Analysis Of Zero-sum, Two-player Games
In some games there are only two players and in the end, one wins while the other loses. This also means that the amount gained by the winner will be equal to the amount lost by the loser. The strategies suggested by game theory are particularly applicable to games such as these, known as zero-sum, two-player games.
Consider the game of matching pennies. Two players put down a penny each, either head or tail up, covered with their hands so the orientation remains unknown to their opponent. Then they simultaneously reveal their pennies and pay off accordingly; player A wins both pennies if the coins show the same side up, otherwise player B wins. This is a zero-sum, two-player game because each time A wins a penny, B loses a penny and visa versa.
To determine the best strategy for both players, it is convenient to construct a game payoff matrix, which shows all of the possible payments player A receives for any outcome of a play. Where outcomes match, player A gains a penny and where they do not, player A loses a penny. In this game it is impossible for either player to choose a move which guarantees a win, unless they know their opponent's move. For example, if B always played heads, then A could guarantee a win by also always playing heads. If this kept up, B might change her play to tails and begin winning. Player A could counter by playing tails and the game could cycle like this endlessly with neither player gaining an advantage. To improve their chances of winning, each player can devise a probabilistic (mixed) strategy. That is, to initially decide on the percentage of times they will put a head or tail, and then do so randomly.
According to the minimax theorem of game theory, in any zero-sum, two-player game there is an optimal probabilistic strategy for both players. By following the optimal strategy, each player can guarantee their maximum payoff regardless of the strategy employed by their opponent. The average payoff is known as the minimax value and the optimal strategy is known as the solution. In the matching pennies game, the optimal strategy for both players is to randomly select heads or tails 50% of the time. The expected payoff for both players would be 0.