Normal-Form Games

Normal-form games are games where:

• there is a finite set of agents $N=\{1,2,...,n\}$$\large N = \{1, 2, ..., n\}$ that act simultaneously without the knowledge of other agents' actions
• every agent $i$$\large i$ has a set of available actions, denoted by $A_i$$\large A_i$, where the action taken by agent $i$$\large i$ is denoted by $a_i\in A_i$$\large a_i \in A_i$
• the outcome is defined by an action profile $a=(a_1,a_2,...,a_n)$$\large a = (a_1, a_2, ..., a_n)$, i.e. the set of actions taken by all agents (often also denoted by $a=(a_i,a_{-i})$$\large a = (a_i, a_{-i})$)
• the set of all action profiles is denoted by $A=\prod _i{A}_i$$\large A = \prod\limits_i A_i$
• every agent $i$$\large i$ has a utility function $u_i$$\large u_i$ that maps an action profile to real numbers
• an action profile of all agents except agent $i$$\large i$ is denoted by $a_{-i}$$\large a_{-i}$
• a set of action profiles of all agents except agent $i$$\large i$ is denoted by $A_{-i}=\prod _{i\ne j}{A}_j$$\large A_{-i} = \prod\limits_{i \ne j} A_j$