Payoff Matrix: Overview and Game Theory Explaination

Game theory is defined as the science of strategy. In decision making situations, individuals are faced with conflicting and cooperative methods of strategy against rational opponents in which different combinations of strategies result in different payouts (Dixit, Nalebluff). Payouts differ depending on the type of game being played, however, they generally follow a trend of being positive for both players, negative for both players, or positive for one and negative for the other. Matrices are constructed to calculate and present these different payouts and serve as the rules for a particular instance of game theory.

A simple payoff matrix to read is one of a two person zero sum game. In this payoff matrix, the trace of the matrix is all zeroes. The rest of the triangle consists of ones and negative ones that represent a win or loss for one of the players. Also, the rows and columns of the matrix contain the same elements in different order so the zero vector is a linear combination of both the rows and the columns (Waner).

Payoff matrices can be used for analyzing phenomena such as dominant strategies. A strategy is dominant if no matter what the player chooses, the payoff will be equal to or greater than any other option available given a certain strategy from the opponent. For example, let’s say player 1 is given the choices (v1,…,vk) and player 2 is given the choices (w1,…,wn). If the payoff v1wn is equal to or better than any payoff vkwn, v1 is player 1’s dominant strategy. Likewise, if the payoff vkw1 is equal to or better than any payoff vkwn, w1 is player 2’s dominant strategy (Sönmez).

There is also a phenomenon known as the dominant strategy equilibrium where both players have a dominant strategy. In this case, it is very likely they both choose their dominant option. This is the dominant strategy equilibrium. When a player has a dominant strategy, we can assume that they will choose the dominant option. In this case, the kxn matrix of the payoffs will reduce in favor of the dominant player. Therefore, if player 1 has the dominant strategy but player 2 does not, the original kxn matrix of choices is transformed into a 1xn matrix with the assumption that player 1 will only choose the dominant strategy. This is called iterated elimination of dominated strategies (Sönmez).

If there are no payoffs that result in this manner, the strategies are non-dominant. A Nash equilibrium occurs when deviating from a given payoff will always result in a lesser payoff. This option is only present where there are no dominant strategies. In this case, for the Nash equilibrium vkwn, vk is the greatest payoff in vector v and wn is the greatest payoff in vector w (Sönmez).

Payoff matrices are also used to calculate what is known as an expected value. Expected values can be found when players decide to use mixed or pure strategies. A mixed strategy is when a player decides to play their strategies at predetermined frequencies. A pure strategy is when a player decides to play only one strategy. A strategy is fully mixed if all frequencies are greater than zero. Expected value e is found by multiplying the row frequency matrix, the column frequency matrix, and the payoff matrix. The expected value represents the average payoff per round given that the players stick to their mixed strategies (Waner).

The fairness of a game can be determined by its saddle-point entry. The saddle-point entry is the point in which the row minimum is also the column maximum. A matrix can have multiple saddle-point entries but they will result in the same payoff. A game is strictly determined if there is at least one saddle-point. If the saddle-point is zero, the game is said to be fair. If the saddle-point is non-zero, the game is unfair or biased (Waner).

Payoff matrices are essential to understanding Game Theory and its outcomes. With that in mind, Linear Algebra is directly essential to the understanding as well. Through mathematical analyzation and visual representations, we are able to navigate the complexities of Game Theory in a simple way. Without Linear Algebra, it would be difficult to see the little details that allow these strategies to work out in the way that they do.