Game theory is a branch of mathematics that studies strategic decision making in various situations.
“A game is the complete exploration of freedom within a restrictive environment”
Game theory is a branch of mathematics that studies strategic decision making in situations where multiple parties (called "players") are interacting with each other and trying to maximize their own outcomes. Game theory can be used to analyze a wide range of situations, including economics, politics, and biology.
In game theory, each player has a set of possible actions (called "strategies") to choose from, and the outcome of the game depends on the choices made by all of the players. Players try to anticipate the actions of their opponents and choose their own strategies accordingly in order to maximize their own payoffs or outcomes.
There are several different types of games that can be analyzed using game theory, including:
Zero-sum games: In a zero-sum game, one player's gain is exactly balanced by the other player's loss. An example of a zero-sum game is poker, where one player's winnings come at the expense of the other player's losses.
Non-zero-sum games: In a non-zero-sum game, both players can gain or lose, and the total gain or loss is not necessarily zero. An example of a non-zero-sum game is negotiating a salary, where both the employer and the employee can benefit from a mutually agreeable outcome.
Cooperative games: In a cooperative game, players can form coalitions and make joint decisions in order to maximize their collective payoffs. An example of a cooperative game is a group of countries negotiating a trade agreement.
Non-cooperative games: In a non-cooperative game, players do not have the option to form coalitions and must make their own individual decisions. An example of a non-cooperative game is a price war between two companies.
There are several different mathematical techniques that can be used to analyze games and determine optimal strategies, including the Nash equilibrium, the minimax theorem, and the prisoner's dilemma. These techniques can be used to compute the expected payoffs for different strategies and identify the strategy that will maximize a player's payoff.
Game theory has a wide range of applications, including economics, political science, psychology, biology, and computer science. It is used to study and analyze a variety of phenomena, including market competition, political negotiations, and social interactions.
Concepts
There are several different mathematical techniques and formulas that can be used in game theory to analyze strategic decision making and determine optimal strategies. Some of the most commonly used formulas and concepts in game theory include:
Payoff matrix: A payoff matrix is a table that shows the payoffs for each player for each possible combination of strategies. For example, if there are two players and each player has two possible strategies, the payoff matrix might look something like this:
|Strategy A |Strategy B|
Player 1 | 5 | 3 |
Player 2 | 3 | 5 | In this example, player 1's payoffs are shown in the rows, while player 2's payoffs are shown in the columns. The numbers in the cells represent the payoffs for each player given their chosen strategies.
Nash equilibrium: A Nash equilibrium is a state of the game where no player has an incentive to change their strategy, given the strategies of the other players. In other words, it is a stable state of the game where no player can improve their payoff by changing their strategy.
Minimax theorem: The minimax theorem states that in a zero-sum game (where one player's gain is exactly balanced by the other player's loss), each player can maximize their own payoff by minimizing the maximum payoff that their opponent can achieve.
Iterated deletion of dominated strategies: This is a technique for simplifying a game by eliminating dominated strategies (strategies that are never the best choice, no matter what the other player does). By iteratively deleting dominated strategies, it is often possible to reduce a complex game to a much simpler form that is easier to analyze.
Regret minimization: In a repeated game (a game that is played multiple times), players may use a strategy called regret minimization, where they choose their actions based on the expected regret they would feel if they chose a different action. This can help players to learn and adapt over time.
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