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obj: concept


wiki: https://en.wikipedia.org/wiki/Game_theory


rev: 20240529







# Game Theory


Game theory is a mathematical framework designed for analyzing competitive situations where the outcomes depend on the actions of multiple agents, each with their own interests. It is widely used in economics, political science, [psychology](../../psychology/Psychology.md), [biology](../../biology/Biology.md), and computer science to model and study interactions among rational decisionmakers.




## Key Concept




### Players


Players are the decisionmakers in a game. Each player aims to maximize their own payoff, which depends on the strategies chosen by all players.




### Strategies


A strategy is a complete plan of action a player will follow in a game, given the possible situations they might face. Strategies can be pure (deterministic) or mixed (probabilistic).




### Payoffs


Payoffs are the rewards or outcomes that players receive as a result of the strategies they choose. Payoffs are often represented in a matrix form in simpler games.




### Games


Games can be classified based on various criteria:


 **Cooperative vs. NonCooperative:** In cooperative games, players can form binding commitments, while in noncooperative games, they cannot.


 **Symmetric vs. Asymmetric:** In symmetric games, the payoffs depend only on the strategies employed, not on who is playing. In asymmetric games, the identities of the players matter.


 **ZeroSum vs. NonZeroSum:** In zerosum games, one player's gain is another player's loss. In nonzerosum games, the total payoff can vary, allowing for winwin situations.




## Types of Games




### SimultaneousMove Games


In simultaneousmove games, players choose their strategies without knowing the choices of the other players. A common representation is the payoff matrix. An example is the Prisoner's Dilemma.




#### Prisoner's Dilemma


The Prisoner's Dilemma is a standard example in game theory. Two prisoners must decide whether to confess or remain silent without knowing the other's decision. The optimal strategy leads both to confess, resulting in a worse outcome for both.




  Prisoner B Stays Silent  Prisoner B Confesses 


      


 **Prisoner A Stays Silent**  A: 1, B: 1  A: 10, B: 0 


 **Prisoner A Confesses**  A: 0, B: 10  A: 5, B: 5 




### SequentialMove Games


In sequentialmove games, players make decisions one after another, with each player observing the previous actions before making their choice. These games are often represented by game trees.




#### Example: Chess


Chess is a classic example of a sequentialmove game where players take turns making moves, and each player's strategy depends on the previous moves made by the opponent.




### Repeated Games


Repeated games occur when players engage in the same game multiple times. Strategies can evolve over time, and cooperation can emerge even in noncooperative settings.




#### Example: Iterated Prisoner's Dilemma


In the Iterated Prisoner's Dilemma, players play the Prisoner's Dilemma multiple times, allowing for strategies like "tit for tat," where a player reciprocates the opponent's previous action.




### Evolutionary Game Theory


Evolutionary game theory studies how strategies evolve over time, often using concepts from [biology](../../biology/Biology.md). It focuses on populations of players who may adopt different strategies based on their success.




#### Example: HawkDove Game


The HawkDove game models the behavior of animals in conflict over resources. Hawks are aggressive, while Doves are peaceful. The game's outcomes depend on the costs and benefits of fighting versus sharing.




## Nash Equilibrium


A Nash Equilibrium is a set of strategies where no player can benefit by unilaterally changing their strategy, given the strategies of the other players. It represents a state of mutual best responses.




### Example: Battle of the Sexes


In the Battle of the Sexes, two players (typically representing a couple) prefer different activities but want to do something together. Multiple Nash Equilibria exist, where each player's best response depends on the other's choice.




  Player B: Opera  Player B: Football 


      


 **Player A: Opera**  A: 2, B: 1  A: 0, B: 0 


 **Player A: Football**  A: 0, B: 0  A: 1, B: 2 




## Applications of Game Theory


Game theory has applications across various fields:


 **Economics:** Market competition, auctions, and bargaining.


 **Political Science:** Voting systems, coalition formation, and international relations.


 **[Biology](../../biology/Biology.md):** Evolution of cooperation, animal behavior, and population dynamics.


 **Computer Science:** Algorithm design, network security, and artificial intelligence.


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#🐇 #notnow




 [Graph Theory](Graph%20Theory.md)


 [Game Theory](Game%20Theory/Game%20Theory.md)




## Number Systems


 [Binary System](Binary%20System.md)




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