The decision-making process of a person has become the subject of statistical and logical analysis by scientists as they try to determine what drives decisions or why individuals make decisions that may not be in their best interests. One can view the example of regular decisions that people make, such as asking for marriage, having children, or probably getting a divorce. However, other decisions are not so regular. For example, one doesn’t decide to ask their date for a nightcap so often, or even ask what kind of gift they should get their partner. Every time a person is faced with a decision, there are enormous amounts of data and variables to be considered. Nonetheless, decisions cannot entirely be calculated mathematically due to the depth of information that remains unavailable to the mathematician (Myerson, 1991). For example, one cannot compute the joy of holding a baby as opposed to the costs of raising such a child. The zero-sum game provides an equal playing ground for the examination of the game theory since there sum of all values should be equal to zero. This paper considers the application of game theory with zero-sum games as a form of linear algebra.
A Game
Consider a game of ‘Morra’ between two players, Isaac and Rose. The game has simplistic rules; each player can play with one finger or two fingers at any given time. If the sum of the total fingers is even, Isaac wins and gets ten jellies. If the sum of fingers is odd, Rose wins and gets ten jellies. Here, none of the strategies is better than the other. This means that if both players play rationally, they should end up with a nearly equal number of jellies. If we change the rules so that the sum of fingers is equal to four Isaac gets twenty jellies from Rose’s stash, and we notice that another consciousness comes into the players where they will want to avoid having four fingers. Rose would then play one finger a few times to avoid twenty jellies. The question is: how many times would she play with one finger and successfully avoid losing her jellies? This is the subject matter of game theory.
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Game Theory
The game described above can be called a zero-sum game between two people. Each player contains a strategy profile, S, which is assigned to both of them. This profile provides the total number of strategies available to the players. In the case of our players, it could be expressed as follows:
S I = ; S R =
Depending on the nature of the strategy, the player will normally pick the first or second strategy. Since the game in simultaneous in nature, strategy is reduced in the game (Strang, 1988). Nonetheless, a game that is not simultaneous will increase the strategy by two. In this case example, Isaac winning the jellies means that Rose will lose some jellies and vice versa. At the end of the game, there is no change in the number of jellies. This is the true meaning of a zero-sum game, where the result remains the same as was in the beginning.
Because the game is simultaneous, this reduces the number of strategies in the game. When a game is not simultaneous, a player may choose to do the same thing as the other player, or the opposite. This would increase the number of options by two. Whenever Simon wins some number of jelly beans, Garfunkel loses the same number. So if we add up the total number of jelly beans at the beginning of the game we will find the same number of jelly beans at the end of the game. Zero jelly beans were added and zero jellies were subtracted. This kind of game is called a zero-sum game.
In this case, the gaining and losing of jelly beans is known as the payoff. The payoff function can be expressed as: J (s i , g j ). There is only one payoff expression since the sum of the two action results in a zero-sum. If a unique strategy is used, however, we will say that such a strategy dominates all other strategies. For example, consider Isaac’s gain of twenty jellies: ∃ i o ∈S i 3 J (s o , g r ) > J(s i , g j ) ∀ i ∈ S i , then i strictly dominates. Such a dominant strategy empowers the player to always play it during the game. However, a game where none gains advantage over another brings about an equilibrium that assures a player of success if they make use of one strategy half the times that they will engage in play. As a result, the mixed strategy ensures that none of the players gain unnecessary advantage over another during the pay-offs (Ichiishi, 2014).
Representing in Matrix
This game can be represented in a matrix of M x N, where M is the number of strategies the first player has, while N is the number of strategies that the second player has. The strategy A ij refers to the strategy that player 1 and 2 have for each play. Consider the matrix for the Morra game given as an example above:
| 10 | -10 |
| -10 | 20 |
Player 1 in this case is Isaac, while Rose is player 2. The columns present strategy options available to Isaac while the rows present options available to Rose. If both Isaac and Rose play two fingers each, the payoff is ten jellies to Isaac. However, if Isaac plays one finger and Rose plays two, the payoff becomes ten jellies to Rose, which can equally be interpreted as -10 for Isaac. This shows the application of a randomized strategy to the game to create a payoff matrix.
Operations of Game Theory with Linear Algebra
It is important to note that this study had certain assumptions when considering the application of the game theory and its solution using linear algebra. First, we assumed that players had perfect recall and knowledge about opponent strategies and methods. This makes the game quite unrealistic. People will not always make decisions rationally, thereby limiting the strict application of this method in determining how players would make decisions.
Furthermore, the zero-sum game assumes that the player will not change their strategy. For example, a player in the stock market being analyzed using this method will be assumed to be using one of two methods of trading. Therefore, the player is assumed to be using bounded rationality in their decision-making. On the other hand, we assume that the number of jellies that each player has doesn’t change their strategy to their advantage. In the case of the trader, it is assumed that his strategy remains constant whether he loses money or gains it. Here, we also consider that the player doesn’t take higher risks to get higher payoffs.
In conclusion then, game theory can be used to determine the outcomes of a play. Nonetheless, it is virtually impossible to gain accurate insight into the next decision of an individual due to the numerous variables that need to be considered. Players may be influenced by hunger or some other external or internal factor to make decisions. Nonetheless, it is possible to predict one’s decisions with some level of accuracy if certain deviating factors are considered to remain constant during decision-making. As such, the game theory operates on the premise that certain rules must be in place for the proper application. Thereafter, one’s odds can be provided on linear equations or matrix format.
References
Ichiishi, T. (2014). Game theory for economic analysis. New York: Elsevier.
Myerson, R. B. (1991). Game Theory. Cambridge: Harvard University Press.
Strang, G. (1988). Linear Algebra and its Applications. Fort Worth: Harcourt College Publishers.
