Game Theory

Game theory is a mathematical theory that deals with the general features of competitive cases and places particular emphasis on the decision making processes of the rivals. 

Historically, the game theory was first proposed by the French mathematician Emil Borel in 1921. Then the game theory was motivated by John von Neumann and Oskar Morgenstern to solve problems in economics, although it was developed extensively by John Nash in the 1950s. 

A strategy is a complete description of a player’s course of action during the game. There are two kinds of strategies, pure strategy and mixed strategy.

A pure   strategy is an unconditional decision always to select a particular course of action. For example, in the game of Rock-Paper Scissors, if a player would choose to only play Rock for each interdependent trial, regardless of the other player’s strategy, it would be the player’s pure strategy. 

A mixed strategy is a decision to choose a course of action for each play in accordance with some particular probability. In other words, a player using a mixed strategy incorporates more than one pure strategy in a game. In the game of Rock-Paper-Scissors, if a player’s probability of using each pure strategy is equal, then the probability distribution of strategy set would be approximately 33% for each course of action. 

A pay-off is numerical value which indicates the amount gained or lost by a player at the end of the game contingent upon the course of actions all of other players. Each pay-off can be negative, positive or equal to zero. If a player seeks to maximize his or her own pay-off, then the player is said to be rational. 

A pay-off matrix shows the gains and losses that result from a combination of players’ strategy choices. The entries of pay-off matrix can be negative, positive or equal to zero. If an element of pay-off matrix is positive, the column player pays this amount to the row player. If an element of pay-off matrix is negative, the row player pays the absolute values of this amount to the column player. 

A saddle point is an element of the pay-off matrix that is simultaneously the smallest element in its row and the largest element in its column. Furthermore, saddle point is also regarded as an equilibrium point in the theory of games.

Probably, one of the most well-known strategic game is The Prisoner’s  Dilemma. Its name is come from a story two prisoners that are each suspected of a suspected in a crime. Two suspects in a serious crime are held in separate cells and unable to communicate with each other. Each is invited to confess and contaminate the other.

Another game is called Matching Coins. In this game, each of two players have a coin. They display their coins simultaneously on the table, with either heads or tails facing up. If both players’ coins are heads or tails, then the first player gets both coins. If the coins do not match, the second player gets both coins.

Battle of the Sexes is another two person game in the game theory. In this game, a couple Mark and Kelly wish to go out together, but they have different preferences. Mark prefers to go to the football match while Kelly wants to go to the opera. In this game both players wish to coordinate their behavior, but they have conflict interests. That is, there are two steady states: one in which Mark and Kelly always choose football match and one in which they always choose opera.

Hawk-Dove, Rock, Paper, Scissors, and The Odds and Evens Game are the other stragey games regarded in the game theory.

Games can be classified according to some certain features. One of the most obvious is to classify a game by the number of players. Another basis for classifying games is the goals of players coincide or conflict based on possible winnings.  Another class of games is symmetric and asymmetric games. In symmetric games, all players have the same actions and even in case of interchanging players, the actions of players remain the same.  

Variable-sum games can be divided even further as being either cooperative or non-cooperative       games. Players of cooperative games can communicate and are permitted to make binding agreements, while non-cooperative games players can communicate but they cannot make binding agreements, such as an enforceable contract. 

In symmetric games, all players have the same actions and even in case of interchanging players, the actions of players remain the same. The other words, the actions in a symmetric game depend on the strategies used, not on the players of the game. The Prisoner’s Dilemma is an example of a symmetric game.

A game is defined to be any situation in which • There are at least two players, • Each player has a finite number of strategies, • Each player’s chosen strategies determine the outcome of the game, • Each player acts rationally to maximize his/her gains, • The different strategies of each player and the amount of gain is known to each player in advance.

A two-person zero-sum game has the feature that for any choice of strategies, the sum of the gains for the players is zero. In these games, every dollar one player wins comes out of the other player’s pocket. Thus, two players have totally conflicting interest and there would be no cooperation between the players. 

• There are two players which have m and n strategies, respectively, can be represented by a m×n matrix. First player’s strategies can be represented by rows and second player’s strategies can be represented by columns. First player is called the row player and second player is called the column player.
• The row player must choose 1 of m strategies and the column player must choose 1 of n strategies, simultaneously.
• If the row player chooses his ith strategy and the column player chooses his jth strategy, the row player gains a_ij and the column player gains, b_ij.

Two-person zero-sum games are also called matrix games. The pay-off matrix of these games can be represent in a matrix form.

The maximin and minimax strategies involve choosing the course of action with the best worst case scenario.

In a matrix game, if the maximin value of row player is equal to the minimax value of column player, then this value is called a saddle point. If a matrix game has a saddle point, both players should play a strategy that contains saddle point.

In a two-person zero-sum game, there are only two players called Row Player and Column Player, and the losses of one player are equivalent to the gains of the other player so that the sum of their net gains is zero. 

The deterministic two-person zero-sum games have saddle points and pure strategies, while the probabilistic ones have no saddle points. Probabilistic games have mixed strategies are obtained with the help of probabilities. To find the optimal strategies in a two-person zero-sum game, the row player tries to maximize his winnings and the column player tries to minimize his losses. 

One of the ways to find optimal mixed strategies is a graphical solution method. This method may be used whenever one of the players has only two pure strategies. After dominated strategies are eliminated from 2×n or m×2 sized games, then the optimal solutions of 2×2 sized games are found with as discussed before.