Statement 3. The proposed mathematical apparatus for describing the market dynamics is built on using orders or quotations of market agents; therefore, it automatically takes into account all the principles of theory, since market agents take into account all the information coming to the market at any given moment in time when choosing the quotations. In other words, they are under constant influence of all forces and influences acting on the market at a given moment: this includes the influence of other agents, assets and markets; as well as the effect of institutional and environmental factors, etc., which is reflected in regular changes in their quotations.

In the next two sections we will describe in detail the mathematical body of the probabilistic economy based on the actions of agents and illustrate its work on the example of a simple model market with one buyer, one seller, and one traded commodity. It will be shown that the most specific features and regularities in the behavior of markets are already evident in such a simple two-agent model. An extension of this theory to multi-agent markets with one traded commodity will be presented in subsequent chapters.

Note that since we neglect all probabilistic effects in classical theory, or classics, we do not consider the uncertainty and probability principle in classics, although it is clear that it plays an important role in probabilistic theory. It is hardly worth seriously discussing which of these two theories is better. As in the case of classical and quantum mechanics, it is preferable to talk about different applications of classical (in a certain sense deterministic) and probabilistic theories, as we will demonstrate more than once below. Let us remind you that the classical theory in this book refers simply to an initial approximation of the probabilistic theory in which the principle of uncertainty and probability are not explicitly taken into account.

Thus, we will thoroughly describe this approach to the study of the economy dynamics, or evolution, within the framework of the classical economy using the example of the simplest model, namely, a market with one buyer and one seller selling one commodity, such as grain. The economic space in this case is obviously two-dimensional.

Let’s consider a typical situation in a market, which has a real potential buyer and seller of a certain good, say, grain. The buyer wants to buy goods in quantity q^>D at price p^>D, and the seller wants to sell goods in quantity q^>S at price p^>S. These four parameters fully characterize the state of the market in the classical economy at each point in time. It is commonplace in the market that both prices and quantities of buyer and seller do not coincide. Therefore, if they both insist on their bid and ask, respectively, there will obviously be no deal. The oldest, well-established mechanism for resolving such trade disputes over the years since the emergence of markets is that the buyer and seller enter into trade negotiations with the aim of getting them to agree to a sale and purchase deal on terms that suit both parties. Let us describe this negotiation process in mathematical language as follows. Let the functions p^>D (t) and q^>D (t) denote the price and quantity of goods desired and offered by the buyer for buying during negotiations with the seller at a certain time t. Similarly, let the functions p^>S(t) and q^>S(t) denote the price and quantity of the good desired and offered by the seller for sell during negotiations with the buyer in the market. In their meaning, the values of prices and quantities introduced above are the main content of agent proposals to buy or sell the goods. Below, for brevity, we will denote these desired and offered values as buyer’s and seller’s quotations. And such a line of agents' behavior in the market will be called a discrete or point strategy, since at each time