Pure strategies in Game theory

14/03/2023 0 By indiafreenotes

In game theory, a pure strategy is a specific, predetermined choice of action that a player will take in a game. It represents the player’s complete plan of action, given all possible scenarios and the choices available to them.

For example, in the game of rock-paper-scissors, a player could use a pure strategy of always playing “rock.” This means that no matter what the opponent plays, the player will always play “rock” as their predetermined choice.

In a game with multiple players, each player can have their own set of pure strategies. The combination of all players’ pure strategies determines the possible outcomes of the game.

The use of pure strategies is often used as a simplifying assumption in game theory, as it allows for straightforward analysis of the game’s equilibrium outcomes. However, in many real-world situations, players may use mixed strategies, which involve a randomized choice of actions with certain probabilities, rather than predetermined, fixed actions.

Here is an example of pure strategies in game theory:

Consider a simple game where two players, Alice and Bob, each have the choice to either cooperate or defect. If both players cooperate, they each receive a payoff of 3. If both players defect, they each receive a payoff of 1. If one player cooperates and the other defects, the defector receives a payoff of 4 and the cooperator receives a payoff of 0.

To analyze this game, we can consider the possible pure strategies for each player:

Alice’s pure strategies: Cooperate or Defect

Bob’s pure strategies: Cooperate or Defect

There are four possible outcomes of the game, depending on the choices of Alice and Bob:

  • Both players cooperate: Payoffs = (3, 3)
  • Alice cooperates, Bob defects: Payoffs = (0, 4)
  • Alice defects, Bob cooperates: Payoffs = (4, 0)
  • Both players defect: Payoffs = (1, 1)

To find the equilibrium outcome of the game, we can use the concept of Nash equilibrium. A Nash equilibrium is a set of strategies where no player can unilaterally improve their payoff given the other player’s strategy.

In this game, the only Nash equilibrium is for both players to defect. If Alice chooses to cooperate, Bob has an incentive to defect and receive a higher payoff. Similarly, if Bob chooses to cooperate, Alice has an incentive to defect and receive a higher payoff.

Thus, the Nash equilibrium outcome of the game is (Defect, Defect), with payoffs of (1, 1). This is the only outcome that satisfies the criteria of Nash equilibrium, where neither player can improve their payoff by changing their strategy, given the other player’s strategy.