**1. Introduction**

Shell Game Optimization (SGO) simulating the rules of a game known as shell game to design an algorithm for solving optimization problems in different fields of science [1]. The key idea of the SGO is to find the ball hidden under one of the three shells, which should be guessed by players [2]. There are many real-life cases for the basic thought of game theory [3]. We are uniformly in the game of our life that reshaped by the actions and decisions made by others.

**2. Inspiration of SGO**

In this game, players try to find a ball that is hidden under one of the three Shells. Shell game is an old game, in which the operator provides three shells and a small ball. In this game, the curiosity of players is stimulated, which helps to increase the accuracy of the players [4]. First, the operator invites several persons as players. Then the operator shows the ball to the players. After that, puts the ball under one of the shells. The operator moves the shells on the table using hand. Now the operator asks the players to guess the shell under which the ball is hidden. Each player may choose the correct or wrong shell, depending on the degree of accuracy and intelligence. More points are awarded to the player that recognizes the correct shell [5].

**3. Game Theory in Artificial Intelligence (AI)**

Game Theory in terms of AI basically helps in making decisions. This is not very difficult considering the fact that “Rationality” is the foundation of Game Theory. As a matter of fact, Game Theory has already started establishing its place in Artificial intelligence. Game Theory helps the concept of Generative Adversarial Networks (GANs) [6]. Game theory also has various applications in machine learning that impacts everyday life and real-world implementations [7].

**4. The Nash Equilibrium in Game Theory**

Nash equilibrium is one of the fundamental concepts in game theory which determines the optimal solution in a non- cooperative game in which each player lacks any incentive to change his/ her initial strategy [8]. Under the Nash equilibrium, a player does not gain anything from deviating from their initially chosen strategy and assuming the other player also keeps their strategies unchanged [9].

**5. Shell Game Optimization (SGO)**

For the Shell Game Optimization (SGO), the following assumptions are considered;

In this game, a person is considered as the game’s operator

- Three shells and one ball are available to the operator.
- Each player has only two opportunities to guess the correct shell

Now, initialize a set of n person is assumed as the game’s players.

*Z _{n}* is actually a random value for the problem variables, the position ‘a’ of player ‘n’ is shown as .

After calculating the fitness function value for each player, the game’s operator chooses three shells that one of the shells is related to the position of the best player and two other shells is chosen randomly.

Where, is the position of minimum or maximum problems of fitness, and are positions of two members of the population. Accuracy and intelligence of each player are simulated according to the fitness function value by;

Where is the accuracy and intelligence of player i and is the position of minimum or maximum of fitness (*F*). Now, the player is ready to guess the ball. The guess vector specified for each player.

The probability of choosing one of the states for shell selection is

Where is the possibly of correct guess at the first selection and denotes the possibly of correct guess at the second time. Finally, Z_{n} vector, which is assumed as the location of each member of population, is updated according to the below following equations;

**6. Advantages of SGO**

**7. Pseudo Code of SGO**

**8. Applications of SGO**

SGO is mathematically modeled and implemented on 23 well-known benchmark test functions as well as on a real life-engineering problem entitled pressure vessel design problem [11].

**Reference**

[1]M. Dehghani, Z. Montazeri, O. Malik, H. Givi and J. Guerrero, “Shell Game Optimization: A Novel Game-Based Algorithm”, International Journal of Intelligent Engineering and Systems, vol. 13, no. 3, pp. 246-255, 2020. Available: 10.22266/ijies2020.0630.23.

[2]T. Satterfield, “The ‘Shell Game’: Why Children Never Lose”, Syntax, vol. 2, no. 1, pp. 28-37, 1999. Available: 10.1111/1467-9612.00013.

[3]J. Caspermeyer, “Shell Game: Understanding Gene Patterns Behind Mollusk Diversity”, Molecular Biology and Evolution, vol. 34, no. 4, pp. 1023-1023, 2017. Available: 10.1093/molbev/msx079

[4]”A Disorder by Any Other Name: Excessive Computer Game Playing”, AMA Journal of Ethics, vol. 10, no. 1, pp. 30-34, 2008. Available: 10.1001/virtualmentor.2008.10.1.jdsc1-0801…

[5]M. DEHGHANİ, Z. MONTAZERİ, A. DEHGHANİ and O. MALİK, “GO: Group Optimization”, GAZI UNIVERSITY JOURNAL OF SCIENCE, vol. 33, no. 2, pp. 381-392, 2020. Available: 10.35378/gujs.567472.

[6]A. Kaveh and S. Talatahari, “A novel heuristic optimization method: charged system search”, Acta Mechanica, vol. 213, no. 3-4, pp. 267-289, 2010. Available: 10.1007/s00707-009-0270-4.

[7]O. Erol and I. Eksin, “A new optimization method: Big Bang–Big Crunch”, Advances in Engineering Software, vol. 37, no. 2, pp. 106-111, 2006. Available: 10.1016/j.advengsoft.2005.04.005.

[8]S. Mirjalili, S. Mirjalili and A. application”, Advances in Engineering Software, vol. 105, pp. 30-47, 2017. Available: Lewis, “Grey Wolf Optimizer”, Advances in Engineering Software, vol. 69, pp. 46-61, 2014. Available: 10.1016/j.advengsoft.2013.12.007.

[9]G. Khan, J. Miller and D. Halliday, “Evolution of Cartesian Genetic Programs for Development of Learning Neural Architecture”, Evolutionary Computation, vol. 19, no. 3, pp. 469-523, 2011. Available: 10.1162/evco_a_00043.

[10]N. Krasnogor, G. Nicosia, M. Pavone and D. Pelta, “Special issue on Nature Inspired Cooperative Strategies for Optimisation (NICSO)”, Natural Computing, vol. 9, no. 1, pp. 1-3, 2010. Available: 10.1007/s11047-009-9177-1.

[11]A. Gandomi, X. Yang and A. Alavi, “Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems”, Engineering with Computers, vol. 29, no. 1, pp. 17-35, 2011. Available: 10.1007/s00366-011-0241-y.

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