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Optimal strategies for two-person normalized matrix game with variable payoffs

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Abstract

This paper considers a two-person zero-sum game model in which payoffs are varying in closed intervals. Conditions for the existence of saddle point for this model is studied in this paper. Further, a methodology is developed to obtain the optimal strategy for this game as well as the range of the corresponding optimal values. The theoretical development is verified through numerical example.

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Acknowledgments

The authors are thankful to the referees whose suggestions have improved the presentation considerably.

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Science and Technology, Palur Hills, Berhampur, Odisha, 761008, India

    Ajay Kumar Bhurjee

  2. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, W.B., 721302, India

    Geetanjali Panda

Authors
  1. Ajay Kumar Bhurjee
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  2. Geetanjali Panda
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Corresponding author

Correspondence to Ajay Kumar Bhurjee.

Appendix A: For player \(P_1\)

Appendix A: For player \(P_1\)

One may observe that an optimal strategy \({\mathbf {y}}^* \in Y\) and the value \(\varvec{\nu }=\varvec{\nu }({\mathbf {y}}^*)\) for player \(P_1\) is equivalent to the solution of the following interval linear programming problem:

$$\begin{aligned} \left. \begin{array}{ll} \text{ Maximize }~~\left[ \nu ^L,\nu ^R\right] \\ \text{ subject } \text{ to } \text{ the } \text{ constraints }\\ \underset{i\in {\varLambda }_m}{\sum }\left[ a_{ij}^Ly_i,a_{ij}^Ry_i\right] \succeq \left[ \nu ^L,\nu ^R\right] ,\quad ~j\in {\varLambda }_n \\ \underset{i\in {\varLambda }_m}{\sum }y_i=1,\quad y_i\ge 0,i \in {\varLambda }_m. \end{array} \right\} \end{aligned}$$
(15)

This is equivalent to the following interval programming problem:

$$\begin{aligned} \left. \begin{array}{ll} \text{ Maximize }[\nu ^L,\nu ^R]\\ \text{ subject } \text{ to } \text{ the } \text{ constraints }\\ \underset{i\in {\varLambda }_m}{\sum }\left( a_{ij}^L+t\left( a_{ij}^R-a_{ij}^L\right) \right) y_i\ge \left( \nu ^L+t\left( \nu ^R-\nu ^L\right) \right) ,\quad ~j\in {\varLambda }_n \\ \underset{i\in {\varLambda }_m}{\sum }y_i=1,\quad y_i\ge 0,\quad i \in {\varLambda }_m \end{array} \right\} \end{aligned}$$
(16)

For positive weight function \(w:[0,1]\rightarrow R_+\), we construct the following deterministic problem:

$$\begin{aligned} \left. \begin{array}{ll} \text{ Maximize }\int _0^1w(t)(\nu ^L+t(\nu ^R-\nu ^L))dt\\ \text{ subject } \text{ to } \text{ the } \text{ constraints }\\ \underset{i\in {\varLambda }_m}{\sum }a_{ij}^Ly_i\ge \nu ^L,\underset{i\in {\varLambda }_m}{\sum }a_{ij}^Ry_i\ge \nu ^R,\quad ~j\in {\varLambda }_n \\ \underset{i\in {\varLambda }_m}{\sum }y_i=1,y_i\ge 0,\quad i \in {\varLambda }_m. \end{array} \right\} \end{aligned}$$
(17)

From Theorem 1, optimal solution of problem (17) is an efficient solution of problem (16).

In particular, for weight function \(w(t)=1,\) problem (17) becomes,

$$\begin{aligned} \begin{array}{ll} \text{ Maximize }~\frac{1}{2}(\nu ^L+\nu ^R)\\ \text{ subject } \text{ to } \text{ the } \text{ constraints }\\ \underset{i\in {\varLambda }_m}{\sum }a_{ij}^Ly_i\ge \nu ^L,\underset{i\in {\varLambda }_m}{\sum }a_{ij}^Ry_i\ge \nu ^R,\quad ~j\in {\varLambda }_n \\ \underset{i\in {\varLambda }_m}{\sum }y_i=1,y_i\ge 0,\quad i \in {\varLambda }_m \end{array} \end{aligned}$$

1.1 Appendix B: For player \(P_2\)

One may observe that an optimal strategy \({\mathbf {z}}^* \in Z\) and the value \(\varvec{\omega }=\varvec{\omega }({\mathbf {z}}^*)\) for player \(P_2\) is equivalent to the solution of the following interval linear programming problem:

$$\begin{aligned} \left. \begin{array}{ll} \text{ Minimize }~~\left[ \omega ^L,\omega ^R\right] \\ \text{ subject } \text{ to } \text{ the } \text{ constraints }\\ \underset{j\in {\varLambda }_n}{\sum }\left[ a_{ij}^Lz_j,a_{ij}^Rz_j\right] \preceq \left[ \omega ^L,\omega ^R\right] ,\quad i\in {\varLambda }_m \\ \underset{j\in {\varLambda }_n}{\sum }z_j=1,z_j\ge 0,\quad j \in {\varLambda }_n. \end{array} \right\} \end{aligned}$$
(18)

This problem is equivalent to the following interval programming problem:

$$\begin{aligned} \left. \begin{array}{ll} \text{ Minimize }~~[\omega ^L,\omega ^R]\\ \text{ subject } \text{ to } \text{ the } \text{ constraints }\\ \underset{j\in {\varLambda }_n}{\sum }\left( a_{ij}^L+t\left( a_{ij}^R-a_{ij}^L\right) \right) z_i\le \left( \omega ^L+t\left( \omega ^R-\omega ^L\right) \right) ,\quad i\in {\varLambda }_m\\ \underset{j\in {\varLambda }_n}{\sum }z_j=1,\quad z_j\ge 0,j \in {\varLambda }_n \end{array} \right\} \end{aligned}$$
(19)

For positive weight function \(w:[0,1]\rightarrow R_+,\) we construct the following deterministic linear optimization problem:

$$\begin{aligned} \left. \begin{array}{ll} \text{ Minimize }~\int _0^1w(t)\left( \omega ^L+t\left( \omega ^R-\omega ^L\right) \right) dt\\ \text{ subject } \text{ to } \text{ the } \text{ constraints } \\ \underset{j\in {\varLambda }_n}{\sum }a_{ij}^Lz_j\le \omega ^L,\underset{j\in {\varLambda }_n}{\sum }a_{ij}^Rz_j\le \omega ^R,\quad i\in {\varLambda }_m\\ \underset{j\in {\varLambda }_n}{\sum }z_j=1,\quad z_j\ge 0,j \in {\varLambda }_n \end{array} \right\} \end{aligned}$$
(20)

From Theorem 1, optimal solution of problem (20) is an efficient solution of problem (19).

In particular, for weight function \(w(t)=1,\) problem (20) becomes,

$$\begin{aligned} \begin{array}{ll} \text{ Minimize }~\frac{1}{2}(\omega ^L+\omega ^R)\\ \text{ subject } \text{ to } \text{ the } \text{ constraints }\\ \underset{j\in {\varLambda }_n}{\sum }a_{ij}^Lz_j\le \omega ^L,\underset{j\in {\varLambda }_n}{\sum }a_{ij}^Rz_j\le \omega ^R,\quad i\in {\varLambda }_m\\ \underset{j\in {\varLambda }_n}{\sum }z_j=1,\quad z_j\ge 0,j\in {\varLambda }_n \end{array} \end{aligned}.$$

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Bhurjee, A.K., Panda, G. Optimal strategies for two-person normalized matrix game with variable payoffs. Oper Res Int J 17, 547–562 (2017). https://doi.org/10.1007/s12351-016-0237-x

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