Game Theory
A theoretical framework for studying strategic interactions between rational decision-makers.
Game theory provides a mathematical basis for analyzing situations where the outcome for each participant (or “player”) depends not only on their own choice but also on the choices made by all other participants.
Core Components of a Game
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Players
The rational decision-makers in a game. These can be individuals, companies, political parties, or countries.
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Strategy
A complete plan of action a player will follow, specifying their choice in every possible situation that could arise in the game.
- Pure Strategy: A player makes a specific choice or takes a specific action.
- Mixed Strategy: A player randomizes between two or more choices based on a set of probabilities.
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Payoff
The outcome or reward a player receives from a particular combination of strategies chosen by all players. It is typically represented by a numerical value (measuring utility, money, or some other benefit).
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Strategy
A complete plan of action a player will follow, specifying their choice in every possible situation that could arise in the game.
- Pure Strategy: A player makes a specific choice or takes a specific action.
- Mixed Strategy: A player randomizes between two or more choices based on a set of probabilities.
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Payoff
The outcome or reward a player receives from a particular combination of strategies chosen by all players. It is typically represented by a numerical value (measuring utility, money, or some other benefit).
Central Concepts & Examples
Nash Equilibrium
A foundational concept in non-cooperative game theory. A Nash equilibrium is a set of strategies (one for each player) where no player can benefit by unilaterally (independently) changing their strategy, assuming all other players keep their strategies unchanged. It represents a stable outcome where everyone is doing the best they can, given what everyone else is doing.
The Prisoner’s Dilemma
The most famous example of a non-cooperative, non-zero-sum game. It demonstrates why two rational individuals might fail to cooperate, even when it appears to be in their best collective interest to do so.
Scenario:
Two prisoners, held separately, must choose to either “Cooperate” (stay silent) or “Defect” (betray the other).
- If both cooperate, they both serve a minimal sentence (e.g., 1 year).
- If one defects and the other cooperates, the defector goes free, and the cooperator serves a long sentence (e.g., 10 years).
- If both defect, they both serve a medium sentence (e.g., 5 years).
The Nash equilibrium is for both players to defect, even though the best *collective* outcome is for both to cooperate.
Key Classifications of Games
1. Cooperative vs. Non-Cooperative
- Cooperative: Players can form binding agreements and coalitions. Analysis focuses on fair payoff distribution (e.g., Shapley value).
- Non-Cooperative: Players act in self-interest without binding agreements. Analysis focuses on predicting strategies (e.g., Nash equilibrium).
2. Zero-Sum vs. Non-Zero-Sum
- Zero-Sum: One player’s gain is exactly equal to another’s loss (e.g., Chess).
- Non-Zero-Sum: Gains/losses are not balanced. All can win or all can lose (e.g., Prisoner’s Dilemma).
3. Simultaneous vs. Sequential
- Simultaneous: Players decide at the same time, unaware of others’ choices (e.g., Rock-Paper-Scissors).
- Sequential: Players take turns, with later players knowing previous moves (e.g., Chess).
4. Perfect vs. Imperfect Information
- Perfect: All players know all previous moves (e.g., Chess).
- Imperfect: Players lack full knowledge of past moves or current choices (e.g., Poker).
Other Common Game Types
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Repeated Games
The same basic game is played multiple times, allowing for strategies based on reputation, trust, and retaliation.
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Bargaining Games
Players must negotiate to divide a resource. (e.g., The Ultimatum Game, where one player proposes a split and the other accepts or rejects).