N-Person Noncooperative Games
Definition
An n-person strategic game is a $2n+1$-tuple
$$G = (N,S_1,\dots,S_n,u_1,\dots,u_n)$$Where
- $N = \{1,\dots,n\}$, with $n \geq 1$, is the set of players
- $S_i$ is the strategy set of player $i \in N$
- $u_i:S=S_1\times\dots\times S_n \rightarrow \mathbb{R}$ is the payoff function of player $i$ defined on the Cartesian product of the players' strategy sets $S = \prod_{i=1}^n S_i$ (The set of situations in the game); that is, for every strategy combination, also known as a strategy profile or a situation, $(s_1,\dots,s_n)\in S$ where $s_1\in S_1,\dots,s_n\in S_n$, $u_i(s_1,\dots,s_n)\in \mathbb{R}$ is player $i$'s payoff.
Each player $i$ chooses his strategy $s_i$ simultaneously and independently from the strategy set $S_i$. Therefore a strategy profile or a situation $s = (s_1,\dots,s_n)$ is generated, where $s_i \in S_i$. Then, each player receives his payoff $u_i(s)$, whereupon the game ends.
Important Concepts
Best Reply
A best reply of player $i$ to the strategy combination $(s_1,\dots,s_{i-1},s_{i+1},\dots,s_n)$ of other players is a strategy $s_i \in S_i$,
$$ u_i(s_1,\dots,s_{i-1},s_i,s_{i+1,\dots,s_n}) \geq u_i(s_1,\dots,s_{i-1},s_i',s_{i+1,\dots,s_n}) ~~\forall ~s_i'\in S_i $$Nash Equilibrium
A Nash equilibrium is a strategy combination $(s_1^*,\dots,s_n^*) \in S$ such that for each player $i$, $s_i^*$ is a best reply to $(s_1^*,\dots,s_{i-1}^*,s_{i+1}^*,\dots,s_n^*)$
Domination
A strategy $s_i' \in S_i$ of player $i$ is strictly dominated by $s_i\in S_i$ if
$$u_i(s_1,\dots,s_{i-1},s_i,s_{i+1},\dots,s_n) > u_i(s_1,\dots,s_{i-1},s_i',s_{i+1},\dots,s_n)$$For all $(s_1,\dots,s_{i-1},s_{i+1},\dots,s_n) \in S_1\times \dots \times S_{i-1} \times S_{i+1} \times \dots \times S_n$; that is, for all strategy combinations of players other than $i$. Logically, a strictly dominated strategy is never used in a Nash equilibrium.
By replacing the condition in last inequality from a greater than to a greater than or equal, then the relationship between $u_i(s_1,\dots,s_{i-1},s_i,s_{i+1},\dots,s_n)$ and $u_i(s_1,\dots,s_{i-1},s_i',s_{i+1},\dots,s_n)$ becomes a weak domination.