Introduction to Game theory
Game theory is a branch of economics and mathematics that studies strategic interactions where the outcome for each participant depends on the choices of others. It applies to various fields, including economics, political science, psychology, and even biology.
The core idea is that individuals (or firms, countries, etc.) make decisions while considering the actions of others to maximize their own benefits.
Key concepts in Game theory
1. Players
- The decision-makers in a game.
- Can be individuals, firms, governments etc.
2. Strategies
- A complete plan of action a player will follow in every possible situation.
- Can be pure strategies (always making a specific choice) or mixed strategies (combining choices).
3. Results
- The outcome or reward received based on the chosen strategies.
- Can represent profits, utility, votes, etc.
4. Types of Game
- Cooperative vs Non Cooperative
- Cooperative games: Players can form binding agreements.
- Non-cooperative games: No enforceable agreements; each player acts independently.
2. Zero sum vs Non zero sum
- Zero-Sum Game: One player’s gain is exactly another’s loss. (e.g., poker)
- Non-Zero-Sum Game: Players can both benefit or suffer (e.g., trade negotiations).
3. Simultaneous vs. Sequential
- Simultaneous Game: Players choose actions at the same time (e.g., rock-paper-scissors).
- Sequential Game: Players take turns, and later players can observe previous actions (e.g., chess).
Nash Equilibrium
In economics, Nash equilibrium is a game theory concept that describes a situation where no player can improve their outcome by changing their strategy. It’s a set of strategies that’s the best response for each player given the other players’ choices.
The Nash Equilibrium, named after John Nash, occurs when “No player can improve their payoff by unilaterally changing their strategy“.
In other words, once an equilibrium is reached, no player has an incentive to deviate.
The Prisoner’s Dilemma
The prisoner’s dilemma is a common situation analyzed in game theory that can employ the Nash equilibrium. In this game, two criminals are arrested and each is held in solitary confinement with no means of communicating with the other. The prosecutors do not have the evidence to convict the pair, so they offer each prisoner the opportunity to either betray the other by testifying that the other committed the crime or cooperate by remaining silent.
If both prisoners betray each other, each serves three years in prison. If A betrays B but B remains silent, prisoner A is set free and prisoner B serves five years in prison, or vice versa. If each remains silent, then each serves just one year in prison.
In this example, the Nash equilibrium is for both players to betray each other. Even though mutual cooperation leads to a better outcome if one prisoner chooses mutual cooperation and the other does not, one prisoner’s outcome is worse.
Types of Game Theory Strategies
- Dominant Strategy
- A strategy that is best regardless of what the opponent does.
- Example: In the Prisoner’s Dilemma, betraying is a dominant strategy because it always gives a better payoff.
2. Mixed Strategy Nash Equilibrium
- Players randomize between different actions to keep opponents uncertain.
- Example: In penalty kicks in football, the kicker and goalkeeper choose directions randomly to avoid being predictable.
3. Minimax Strategy
- Players minimize the worst possible loss they might face.
- Used in chess and poker to prevent opponents from exploiting weaknesses.
Limitations of Game theory
- Simplifying assumptions
It often relies on highly simplified models of real-world situations. In order to make the mathematically complex calculations involved in game theory feasible, certain assumptions must be made about the behavior of the players involved and the structure of the game itself. However, these assumptions may not always hold true in practice, leading to inaccurate predictions and recommendations.
- Rationality assumptions
All players involved are rational and act in their own self-interest. However, this assumption may not always hold in reality, as individuals may have other motivations or biases that influence their decisions. For example, in a workplace setting, an employee may prioritize their loyalty to their team or manager over their own individual self-interest, leading to behavior that is not accurately captured by game theory models.
- Information limitations
It assumes that all players have access to complete and accurate information about the game they are playing and the other players involved. However, in many real-world situations, information may be imperfect or asymmetrical, meaning that some players have more information than others. This can lead to situations where game theory fails to accurately predict behavior, as players may make decisions based on incomplete or inaccurate information.
- Game complexity
It is best suited to analyzing situations with a limited number of players and a relatively simple structure. However, in more complex games with many players and/or multiple stages, the mathematically complex calculations involved in game theory become much more difficult to manage. In these situations, it may be necessary to use alternative approaches or simplifying assumptions to make the problem tractable.
Conclusion
Game theory is a powerful tool for understanding strategic decision-making in economics, business, politics, biology, and AI. While its assumptions may not always hold, it provides valuable insights into competition, cooperation, and conflict resolution.
