There is no need to use a number of axioms to explain the phenomenon of the "golden section" (0.618) in human binary choice. This parameter reflects the mean value of the majority of voices under conditions of uncertainty. A random distribution of choices -- which may be equiprobable (rectangular), needle-like, or, in the most common case, near-Gaussian (bell shaped) -- determines the mean value of dominant choices which differ just slightly from the "golden section". The negligible difference is about one per cent. Lefebvre's axiomatic theory stimulates the search for general quantitative regularities in human choice. The search, which is based on neurophysiological data like ours, is an alternative one. It explains certain peculiarities of random choice distributions.
Y=a0+a1x1+a2x2+a3x3+a4x1x2+a5x1x3+a6x2x3+a7x1x2x3, (1)
and found, with a number of postulates and axioms, that
Y=x1+x3(1-x1-x2+x1*x2), (2)
where x1, x2, x3 are variables extracted from explained psychological data. Their function Y may be equal in part to one of them.
2. Equation (2) is the main output of Lefebvre's theory for quantitative explanation of psychological data relating to human choice.
3. It is very interesting to investigate human behavior in the situation of uncertainty, when available stimuli, which are needed to choose only one act among others, are absent or not different in their intensity. If probability Pmin to do a certain single selection is less than 0.5, then instead of it one must keep in mind an alternative choice, probability P of which is
P=1-Pmin. (3)
4. According to the condition of uncertainty, one can assert that the most probable value is P=0.5, and the distribution of probabilities at first approximation is a bell curve with its maximum at 0.5. One extreme case is a needle-like distribution, the so called delta distribution. Another extreme case is rectangular distribution with a just noticeable maximum at the 0.5 level. In the first case, probability P=0.5. In the second case, mean value does not exceed (1-0.5)/2 = 0.75, again in accordance with the condition of uncertainty. In all other cases 0.5 < P < 0.75. Therefore, at first approximation, its most probable value is equal to (0.5+0.75)/2 = 0.625. I underline that it is only a first approximation. The second approximation is based on most probable near Gaussian distribution (i.e., neither rectangular nor needle-like) of probabilities in the same range. I mean that the distribution of searched probabilities differs not significantly from Gaussian law with minimal probability -1.0 corresponding to -3*sd, and maximal probability +1.0 corresponding to +3*sd, where sd is the standard deviation, and asterisk means multiplication. In this case, the majority (3*sd+0.67*sd) of results divided by the range of all results of votes (6*sd) is equal to 0.612. This means that if every subject votes randomly with probability 0.5, then the searched mean value of the majority of voices is equal to 0.612. Lefebvre has found other theoretical values using formula (2). Substituting in it the values x1=x2=1-x3 and supposing that Y=x3 he has found the majority of voices to be Y=(SQRT5-1)/2=0.618. Our theoretical result differs a bit from 0.618 which is equal to the "golden section." The negligible difference is about one per cent.
5. In accordance with the data of many authors in Table 1 (Lefebvre, 1995), its real values are equal to 0.60, 0.61, 0.62, 0.66, 0.63, 0.62, 0.61, 0.63, and 0.62, with mean, 0.622. The difference is 0.625 - 0.622 = 0.003.
6. Lefebvre has found an alternative solution using formula (2). Substituting in it the values x1=x2=1-x3 and supposing that Y=x3, he has found that Y=(SQRT5-1)/2=0.618. The difference is just the same and may be a bit worse, 0.622 - 0.618 = 0.004.
7. Until now, both explanations are approximately equal in the sense of correctness of prediction, but our explanation requires fewer different postulates, axioms, and limits.
8. Nevertheless it is interesting to analyze concrete possible cases of distribution of probabilities P in the range 0.5 and 1. Earlier, after Zipf (1935), we found (Lebedev, 1983) using our own neurophysiological premises that probabilities of actualization of different memory items correspond to the harmonic row
1 : 1/2 : 1/3 : 1/4 ... 1/M, (4)
where M is the number of different items (the number of memory images) which are in active state simultaneously, like conceptual or perceptive codes in the model of Atkinson (1974).
9. The harmonic row is determined by cyclic structure of supposed neuronal codes of human memory (Lebedev, 1990).
10. It follows from (4) that in case of two memory images which are in an active state simultaneously, one of them dominates with probability 0.667. Another is in an active state with P=0.33 in correspondence with the harmonic row
1: 1/2 = 2/(1+2) : 1/(1+2) = 2/3 : 1/3.
11. By the way, because the second probability is less than 0.5, one must use, in accordance with condition (3), the formula 1 - 1/(1+2). Keeping this condition in mind, we have found the probabilities of actualization of memory images in case of 3 active memory images to be:
6/(6+3+2): 1-3/(6+3+2): 1-2/(6+3+2) = 0.56 : 0.73 : 0.82. (5)
In case of 4 active images, the row of computed probabilities is
13/25 : 19/25 : 21/25 : 22/25 = 0.52 : 0.76 : 0.84 : 0.88. (6)
12. In total, keeping in mind only these possible cases, we have found that peaks of distribution of probabilities in the range 0.5 and 1 are equal to 0.52, 0.56, 0.67, 0.76, 0.82, 0.84, 0.88. In reality, they are equal to 0.51, 0.55, 0.61, 0.64, 0.66, 0.71, 0.74, 0.76, 0.79, 0.82, 0.85, about 0.89 and 0.93 in accordance with graph 3.2.1 in the Lefebvre's book (1991, p.46). The graph shows real distributions of results of numerous votes in California during referendums in the years 1884-1990.
13. One can see that all computed peaks correspond to proper ones among observed peaks with a minimal difference of about 0.01. One of the real peaks at 0.61 corresponds to the above mentioned theoretical values of 0.625 and 0.618.
14. Lefebvre uses rows of parameters in his model such as
1/M : 1/(M-1): .. : 1/3 :1/2 : 1:2:3:4 .., (7)
and this solution may not only be philosophically based, but neurophysiologically based as we shall see.
15. A few suppositions, based on neurophysiological data related to cyclical structure of memory codes, explain a peculiarity of human choice in uncertain circumstances as well as some peculiarities of language, memory, and sensations (Lebedev, 1983; 1993).
16. Lefebvre explains the phenomenon of categorical scaling. For this purpose he derives equation (8) from formula (2)
Y= x1/(x1+x2-x1*x2), (8)
where Y = (R-1)/(M-1) is the subjective categorical estimation of a stimulus in relative units. M is equal to the upper limit of the categorical scale and R is the current categorical estimation; x1 is an absolute estimation on Stevens's magnitude scale, and x2 is a parameter which determines the level of convexity of the curve which is to be predicted.
17. Another solution is derived from our model of neuronal mechanism of subjective estimations (Zabrodin & Lebedev, 1976). If R is an estimation of a stimulus with intensity S, and d is the differential, then, in accordance with Zabrodin, one can write
dR/RZ =dS/S, (9)
where denotes "to the power of", Z is a parameter connected with neurophysiological constants like dominant frequency of alpha rhythm in human electroencephalogram and step-like difference between phases of alpha waves as well as with some conditions of measurement (Lebedev & Myshkin, 1988; Lebedev, 1993). If Z=0, then one can observe Fechner's logarithmic scale. If Z=1, then Stevens's magnitude scale exists. But when this parameter is about Z=0.5, one can see categorical scale.
18. Our solution differs from Lefebvre's formula (8), because ours explains not only categorical estimations, but direct magnitude estimations as well as just noticeable differences without suppositions, which can not be derived from experiment. Is it good or bad that "without"?
19. About the comparison of a human brain with a machine: In my opinion the brain is not like a heat machine. Highly organized in time and space, cyclic neuronal processes constitute the basis of human cognition, emotions, and will, as well as the human internal world in total.
20. There is no doubt that in the 21st century the neuronal code of the human subjective world will be discovered to be cellular, as subatomic, atomic, and molecular codes have been discovered in physics, chemistry and biology.
21. As well, new broad and simple laws of human cognition, together with a small number of easy understable and measurable but universal neuronal constants, will be discovered.
22. Together with Luce (1986), Link (1992), Geissler (1990), and other scientists of the same ilk, Lefebvre strongly stimulates with his publications the search for universal principles of brain functioning. I was witness to great interest in his scientific paper presented at the 23rd Meeting of the European Mathematical Psychology Group in Moscow in 1993.
23. Hence, despite my critical remarks, I highly esteem the paper of Lefebvre overall.
Atkinson R.C., Herrmann, D.J. and Wescourt K.T. (1974) Search process in recognition memory. In: R.I. Solso (ed.). Theories in cognitive Psychology. The Loyola Symposium. Hillsdale, New Jersey, Erlbaum Associates, p.101-146.
Geissler H.-G. (1990) Foundation of quantized processing. In: Psychophysical explorations of mental structures. Edited by H.-G. Geissler. Hogrefe & Huber Publication, p. 303-310.
Lebedev A.N. (1983) The regularities of words reappearances within texts. (in Russian). Psychologichesky journal, vol. 4, No. 5, pp. 11-22.
Lebedev A.N. (1990) Cyclical neural codes of human memory and some quantitative regularities in experimental psychology. In: Psychophysical explorations of mental structures. Edited by H.-G. Geissler. Hogrefe & Huber Publication, pp. 303-310.
Lebedev A.N. (1993) Derivation of Stevens's exponent from neurophysiological data. - Behavioral and Brain Sciences, vol. 16, N81, pp. 152-153.
Lebedev A.N. and Myshkin I.Yu. (1988) Neurophysiological account for some regularities of acoustic and visual perception. In: Psychophysiology of cognitive processes. Proceedings of 3-rd Soviet-Finnish Symposium on Psychophysiology, Moscow., Institute of Psychology RAS, pp. 173-177.
Lefebvre, Vladimir A. (1995) The Anthropic Principle In Psychology and Human Choice. PSYCOLOQUY 6(29) human-choice.1.lefebvre.
Lefebvre, Vladimir, A. (1991) Formula of a man (in Russian). Moscow. Edition "Progress", p. 107.
Link, Stephen W. (1992) The wave theory of difference and similarity. Lawrence Erlbaum associates, Publishers, p. 373.
Luce, R.D. (1986) Response times. Oxford Science Publication, p. 562.
Zabrodin, Yu.M. and Lebedev, A.N. (1976) On the relationships between psychophysical laws. In: Advances of Psychophysics. Ed. by H.-G. Geissler and Yu. M. Zabrodin. VEB Deutscher Verlag der Wissenschaften, Berlin, pp. 399-410.
Zipf, G.K. (1935) The psycho-biology of language. Boston.
psycoloquy.97.8.09.human-choice.9.lefebvre Wednesday 9 July 1997 ISSN 1055-0143 (15 paragraphs, 8 equations, 2 references, 170 lines) PSYCOLOQUY is sponsored by the American Psychological Association (APA)
Copyright 1997 Vladimir Lefebvre
THE PRINCIPLE OF FREEDOM
Reply to Bulitko on Human-Choice
Vladimir A. Lefebvre
School of Social Sciences
University of California, Irvine
Irvine, California
VALEFEBV@UCI.EDU
ABSTRACT: Bulitko (1997) has suggested expanding my model by
replacing Axiom 1 with a more general assumption. It is shown in
this response that such an expansion is incompatible with the
principle of freedom. It is also proven that Axiom 1 can be
replaced by a formal analogue of the statement that the subject has
free will.
1. Before analyzing the possibility of the expansion of my model (Bulitko, 1997, par. 4) I would like to briefly outline my approach. I begin constructing the model by formulating a general abstract statement which I call the "principle of freedom": we humans have free will and also, under certain circumstances, freedom of choice.
2. I chose the following equation as a formal basis for the model of man:
X1=f(x1,x2,x3), (1)
where variable x1 corresponds to the influence of the external world and variable x2 to the subject's image of this influence. Variable x3 reflects the subject's intention: the value of x3 is the probability of his intention to choose the positive pole. And variable X1 reflects the subject's behavior: it is the probability of his choosing the positive pole in reality.
3. It is clear that the values of x3 and X1 belong in the interval [0,1]. I assume that the values of x1 and x2 also belong in this interval.
4. Further, equation (1) is specified in such a way that it reflects the principle of freedom. First, I introduce a formal analogue of the statement "we have free will." The meaning of this statement is that our intentions have no restrictions. In relation to the representation of a subject by equation (1), this means that variable x3 can take on any value in the interval [0,1], independently of the values taken on by x1 and x2.
5. This definition is not trivial. Suppose we choose a function X1=f(x1,x2,x3) such that for a certain pair of values x1 and x2 there exists x3=c such that X1<0. Since the probability is never negative, we conclude that the subject cannot have intention x3=c, that is, he does not have free will. Therefore, we will have to reject this particular function.
6. Second, I introduce the formal analogue of the statement that "under certain circumstances we have freedom of choice." In relation to equation (1) this means that there must be at least one pair of values x1=a and x2=b such that any subject's intention turns into an action, that is, for any x3 equation f(a,b,x3)=x3 holds. Therefore, in the framework of our approach the concepts of free will and freedom of choice are different.
7. The statement that we have freedom of choice only under certain circumstances implies the assumption that there might be circumstances under which we do not have freedom of choice. In the target article (Lefebvre, 1995) I introduced formal analogues for such circumstances and specified the principle of freedom in the framework of the model of man represented by equation (1). As a result three axioms were formulated:
Axiom 1: f(0,0,x3)=x3, Axiom 2: f(0,1,x3)=0, Axiom 3: f(1,x2,x3)=1,
where x2 and x3 are any numbers from [0,1].
8. These limitations are not enough to find a specific function X1=f(x1,x2,x3). So I assumed that this function becomes linear if any pair of variables are fixed. I justified this choice by the desire to find the simplest function. Bulitko (1997) criticizes me for choosing the criterion of simplicity in this way. I agree that other criteria might also exist, but I will hold to this one because it reflects my intuitive understanding of a function's simplicity. This criterion allows me to represent the right-hand part of equation (1) in the form
X1=a0+a1x1+a2x2+a3x3+a4x1x2+a5x1x3+a6x2x3+a7x1x2x3 (2)
and then, with the help of axioms 1, 2, and 3, to find the value of ai. As a result we obtain the following function:
X1=x1+(1-x1)(1-x2)x3. (3)
The subject represented by this function has freedom of choice when x1=x2=0. He also has free will because for any x3, independent of x1 and x2, the value of X1 belongs in the interval [0,1].
9. Axiom 1 determines the values of parameters a0=0 and a3=1. Bulitko (1997, par. 4) attempted to expand the model by lifting the limitations given by Axiom 1 and providing a subject with freedom of choice at values of a0 and a3 other than a0=0 and a3=1. I will show later that such an expansion is incompatible with the principle of freedom.
10. By applying Axioms 2 and 3 (without Axiom 1) to equation (2) we obtain the function
X1=x1+a0(1-x1)(1-x2)+a3(1-x1)(1-x2)x3. (4)
The subject represented by equation (4) has freedom of choice, if the values of a0,a3,x1, and x2 are such that for any x3 from [0,1] the condition X1=x3 holds.
11. It follows from (4) that the necessary and sufficient conditions for the identity X1=x3 to hold are the following:
x1+a0(1-x1)(1-x2)=0, (5)
a3(1-x1)(1-x2)=1. (6)
It follows from these equations that a0<=0 and a3>=1.
12. To clarify the main idea of the further analysis, we will consider an example. Suppose we want the identity X1=x3 to hold for x1=x2=(1/2). Substituting these values into (5) and (6) we find that a0=-2 and a3=4. So for this case the equation (4) looks as follows:
X1=x1-2(1-x1)(1-x2)+4(1-x1)(1-x2)x3. (7)
If x1=x2=0 the subject is represented by equation
X1=-2+4x3. (8)
It is easy to see that this subject cannot have intention x3<(1/2), since it would make X1<0, and probability cannot be negative. Therefore, although the subject represented by equation (7) has freedom of choice at x1=x2=(1/2), he does not have free will, because at x1=x2=0 a set of "forbidden" intentions x3<(1/2) appears. Thus, equation (7) must be rejected.
13. Let us consider now a general case given by equation (4). (A) Let a0<0 and x1=x2=0. With x3=0 the value of X1=a0<0. Therefore, at a0<0 the subject does not have free will, and the only acceptable value of a0 is 0. (B) Let a0=0, a3>1 and x1=x2=0. We choose x3=1 and find that X1=a3>1, but probability cannot be greater than 1, therefore, at a3>1 the subject cannot have intention x3=1, that is, he does not have free will, and the only acceptable value for a3 is 1.
14. Thus we have proven that any expansion of the model by allowing the variables a0 and a3 to take on values different from a0=0 and a3=1 leads to the subject's losing free will. Moreover, we have demonstrated that our Axiom 1 is superfluous from the formal point of view. It can be replaced by the condition that the subject has free will. To include this condition, it is enough to assume that the function X1=f(x1,x2,x3) is such that for any three values of x1, x2, x3 in the interval [0,1] the value of X1 also belongs in the interval [0,1]. With such introduction to this model, Axiom 1 acquires the status of a theorem.
15. In conclusion I would like to emphasize that the principle of freedom, as well as the anthropic principle, is a means for the rejection of models rather than for their deduction.
Bulitko, V.K. (1997) Lefebvre's Principle of Freedom and One Alternative Approach. PSYCOLOQUY 8(05) human-choice.8.bulitko.
Lefebvre, V.A. (1995) The Anthropic Principle in Psychology and Human Choice. PSYCOLOQUY 6(29) human-choice.1.lefebvre.