Go directly to the solver
What is a solution to a zero-sum game?
In a two-person zero-sum game, the payoff
to one player is the negative of that going
to the other. Although zero-sum games are
not terribly interesting to economists, who
typically study situations where there are
gains to trade, most common parlor games
such as poker and chess are zero sum: one
player wins, one loses. According to Von-Neumann's
theory, every zero sum game has a value. Each player can guarantee himself this
value against any play of his opponent, and
can prevent the other player from doing any
better than this. We typically write a zero-sum
game by forming a matrix and allowing one
player to choose the rows and the other the
columns. The entries in the matrix are the
payoffs to the row player. For example in
the game of matching pennies, we can write
the payoff matrix
so that the row player is trying to match
the column player and the column player is
trying to guess the opposite of the row player.
The value of the game may be calculated as
either the minimum of what the row player
can achieve knowing the strategy of the column
player (the minmax for the row player) or
the maximum of what the column player can
hold the row player to, knowing the strategy
of the row player (the maxmin for the row
player). Von Neumann's famous minmax theorem
shows that these two quantities are the same.
It is possible to solve a zero-sum game using
the simplex algorithm or any other algorithm
that can solve a linear programming problem.
This is implemented below. To solve a zero
sum game, fill in the payoffs to the row
player in the blank area below separated
by commas. Do not enter blank lines. The program will then
find the strategy for the column player that
holds the row player's payoff to a minimum.
For example in the game of matching pennies:
Enter the payoffs to the row player:
The program finds the column player strategy
that holds the row player's payoff to a minimum,
and reports the value of the game to the
If you have questions about the program or
about zero-sum games, you should check out
discussion on the forum.