and physical wave theory, I derive elsewhere (Wang, preprint) that

P(x, t) = |Y(x, t)|2, (3)

where Y(x, t) is a complex scalar wavefunction analogous to quantum physical wavefunctions obeying Schrödinger’s equation under influence of arbitrary U(x) (Feynman and Hibbs, 1965; Landau and Lifshitz, 1977). Underlying P(x, t) of non-negative magnitude, Y(x, t) is thus a wave intensity, summed directly when superposing two or more initial component behavioral wavefunctions.

Wavefunction continuity simultaneous with variance implies practical absolute limits on predictability and control even of mundane behavior, (Cox, 1969; Grobstein, 1988, 1994; McKelvey and Palfrey, 1995, 1998). Anomalous behavioral interference frequency patterns arise not from errors but from balancing conflicting requirements for efficiency and minimal error given limited resources available in evaluating new situations and information. In two of the examples in this paper, scientific paradigms and insurance reserve accounting, increased conceptual resources, data and computational power have eventually exceeded initial limits, reducing anomalous behavioral interference effects. In the other two examples, however, limits have not been so readily overcome. Fundamental cultural values and priorities can significantly influence resource allocation towards consumer decisions. Intellectual resources are finite in extending knowledge of an infinite God. Such intrinsic limitations can be ignored in social, economic, and educational planning only at great cost and inefficiency. Recognition and quantitative understanding of these limitations can aid towards minimal unnecessary conflict and discomfort and maximal enhancement of individual satisfaction and productivity in societies.

I wish to acknowledge invaluable discussions with and/or helpful criticism from John Kao, Scott Schaffer, Susan C. S. Wang, and the reviewers of this paper for JMB.

Works Cited

Allais, M. “Le Comportement de l’Homme Rationnel devant le Risque, Critique des Postulats et Axiomes de l’Ecole Americaine.” Econometrica 21 (1953): 503-46.

Anderson, P. W. Basic Notions of Condensed Matter Physics. Menlo Park: Benjamin/Cummings, 1984.

Black, Jr., K. and H. D. Skipper, Jr. Life Insurance, 12th ed. Englewood Cliffs: Prentice Hall, 1994.

Briley, Donnel A., Michael W. Morris, and Itamar Simonson. “Reasons as Carriers of Culture: Dynamic versus Dispositional Models of Cultural Influence on Decision Making.” Journal of Consumer Research 27 (September, 2000): 157-78.

Chidester, David. Christianity: A Global History. San Francisco: Harper, 2000.

Costa-Gomes, M. and K. G. Zauner, ‘Learning, Non-equilibrium Beliefs, and Non-pecuniary Payoffs in an Experimental Game,’ to be published.

Cox, Harvey. The Feast of Fools: A Theological Essay on Festivity and Fantasy. Cambridge: Harvard University, 1969.

Day, R. H. “Complex Dynamics, Market Mediation and Stock Price Behavior.” North American Actuarial Journal 1 (1997): 6-23.

Donahue, Paul J. “The Stable Value Wrap: Insurance Contract or Derivative? Experience Rated or Not?.” Risks and Rewards 37 (2001): 18-25.

Festinger, L. A Theory of Cognitive Dissonance. Stanford: Stanford University, 1957.

Feynman, R. P. and A. R. Hibbs. Quantum Mechanics and Path Integrals. New York: McGraw-Hill, 1965. In quantum electrodynamics a fundamental ansatz extends Schrödinger’s equation to massless electromagnetic field quanta, i.e., photons; see pp. 230-1.

Fitzsimons, Gavan J. and Baba Shiv. “Non-Conscious and Contaminative Effects of Hypothetical Questions on Subsequent Decision-Making.” Journal of Consumer Research, to be published (September, 2001).

Goetzmann W. and M. Spiegel (1997), ‘A Spatial Model of Housing Returns, and Neighborhood Substitutability,’ Journal of Real Estate Finance and Economics, p.14.

Grobstein, P. “From the head to the heart: some thoughts on similarities between brain function and morphogenesis, and on their significance for research methodology and biological theory.” Experientia 44 (1988): 960-71.

Grobstein, P. “Variability in Brain Function and Behavior.” The Encyclopedia of Human Behavior, Vol. 4. Ed. V. S. Ramachandran. London: Academic Press, 1994.

Hardy, Thomas. The Return of the Native. New York: Washington Square Press, Inc., 1955: 434.

James, W. Principles of Psychology. New York: Dover Publications, 1890 (reprinted 1950).

Kahneman, D. and A. Tversky. “Prospect Theory: An Analysis of Decision Theory Under Risk.” Econometrica 47 (1979): 263-91.

Kahneman, D. and A. Tversky (eds.). Choices, Values and Frames. New York: Cambridge University and the Russell Sage Foundation, 2000.

Kiefer, Otto. Sexual Life in Ancient Rome. London: Abbey Library, 1971.

Kuhn, T. S. The Structure of Scientific Revolutions. Chicago: University of Chicago, 1996.

Küng, Hans. On Being a Christian. Trans. from Christ Sein by Edward Quinn. Garden City: Doubleday and Company, 1976.

Landau, L. D. and E. M. Lifshitz. Quantum Mechanics. Oxford: Pergamon, 1977.

Lévi-Strauss, C. The Savage Mind. Chicago: The University of Chicago, 1966.

McKelvey, R., and T. Palfrey. “Quantal Response Equilibria for Normal Form Games.” Games and Economic Behavior 10 (1995): 6-38.

McKelvey, R., and T. Palfrey. “Quantal Response Equilibria for Extensive Form Games.” Experimental Economics 1 (1998): 9-41.

Myers, John and Itamar Simonson. “The Compromise Effect in a Cross-Cultural Context.” Unpublished, 1992.

Savage, L. J. “The Sure-Thing Principle.” The Foundations of Statistics. Leonard J. Savage. New York: John Wiley, 1954.

Shafir, E. and A. Tversky. “Thinking through Uncertainty. Nonconsequential Reasoning and Choice.” Cognitive Psychology 24 (1992): 449-74.

Shiller, Robert J. “Human Behavior and the Efficiency of the Financial System.” Handbook of Macroeconomics. Eds. John B. Taylor and Michael Woodford. New York: Elsevier Science, 1997.

Skinner, B. F. “Superstition in the Pigeon.” Journal of Experimental Psychology 38 1948: 168-72. (reprinted in same journal, 121 (3) 1992: 273-4)

Thaler, Richard H. The Winner’s Curse: Paradoxes and Anomalies of Economic Life. Princeton: Princeton University, 1992.

Tversky, A. and D. Kahneman. “Judgment Under Uncertainty. Heuristics and Biases.” Science 185 1974: 1124-31.

Wang, J. K. “Discussion on ‘Complex Dynamics, Market Mediation and Stock Price Behavior,’ by Richard H. Day.” North American Actuarial Journal 2 1998: 117-8.

Wang, J. K. “A Wave Mechanical Theory of Decision-Making.” Working paper.

Author: Joseph Wang originally obtained M. A. and Ph. D. degrees at Princeton University in condensed matter physics. Following interests in mathematical modeling in social sciences, he completed actuarial examination work for an F. S. A. designation with specialization in Investments and is currently a pension investment product actuary. Research interests are in mathematical modeling of behavioral phenomena in social sciences, economics and finance theory in particular.