Every strategic situation has three things in common. First there are the players, or the economic actors, involved in decision making. These players can be individuals, firms, or even countries. Second, each of these players has options, or strategies, to choose from. These strategies may be the same or different across players. Last, each player expects some outcome as a result of playing the game, which is the expected payoff from choosing a particular strategy. Note that these payoffs depend not only on the player's choice, but on the choices made by other players as well.
As a simple example, consider a coin flip game with two players who can each choose heads or tails, and the payout of the game is $10 for a correct choice. If the two players make the same choice, then they split the $10 if they are right. If they are wrong, neither player gets any money. However, if they make different choices, then the player who chose correctly gets $10, and the other player gets nothing. The strategy here is the choice of heads or tails, and the payoff is the amount of money the person receives.A payoff matrix is a table that can be used to describe the three components of a game: the players, the strategies, and the payoffs. In an example payoff matrix for the coin flip game, the columns would be the strategies for player 1 and the rows would be the strategies for player 2. Option A would be for choosing tails, and Option B would be for choosing heads, creating a 2-by-2 matrix. Each entry in the table is then a payoff for that outcome. For example, the entry in row 1 and column 2 (the upper right corner) would be the payoff if player 1 chose heads and player 2 chose tails. The upper left and lower right entries in the payoff matrix would both correspond to the two players choosing the same outcome, and therefore the entries in those spaces would be $5 for each player. In the case that the coin flip showed heads, the payoff matrix would have $10 for player 1 and nothing for player 2 in the upper right corner. The lower left entry would have $10 for player 2 and nothing for player 1. In the case that the coin flip showed tails, these two entries would be exchanged.