Move
In each match situation every player has the possibility to decide how to move.
Strategy
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.
Winning strategy
If a strategy leads to a positive payoff not depending on the other player’s moves it might be considered as a winning strategy.
Player
All involved people in a game are called players.
Game
A game consists of players, rules and payoff values.
Match
It is called a match when players play a game once.
Complete information
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.
Incomplete information
If a game contains information that is not avaiable to all of the players it is called a game with incomplete information.
Tree presentation
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.
Payoff
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.
Matrix 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.
Minimax Strategy
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.
Equilibrium
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.
Fair game
The payoff is zero in the equilibrium.
Mixed strategy
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 strategy
Pure strategies are the strategies that determine the matrix.
Nash equilibrium
There is a Nash equilibrium if a player can only degrade his payoff changing the strategy.
optimal 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.