A game can have many other structural properties. It can be represented as a single-shot or multi-stage decision problem, or it can include simultaneous or stochastic moves. We start with games in strategic form without stochastic moves, and will introduce more sophisticated games as we go along in the entry. In a strategic game, each player \(i\) can choose from a (finite) set \(S_i\) of options, also called actions or strategies. The combination of all the players’ choices, denoted \({\mathbf{s}}\), is called a strategy profile, or outcome of the game. We write \({\mathbf{s}}_i\) for \(i\)’s component in \({\mathbf{s}}\), and \({\mathbf{s}}_{-i}\) for the profile of strategies for all agents other than \(i\). Finally, we write \(\Pi_{i \in \Agt} S_i\) for the set of all strategy profiles of a given game. Putting everything together, a strategic game is a tuple \(\langle \A, \{S_i, u_i\}_{i\in\A}\rangle\) where \(\A\) is a finite set of players, for each \(i\in\A\), \(S_i\) is a finite set of actions and \(u_i:\Pi_{i\in\A} S_i\rightarrow\mathbb{R}\) is player \(i\)’s utility function.