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4.4 Financial and social systems The methods of statistical physics and thermodynamics have been widely used for economic, financial, and social systems, as can be inferred from the reviews (Baumgärnter 2004; Smith and Foley 2008; Castellano et al. 2009; Yakovenko and Rosser 2009). A statistical description, characteri... |
disagreement function , one often uses Hamiltonians of spin systems (Chowdhury and Stauffer 1999; Zhou and Sornette 2007; Stauffer 2008; Harras and Sornette, 2011). In applications to social systems, one uses a constraint that is equivalent to the system energy and is called |
system frustration , or conflict (Galam and Moscovici 1991; Galam 1996; Florian and Galam 2000; Gallo et al. 2009). Markets or social groups are, certainly, finite systems, hence, they can be characterized by the distributions of type ( 27 ). The free energy for financial systems can be defined (Smith and Foley 2008) a... |
Extending the definition of free energy to social systems, one defines it in the following way. The system energy [MATH] of a state [MATH] can be termed the state cost . The society temperature [MATH] has the meaning of the intensity of noise produced by the surrounding playing the role of a thermostat. The noise energ... |
[EQUATION] Being a finite system, a society cannot be in a single pure state, but always possesses finite probabilities of being in different states. Phase transitions occur from one dominant state to another, as is described in Sec. 4.3. Continuous transitions correspond to fast evolutions, while discontinuous transit... |
4.5 Biological and ecological systems For biological and ecological systems, [MATH] can correspond to a type of species characterized by fitness [MATH] . The intensity of external noise is described by selection temperature |
[MATH] . As constraint ( 16 ), one defines the average fitness [EQUATION] Then, from the principle of minimal information of Sec. 3, one finds the distribution called the relative reproduction rate |
[EQUATION] This exponential form of the reproduction rate is often used in the biological literature (Manly 1976; Crozier and Pamilo 1979; Russell 1996; Arias et al. 2001; Cowperthwaite et al. 2005; Martin and Lenormand 2008; Saakian et al. 2010). Expression ( 37 ) shows that, among a variety of different species, the ... |
The goal of this section has been the demonstration of the main idea that rather different systems can be described in a general probabilistic way enjoying the same mathematical characterization. |
Self-organization in dynamical systems In dynamical systems, self-organization is usually accompanied by the appearance of spatial structures or patterns (Glansdorff and Prigogine 1971; Haken 1983, 2005; Nicolis 1986) or it is connected with critical transitions (Kuehn 2011) when the system behavior changes qualitative... |
The main message of the present section is twofold. First, we demonstrate that nonequilibrium systems, similarly to equilibrium ones, can be characterized by probabilities derived from the principle of minimal information, that is, from conditional entropy maximization. Second, we prove that the notion of stability is ... |
5.1 Probabilistic pattern selection Suppose a dynamical system can acquire several different spatial structures, with the type of the [MATH] -th structure being denoted by [MATH] . To make the description of the probabilistic approach transparent, let us consider a one-dimensional dynamical system, whose evolution is g... |
[EQUATION] where [MATH] represents an observable quantity. A generalization to dynamical systems of any dimensionality is straightforward (Yukalov 2001a, 2001b, 2003b). |
Self-organization of a dynamical system can be interpreted as the system search for stability. The latter is characterized by the map multipliers |
[EQUATION] If [MATH] , the structure [MATH] is locally stable at time [MATH] When [MATH] , the structure is locally neutral, and when |
[MATH] , the structure is locally unstable. The map multiplier can be expressed through the Jacobian [EQUATION] in the form [EQUATION] |
It is convenient to introduce the expansion exponents [EQUATION] These exponents show how quickly a deviation from an initial condition varies in time, either converging to or diverging from this initial condition according to the relation |
[EQUATION] The expansion exponent is connected with the Jacobian by the equation [EQUATION] Our aim is to find an expression for the structure probability [MATH] |
that should satisfy the standard normalization condition [EQUATION] at each moment of time. Following the general prescription of Sec. 3, we define the information functional |
[EQUATION] By the assumption that the system searches for the most stable structure, the trial distribution [MATH] can be taken to be inversely proportional to the modulus of the map multiplier [MATH] . Then, from the principle of minimal information, we get the structure probability |
[EQUATION] with the partition function [EQUATION] In view of relation ( 42 ), the structure probability can be expressed through the expansion exponent as |
[EQUATION] where the partition function is [EQUATION] Thus, the dynamical system, in general, can exhibit different structures, with the corresponding probabilities ( 47 ). The system tries to self-organize acquiring the most stable structure. At the same time, other less stable structures are also admissible, though w... |
The importance of the present section is the direct demonstration of the intimate relation between stability and probability for nonequilibrium systems. |
The most stable state is the most probable 5.2 Turbulent photon filamentation In order to show that the results of the previous section provide a practical tool for treating concrete nonequilibrium systems, let us consider the effect of turbulent photon filamentation. This is the phenomenon in which an assembly of reso... |
The microscopic description for a system made of two-level resonant atoms starts with the Hamiltonian [EQUATION] consisting of the atomic Hamiltonian [MATH] , field Hamiltonian |
[MATH] , and the Hamiltonian of atom-field interactions, [MATH] The atomic Hamiltonian is [EQUATION] where [MATH] is the number of atoms, [MATH] is the atomic transition frequency, and [MATH] is the [MATH] - component of the pseudospin operator of the [MATH] -th atom. The field Hamiltonian is |
[EQUATION] with electric field [MATH] , magnetic field [MATH] and vector potential [MATH] . The atom-field interaction is given by the Hamiltonian |
[EQUATION] in which the short-hand notation [MATH] is used and the transition current has the form [EQUATION] where [MATH] is the transition dipole and [MATH] are the ladder operators. |
The dynamical system is composed of the evolution equations for the average vector potential [MATH] and the pseudospin averages describing dipole transitions, |
[EQUATION] coherence intensity [EQUATION] and the population difference [EQUATION] The arising filaments can possess different radii [MATH] that correspond to different structures. Employing the method of Sec. 5.1, it is possible to find (Yukalov 2000, 2001a, 2001b) that the probability of a filamentary structure with ... |
[EQUATION] with [EQUATION] where [MATH] are the longitudinal, transverse, and dynamical attenuations, respectively, [MATH] is the effective atomic interaction in the related structure with the filament radius [MATH] , and |
[MATH] is the average population difference ( 55 ) in a filament of that structure. The maxima of the above probability define the filament radii corresponding to the zeroes of the integral sine: |
[EQUATION] where [MATH] is the radiation wavelength and [MATH] is the length of the sample. The optimal radius is given by the absolute maximum of the structure probability, yielding |
[EQUATION] The majority of the arising filaments have this radius ( 57 ), although the filaments with other radii, satisfying Eq. ( 56 ), are also present. These theoretical results have been found to be in very good agreement with experiments (Encinaz-Sanz et al. 2000, Leyva and Guerra 2002). |
Decision making as self-organization In the examples treated above, we have shown that practically any system, whether natural, social, financial, biological or ecological, can be characterized by a probability measure prescribing a weight to each admissible system state or structure. The larger the state probability, ... |
6.1 Classical utility theory Classical decision theory is based on the notion of expected utility (von Neumann and Morgenstern 1953; Savage 1954). We briefly recall the basic definitions that will be used in what follows. |
The consequences of actions are measured by outcomes, or payoffs, composing a set [EQUATION] The payoffs can be weighted in different ways, by means of different probability measures over the set ( 58 ), enumerated with the index |
[MATH] , with the probabilities [MATH] satisfying the standard normalization condition [EQUATION] A lottery, is the set of payoffs and their weights, |
[EQUATION] One defines the lottery mean [EQUATION] and the lottery variance [EQUATION] One calls the lottery uncertain when its variance is not zero, and it is certain if the variance vanishes. |
On the set of payoffs, one defines a utility function [MATH] , which is non-decreasing and concave (Bernoulli 1738). The cardinal expected utility reads as |
[EQUATION] The expected utility serves as a characteristic of the lottery usefulness. One says that a lottery [MATH] is more useful than [MATH] , if and only if |
[EQUATION] Two lotteries are equally useful, when [EQUATION] And a lottery [MATH] is not less useful than [MATH] if [EQUATION] The action of choosing a lottery under uncertainty is termed a prospect The prospects are analogous to the system states considered in Sec. 2. The set ( ) of all admissible prospects, which are... |
[EQUATION] And there is the most useful prospect [MATH] , with the largest expected utility: [EQUATION] Because of this, the prospect set corresponding to ( 60 ) forms a |
complete lattice In this formulation, classical decision theory is deterministic since a decision maker is supposed to necessarily prefer the most useful prospect. |
6.2 Probabilistic utility theory The classical normative utility theories as well as different descriptive behavioral utility theories, such as prospect theory (Tversky and Kahneman 1973; 1980; 1983), are all deterministic, requiring, with certainty to prefer the prospect characterized by the largest functional quantif... |
Our aim is to describe the process of decision making as an intrinsically probabilistic procedure. The first step consists in evaluating consciously and/or subconsciously the probabilities of choosing different actions from the point of view of their usefulness and/or appeal to the choosing agent. We transform the abov... |
[MATH] , with the usual normalization condition [EQUATION] The weight [MATH] can be called the utility factor , since it describes the usefulness of the prospect [MATH] . According to this meaning, the usefulness of a prospect with zero utility has to be zero, which imposes the limiting condition |
[EQUATION] Being a random quantity, the utility [MATH] is assumed to be normalized as [EQUATION] This practical condition guarantees that the involved lotteries are well defined, having finite expected utilities. |
To find the distribution [MATH] , we resort to the principle of minimal information of Sec. 3, introducing the information functional |
[EQUATION] [EQUATION] with the Lagrange multipliers [MATH] and [MATH] taking into account conditions ( 69 ) and ( 71 ). To satisfy condition ( 70 ), the trial distribution [MATH] can be defined as the likelihood ratio proportional to [MATH] . Then the minimization of the information functional leads to the utility fact... |
[EQUATION] with the partition function [EQUATION] Note that the utility factor specified by ( 73 ) satisfies the limiting condition ( 70 ). |
A probabilistic representation of decisions is not new, since it is at the core of classical choice theory (Anderson et al. 1992). Classical choice theory assumes that the probability to choose between different alternatives can be written similarly to expression ( 73 ) but without the [MATH] prefactor, which is called... |
The Lagrange multiplier [MATH] plays the role of a parameter capturing the level of confidence or belief in selecting the prospects, hence, [MATH] |
can be called the belief parameter or confidence parameter Requiring that the utility factor be an increasing function of utility makes the belief parameter non-negative, [MATH] . The limiting values of this parameter characterize decision making in the situations of underconfidence or overconfidence (Griffin and Tvers... |
[EQUATION] In the opposite case of extreme confidence, we get [EQUATION] The latter situation recovers the deterministic formulation of utility theory of Sec. 6.1. |
The ordering of prospects by their usefulness can be done by means of the utility factors. A prospect [MATH] is deemed more useful than [MATH] if and only if |
[EQUATION] Two prospects are equally useful when [EQUATION] And a prospect [MATH] is not less useful than [MATH] if [EQUATION] This ordering is in agreement with that of Sec. 6.1, based on the comparison of expected utilities. |
The application of this approach to time-dependent processes is straightforward. This simply requires including time dependence into the definition of expected utility by incorporating in it a temporal discount rate (Samuelson 1937; Loewenstein and Thaler 1989; Frederick et al. 2002; Rambaud and Torrecillas 2005; Berns... |
This probabilistic formulation of utility theory puts it in the same frame as the description of any self-organizing system presented in previous sections. In this framework, the process of decision making can be understood as the search for the most preferable prospect that enjoys the largest probability. This can be ... |
6.3 Behavioral and quantum decision making Decision theory, based on utility theory, even in the probabilistic variant, characterizes the objective features of the involved prospects, leaving aside all subjective effects connected with decision makers. Classical utility theory assumes that decision makers are rational ... |
Sometimes, one says that realistic decision making contains generic indeterminism (Nichols 2011). Remembering that similar indeterminism is typical of quantum theory, this hints on the possibility of characterizing behavioral decision making by means of quantum probability (Lehrer and Shmaya 2006). Actually, Bohr (1933... |
In the present subsection, we wish to emphasize that the probabilistic way of constructing decision theory can be extended to behavioral decision making taking into account such subjective features as emotions and biases. We shall not go into details of this approach involving quantum techniques, which can be found in ... |
By construction, quantum theory is probabilistic. The scheme of calculating the prospect probabilities follows the rules of defining the observable quantities in the quantum theory of measurement. The space of mind of a decision maker is described by a Hilbert space on which prospect operators [MATH] |
are defined. These operators play the role of the operators of observables, whose averaging yields the observable quantities corresponding to the prospect probabilities: |
[EQUATION] where [MATH] is a statistical trace-one operator characterizing the decision maker, and the trace is taken over the decision-maker space of mind. The prospect probabilities are normalized as in condition ( ). |
It is straightforward to show that the prospect probability ( 81 ) is the sum of two terms: [EQUATION] The first term corresponds to the utility factor characterizing objective features, while the second term is due to quantum effects of coherence and interference, which corresponds to subjective features. |
As is known (Zurek 2003), classical theory is a particular case of quantum theory, corresponding to the situation when coherence effects disappear, which is called decoherence . In the present case, in the same way as for any observable in quantum theory, decoherence implies the disappearance of the quantum coherence t... |
has to correspond to the classical utility factor described in the previous subsection. In this way, classical decision theory is obtained as a limiting case of the quantum decision theory, when decoherence occurs. |
The quantum coherence, or interference, term is what distinguishes quantum prospect probabilities from their classical counterparts. From the point of view of quantum theory, the arising coherence term can be ascribed to quantum indeterminacy, being contextual. Interpreted in the contexts of behavioral decision theory,... |
Thus, the quantum prospect probability ( 82 ) is of dual nature, containing the objective utility factor [MATH] , defined in terms of the prospect utility, and the attraction factor [MATH] , characterizing the subjective attractiveness of the prospect for the decision maker. |
Despite the fact that the attraction factor embodies subjective and unconscious components of the decision making process, it enjoys several quantitative properties making it possible to give quantitative predictions for the prospect probabilities. |
First of all, the attraction factor lies in the interval [EQUATION] The normalization conditions lead to the alternation property |
[EQUATION] And the following average estimate holds: [EQUATION] These properties allow one to make quantitative predictions for aggregate groups of decision makers, which are found to be in excellent agreement with empirical data, as has been shown in our previous publications (Yukalov and Sornette 2009a, 2009b, 2010, ... |
The basic idea of the present subsection is to emphasize two important facts: (i) First of all, subjective behavioral phenomena in decision making cannot be described by minimizing an information functional, whose minimization can provide only the objective part of the total quantum probability. But taking into account... |
(ii) Nevertheless, even quantum systems in nature, as well as subjective effects in decision making, allow for a unified general procedure of calculating both the state probabilities of quantum systems as well as the prospect probabilities for behavioral decision makers. In that way, we see again that there is no princ... |
According to the behavioral interpretation of quantum decision theory (Yukalov and Sornette 2008, 2009a, 2009b, 2010, 2011), the process of decision making goes through the following steps. One fixes a set of prospects, then evaluates their utility and attractiveness, resulting in the evaluation of the prospect probabi... |
[EQUATION] But the same sequence is typical of self-organization of any system. It is just a matter of terminology, whether one talks of decision prospects or system states. Decision making and self-organization are the same processes, sometimes occurring in different systems and often times happening in the same syste... |
Summary Self-organization in different systems is described as a process of evaluation of the state probabilities in the search for the most stable state, hence for the state with the largest probability. Natural systems evaluate the admissible states by means of fluctuations. The explicit expression for the probabilit... |
Decision making, formulated in a probabilistic representation, is also a process of evaluating the prospect probabilities, in the search for an optimal prospect, having the largest probability. Decision makers evaluate the admissible prospects by deliberations. In classical decision theory, the prospects are classified... |
In all cases, the procedure of self-organization is analogous to that of decision making, both being characterized by the same mathematical scheme. It is only the language that is slightly different. But there is a direct translation of one language onto another, which is exemplified in the following dictionary. |
Complex system Decision maker System states Decision prospects System fluctuations Decision-maker deliberations State probability |
Prospect probability System stability Prospect preferability Most stable state Most preferable prospect Quantum fluctuations Behavioral biases |
Self-organization Decision making It is possible to state that self-organization and decision making are equivalent processes. This conclusion is not merely important from the general descriptive point of view, but it has far-reaching practical consequences. For instance, these analogies may suggest the way of creating... |
The processes of self-organization and decision making can be treated from two different points of view, complementing each other. First, it is possible to analyze the actual process of the appearance of structures in a complex system consisting of many agents that are characterized by their typical features and by the... |
The second part is the choice of the best way of presenting the results of solving the complicated dynamical systems, allowing for a convenient description and classification of the found solutions. This final stage is necessary for a clear understanding of the obtained results and for their correct interpretation. Our... |
Our main aim here has been to show that the processes of self-organization in complex systems and of decision making by alive beings can be represented in the same mathematical language of the search for the highest probability corresponding to the most stable state or to the most preferable prospect. Several examples ... |
Acknowledgment The authors acknowledge financial support from the Swiss National Science Foundation. We are grateful to M. Favre and E.P. Yukalova for many useful discussions and advice. |
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