A mathematical game specifies participants, available choices, information and consequences of combined choices. A normal-form game records complete strategies and payoffs; an extensive-form game records the sequence of decisions and what each player knows at them.
For example, matching pennies gives each player two actions and opposing payoffs depending on whether their choices agree. Its equilibrium requires mixed strategies. A puzzle with one decision maker is often called a game informally, but strategic interaction and game theory enter when outcomes depend on several decision makers' choices. The rules determine the mathematical object; a physical board alone does not.
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