The article provides an analysis of the optimum equilibrium to a game and theory model of hierarchic game under conditions of a mixed joint project management. The authors considered a situation providing for a project being managed both by the head organization and by several contractor companies. They proposed a mathematic model enabling determination of the optimum strategies of each party and finding the Nash equilibrium in the game. The analysis results can be useful for efficient management of joint projects under conditions of competition and cooperation between contractors and the head organization. The paper shows an approach to analysis and synthesis of management in matrix structures of the project management based on the game and theory modeling in hierarchic game systems.
depending on several participants. Under such conditions, it is important to have efficient models of management taking account of the hierarchic structure and interaction between different participants. The project management under modern conditions includes elements of competition between various contractors or partners. The use of the method of optimum equilibrium of a game and theory model of hierarchic game under conditions of a
0000- 0001-7846-9034 (L. Chernova); 0000-0001-8772-9889 (S. Titov); 0000-0002-9069-0409 (S. Chernov); 0000-0002-0666-0742 (L. Chernova); 0000-0003-3007-2462 (N. Kunanets)
© 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). mixed joint project management enables successful implementation of complicated projects based on joint initiatives.
Creation of a team became the basic method of combining resources and experience for
attaining a particular purpose with distribution of risks and win at the same time. The team
management is important for a proper and efficient settlement and management of
conflicts. The advantages and success in conflicts settlement enables enlarging the project
scale [
participants helps to efficiently implement them and ensures competitive advantages. Using the method of game and theory model analysis enables determining the optimum solutions for project managers and companies participating in joint projects. This facilitates avoiding conflicts and safeguards the optimum usage of project resources.
The mixed joint project management is an approach including combined elements of various methodologies and approaches to project management aimed at optimization of results and assurance of a successful project result. This approach is distinguished with its flexibility and adaptiveness to enable using various methods and approaches depending on certain project needs and conditions. Particularly, it enables the project teams to be more flexible and adaptive to changes that may happen throughout the project period. Mixed management allows using the best practices from various methodologies, such as Agile,
balancing the requirements to the product development period and its quality, budget and risks, which assures successful project implementation with minimum expenses and risks. Using various methods and approaches can contribute to increase of the project efficiency providing more exact reflection of the client needs and better utilization of resources. It also facilitates development of the cooperation and communication culture in the team. The teams learn to interact and to share their knowledge and experience, which helps increasing the efficiency of work. The main objective of the mixed management consists in achieving successful project results by means of combining various methods and approaches corresponding to the project conditions and needs in the best way. Taken in general, mixed management of a joint project permits the teams to be more flexible, adaptive and efficient in project management ensuring the attainment of results desired.
can take the key part in a mixed management of joint project, to ensure the following:
consequences of various strategies for all project participants. This allows avoiding unexpected problems and solving conflicts before their escalation.
conflicts and developing strategies of their management. This can include a search for compromise solutions or development of conflicts solving mechanisms. Therefore, the analysis of the optimum equilibrium to a hierarchic game theory and game model is an important tool for efficient management of a joint project, which enables avoiding conflicts, optimizing utilization of resources and attaining the joint objectives.
determining the optimum equilibrium between the participants. These can be Nash analysis, a decision based on dominating strategies or other methods.
results is to be assessed. Such factors are to be taken into account as the resources utilization efficiency, risks minimization and participants’ needs satisfaction.
Let us consider a problem of mixed management by project managers of the actions of team members presented from the game and theory side. We might have a two-level system U , which consists on the top hierarchic level of n project managers C , C2 , …, Cn , whose 1 actions are aimed at the controlled objects, team members having different functions, A , j j = 1,
, m . A hierarchic game contains two types of participants to this game, project managers
Ci and executing team members
A , j U = G = C1,C2 , ,Cn , A1, A2 , , Am . (Fig. 1)
In multilevel systems, one and the same player can be the project manager and an executive at the same time, i.e. it can fulfil instructions of the game participants occupying higher hierarchic levels. Depending on their behavior strategies selected, game participants can be characterized by the level of activity having effect on the position and behavior strategy selection of all hierarchic system participants. Participants’ behavior strategies are represented on the set of their positions by target functions putting their wins in correspondence with strategy vectors.
At the same time, the system participants’ behavior strategies are subordinated to the behavior reasonability, in other terms – to their target function maximization. This approach separates the subset of priority actions from the set of all possible actions. In the system equilibrium concept selected, the game participants, while acting without cooperation, i.e. selecting their optimum behavior strategies without cooperation with other players, have to move to the Nash equilibrium point.
C1 A1
C 2 A2
X A1 == x A11 , x A21 , , x Am1 .
X C == x C11 , x C21 , , xC1l , 1 x == x C1l , x A1 = x1, x2 . m
WC1 == WC1 ( x C1l , x Am1 ) - target function of project manager C1 ,
WA1 == WA1 ( x C1l , x Am1 ) - target function of player A1 . It consists of one project manager C1 as a managing body on the top hierarchic level, and one executive A1 - on the bottom level. We interpret our first base game and theory model G1 as a normal game U = C1, A1 G1 , in which two players C1 and A1 are participating. For each participant of game G1 , sets of all possible behavior strategies are determined as
A certain real number being the player’s win is put by target functions in correspondence with each solution vector x1, x2 .
The full normal form of hierarchic game G1 , which models this simplest linear system, contains the following set: players, strategies and target functions
G1 == C1, A1, X C1 , X A1 , WC1 , WA1 .
In hierarchic game G1 , strategy x C1l is chosen by project manager C1 first, and only after that, subject to the project manager selected already known, the second player – member of team A1 selects its strategy x Am1 .
The target function of center WC1 == WC1 ( x C1l , x Am1 ) depends on both the selected own strategy x C1i X C1 == x C11 , x C21 , , xC1l and the strategy of player A1 x A1j X A1 == x A11 , x A21 , , x Am1 . The value of the win of executive A1 is defined by its target function WA == WA ( x C1l , x Am1 ) , which depends on the same variables in the same 1 1 way. In view of this, we have a hierarchic game of two players in normal form. Should there be no additional conditions of selecting the strategies, the game is to be solved by selecting a Nash equilibrium.
to executive A1 , the respective hierarchic game is called game G1 . Let us consider possible behavior of the second player subject to the first player’s strategy being known.
The set of actions, on which the executive’s target function maximum is acquired with the fixed selection of center, is defined under the formula:
WA1 == arg max WA1 ( x A1jx C1l , x Am1) .
The set of solutions to game G1 depends on strategy X C == x C11 , x C21 , , xC1l of 1 the center’s behavior. If center C1 and executive A1 are aware of the supposed sets and target functions, the center can forecast the reaction of player A1 upon its action. Having the possibility to use its own target function WC == WC ( x C1l , x Am1 ) , the center can 1 1 confidently forecast the behavior of player A1 from the supposed set of its strategies X A == x A11 , x A21 , , x Am1 . In the vast majority of cases, there are several variants of 1 such behavior strategies of the second player. Therefore, canonical assumptions of the decision-taking theory are to be introduced into consideration. These assumptions consist in optimistic and pessimistic criteria.
A1 will have a positive attitude to managing center C1 in game G1 . Accordingly, the second player A1 will select such actions within the set of supposed actions that maximize the target function of the project manager. At the same time, the project manager will also maximize the target function as a reasonable player. Therefore, the optimum management in game G1 will consist in such a project manager strategy that implements the maximum on the set of supposed actions of such a function, which the values of maximums on the set of supposed actions of the second player (the managed object) have been applied to. For the optimistic criterion, we’ll have the following solution:
x C1i arg mxXaCx1 mxXaAx1 WC1 ( x A1jx C1l , x Am1) .
Taking the pessimistic criterion into consideration, it’s not difficult to obtain the maximin concept with respective solution as given below:
x C1i arg mxXaCx1 mxXinA1 WC1 ( x A1jx C1l , x Am1)
The base hierarchic game G1 can be considered under decision-taking criteria with two solutions. The first is optimistic: arg mxXaCx1 mxXaAx1 WC1 arg max min W ( x A1x C1l , x A1) is the maximum guaranteed win of the project xXC1 xX A1 C1 j m manager.
strategy depending on the strategy selected by A1 team member, will be provided by the ( x A1x C1l , x A1) , and the second: j m second base game G2 . (Fig. 3)
G 2 Project manager x C1 ( x A1 ) m l C
1 A 1 x A1 m
Team member
The set of project manager’s strategies will already be a function of the second player’s strategies solutions:
x == x 1 ( x Am1 ), x A1 = x1 ( x2 ), x2 .
C
l m x C1 arg max max W i xXC1 xX A1
C1 ( x == x 1 ( x Am1 ), x Am1) та
C l x C1i arg mxXaCx1 mxXinA1 WC1 ( x == x 1l ( x Am1 ), x Am1) де C x A1j arg max WA 1 (x 1 ( x Am1 ), x Am1) .
C l
a team member. Let the project manager’s strategies set already be a function of the second player’s strategies. During the game analysis by complete analogy, we obtained two possible solutions: G1 and G2, where G1 represents the reward mode and G2 - the punishment mode.
management function that consists of two modes: the reward mode and the punishment mode. This mode is applied depending on whether the team member completes the tasks required for successful project implementation or fails to complete them.
hierarchic game be applied for solving this problem, it can be expected that each player would maximize its win with account taken of the other player’s actions and its own possibilities.
mode) and G2 (punishment mode). Player A (a team member) also has its own strategies relative to the project manager’s actions.
Analyzing possible combinations of players’ strategies, we can find the optimum equilibrium that maximizes the total win of both the parties. This equilibrium can be achieved by means of agreement upon both players’ actions in such a way that nobody can get a bigger win by changing its own strategy.
finding the optimum strategies of the project management and team’s participation, which will ensure successful project implementation under mixed management conditions.
game theory and game model can be an efficient tool for solving problems of joint projects management. Using the game theory in the context of a mixed joint project management enables taking account of the differing interests of various project participants and finding the optimum strategies of cooperation between them.
achieved by means of agreement upon all project participants’ actions, which facilitates successful implementation of the project tasks. The research results can be useful for project managers and team members while taking strategic decisions and setting up efficient management mechanisms under conditions of difficult project environments.
[10] D. S. Kobylkin and O. B. Zachko, "Structural models of safety-oriented management of infrastructure projects decomposition," in Materials of 2020 IEEE 15th International
Technologies (CSIT 2020), vol. 2, Lviv-Zbarazh, 2020, pp. 131–134. [11] O. Kovalchuk, O. Zachko, D. Kobylkin, and H. Tanaka, "IT development of HR system in the field of human safety," ITPM 2021, pp. 314–323. [12] U. Faigle and B. Peis, "A Hierarchical Model for Cooperative Games," in Algorithmic
vol. 4997, DOI: 10.1007/978-3-540-79309-0_21. [13] O. Kovalchuk, D. Kobylkin, and O. Zachko, "Digitalization of HR-management processes of project-oriented organizations in the field of safety," ITPM 2022, pp. 183–195. [14] O. Kovalchuk, O. Zachko, and D. Kobylkin, "Criteria for intellectual forming a project teams in safety oriented system," International Scientific and Technical Conference on
[15] V. Pіterska, A. Shakhov, "Development of the Methodological Proposals for the Use of
Engineering & Technology (UAE), vol. 7, no. 4.3, pp. 257–261, 2018. DOI: 10.14419/ijet.v7i4.3.20129. [16] V. Piterska, A. Shakhov, O. Lohinov, L. Lohinova, "The Method of Transfer of Research
80. DOI: 10.1109/STC-CSIT.2019.8929887. [17] P. Fedorka, N. Kunanets, S. Chernov, S. Titov, L. Chernova, L. Chernova, E. Trushliakov, "The project of information system for students knowledge evaluation," CEUR Workshop Proceedings, vol. 3453, pp. 117–127, 2023. [18] N. Kunanets, O. Artemenko, V. Pasichnyk, V. Kut, A. Ivanchov, "Using react and fuzzy expert systems for better travel experience in local route planning," CEUR Workshop Proceedings, vol. 3426, pp. 372–384, 2023. [19] V. Pasichnyk, N. Kunanets, V. Yunchyk, M. Khomyak, S. Yatsyuk, V. Muliar, A. Fedonuyk, "Model of the recommender system for the selection of electronic learning resources," CEUR Workshop Proceedings, vol. 3426, pp. 344–355, 2023..