On the Representation of Winning Strategies of Finite Games by Groups and Neutrosophic Groups

Mikail Bal, Gaziantep University, Turkey

Mohammad Abobala, Tishreen University, Syria

Mohammadabobala777@gmail.com

Abstract: In this paper, we show that for a finite game with two players A , B:

Each winning strategy of the first player A can be represented by a neutrosophic subgroup of the neutrosophic group ( , and each winning strategy of the second player B can be represented by an elementary abelian group .

Also, we introduce the concept of algebraically relative  games and present some examples on it.

Keywords: Group , Neutrosophic group , Winning strategy

1.Introduction

Groups are always very useful in representations of algebraic structures, and finite games as a finite steps can be considered.

 Neutrosophy as a branch of philosophy introduced by F.Smarandache has many applications in the real world and the mathematical concepts. The concept of neutrosophic group had been defined in[2]  as a generalization of classical groups, subgroups and normal subgroups also were defined and studied.

The most useful understanding of neutrosophic group has been written in [3], we consider N(G) as a union of G and GI  i.e N(G)=  

We will use a neutrosophic subgroup to represent every winning strategy of player A, and a classical group to represent every winning strategy of player B.

This research maybe very useful in the progression of game theory by algebraic views.

2. Preliminaries :

Definition 2.1 :[2]

  Let  (G,*) be a group . Then the neutrosophic group is generated by  G and  I under * denoted by N(G)= .

I is called the indeterminate element (neutrosophic element)  with the property .

The most useful understanding of this definition has been written in [3], we consider N(G) as a union of G and GI  i.e N(G)= .

Definition 2.2 :[2]

Let N(G) be a neutrosophic group, then a neutrosophic subgroup is a subset of N(G) contains a proper subgroup of G.

Remark2.3 :[2]

Neutrosophic subgroup is not a group but contains a group.

Definition 2.4 :[5]

An abelian group G is called elementary abelian if it is isomorphic to  for such a positive integer n.

For concepts like game, analyzing game, and combinatorial game see [4].

3. Main results and discussion :

Suppose that G is a game with finite steps. Two players A , B play this game, they make their steps alternately,  i.e (their choices) from a finite set of objects S={ .

If we reach to a position which A cannot chose any object then B is the winner, and conversely A is the winner.

Without affecting the generality we can suppose that the alternating choices of two players can be realized as :

We say that a step i is complete if both players were able to chose objects without being losers.

For each complete step, we can represent it by a bijective map f  wich permutes the chosen objectives in this step an fixes the rest of unchosen objectives, i.e if the player A chooses the element  and B chooses , then we represent this complete step by the map : f  with f( )=  and f( )=  and f( )=  for each  , we can use algebraic symbol as: f = (  .

Theorem 3.1 :

Let  be the representation of the complete step i, then =I (of order 2) , where I is the identity map on S.

Proof :

It is easy to see that .

We represent the beginning position of the game by I (identity map).

Theorem 3.2 :

Each winning strategy of second player B can be represented by a group with type

Proof :

If B has a winning strategy, then we will reach to a position that B can choose and player A cannot, as follows :

We assume that the number of steps is k, we remark that all steps are complete and each step's representation is a bijective with order 2, so the group generated by all representations is .

We call the previous group by a strategy representation.

Definition 3.3 :

If we reach to a position which A can chose and B cannot, we represent it by the indeterminate map J, which it means that A can pick an object and B cannot.

Remark: The indeterminate map J has the property  , we mean by this property that if we reach to a winning position of player A , then the next position is the same.

 Theorem 3.4 :

Each winning strategy of first player A can be represented by a neutrosophic subgroup with type .

Proof :

If A  has a winning strategy then we will reach to a position that A can choose and then B cannot , as the following :

We assume that the number of steps is k+1 , we remark that all steps are complete unless the last step. The group generated by all steps unless the last one is .

For the last step we can represent it by the indeterminate J, thus the strategy representation is the neutrosophic subgroup of N(  which is set :

Result 3.5 :

If A is the winner then the strategy representation is a neutrosophic group , and if B is the winner then the strategy representation is a classical group
Definition 3.6 :

(a) If the player B has a winning strategy, then the winning strategy with minimum representation group order is called the perfect strategy of B.

 (b) If the player A has a winning strategy , then the winning strategy with minimum representation group order is called the perfect strategy of A.

Definition 3.7 :

If H , K are two finite games , we say that H is algebraically relative or (H-ar-K), if there is a perfect strategy of the Player A in both games with the same representation neutrosophic group, or a perfect strategy of the player B in both games with same representation group .

Remark : The essential meaning of algebraically relative games is that they have winning strategies with the same number of steps.

Example 3.8 : 

Suppose that we have two players A , B which they are playing Wythoff  game with (3,2) as a beginning position , A at least needs two steps to win , we can clarify it by the following example:

 The representation neutrosophic subgroup is

Let the same players play the HIM-Game defined in [4]. The beginning position is

(2,4,5,10) , A has a perfect strategy as

The representation neutrosophic subgroup is  , thus the previous two games are algebraically relative.

4. Conclusion

In this research, we have introduced a representation of winning strategies of finite alternating games by groups and neutrosophic groups. Also, we have introduced the notion of algebraically relative games and gave many examples.

References

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[2] Kandasamy .V and Samarandache ,F , some neutrosophic algebraic structures and neutrosophic N-algebraic structures , Hexis , Phonex , Arizona 2006 , p.p 219

[3] Chalapathi .T and Kumar.K , neutrosophic graphs of finite groups , neutrosophic sets ans systems , vol 15 , (2017)

[4]Dabash.M , Al-Najjar.H and Barbara.H , Create HIM , the adjusted NIM game , and analyses its winning strategy , PH.D thesis , university of Aleppo press , 2013 , pp20-130

[5] Haushi M , Algebraic structures , Tishreen university press , 2004 , p.p 112-140.

[6] Prem Kumar Singh, Three-way n-valued neutrosophic concept lattice at different granulation, International Journal of Machine Learning and Cybernetics, 2018, Vol 9, Issue 11, pp. 1839-1855