The Application of Game Theory in Online Multiplayer Games

In the field of mathematics, game theory is the study of decision making based on strategy. Developed in the 18th century, Game Theory was initially intended for addressing zero sums games (Bram 15). In these games, there are two participants whereby the loss of one results in the gain of the other individual. For a group of people to be considered in a game, their decisions on a particular issue must be interdependent. Interdependence means that the action taken by each one of them influences the outcomes of every participant.

In the recent past, game theory has found application in various games, primarily the online multiplayer gaming. Regardless of the field of use, the primary emphasis of the game theory is how strategic behavior impacts the interaction among rational participants in a game (Bram 11). One can incorporate two viewpoints to elaborate the Game theory; mathematical point of view, or experience from players of online games. Experience perspective is the best way to flesh out the game theory to a layperson, and games such as the Battle of Sexes, Prisoner's Dilemma, and Cake Cutting are some of the examples in this point of view.

Battle of the Sexes

In most of the online games which implement the game theory in their solution solving mechanism, players have dominant strategies. Nonetheless, in the battle of sexes, the gameplay is not as direct as the others. In this game, a man (player 1) and his girlfriend (player 2) drive out in their cars on a Saturday night to meet each other. This game was based in an era where there were no wireless communication devices, and thus this couple cannot communicate to each before they meet. Previously, they had an arrangement which involved either attending an opera performance or a football match. The lady loves opera while her boyfriend is a football fanatic. However, while driving, none of them can recall where they agreed to meet. Furthermore, they love each other, and neither of them wishes to spend a night without the other at the approved location. If the lady ends up in an opera performance with her boyfriend, her payoff will be three. The payoff will be zero if she ends up alone in the opera. However, if she ends up in a football match and her boyfriend turns out to be there, her payoff will be one. But if she ends up in a football match without him, her payoff will be zero. The same payoff scenarios will also be applied if the man makes the same decisions. The table below illustrates the Battle of the Sexes game in each situation.

MAN

Football

Opera

WOMAN

Football

1, 3

0, 0

Opera

0, 0

3, 1

The first thing to be noted in this game is that there are no dominant strategies since each player intends to maximize his or her outcome. An equilibrium concept used in the battle of the sexes is known as Nash Equilibrium (Serrano and Feldman 8). In the pure strategy Nash equilibria there are two approaches; the first is where both attend the opera performance, and the second strategy they both end up in a football match. In the mixed approach Nash equilibrium, both players attend the shows they prefer. In the pure strategy, one player will always do better than the other, while in mixed approach there is inefficiency. The best solution to such a scenario is for the couple to agree before making plans to base their decision strategy on a randomizing device like flipping a coin (Serrano and Feldman 8). This method is known as correlated equilibrium and is meant to reduce the complexity of solving Nash equilibria.

Cake Cutting

Also known as the fair division problem, cake cutting involves a resource which can be shared equally among the individuals without destroying its value. For instance, a cake to be shared may be comprised of different toppings, and the players have unique preferences regarding their pieces. Despite that the cake topping is not evenly distributed, it has to be shared in a fair manner. To divide the cake equitably, every person should have an equal level of satisfaction. In most of the algorithms used to subdivide the pie among three people, the outcome results in an unending cycle of cake trimming. However, the best strategy to share a cake between three people is known as envy-free cake cutting algorithm (Klarreich). For example, if John, Mary, and Tom want to share a sugar cake with a topping, Tom is asked to cut it into three equal parts according to the preferences of each. John and Mary are then asked to choose their favorite slices. If they choose differently, the cake is equally subdivided, and Tom picks the last piece. However, if Mary and John want the same piece, John is asked to trim the slice so that it is equal to his second-best portion. The cut cake is set aside for later, and Mary gets to choose among the remaining three "equal slices." If Mary does not choose the slice which was trimmed, John must take it. At this stage, neither of them is envious of the pieces picked. Moreover, Tom is happy with his piece, regardless of whether Mary and John decide to share the trimmed slice set for later or not since his original measurement was "fair."

The Prisoner's Dilemma

The prisoner's dilemma is one of the less sophisticated game theory illustration. In this game theory, two offenders have been arrested by the law enforcers. The two suspects are put into different cells which are dirty and cold. The police decide to interrogate them separately without the presence of a lawyer. The terminologies used in this scenario are: defect which means one chooses to testify and cooperate if either refuse to confess (Serrano and Feldman 2). The choices offered by the police to each offender are; if both refuse to plead guilty, they will be convicted of minor charge which results in six months jail for each. If both acknowledge their mistake, they will be jailed for five years. However, if one of them chooses to defect and the other cooperates, the one who owned up will be freed while the other will be jailed for ten years. The suspects have a choice to defect or work with each other, decisions which all result in either one or both getting jailed for six months, five, or ten years. In a table form, the payoffs are as shown below.

Cooperate

Defect

Cooperate

Six months, six months

Ten years, None

Defect

None, ten years

Five years, five years

Before choosing what to do, player one will consider what approach player two might select. For instance, if player two decides to defect while player one has cooperated, player one will be jailed for ten years. Hence, the most likely scenario is that player one will opt to defect since this is the outcome that favors the choices made by player two. This type of a decision is known as dominant strategy (Serrano and Feldman 4). Regardless of the fact that the consequence of confession results in a longer jail time than if they had chosen to cooperate, it is the best strategy for safeguarding the choices made by each.

Conclusion

Game theory is crucial in determining the best strategies to utilize while dealing with various dilemmas. Understanding the dominant approach is useful in circumventing the issues which might arise due to particular decisions. However, in some situations, there are no dominant strategies, and thus the best solution is to utilize correlated equilibrium. A correlated equilibrium aims to reduce the number of computational steps involved while solving Nash equilibria. If a dominant strategy exists, one is advised to select it since it minimizes consequences which might arise if the other players refuse to cooperate.

Works Cited

Bram, Uri. The Game Theory. 1st ed., CreateSpace, 2013, pp. 10-57.

Klarreich, Erica. "New Algorithm Solves Cake-Cutting Problem". Quanta Magazine, 2016, www.quantamagazine.org/new-algorithm-solves-cake-cutting-problem-20161006/. Accessed 13 Apr 2018.

Serrano, Roberto, and Allan M. Feldman. "A Short Course in Intermediate Microeconomics with Calculus". 2010. Cambridge University Press, doi:10.1017/cbo9781139084093. Accessed 13 Apr 2018.