Accounting for uncertainty in agents' strategies should obviously lead to «blurring» of these functions and turning them into continuous dome-shaped functions with maxima at the points p^>D, p^>S, q^>D и q^>S and agent widths Г^>DP, Г^>SP, Г^>SQ and Г^>DQ respectively. It seems reasonable, both from the economic and technical point of view, in the first approximation to use normal, or, simply, Gaussians distributions [Kondratenko, 2015]. Then the demand function has the following form in this approximation:

where the parameters w^>DPand w^>DQ (agent frequency parameters below) are related to the agent widths as follows:

Formulas for demand have a similar structure naturally:

To avoid misunderstandings, note that formulas (1.17) and (1.21) express the relationship known for Gaussians between their agent frequency parameters and the widths, more precisely, the widths of Gaussians at half-height. The numerical values of these widths are set or selected explicitly or implicitly by the agents themselves, just as prices and quantities are set by them. But, very importantly, unlike price and quantity quotations, the «quotations» of widths are not explicitly exhibited either in negotiations or in organized markets. The values of these widths may not even be accurately realized by the agents themselves; in this respect, agents may act purely intuitively, depending on the market situation.

Summing up the intermediate results, we can briefly say that in this version of the theory the buyer's probabilistic demand function is described by four parameters, the price p^>D, the quantity q^>D and two widths Г^>DР and Г^>DР. The same statement is of course true for the seller's probabilistic supply function. It is these eight parameters that take into account all of the relevant market information that the buyer and seller use before they put up quotations at any given time in the process of trading in the market. And let us emphasize for clarity that usually both buyers and sellers declare or announce publicly and unambiguously only their price and quantity quotations, leaving the information about their widths "behind the scenes".

For our model grain market the probability functions S&D are presented graphically in Figs. 1.6–1.9.

Obviously, for a two-agent economy, all S&D market function surfaces have a simple smooth structure with one maximum. Of course, for more complex economies the structure of the surfaces will be much more complex.

Fig. 1.6. Graphical representation in the rectangular two-dimensional coordinate system [P, S& D] of one-dimensional price functions d^>P (p) and s^>P (p) as two-dimensional curves with maxima at prices p^>D and p^>S and widths Г^>DР and Г^>SР respectively. The values used for the widths are: Г^>DР = 23.8 $/ton, Г^>SР = 37.0 $/ton.

Fig. 1.7. Graphical representation in the rectangular two-dimensional coordinate system [P, S& D] of one-dimensional quantity functions d^>Q(q) and s^>Q(q) as two-dimensional curves with maxima at quantities q^>D and q^>S and widths Г^>DQ and Г^>S>Q respectively. The values used for the widths are: Г^>DQ = 26.4 ton / year, Г^>SQ = 6.8 ton/year.

Below we will discuss in detail all new concepts, main features and calculation details for our simplest two-agent system, so that we will not be distracted by their discussion in further consideration of more complex issues concerning the exchange. So, by definition and in its essence, the probabilistic function of demand