Normal form game
In game theory, normal form is a way of describing a game. It differs from the extensive form in that it is not graphical per se, but it can be of greater use in identifying strictly dominated strategies and Nash equilibria.
In static games of complete, perfect information, a normal form representation of a game is a specification of players' strategy spaces and payoff functions. A strategy space for a player is the set of all strategies available to that player, where a strategy is a complete plan of action for every stage of the game, regardless of whether that stage actually arises in play. A payoff function for a player is a mapping from that the cross-product of players' strategy spaces to that player's set of payoffs (normally the set of real numbers, where the number represents a cardinal or ordinal utility - often cardinal in the normal form representation) of a player, i.e. the payoff function of a player takes as its input a strategy profile (that is a specification of strategies for every player) and yields a representation of payoff as its output.
Payoff matrices
It is often convenient to represent a normal form game with a matrix. For example:
| Player 2 plays action 2a | Player 2 plays action 2b | |
|---|---|---|
| Player 1 plays action 1a | 1, 1 | 0, 2 |
| Player 1 plays action 1b | 0, 0 | 1, 2 |
The above constitutes a normal form representation of a game in which players move simultaneously (or at least do not observe the other player's move before making their own) and receive the payoffs as specified for the combinations of actions played. For example, if player 1 plays 1a and player 2 plays 2b, player 1 receives 0 and player 2 receives 2.
This way of representing games is a useful way of presenting the information. From the above matrix, it is easy to see that 2a is strictly dominated by 2b. There is no situation in which player 2 is better off (or even at least as good off) by playing 1a. Given common knowledge, player 1 knows this and so knows that player 2 will always play 2b. In this case, however, player 1 is always better off playing 1b.
The above is an example of iterated elimination of strictly dominated strategies, a solution concept facilitated in its application by a matrix.
It is also useful to spot (pure strategy) Nash equilibria.
| Left | Right | |
|---|---|---|
| Up | -1, 2 | 1, 2 |
| Down | 0, 2 | -1, 3 |
For example, in the game matrix on the left if a payoff is emboldened, it means that player whose payoff it is has a best response to the corresponding strategy played by the other player by playing that strategy.
If player 2 plays Left, player 1's best response is Down. If player 2 plays Right, player 1's best response is Up.
If player 1 plays Up, player 2 is indifferent between Left and Right (they are both best responses) and if player 1 plays Down, player 2's best response is Right.
Hence the Nash equilibrium, a best response to a best response (and shown by both players having bold payoffs), is (Up, Right).
However, note that these matrices only represent games in which moves are simultaneous (or more generally information is imperfect. The above matrix does not represent the game in which player 1 moves first observed by player 2 and then player 2 moves because it does not specify each of player 2's strategies in this case, which must specify all of player 2's actions, even in contingencies that can never arise in the course of the game. In this game, player 2 has actions, as before, Left and Right, but he has strategies, unlike before where his strategy space and action space were identical, Left if player plays Up and Left otherwise, Left if player plays Up and Right otherwise, Right if player plays Up and Left otherwise, and Right if player plays Up and Right otherwise. The above matrix, although it is also a payoff matrix for this game, is not a game matrix because it does not specify all of player 2's strategies.
| Left, Left | Left, Right | Right, Left | Right, Right | |
|---|---|---|---|---|
| Up | -1, 2 | -1, 2 | 1, 2 | 1, 2 |
| Down | 0, 2 | -1, 3 | 0, 2 | -1, 3 |
In the matrix on the left, payoffs are emboldened as before to show the best response of the player whose payoff it is (note that a best response is the strategy according to that payoff, not the payoff). Player 2's strategies are represented as follows: 'A, B' means 'play A if player 1 plays Up and play B otherwise'.
There are four (pure strategy) Nash equilibria. (Up; Right, Left) and (Up; Right, Right) correspond to (Up, Right) in the previous game. Whatever player 2 does if player 1 plays Down is not relevant in these equilibria, since a best response of 1 is to play Up if player 2 plays Right when player 1 plays Up.
The equilibria (Up; Left, Right) and (Down; Left, Right) arise from the fact that player 1 is indifferent playing Up and Down if player 2 will play Left when player 1 plays Up and Down otherwise. If player 1 thought correctly that for some reason player 2 would play the strategy Left, Right, it would not matter to his payoff what he played, i.e. Up and Down are both best responses to player 2's strategy. However, Left, Right is a best response both to Up and to Down; hence these Nash equilibria occur.
Importantly, these Nash equilibria demonstrate the difference between a simultaneous game and sequential game that have the same action spaces (but not strategy spaces) and payoff matrices (but not game matrices).