## Abstract

In an optimization problem , we have a single decision maker, his feasible decision alternative set, and an objective function depending on the selected alternative. In game theoretical models, we have several decision makers who are called the *players* , each of them has a feasible alternative set, which is called the player’s *strategy set*, and each player has an objective function what is called the player’s *payoff function.* The payoff of each player depends on the strategy selections of all players , so the outcome depends on his own decision as well as on the decisions of the other players . Let *N* be the number of players , \(S_k\) the strategy set of player \(k (k=1, 2, \ldots , N)\) and it is assumed that the payoff function \(\phi _k\) of player *k* is defined on \(S_1 \times S_2 \times \cdots \times S_N\) and is real valued. That is, \(\phi _k : S_1 \times S_2 \times \cdots \times S_N \mapsto \mathbb {R}\). So if \(s_1, s_2, \ldots , s_N\) are the strategy selections of the players , \(s_k \in S_k (k=1, 2, \ldots , N)\), then the payoff of player *k* is \(\phi _k(s_1, s_2, \ldots , s_N)\). The game can be denoted as \(\varGamma (N; S_1, S_2, \ldots , S_N; \phi _1, \phi _2, \ldots , \phi _N)\) which is usually called the *normal form* representation of the game.