What is Game Theory?
Game theory is a method of using mathematics to understand how people make choices when their outcomes depend on the choices of themselves as well as others. The idea is to study situations in which one person’s actions affect another person’s actions and vice versa. The game can be as superficial as rock paper scissors or a game of chess, or as significant as the trade war between China and the U.S.A. Heard of the Cuban Missile Crisis? That is one of the most classic political games!
The four core elements of a game are the players, payoffs, actions, and information. Players make choices in the game based on the information they know about the payoff and other players. The differences between the expected rewards and costs are called payoffs. The choices the players make are called actions or strategies. Using game theory, we can predict how people would act in different situations and maximize our own payoffs.
The most important assumption of game theory is that the players are completely rational. All the models and conclusions of game theory are based on this assumption. This is why it is hard to apply game theory to politics and business, as people in these areas are much less likely to be completely rational.
Nash Equilibrium
One of the most important concepts in game theory is Nash equilibrium, which is a concept that describes a state in which each player in a game chooses a strategy that is optimal for themselves, given the strategies chosen by all other players. In other words, if all players are playing their optimal strategy, no player can improve their outcomes by changing their own strategy.
Let’s take the “prisoner’s dilemma” as an example. Imagine you and your friend were arrested for not doing your homework. You can either choose to confess or deny your “crime”. If both of you confess, you will both be imprisoned for 3 years; if you both deny the crimes, then the police will not be able to find evidence, and you will both be imprisoned for 1 year. If one of you confesses while the other one denies the crimes, the one confessing will be released immediately, and the other will be imprisoned for 5 years. However, you are separated into different rooms, hence both of you are unaware of the other’s choices. The situation can be visualized by the following payoff matrix, where C represents “confess”, and D represents “deny”:
Let’s assume the horizontal “CD” represents player 1’s choices. The first position of each number pair represents the payoff of player 1, and the second represents player 2. From the perspective of player 1, when player 2 chooses to confess, I’d better confess as well, since 3 years is better than 5. When player 2 chooses to deny, I’d better deny as well, so I don’t have to be imprisoned at all. Circling player 1’s choices in each circumstance, the matrix looks like this:
As for player 2, the analysis is completely the same. Hence, this is what we have now:
The only circumstance in which players 1 and 2 would reach an equilibrium is if they both confess. In this situation, neither of them can shorten their year of imprisonment by choosing to deny. Unfortunately, this means that players should always betray their partners. From the diagrams above, the best strategy is, in fact, for both players is to deny (1,1), while the Nash equilibrium leads to the sub-optimal solution (3,3). In fact, for zero-sum games (in which one person's gain is equivalent to another's loss) where players act rationally, Nash equilibrium states that the optimal group solution can never be achieved.
Watch this video to view another interesting example.
Works Cited
Economics (2023). Game theory is a branch of economics that studies how people and organizations make strategic choices in situations of conflict or cooperation. By applying game theory to your decision making and problem solving skills, you can improve your understanding of the incentives, payoffs, and best response. [online] Linkedin.com. Available at: https://www.linkedin.com/advice/1/how-do-you-use-game-theory-improve-your-decision-making [Accessed 27 Sep. 2023].
Investopedia. (2023). Zero-Sum Game Definition in Finance, With Example. [online] Available at: https://www.investopedia.com/terms/z/zero-sumgame.asp [Accessed 27 Sep. 2023].
Nash Equilibrium (2023). Nash Equilibrium: Definition, Example & Graph | StudySmarter. [online] StudySmarter UK. Available at: https://www.studysmarter.co.uk/explanations/microeconomics/imperfect-competition/nash-equilibrium/ [Accessed 27 Sep. 2023].
ResearchGate. (2013). FIG. 1. The pay-off matrix in the Prisoners’ Dilemma game. The first... [online] Available at: https://www.researchgate.net/figure/The-pay-off-matrix-in-the-Prisoners-Dilemma-game-The-first-entry-refers-to-Alices_fig1_2184338 [Accessed 27 Sep. 2023].
TED-Ed (2012). Why do competitors open their stores next to one another? - Jac de Haan. YouTube. Available at: https://www.youtube.com/watch?v=jILgxeNBK_8 [Accessed 27 Sep. 2023].
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