Zastosowanie teorii gier w ekonomii

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The application of game theory in economics

The considerations presented in the work The application of game theory in economics presehe possibility of using the mathematical apparatus of game theory to certain economic problems. The importance of this use is evidenced by the fact that J. Nash, whose equilibrium we were considering (a mathematician and economist), received the Nobel Prize in Economics in 1994, together with Reinhard Selten and John Harsan, for the use of game theory in this discipline. The paradox is that in 2015 J. Nash received an equally prestigious award in mathematics, the so-called Abel Prize for his contributions to the theory of partial differential equations, and died in a car accident together with his wife while returning from the ceremony to receive the award. Similarly, Lloyd Shapley (mathematician and economist) whose value of game we discussed was awarded the 2012 Nobel Prize in Economics, together with Alvin E. Roth, for the use of game theory in economics. The outstanding American mathematician of Hungarian origin, John von Neumann, was also intensely interested in game theory; his theorem quoted and proven is of great importance in the application of game theory in economics. The paper also shows the possibility of using some theorems from number theory in game theory, which can be used in so-called Dutch auctions. The author of the work tried to present the material in an accessible way so that it could be used by a larger group of readers. Examples of applications of game theory in economics were taken mainly from the author’s own research. In the first chapter, entitled Application of game theory in economics. Literature review, the most important English-language publications on the application of game theory in economics are cited and discussed. In the second chapter, entitled Introductory information to game theory, the author presents the basic concepts used in game theory. Chapter three entitled The minimax theorem, John von Neumann’s theorem is divided into two subchapters. In section 3.1 the minimax theorem and its proof are presented. Similarly, in section 3.2 John von Neumann’s theorem was presented along with its proof. Chapter four entitled Different types of games is divided into five subchapters. In section 4.1, information about the 2 x 2 game is presented, along with examples and a theorem with a proof concerning the solution of this type of game.

Three-player zero-sum games are discussed in section 4.2. Section 4.3 discusses the 2 x n type of game using the 2 x 4 game as an example. Section 4.4 is an empirical subsection in which a specific 5 x 5 game is presented and its solution. The last subsection 4.5 in chapter four deals with two-matrix games. In this section we state and prove the important theorem that every finite two-matrix game has at least one equilibrium point. In the fifth chapter entitled The dendrite of the game, basic information about the dendrite of the game is given and the theorem about the decomposable game and its n strategies in equilibrium is proved.

Chapter six is titled The tendering problem, J. Nash’s axioms, J. Nash’s theorem. This chapter discusses the situation in which cooperation between players is allowed. To find a tender solution, John Nash’s axioms are used. The chapter ends with John Nash’s theorem. It proves that for all bidding problems there is exactly one function for these problems that satisfies the Nash axioms. Chapter seven is titled N-person games. This chapter introduces, among others, the concepts of coalition, imputation, characteristic function of the game, game core, and game isomorphism. In addition, a number of definitions and 7 theorems are given, 3 of which are proven. Chapter eight, entitled Market games, presents market games according to Edgeworth. Chapter nine is entitled Game theory according to L. S. Shapley. This chapter presents Shapley’s axioms. A theorem with a proof is also given that for all games there is only one function that satisfies the Shapley axioms. The Shapley value has also been defined in various cases, especially for simple games. The next tenth chapter is about infinite games.

In addition to the definition of infinite games, an example of a solution to an infinite game is presented. Furthermore, mention was made of S. Ulam and Banach–Mazur type games used in auctions. An infinite positional game with complete information is defined as well as an infinite determinate game. Chapter eleven is titled Statistical games. These are the main models of decision-making. This chapter defines the concepts of risk function, statistical game, decision function, Bayesian risk of decision function, and extended statistical game. In addition, the synthetic quality index and its use are discussed. Chapter twelve concerns stochastic games. This chapter defines the concept of a stochastic game, the player’s strategy in a stochastic game, and gives two theorems with proofs concerning stochastic games. Chapter thirteen is titled Differential games. This type of games is a certain generalization of stochastic games. In addition to the definition of a differential game, the basic equation of the theory of differential games is derived here. The chapter ends with an empirical example on the application of differential games. The last chapter of the monograph entitled Examples of application of game theory in economics contains fourteen examples of applications. Example 1 discusses the problem of optimal investment strategies for investors producing products in the same industry. For this purpose, a tree illustrating a small and a large market was used. Example 2 discusses a two-goods market in which two producers operate. John von Neumann’s theorem was used to find the optimal strategies of both players. Example 3 shows how to find the bargaining set and the John Nash equilibrium in a specific case. Example 4 discusses the tendering problem using an auction as an example. Example 5 discusses the duopoly problem in terms of achieving various types of equilibria, from classical equilibrium, Nash-Carnot equilibrium, Nash equilibrium, to the equilibria of producers as monopolists. Example 6 discusses the Stock Exchange as a case of a multiplayer game. The subject of Example 7 is the calculation of the Shapley value using the example of a joint-stock company consisting of four shareholders. Example 8 shows what a game is in the sense of S. Ulam. Example 9 discusses a Banach-Mazur type game. Examples 10 and 11 show how to use elements of number theory for game theory. Example 12 discusses the use of a synthetic quality measure to solve specific problems. The problem discussed is the evaluation of a pair of stockings produced by a knitting company. Example 13 concerns the selection and purchase of a license for the purchase of a truck trailer. The last example, No. 14, concerns the important problem of exploitation of valuable deposits.

The practical and theoretical goal set in the introduction to the monograph has been achieved.

It is hard to overestimate the importance of game theory in contemporary economics. There is constant research into the even broader use of this discipline in the social sciences. This makes sense because economics is a social science and human behavior plays a very important role in it.