Player 1}}"},"2":{"wt":"{{color#900Player 2}}"}},"i":0}}]}" id="mwFw">
The matrix provided is a normalform 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 top and player 2 plays left, player 1 receives 4 and player 2 receives 3. In each cell, the first number represents the payoff to the row player (in this case player 1), and the second number represents the payoff to the column player (in this case player 2).
Often, symmetric games (where the payoffs do not depend on which player chooses each action) are represented with only one payoff. This is the payoff for the row player. For example, the payoff matrices on the right and left below represent the same game.


The topological space of games with related payoff matrices can also be mapped, with adjacent games having the most similar matrices. This shows how incremental incentive changes can change the game.
Player 2 Player 1  Cooperate  Defect 

Cooperate  −1, −1  −5, 0 
Defect  0, −5  −2, −2 
The payoff matrix facilitates elimination of dominated strategies, and it is usually used to illustrate this concept. For example, in the prisoner's dilemma, we can see that each prisoner can either "cooperate" or "defect". If exactly one prisoner defects, he gets off easily and the other prisoner is locked up for a long time. However, if they both defect, they will both be locked up for a shorter time. One can determine that Cooperate is strictly dominated by Defect. One must compare the first numbers in each column, in this case 0 > −1 and −2 > −5. This shows that no matter what the column player chooses, the row player does better by choosing Defect. Similarly, one compares the second payoff in each row; again 0 > −1 and −2 > −5. This shows that no matter what row does, column does better by choosing Defect. This demonstrates the unique Nash equilibrium of this game is (Defect, Defect).
Player 2 Player 1  Left, Left  Left, Right  Right, Left  Right, Right 

Top  4, 3  4, 3  −1, −1  −1, −1 
Bottom  0, 0  3, 4  0, 0  3, 4 
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. In order to represent this sequential game we 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. Unlike before he has four strategies, contingent on player 1's actions. The strategies are:
On the right is the normalform representation of this game.
In order for a game to be in normal form, we are provided with the following data:
A pure strategy profile is an association of strategies to players, that is an mtuple
such that
A payoff function is a function
whose intended interpretation is the award given to a single player at the outcome of the game. Accordingly, to completely specify a game, the payoff function has to be specified for each player in the player set P= {1, 2, ..., m}.
Definition: A game in normal form is a structure
where:
is a set of players,
is an mtuple of pure strategy sets, one for each player, and
is an mtuple of payoff functions.
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Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The possible outcomes are:
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