Fig. 1.4. Diagram of functions S&D, reflecting the dynamics of the classical two-agent market economy in the coordinate system [T, S & D] in the first time interval [t₁,t₁^E].

Further, similarly to classical mechanics, we can consider prices and quantities of market agents as trajectories of market agents in two-dimensional economic space using the coordinate system [P, Q] as shown in Fig. 1.5. Let us clarify that time t in this parametric representation of functions S&D is an implicit parameter. Generally, this parametric representation gives nothing new compared to Figs. 1.3 and 1.4. Nevertheless, there is one interesting nuance here – the similarity of this diagram with the traditional picture in the neoclassical S&D model, namely the Marshall cross. We will touch on this issue a little later, but for now let's look at some of the features in Fig. 1.5. First, as the arrows show, the buyer and the seller move toward each other in terms of price: the seller lowers it, and the buyer, on the contrary, raises it. Thus the figure reflects normal market negotiation processes. Second, usually during negotiations, quantity quotations are reduced by both agents, i.e. both the buyer and the seller.

Fig. 1.5. Dynamics of the classical two-agent market economy in the two-dimensional economic space of price-quantity in the first time interval [t₁,t₁^E].

Clearly, all agents want to buy or sell less goods at a compromise market price than at the desired prices they stated at the beginning of the trade. These factors together determine that the slope of the demand curve q^D(p^D) is negative and the slope of the supply curve q^S(p^S) is positive, just as the S&D functions in the neoclassical model «should» be. But this visual similarity is incomplete, because the economic meaning of these pictures in the two theories differs significantly: in the classical model it is a description of the actual process of negotiations in order to reach a deal, and in the neoclassical model it is a description of strategies of behavior of agents in the market in terms of neoclassical supply and demand curves, q^D(p^D) and q^S(p^S). We emphasize that while in neoclassics these curves, by definition, represent as it were the actual functions of supply and demand, in classics these curves are simply a graphical representation of the price and quantitative time trajectories in the form of one trajectory in the course of trading. Thus, the classic economic theory does not assume the existence of any definite dependence of the agents’ quantitative quotations on price quotations, i.e. the existence of any definite functions q^D(p^D) and q^S(p^S).

In conclusion, we would like to emphasize that, as we have seen, if agents insist on their initial offers and show no willingness to bargain and compromise, the volume of bargaining will be zero. It is the willingness of the buyer and seller to modify their initial offers that leads to bargains. Thus, we can argue that market agents should initially include into their strategies a certain possible range of prices and quantities for their quotations. From this point, only one important step remains to build a better probabilistic model.

1.7. PROBABILISTIC THEORY OF THE TWO-AGENT MARKET

In order to achieve greater transparency of the presentation, we will also reserve ourselves in this section to describing the details of probabilistic theory on the example of the two-agent model of the grain market.

1.7.1. PROBABILISTIC STRATEGY OF MARKET AGENTS AND CONCEPT OF SUPPLY AND DEMAND IN PROBABILISTICS

We have come to the most intriguing point in the presentation of probabilistic economics, namely, we will now include the sixth principle – uncertainty and probability – to the theory. We will proceed as follows: first, for the analogy with theoretical physics, or more precisely, with the procedure of transition from classical mechanics to quantum mechanics to be clearly visible, and, second, we will try not to lose key aspects of describing the economic character, i.e. meaningful and rational behavior of agents in the market. The latter concerns, first of all, the process of agents' decision-making about the strategy of behavior in the market as well as the method of mathematical representation of market actions implementing these agents’ decisions. Obviously, taking into account the principle of uncertainty and probability should in one way or another lead us from a point strategy of agents to some continuous strategy. Mathematically we will make this transition in exactly the same way as in theoretical physics we make the transition from temporal trajectories of particles to probability distributions of particles in space. Namely, let us move from a description of economic dynamics in classical economic theory in terms of temporal price and quantity agent trajectories, p^D(t), q^D(t), etc., to a description of dynamics in probabilistics using continuous agent distributions of price and quantity probabilities D(p, q) и S(p, q), which we will call probabilistic agent functions S&D. For certainty, let us note that these distributions themselves depend on trajectories, p^D(t), q^D(t), etc., so they are themselves functions implicitly dependent on time. In order not to «obfuscate» the formulas by specifying this time dependence everywhere, we will often omit the time variable t in the formulas.

Let us briefly elaborate again on the rationale for this approach to physical-economic modeling. Market agents, forced to constantly work on the market in a continuously changing situation, are aware that the prices and quantities declared by them in the point strategy may not suit the counterparty at a given time. Based on previous experience in the market, they are well aware that they cannot know exactly how prices and quantities will develop in the market even in the near future. They are already used to operating in a market with great uncertainty, entailing high risks and the resulting potential costs. As a result, they realize that in the market they should always consider all their decisions and actions as possible with a certain degree of probability. This probabilistic aspect of the process of market decision-making is of great importance for understanding the behavior of agents in the market and the market as a whole [Mises, 2005; Gilboa et al., 2008]. Market agents think and act as homo oscillans. That is why they are forced to enter the real market not with discrete strategies, but with continuous strategies which can be represented by continuous probability distributions with certain widths correlated with the amount of uncertainty in the market situation in a given period of time.

1.7.2. PQ-FACTORIZATION OF AGENT S&D FUNCTIONS

As already noted, the description of continuous strategies requires the use of two-dimensional functions S&D, D(p, q), and S(p, q). Of course, it is rather tedious to calculate and analyze two-dimensional functions representing three-dimensional surfaces already in the case of a one-commodity market. For this reason, we will take one more step in simplifying our models, which will make it possible to perform economic calculations of real multi-agent markets and to analyze the results obtained by our method at the highest scientific level. Thus, we a priori assume that we can factorize the agent functions S&D with a sufficient degree of accuracy, i.e. we can approximate their representation as a product of one-dimensional functions as follows:

This type of factorization and the corresponding approximation, in which the price and quantitative variables are separated, will be called PQ-factorization and PQ-approximation, respectively. Here d^P(p) ands^P(p) are one-dimensional price functions, d^Q(q) and s^Q(q) are one-dimensional quantity functions of S&D normalized by definition to 1:

C^D and C^S are simply normalization factors. They are derived from the condition of such a natural normalization selection of the agent functions S&D:

Here D⁰ and S⁰ are obviously total demand and complete supply of the buyer and seller, respectively. Below we will also omit the word «total» for the sake of brevity. It is easy to show that the agent S&D-functions, D(p, q) and S(p, q), normalized in this way, are dimensionless functions. It is also obvious that in the point or discrete strategy described above, one-dimensional functions are represented by the so-called Dirac delta functions as follows:

Keep in mind, that the special Dirac function by definition is zero everywhere except at the zero point, where it is equal to infinity, and its integral from minus infinity to plus infinity is 1. By the way, these functions can be applied to describe the probability functions of monopolist and monopsonist supply and demand in real markets.

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".

1.7.3. GRAPHICAL REPRESENTATION OF AGENT S&D FUNCTIONS IN PQ-SPACE

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^SQ respectively. The values used for the widths are: Г^DQ = 26.4 ton / year, Г^SQ = 6.8 ton/year.

1.7.4. PROBABILISTIC MECHANISM OF MARKET PRICING

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 D(p,q) (supply S(p,q)) is the probability of the buyer (seller) concluding a deal to buy and sell the traded goods in quantity q at price p. If this is so, then, according to the standard concepts of probability theory, it is natural to define the probability of the transaction under these conditions as the multiplication of these probabilities:

We call this probability of making a deal a market deal function, and, for convenience, we also refer it to the market functions of supply and demand. Like the market functions D(p, q) and S(p, q), it is dimensionless. For the sake of certainty, let us explain that, generally speaking, purchase and sale transactions can occur in the market at any time, at any price and in any quantity, within reasonable limits, but with varying degrees of probability. But if the transaction function is a sufficiently high and narrow bell with a single maximum with the parameters p^M and q^M, then almost all transactions will occur in the proximity of these values, so it is reasonable to consider these very values to be market prices and quantities. If the function of transactions looks otherwise, of course, these definitions are somewhat meaningless, and one should consider the mechanism of probabilistic pricing in detail. Below we will always assume that the function of transactions is such as to allow market prices and quantities to be determined in a fairly simple way. This is exactly the case we have graphically presented in Fig. 1.10 for our two-agent model of the grain market.

Fig. 1.8. Graphical representation in a rectangular three-dimensional coordinate system [P, Q, S&D] of the two-dimensional buyer demand function as a three-dimensional surface D (p, q) with a maximum at the point A (p^D, q^D) in the plane (P, Q).

Fig. 1.9. Graphical representation in the rectangular three-dimensional coordinate system [P, Q, S&D] of the two-dimensional seller's supply function as a three-dimensional surface S(p, q) with a maximum at point B (p^S, q^S) in the plane (P, Q).

As expected, the surface of the market transaction function F(p, q) has only one maximum. For multi-agent economies, the structure can be much more complex.

Fig. 1.10. Three-dimensional graphical representation in a rectangular three-dimensional coordinate system [P, Q, F] of the three-dimensional deal surface F(p, q) in the form of a high and narrow bell with one maximum at the point C (p^M, q^M) in the plane (P, Q). The graphical method of calculation gives the following results for market prices and quantities: p^M = 281.4 $/ton, q^M = 51.9 ton/year.

Let us now turn to the question of calculating market prices and quantities within the framework of probabilistic economics. It is well known from the standard course of mathematical analysis that extrema of a multidimensional function should be defined as points on the corresponding surface in which the total differential of this function is 0. In our situation this condition leads to the following equation:

This equation is equivalent to the following two partial derivative equations:

In terms of S&D functions, this system is transformed as follows:

At this point it makes sense to introduce a new concept into theory, namely the concept of S&D market forces with such definitions:

In terms of market forces we can write the system of equations (1.25) more compactly as follows:

Obviously, this system of equations looks like a system of equality of S&D market forces at values of market prices and quantities. And it is similar to the system of forces equality at the static equilibrium point in classical mechanics. In other words, the system of economic equations (1.27) looks like a formulation of Newton's third law in classical mechanics. Substituting specific S&D functions from equations (1.16) and (1.20) into the system of equations (1.26), we obtain such simple and clear formulas for calculating market forces:

As we can see, all forces have become one-dimensional functions in this model. Then, using these equations, we obtain a very elegant system of two independent linear equations to determine market prices and quantities:

This system is so simple that you don't even have to solve it in the usual sense to get a very nice looking solution for market prices and quantities:

Thus, probabilistic market prices and quantities in a two-agent economy, when using factorized agent functions in the form of Gaussians, are determined by averaging the corresponding agent parameters, with the frequency parameters of the agents serving as weights in this averaging. The fundamental point here is that these two simple, and independent, algebraic formulas, which include only four buyer parameters (p^D, q^D, w^DP and w^DQ) and four seller parameters (p^S, q^S, w^SP и w^SQ), uniquely determine the market price p^M and the market quantity q^M. Let us discuss some of the most prominent features of the resulting formulas.

Feature 1. Market price and market quantity do not depend on each other in this model at all. Keep in mind that p^M and q^M in their meaning are the most probable price and quantity, the two most important conditions under which a deal is most likely to be concluded. This means that these two formulas unambiguously set the trend in the development of market dynamics at a given moment in the direction of these very values. There is practically no need to make any complex calculations. This fact is the advantage of this model.