In game theory, the strategic form (or normal form) is a way of describing a game using a matrix. The game is defined by exhibiting on each side of the matrix the different players (here players 1 and 2), each strategy or choice they can make (here strategies A and B) and sets of payoffs they will each receive for a given strategy (p1A,p2A ; p1A,p2B ; p1B,p2A ; p1B,p2B).
The strategic form allows us to quickly analyse each possible outcome of a game. In the depicted matrix, if player 1 chooses strategy A and player 2 chooses strategy B, the set of payoffs given by the outcome would be p1A,p2B. If player 1 chooses strategy B and player 2 chooses strategy A, the set of payoffs would be p1B,p2A.
The strategic form is usually the right description for simultaneous games, where both players choose simultaneously, as opposed to sequential games for which is better to describe the game using the extensive form (or tree form). It’s worth mentioning that simultaneous games imply there is complete and imperfect information, and the rules of the game as well as each player’s payoffs are common knowledge.
A well-known example of a simultaneous game described using the strategic form is the prisoner’s dilemma, where two prisoners need to decide whether they are willing to confess a crime or to lie about it. In this game, payoffs are negative values since they represent years of prison. If both prisoners confess, they will get 8 years each, if they both cooperate with each other and lie about the crime, they will get 1 year each. However, if one of them confesses while the other doesn’t, they will get very different sentences: the one who confessed will walk, while the other will make time (10 years). The elimination of dominated strategies shows us that confess-confess is a Nash equilibrium.
Although is not very common, sequential games can also be described using the strategic form. Considering the following example: player 1 has to decide between going up or down (U/D), while player 2 has to decide between going left or right (L/R). In this case, we can represent this game using the strategic form by laying down all the possible strategies for player 2:
-go Right if player 1 goes Up, go Left otherwise;
-go Left if player 1 goes Up, go Right otherwise;
-go Right no matter what;
-go Left no matter what.
We can see how this game is described using the extensive form (game tree on the left) and using the strategic form (game matrix on the left). Since this is a sequential game, we must describe all possible outcomes depending on player 2 decisions, as seen in the game matrix. There is an perfect subgame Nash equilibrium (green) and a subgame imperfect Nash equilibrium (red).