In each match situation every player has the possibility to decide how to move.
A special sequence of moves which determinate the end of the match is called strategy. A strategy can be understood as a comprehensive action plan: It tells the player how to move in each game situation.
Set of Strategy
The totality of all possible strategies in a game is called set of strategies.
If a strategy leads to a positive payoff not depending on the other player’s moves it might be considered as a winning strategy.
All involved people in a game are called players.
A game consists of players, rules and payoff values.
It is called a match when players play a game once.
If a player knows all other player’s possible moves the game can be called a game with complete information. The knowledge of the other player’s strategies is given as well.
If a game contains information that is not avaiable to all of the players it is called a game with incomplete information.
Possible game options can easyly be illustrated as a tree-diagram. The knots symbolize the players, the strategies at a certain point of time are shown as the edges of the diagram.
Receiving a positive payoff is the objective for each player.
two-persons zero-sum games
If the added payoff values of both players equal zero the game is called a zero-sum-game.
The matrix consists of rows that contain the payoffs of the row player, and columns that contain the data of the other player. A cell containing the payoff values exists for each set of two strategies.
Since in a zero-sum-game the payoff value of each player is the negated value of the other, matrixes illustrating zero-sum-games show only one value. That value is associated to the player shown in the lines.
The minimax strategy is the ideal strategy for two-person zero-sum-games. The row player chooses the maximum of the row minima and the colomn player chooses the minimum of the column maxima.
If both values are equal, there is a value of the game.
Do both player in two-person zero-sum-game choose their minimax strategy and the payoffs are equal, there is an equilibrium in the set of strategies.
The payoff is zero in the equilibrium.
There is a mixed strategy, if the decision for the pure strategy is not the same repeating the game and is calculated by a random procedure.
Based on two pure strategies, they will be choosen with the porbability p respectively 1-p, whereby 0 <= p <= 1. According to this there are endless strategies.
Value of a game
The value of a game is the value in the equilibrium.
Pure strategies are the strategies that determine the matrix.
There is a Nash equilibrium if a player can only degrade his payoff changing the strategy.
The equilibrium strategy is the optimal strategy.
non constant-sum game
The cells of the matrix contain two numbers. The sum of those two number is not constant and especially not 0.