Game theory is the study of mathematical configuration of contention and cooperation between reasonable, sound decision makers. The games that are studied in game theory are made up of sets of players, strategies available to the players, and the payoff or reward specific to each combination of strategies. A normal form game can be exemplified by a matrix which shows the players, strategies, and payoffs. Each player has a number of strategies stipulated by the number of rows and columns while the payoffs are expressed in the interior. The focus of game theory is the strategies applied by the players known as equilibria in these games.
1. For two players, A and B engaged in a coin-matching game, each shows a coin as either heads or tails. If the coins match, B pays A $1 and if they differ, A pays B $1. Depicting this game in tabular form provides a visual presentation of the players’ strategies and their respective payoffs. B’s strategies
A’s strategies Heads +1, -1 -1, +1
Tails -1, +1 +1, -1
The payoffs are based on the choices made by the players although this choice is unknown to the other player. The players’ payoffs or utility are balanced in the sense that one player’s gain in utility is an inverse of the second player.
The Nash equilibrium is not exhibited in this instance since there is no available channel for the players to maximize their payoffs. Though the players have adequate aptitude to deduce the solution, the equilibrium remains unknown owing to the intricacy of the game. A zero-sum game as depicted above can be solved by adopting linear optimization.
2. For two players, Smith and Jones playing a number matching game with the choice of the numbers 1, 2 or 3, if the numbers match, Jones pays Smith $3 and if they differ, Smith pays Jones $1. The payoff matrix for this game is shown below:
1 2 3
Jones’ Strategies 2
The payoff of this game exhibits a zero-sum game although the players may adopt different strategies to vary their utility. The matrix does not have a Nash equilibrium strategy pair because a change in strategy by either of the players will benefit that player. There is no equilibrium because each strategy pair offers one of the players an incentive to adopt another strategy. In addition, it is not mandatory that each player will choose the numbers with equal probability.
In a mixed strategy, there exists equal probability of the players to choose the numbers. This can be proved by calculating the mixed strategy Nash equilibrium. If the probability of Jones playing 1 is assigned p while his probability of playing 2 is represented by r while that of playing 3, (1-p+r). The probabilities of Smith playing the numbers 1, 2 and 3 are represented by s, t and (1-s+t) respectively. Thus the payoffs E can be calculated and compared to determine the random selection of the numbers with equal probability of a third. This game is valuable in exhibiting the mixed strategies that can be applied in Nash equilibrium games.