Sociological Mathematics: An Introduction to Game Theory
The field of game theory, popularized by films such as “A Beautiful Mind”, proposes mathematical models to represent social interactions. As the name suggests, game theory sees human interaction as just that: a game. The founding of the field is credited to John Nash and John von Neumann.
Game theory at its foundation aims to represent interactions between agents as games, with strategies, winners and losers, rewards and punishment. Although the field was initially used to describe economic behavior, it has since expanded, emerging in areas from data science to politics.
An easy way to demonstrate the usage of game theory in real life is through popular games such as chess and rock-paper-scissors, where each person attempts to maximize their payoff (win the game).
Mathematically, games are represented in various ways. Most games can be represented in the form of a payoff matrix. Here, 2 players’ choices are shown, with their payoffs marked. From here, game theorists can begin to understand the nature of the game through calculations.
One of Nash’s chief contributions to the field was his discovery of Nash equilibrium. A game is said to be in Nash equilibrium if no player has an incentive to change his game strategy after considering the strategies of all other players. Thus, an individual can receive no incremental benefit from changing actions, assuming other players remain constant in their strategies. A game can have multiple Nash equilibria or none at all.
To better demonstrate this concept, people often turn to the prisoner’s dilemma. Here, two members of a gang of bank robbers, are being interrogated in separate rooms. They are the sole witnesses. The authorities can only prove the case against them if they can convince at least one of the robbers to betray his accomplice and testify to the crime. Each bank robber is faced with the choice to cooperate with his accomplice and remain silent or to ‘defect’ from the gang and testify for the prosecution. If both remain silent, they may get a 3-year sentence. If one defects and the other stays silent, the defector will run free, while the speechless accomplice serves 5 years. If both testify against each other, they will both serve just 1 year. Here, the Nash equilibrium would be for each prisoner to defect regardless of the other’s choice since it will always net them a better result.
While game theory is often touted for its ability to predict human behavior, it is not always so. Game theory, at its foundation, prioritizes the outcome for each agent, assuming that every actor is rational and seeks maximum value from their actions. Hence, game theory fails to account for the fact that in some situations we may not fall into a nash equilibrium, depending on the social context and the nature of the players. It also requires the rules of play to be precise, while in real life, it is often ambiguous. To construct a payoff matrix, you need just that — payoffs, which are often not exactly known in reality. Finally, the theory often provides many equilibria and no way to choose among them.
