Discrete Static Games

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.