Two Person Zero-Sum Game

A two-person zero-sum game is a type of game in game theory where there are only two players involved and the sum of their payoffs is always zero. In other words, whatever one player gains, the other player loses.

In a two-person zero-sum game, the players must make choices simultaneously or sequentially and the outcome of the game is determined by the strategies chosen by each player. Each player’s strategy is chosen with the goal of maximizing their own payoff while minimizing the other player’s payoff.

One example of a two-person zero-sum game is a simple game of rock-paper-scissors. In this game, each player simultaneously chooses one of three options: rock, paper, or scissors. Rock beats scissors, scissors beats paper, and paper beats rock. The winner receives a payoff of +1, while the loser receives a payoff of -1. If both players choose the same option, the payoff is 0 for both.

Another example of a two-person zero-sum game is a game of poker. In this game, each player is dealt a hand of cards and must make bets based on the strength of their hand. The winner of the game is the player with the strongest hand, and they receive the sum of all bets made by both players. The loser receives a payoff of -1 times the sum of all bets made. In this case, the sum of the payoffs is always zero, as one player’s gain is always equal to the other player’s loss.

Two-person zero-sum games can be analyzed using various mathematical techniques, such as the minimax theorem and the Nash equilibrium. These techniques help to identify the optimal strategies for each player and the resulting payoffs.

Pure and Mixed Strategy Games

In game theory, a game can be classified as either a pure strategy game or a mixed strategy game based on the nature of the players’ strategies.

A pure strategy game is a game where players choose a specific action or strategy without any randomness involved. In other words, players choose a single option or decision from their available choices. The most common example of a pure strategy game is the game of tic-tac-toe, where both players choose a single location to place their symbol. There is no randomness involved in this game and the outcome is entirely dependent on the players’ strategies.

On the other hand, a mixed strategy game is a game where players choose their strategies randomly with some probability. In this type of game, players select their actions based on a probability distribution. For example, in the game of rock-paper-scissors, players may choose each option with equal probability. This introduces an element of randomness and uncertainty into the game, making it more complex than a pure strategy game.

In some cases, mixed strategies may be necessary to achieve the best outcome for a player. This is because some games may have multiple Nash equilibria, which are stable outcomes where no player can improve their payoff by changing their strategy. A mixed strategy can help a player to achieve the best possible outcome in situations where there is more than one Nash equilibrium.

In summary, pure strategy games involve players making a single choice without any randomness involved, while mixed strategy games involve players selecting their strategies based on a probability distribution.