A multifactor model for the process of trading operations with Digital Cryptocurrencies (DCC) is introduced. This model is intended to be integrated into the computational core of an intelligent information system for investors in the DCC market. The model demonstrates that the controllability of the process of buying and selling a set of DCCs can be depicted using a game-theoretical approach. Adopting this approach will enable investors to craft strategies that bolster currency stability in the DCC market. What sets the introduced model apart is its ability to accurately depict the buying and selling processes in the multi-currency DCC market. This insight has facilitated the use of a constructive methodology to deduce the buy-sell strategies of market participants by solving bilinear multistep quality games with multiple terminal surfaces. The results from a computational experiment are also presented. In this experiment, various parameter relations describing the DCC trading operations were analyzed. These findings offer all stakeholders, especially the market participants of DCCs, tools to ensure stability not only in the traditional currency market but also the DCC market.
Digital Cryptocurrencies (hereinafter referred
to as DCCs) are swiftly gaining traction
globally. As the interest in DCCs surges, funds
are increasingly being directed into the DCC
market, which remains unregulated in most
countries. Additionally, in many nations,
central banks have yet to exert any influence
over these activities. To retain their ability to
regulate the financial system, address
economic crises, manage inflation rates, and
influence the prices of goods and services,
central banks have started exploring various
means of adapting to this evolving landscape.
One such approach has been the issuance of
their DCCs. As per estimates from [
A vast body of scientific literature is
dedicated to the study of CCE markets and the
analysis and assessment of the efficiency of
investments in DCC. However, the challenges
of managing the process of buying and selling
DCCs in an ambiguous setting—especially
when leveraging both game theory and fuzzy
mathematics—have not been sufficiently
addressed. As will be demonstrated
subsequently, utilizing game-theoretic
modeling in the management of DCC purchase
and sale procedures (while considering
uncertainty, conflict, and the resultant
economic risk for investors) provides a means
to evaluate the reliability of potential
transactions in the DCC market. This, in turn,
can help mitigate economic risks for DCC
investors. This backdrop has underscored the
relevance and interest in the subject of this
study.
2. Literature Review and Analysis
With the swiftly growing interest of investors
in the DCC market, researchers from a diverse
range of scientific fields, from mathematics to
applied psychology, have started to closely
examine the phenomenon of DCC [
Successful economic development [
The review of scientific publications
indicates that the attractiveness of
investments in DCC remains underexplored.
While a majority of studies center on DCC
volatility [
It should be noted that the greatest value is
represented by models that allow to application
of direct methods of maintaining currency
stability. Such models are presented in [
From a review of recent publications, it’s evident that there currently isn’t a comprehensive methodological framework capable of offering a thorough description of DCC markets. More critically, such a framework is needed to provide a clear forecast of the market’s prospects.
All of the above determined the relevance and objectives of our research.
Development of a multifactor model of trading operations with DCCs based on the use of the apparatus of bilinear multistep quality games with multiple terminal surfaces.
Multifactor model of trading operations with digital cryptocurrencies.
We assume that there are two players
involved in trading operations on the DCC
market. Players participate in the process of
buying and selling several DCCs, such as ADA,
BTC, DOT, EOS, ETC, ETH, LINK, LTC, XRP, and
others [
The difference between the considered
multifactor model of the DCC market and those
previously considered in [
Let us describe the procedure of buying and selling DCCs, which is the basis for the creation of the multifactor model of DCCs.
So, Player I, having set DCC1 equal to x(0) = (x1(0),...,xK (0)) , buys (or sells) set DCC2 equal to y(0) = ( y1(0),..., yM (0)) , from Player II. Player II, having DCC2, sells (or buys) DCC1. Before the beginning of the trading session, the spot rates of DCC1 against DCC2 and vice versa are set. Let’s denote the matrix of spot rates of the currencies of the second group to the currencies of the first group. An element ij of the matrix means the spot rate of the currency of j is the second group against i is the currency of the first group (i = 1,...,K; j = 1,...,M ) . Through denote the matrix of spot rates of currencies of the first group to currencies of the second group. An element ij of the matrix means the spot rate of i is currency of the first group against j is currency of the second group (i = 1,...,M; j = 1,..., K) .
At the moment of t = 0 (start of trading) Player I have a set x(0) (DCC1) to buy a set of DCC2. Player II has y(0) a set of DCC2 to buy DCC1.
Here is a description of the model of trade operations with the selected DCCs. At the moment t = 0 Players I and II replenish their sets of DCCs x(0) (DCC1) and y(0) (DCC2) and have the following volumes of DCCs A x(0) and B y(0) . Here, respectively, A and B are the transformation matrices of the DCC1 and DCC2 sets (analogous to the growth rates in the univariate case), respectively. A is an order K matrix with positive elements that are greater than (or equal to) the elements (elements) of the unit order K matrix E ; B is an order M matrix with positive elements that are greater than (or equal to) the elements (elements) of the unit order M matrix E .
Then players allocate, respectively, U(0) A x(0) DCC1 and V (0) B y(0) DCC2 to purchase DCC2 and DCC1. Here U (0) A x(0) — a vector, the components of which characterize the expenditures of this particular DCC of Player I in the process of purchasing DCC of Player II; U(0) a diagonal matrix of order K , consisting of the elements
K ui (0) : ui (0) 0, ui (0) = 1. Besides V (0) B y(0) is i=1 a vector, the components of which characterize the expenditures of this particular DCC of Player II in the process of purchasing DCC of Player I; V (0) a diagonal matrix of order M , consisting of the elements of
M v j (0) : v j (0) 0, v j (0) = 1.
j=1
It is assumed that at the moment of the trading session the players know the matrices * G pok , Dprod , Dprod , where:
G pok is the matrix of purchases of each DCC of the second group of DCCs of the first group. The purchase matrix G pok consists of the elements (gipjok ) . This matrix of dimension K M .
Dprod is the matrix of sales of each DCC of the second group to the DCCs of the first group. The sales matrix consists of the elements qipjrod .
*
Dprod is a matrix with elements of 1/ qipjrod . This matrix is also of dimension M K . Then, the volumes of the sets of DCC1 and DCC2 of Players I and II at the moment t =1 will be x(1) and y(1) respectively, where x(1) and y(1) are determined from the relations: x(1) = A x(0) + + [−E + D*prod ] U (0) A x(0) + + [ − G pok ] V (0) B y(0) y(1) = B y(0) +
Let us explain relations (1) and (2) step by step.
In the beginning, we will describe the actions of Player I, then of Player II.
Step 1. Player I increased the volume of the set of his DCCs1 from x(0) to A x(0) .
Step 2. Player I allocated part of the set of his DCCs1 to buy a set of DCCs2 in the form of a set of currencies U (0) A x(0) .
Step 3. Player I has determined (based on statistics of previous trading sessions) the structure (share) of his investments (expenses) of the sets of his DCC1 in each currency of the second group. This structure is given by diagonal elements j ( j = 1,..., M ) :
M j 0, j = 1 . Let us denote by is the j=1 matrix of diagonal elements j .
If there is a set of currencies U (0) A x(0) of Player I, then if we operate: D*prod U (0) A x(0) , then we get a M dimensional vector which “as if” means the volume of the set of currencies of Player II. However, this product allows us to define only one component of this M-dimensional vector, since the entire vector U (0) A x(0) will be spent to buy only this one component. There is no more of Player I’s currency available for the other components (other currency) of Player I. It has already been used up. Therefore, it is necessary to partition the set of currencies into M parts so that the entire range of Player II’s currencies can be purchased. This is done by introducing the set: j ( j = 1,..., M ) :
M j 0, j = 1 . Note that the selection of these j=1 coefficients can be done in other ways, not necessarily as in Step 3.
Step 4. To determine each currency of Player II (each component) that Player I bought, it is enough to multiply the matrix * Dprod by the vector j U (0) A x(0) , and then the jth currency (component) of the second group will be determined. Thus, the product D*prod U (0) A x(0) will determine the volume of Player I’s currency if the volume of Player I’s currency set is distributed using the coefficients j ( j = 1,..., M ) .
Step 5. Conversion of the set of currencies of Player II into the set of currencies of Player I takes place. For this process a matrix is applied—the matrix of spot rates of the currencies of Player I to the currencies of Player II. This corresponds to the fact that we have the expression: D*prod U (0) A x(0) , which defines the volume of Player I’s currency set in the case when all of Player II’s currencies are converted into one particular currency (one component).
Note. The currency conversion situation is similar to the currency purchase situation (see Explanation, Step 3). This is taken into account in Step 6.
Step 6. In Step 6, the structure of the currencies of the first group is determined by specifying a diagonal matrix of order K , with diagonal elements i , which characterizes the volume of Player II’s i is currency from converting i a fraction of the volume of the set of currencies D*prod U (0) A x(0) of Player I. This matrix characterizes the structure of the exchange rate of the currency set of Player II’s currencies to the currency set of Player I: D*prod U (0) A x(0) .
Step 7. Player II, as well as Player I, allocates to the purchase of currencies of Player I the value of V (0) B y(0) .
Step 8. In this step, we will convert the set of currencies V (0) B y(0) .
Considering remark 1, to determine the structure of the set of currencies of Player I, we define a matrix (see Step 9).
Step 9. Let us denote by is a diagonal matrix of order K with elements
K i i 0, i = 1. Then we obtain:
i=1 V (0) B y(0) , the volume of Player I’s currency set after Player II has allocated a set V (0) B y(0) to buy Player I’s currency set.
Step 10. Let’s describe the payment by Player I to purchase Player II’s set of currencies.
To do this, we will do a multiplication Gpok by the vector
V (0) B y(0) .
G pok V (0) B y(0).
And, taking into account “Explanation 1” in Step 3, we perform Step 11.
Step 11. Let’s denote by the diagonal matrix of order K with elements
K i i 0, i = 1. We obtain: Gpok V (0) B y(0) , i=1 “structured” (i.e., distributed over the components of Player I’s currency set) payment for the volume of Player II’s currency set.
Thus it is possible to determine the values of Player I’s currencies at the time t = 1: x(1) = A x(0) + + [−E + D*prod ]U (0) A x(0) + + [ − G pok ]V (0) B y(0);
Let’s explain the set of actions on the part of Player I.
Step 12. The second player has increased the volume of the DCC set y(0) to B y(0) .
Step 13. Player II allocated a part of the volume of his set of DCC to buy some volume of Player I’s set of DCC in the form of a set of currencies V (0) B y(0) .
Step 14. The second group has determined (based on the statistics of previous trading sessions) the structure (share) of its investments (expenditures) of its currency in each currency of the first group. This structure is given by the diagonal elements ri (i = 1,..., K )
K : ri 0, ri = 1 . These elements form a diagonal i=1 matrix R .
Explanation 2.
If there is a set of currencies V (0) B y(0) of Player I, then if we operate: Gpok V (0) B y(0) , then we get a K-dimensional vector which “as if” means the volume of the set of currencies of Player I. However, this product allows us to define only one component of this K dimensional vector, since the entire vector V (0) B y(0) will be spent to buy only this one component of Player I. There is no more second-group currency for the other components (other currency) of Player I. It has already been used up. Therefore, it is necessary to partition the set K of currencies into parts so that the entire range of Group I currencies can be purchased. This is done by introducing K the set: ri (i = 1,..., K ), ri 0, ri = 1 . The i=1 selection of these ratios can be done in different ways (see “Explanation 1”).
Step 15. To determine each currency of Player I (each component) it is enough to multiply the matrix Gpok by a vector ri V (0) B y(0) and then the ith currency (component) of Player I will be determined. Thus, denoting by R is a diagonal matrix of order, with diagonal elements ri , we obtain that the product R Gpok V (0) B y(0) will determine the volume of the set of currencies of Player I if the volume of the set of currencies of Player II was distributed using the coefficients ri (i = 1,..., K ) .
Step 16. Step 16 converts the set of currencies of Player I to the set of currencies of Player II. This corresponds to the fact that we have the expression: R Gpok V (0) B y(0) .
Considering “Observation 1”, we proceed to Step 17.
Step 17. In Step 17, the structure of Player II’s currency set is determined by specifying a diagonal matrix * of order M , with diagonal elements i* , which characterizes the volume of the second group’s currency from the conversion i* of a fraction of the volume of Player II’s i -currency set R G pok V (0) B y(0) . This matrix characterizes the exchange rate structure of Player I’s set of currencies to Player I’s set of currencies: * R Gpok V (0) B y(0) .
Step 18. At this step, we will convert a set of currencies U(0) A x(0)
Taking into account “Remark 2” to determine the structure of Player II’s currency set, we define a matrix L (see Step 19). structured payment by Player II of the set of currencies of Player I.
Thus it is possible to determine the values of the currencies of Player II at the time t = 1: y(1) = B y(0) +
Consequently, we have that the values of the players’ currency set at the moment t = 1 are written using relations (1) and (2).
The conditions for the end of the trading session at the moment t = 1 will be the fulfillment of conditions (3), (4), or (5):
The following options are possible at the moment t = 1: (x(1), y(1)) S0. (x(1), y(1)) F0. (x(1), y(1)) D0 , (x(1), y(1)) H0 , where S0 , F0 , D0 and H0 as:
M S0 = {(x, y) : (x, y) RK +M , x 0, yi 0} , i=1
K F0 = {(x, y) : (x, y) RK +M , xi 0, y 0} , i=1 (3) (4) (5) (6)
Case (3) is desirable for Player I, and case (4) is desirable for Player II. In case (5), the players also stop interacting since they cannot continue. In case (6), they continue the interaction for moments t 1 .
Due to symmetry, we restrict ourselves to
considering the problem from the position of
allied Player I [
The definition of the pure strategy and the
preference set of Player I was given in [
The set of such states will be called the set of preferences of Player I. Accordingly, the strategies U*(.,.,.) of Player I possessing the above properties are his optimal strategies.
The solution to Problem 1 consists in finding the sets of “preferences” of Player I allies W1 and their optimal strategies. Similarly, the problem is posed from the point of view of allied Player II.
Solution of Problem 1
The solution to the problem depends on the ratio of parameters defining the procedure of confrontation between the allied player and Player II opponent.
All cases of the ratio of parameters defining the multifactor model of the DCC market will be presented in the form of two cases:
Case 1. [L − * D*prod ] 0, [−E + * R Gpok ] 0. [ − Gpok ] 0 , [−E + D*prod ] 0 .
{ [L − * D*prod ] 0, [−E + * R Gpok ] 0. [ − Gpok ] 0 , [−E + D*prod ] 0 }.
Let us consider Case 1 and introduce the notations: H (1) = B, F (1) = [−L + * D*prod ] A; H (k + 1) = H (k) B, F(k + 1) = = {F(k) [ D*prod ] A − − H (k) [L − * D*prod ]} A;
E is a singular matrix of order K; k = 1,...
Let us denote by W1k is the set of such initial states of players, which have the property that if the game starts from them, then Player I can choose his strategy U*(.,.,.) to ensure the fulfillment of condition (3) at the moment k , (k = 1,...) . At the same time, this strategy chosen by Player I contributes to preventing Player II from fulfilling conditions (4) and (5) at previous moments. Through (W1k )i is denotes the set of such initial states of players, which have the property that if the game starts from them, then Player I can choose his strategy U*(.,.,.) to ensure at the moment k= 1,… the fulfillment of condition (3) on the i − th component of Player II. At the same time, this strategy chosen by Player I contributes to preventing Player II from fulfilling conditions (4) and (5) at previous moments.
It is not difficult to see that W1 = k=1W1k , and
M W1k = (W1k ) j .
j=1
If Condition 1 is satisfied, the sets (W1k ) j are written as follows: (W1k ) j = {(x(0), y(0)) : (x(0), y(0) R+K+M , [H (k) y(0)] j [F(k) x(0)] j }.
At such ratios of parameters the optimal strategy of Player I
E, (x, y) W1 : U *(.) :Uo*pt (x, y) =
not determined,(x, y) W1;
Realization of the strategy of counteraction of Player II to Player I is such: Vo*pt (x, y) = 0.
Note that if the conditions of Case 1 are met, the preference sets are finite in number, and for each component of Player I. Hence, we obtain that the procedure of constructing the preference sets of Player I is finite. Let us add that if the conditions of Case 1 are satisfied, the components of the vector of Player I are positive for any moment and any strategies of players. Finding the preference sets and optimal strategies of allied Player I in Case 2 is done similarly. The construction of preference sets and optimal strategies of ally Player II is done similarly.
Note. Since the model uses diagonal matrices that specify the structures of the DCC vectors, it is possible to manipulate them by changing the diagonal elements to obtain the desired result. 5. Computational Experiment The computational experiment concerning the problem of identifying the area of investor preference in DCC2 was conducted using the Matlab environment. The results are depicted in Fig. 1. On the abscissa, investments in the set of DCCs by Player I are represented. Conversely, on the ordinate axis, investments in the set of DCCs by Player II are shown. The red straight line (1) illustrates the balance beam.
The blue dashed line shows the trajectory of the player’s movement in the area of preference of Player II, which is above the equilibrium ray. At present, the results of theoretical deductions and experimental studies have formed the basis of the developed intelligent system for DCC trading.
terminal area. If the trajectory aligns with the equilibrium ray—which demarcates the boundary of Player I’s preference set—it signifies a situation where the DCC rate is at equilibrium. In this somewhat rare case, both players navigate along this ray, employing their optimal strategies, resulting in a state that simultaneously satisfies both parties. Conversely, if the trajectory falls below the equilibrium ray, it demonstrates a scenario where Player I (the buyer of DCC2) holds an advantageous position in the parameter ratio, meaning the situation is within Player I’s preference set. Under these circumstances, Player I, by applying his optimal strategy, will realize his objective—essentially steering the system's state to his terminal zone.
A multifactor game model of the DCC market has been explored. The research demonstrates that the controllability of processes within a trading session can be articulated through a game approach. This approach is grounded in solving a system of discrete bilinear equations with multivariate variables. The model’s distinctiveness lies in its deviation from existing methodologies, specifically by addressing a bilinear multistep quality game with multiple terminal surfaces. For the first time, a solution to this new bilinear multistep quality game, which considers dependent motions, has been identified. We also showcase the outcomes of a computational experiment, wherein diverse parameter relationships that outline the DCC buying and selling process are considered.
The findings detailed in this paper can be instrumental in preventing instances of exchange rate instability in the DCC investment market, which are commonly observed in practice. Consequently, this model might also prove beneficial for predicting situations on trading floors that deal with DCCs. Additionally, the results offer insights into selecting control measures to sustain exchange rate stability in the DCC investment market, especially for major banking entities.