Optimization competition and game models in economics

Aim. The work aimed to analyze the relationship between the optimization competition indicator introduced earlier in the author’s articles and the game models widely used in economics, in particular, zero-sum matrix games.
Objectives. The work seeks to determine the quantitative relationship between the solutions of game models and optimization competition, which allows for a new interpretation of the results of game models in economics; as well as to correlate the optimal strategies in game models with the optimization competition indicator.
Methods. The analysis was used to perform a study of the relationship between the previously introduced competition indicator (optimization competition) and game models. Examples were applied to establish a quantitative relationship between optimization competition and the results of zero-sum matrix games in pure and mixed strategies.
Results. A number of game models involve the use of optimization methods, i.e. linear programming methods, in matrix games with mixed strategies. Since the previously introduced optimization competition indicator was developed specifically for optimization problems, it becomes appropriate to study the relationship between it and the solution of game problems. The idea consists in comparing the calculations of the optimization competition indicator with the results of game models. The examples present the patterns of changes in the optimization competition indicator depending on various types of game models, in particular, matrix games. The differences and features in the cases of matrix games in pure and mixed strategies were determined. 
Conclusions. The work presents the relationship between optimization competition and the results of zero-sum matrix games. “Pure gain” (or “pure strategies” called in game economic models) is possible only with non-zero optimization competition, and the average one, expected, with probability, depending on the payoff matrix, can be accompanied by both zero and non-zero competition. In other words, pure gain requires that competition be greatest, all other things being equal. This approach allows a new approach to interpreting the results of game models, in particular, zero-sum matrix games. Nowadays, the game result is only its price determined in pure or mixed strategies. But the results given indicate that it is advisable to compare the value of the game price with the value of optimization competition, which provides additional information for analysis in game economic models. 

1. Barkalaya O.G. The concept of competition in optimal resource allocation and methods for its assessment. Ekonomika i upravlenie = Economics and Management. 2021;27(9): 734-740. (In Russ.). DOI: 10.35854/1998-1627-2021-9-734-740

2. Barkalaya O.G. Investigating competition in the problems of optimal resource allocation. Ekonomika i upravlenie = Economics and Management. 2022;28(4):359-368. (In Russ.). DOI: 10.35854/1998-1627-2022-4-359-368

3. Wagner H.M. Principles of operations research: With applications to managerial decisions. Englewood Cliffs, NJ: Prentice-Hall, Inc.; 1969. 937 p. (Russ. ed.: Wagner H. Osnovy issledovaniya operatsii. In 3 vols. Vol. 1. Moscow: Mir Publ.; 1972. 335 p.).

4. Trukhaev R.I. Models of decision making under uncertainty. Moscow: Nauka; 1981. 257 p. (In Russ.).

5. Zenkevich N.A., Gubar E.A. Operations research workshop. St. Petersburg: Zolotoe sechenie; 2007. 170 p. (In Russ.).

6. Ackoff R.L., Sasieni M.W. Fundamentals of operations research. New York, NY: John Wiley & Sons, Inc.; 1968. 455 p. (Russ. ed.: Ackoff R., Sasieni M. Osnovy issledovaniya operatsii. Moscow: Mir Publ.; 1971. 534 p.).

7. Saaty T.L. Mathematical methods of operations research. New York, NY: McGraw-Hill Book Co., Inc.; 1959. 421 p. (Russ. ed.: Saaty T.L. Matematicheskie metody issledovaniya operatsii. Moscow: Voenizdat; 1963. 420 p.).

8. Taha H.A. Operations research: An introduction. New York, NY: Macmillan Publishing Co., Inc.; 1987. 876 p. (Russ. ed.: Taha H.A. Issledovanie operatsii. Moscow: Dialektika; 2019. 1056 p.).

9. Danskin J.M. The theory of max-min and its application to weapons allocation problems. Berlin, Heidelberg: Springer-Verlag; 1967. 128 p. (Russ. ed.: Danskin J.M. Teoriya maksimina i ee prilozhenie k zadacham raspredeleniya vooruzheniya. Moscow: Sovetskoe radio; 1970. 200 p.).