Mixed Strategies in Game Theory
In game theory, a pure strategy is a single, specific move that a player can choose to make. Pure strategies are often contrasted with mixed strategies, which involve a player randomly choosing between two or more possible moves with a certain probability.
A Nash equilibrium is a concept in game theory that describes a state in which no player can improve their payoff by unilaterally changing their strategy. In other words, a Nash equilibrium is a set of strategies where each player's strategy is the best response to the other player's strategies.
For example, consider the classic game of rock-paper-scissors. In this game, each player can choose to play rock, paper, or scissors. If both players choose the same move, the game is a tie. If they choose different moves, one of them wins. The game has a unique Nash equilibrium, which is for each player to choose each move with equal probability, i.e., to play rock, paper, and scissors with probability 1/3 each.
Nash equilibrium can be found through a process called iterated elimination of dominated strategies. This process involves eliminating any strategy that is dominated by another strategy for a given player until a unique set of strategies remains. The remaining strategies form the Nash equilibrium.
It is important to note that not all games have a Nash equilibrium, and even when they do, there may be multiple Nash equilibria. In some cases, the Nash equilibrium may not be a desirable outcome for all players. For example, in a prisoner's dilemma game, the Nash equilibrium involves both players defecting, even though mutual cooperation would result in a better outcome for both players.
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