Conclusions and discussions

The article can be concluded by the five main and some additional items:

1) Problems. There are the well-known problems of prospect theories (see, e.g., Hey and Orme 1994, Kahneman and Thaler 2006, Thaler 2016): The choices of the subjects (people) don’t correspond to the expectations of the outcomes.

Some of the typical problems consist in the comparison of sure and probable outcomes (see, e.g., Kahneman and Tversky 1979, Thaler 2016). They are the most pronounced near the boundaries of intervals. Some of them have opposite solutions for different domains. For example, Thaler (2016) states (the boldface is my own):

“We observe a pattern that was frequently displayed: subjects were risk averse in the domain of gains but risk seeking in the domain of losses.”

These problems can be represented in the simplified and demonstrable form by the qualitative and special qualitative problems (or that of the equal expectations for the probable and sure outcomes) that are considered in the present article similar to Harin (2012b). The special qualitative problems are:

First domain. Gains. Choose between:

A) A sure gain of $99.

B) 99% chance to gain $100 and 1% chance to gain or lose nothing.

The expectations are

% 99 100

$ 99

$ 99

$

% 100 99

$ × = = = × .

Second domain. Losses. Choose between:

A) A sure loss of -$99.

B) 99% chance to loss -$100 and 1% chance to gain or lose nothing.

The expectations are

% 99 100

$ 99

$ 99

$

% 100 99

$ × =− =− =− ×

− ,

The expected values are exactly equal to each other in the both domains. A wealth of experiments (see, e.g. Kahneman and Tversky 1979, Starmer and Sugden 1991, Thaler 2016) proves nevertheless that the choices of the subjects are essentially biased. Moreover as is pointed out, e.g., in Thaler (2016), they are biased in the opposite directions for gains and losses. These are the well-known and fundamental problems that are usual in behavioral economics and other sciences.

40

2) Analysis of the problems. A new analysis of these problems was developed in recent years (see, e.g., Harin 2012a, Harin 2012b, Harin 2015). The analysis is founded on the idea of the non-zero forbidden zones studied here and enables at least qualitative explanation of these problems (see, e.g., Harin 2012b).

3) Mathematical support for the analysis. The forbidden zones theorem is proven in the present article. The theorem states that, for a finite interval [a, b]

under the condition of existence of some non-zero minimal variance σ²Min : σ² ≥ σ²Min > 0, the expectation µ of the measurement data is separated from the boundaries a and b of the interval [a, b] by the non-zero forbidden zones

a b b b

a a b

a ^{Min Min} <

− −

≤

− ≤ +

< σ² µ σ²

.

In other words, the theorem proves the possibility of existence of the non-zero forbidden zones that were used in the above analysis. The forbidden zones can exist near the boundaries of the intervals of the measurement data. The theorem also determines the conditions of the existence of the zones and their minimal width.

41

4) Mathematical approach for the analysis. The mathematical approach of the biases of the expectations (or, simpler, approach of biases, or, simple, approach) is founded on the theorem and is to explain not only the objective situations but also and mainly the subjective behavior and choices of subjects.

The two main presuppositions of the approach are:

1. The subjects make their choices (at least to a considerable degree) as if there were some biases of the expectations of the outcomes.

(This presupposition can be supported, at least formally: such biases may be proposed and tested even only from the purely formal point of view)

2. These biases (real biases or subjective reaction and choices of the subjects) can be explained (at least to a considerable degree) with the help of the theorem.

The supposed main general relationships of the approach can be accumulated into the three groups (partially corresponding to the above presuppositions):

1) The relationship (7) of the non-zero difference between the biases for the choices

0

|

|

: >

∃d_Choice d_Choice or sgnd_Choise ≠0.

2) The relationships (8) of the theorem and biases of the choices

2^Min >0

σ

and d_Choise =O(d_Theorem)_.

3) The relationships (9) of the probable and sure outcomes and choices

|

|

|

|∆_Prob > ∆_Sure or sgnd_Choise =sgn∆_Prob.

Here Δ^Prob , Δ^Sure and dChoise ≡ Δ^Prob - Δ^Sure – are appropriately the biases of the expectations of the data for the probable and sure outcomes and their difference, that is required to obtain the data corresponding to these choices; dChoise – is the difference that can be obtained by the theorem.

The first stage of the approach consists in the qualitative mathematical explanation of the qualitative problems.

42 5) Mathematical models for the analysis.

5.1) Basics of general qualitative model. The basics of the general formal preliminary qualitative mathematical model are developed in the present article.

The supposed main general relationships additional to the approach are dµ

d_Choice sgn

sgn ≠ and |d_Choice|≥|d_µ |,

where dµ≡ µProb - µSure – is the observed difference between the expectations.

The general model enables formal solutions of the qualitative problems considered here, but the limits of its applicability need additional research.

5.2) Special qualitative model. The special qualitative mathematical model is intended for the practical analysis of the above problems in the special cases when the expectations for the probable and sure outcomes are exactly equal to each other. The additional relationships (11) of these special cases can be written as

0

sgnd_µ = or d_µ =0 or µ_Probable=µ_Sure.

The model can be considered as the first step of the first stage of the approach.

The special qualitative mathematical model implies the application of the forbidden zones theorem under the additional facilitating supposition:

Due to relationships (9), the bias for the probable outcomes |∆_Probable|>0 should be non-zero but can be as small as possible. Therefore the minimal variance of the measurement data for the probable outcomes can be supposed to be equal to an arbitrary non-zero value that is as small as possible to be evidently explainable in the presence of a common noise and scattering of the data.

43

Numerical examples. In the scope of the special model, suppose that the biases of the expectations are equal, for example, to ΔProb = $2 for the probable outcomes and ΔSure = $1 for the sure outcomes. Then we have:

First domain. Gains. In the case of gains we have 98 directed from the boundary to the middle of the interval and, hence, decrease the modules of the values and the both values themselves. Therefore the biased sure expected value is more than the biased probable one

97

$ 98

$ > .

The sure gain is evidently more preferable than the probable one and this choice is supported by a wealth of experiments.

Second domain. Losses. In the case of losses we have 98 directed from the boundary to the middle of the interval and, hence, reduce the modules of the values but, due to their negative signs, increase the both values.

Therefore the biased sure expected value is less than the biased probable one 97

$ 98

$ <−

− .

The probable loss is evidently more preferable than the sure one and this choice is supported by a wealth of experiments.

So, the special model enables the qualitative analysis and qualitative explanation for the above special problems in more than one domain.

Main mathematical contributions. The four main particular applied mathematical contributions of the present article are the mathematical support, approach and special qualitative mathematical model for the above analysis and the successful uniform application of this model in more than one domain.

The author has not revealed in the literature such a natural, uniform and successful application of any model in more than one domain of the discussed problems. Therefore, instead of seeming simplicity, the successful natural and uniform application of the special qualitative mathematical model in more than one domain belongs also to the main contributions.

44

Possible additional contributions. The two more possible additional general applied mathematical contributions can be preliminary mentioned:

Possible general addition. Noise. In addition, possible general consequences and applications of the theorem for a noise are preliminary considered.

In particular, suppose that some type of noise leads to a non-zero minimal variance σ²Min : σ²> σ²Min > 0 of a random variable. Then the theorem predicts (6) the existence of the forbidden zones having the width rNoise which is not less than

a r

_Noise

b

^Min

≥ σ −

²

.

The future goal of this consideration is a general mathematical description of the possible influence of a noise near the boundaries of finite intervals.

Possible general addition. Biases. In addition, possible general applications of the theorem for biases of measurement data are preliminary considered.

In particular, if the minimal variance of the data σ²Min is non-zero, that is if σ²> σ²Min > 0, then the theorem predicts the biases of measurement data in general cases. The biases have the opposite signs near the opposite boundaries, are maximal near the boundaries and tend to zero in the middles of the intervals. Right against the boundaries of intervals, the absolute value ΔBias of the biases (6) is not less than

a b

Min Bias

≥ −

∆ |

²

| σ

.

The future goal of this consideration is a general mathematical description of the biases of measurement data that can be caused by the above forbidden zones.

Two main future questions. The first main question for future research is to analyze the widths of the forbidden zones for various types of distributions. The second main future question is to define rigorously the term “non-negligible noise”

of measurements and prove that any non-negligible noise of measurements causes some non-zero minimal variance of the measurement data or, at least, to rigorously determine such types of a noise.

Acknowledgements

The author wishes to say many words of gratitude to Professor A. A.

Novosyolov for his long-term support and methodological tutorship.

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Appendix. Lemmas of variance maximality conditions Preliminaries

The initial particular need is the mathematical support for the analysis (see, e.g., Harin 2012a, Harin 2012b and Harin 2015) of the problems of behavioral economics. These problems take place for the discrete finite random variables. The support for the discrete distributions is given in Bhatia and Davis (2000). Let us give an alternative support for the general case.

In the general case, we have for the random variable of subsection 2.1

2

under the condition (1) that either the probability mass function or probability density function or alternatively both of them are not identically equal to zero

1

Pairs of values whose mean value coincides with the expectation of the random variable were used, e.g., in Harin (2013). More arbitrary choice of pairs of values was used in Harin (2017). Here every discrete and infinitesimal value will be transformed, namely divided into the pair of values in the following manner:

Let us divide every value p(xk) into the two values located at a and b

The total value of these two parts is evidently equal to p(xk). The center of gravity of these two parts is evidently equal to xk.

The total value of these two parts is evidently equal to f(x). The center of gravity of these two parts is evidently equal to x. So these divisions (transformations) do not change the expectation of the random variable.

Let us prove that the variances of the divided parts are not less than those of the initial parts.

51

A1. Lemma 1. Discrete part

Lemma 1. Discrete part lemma. If the support of a random variable X, is an interval [a,b]: 0<(b−a)<∞ and its variance can be represented as

then the inequality

∑

Proof. Let us find the difference between the transformed

∑

= 

expressions for the variance.

Let us consider separately the cases of xk≥ µ and xk≤ µ.

52 the inequality

µ expressions for the variance is non-negative.

53 expressions for the variance is non-negative as well.

54

A.1.3. Maximality So the difference



is non-negative.

Let us calculate the difference between the transformed and initial expressions of the discrete part of the variance

∑

Every member of a sum is non-negative, as in the above expression. Hence the total sum is non-negative as well. The lemma has been proven.

So for the discrete case the variance is not more than that for the probability mass function which is concentrated in the two boundary points a and b.

55

A.1.4. Theorem of Huygens-Steiner

Besides, in the initial expression of the discrete part of the variance

∑

= 

the expression

) can be identically rewritten to

) can be identically rewritten to

Im Dokument Forbidden zones for the expectation. New mathematical results for behavioral and social sciences (Seite 40-62)