Let the elements in a Payoff Matrix be denoted
, where the
s are player A's Strategies
and the
s are player B's Strategies. Player A can get at least
![\begin{displaymath}
\min_{j\leq n} a_{ij}
\end{displaymath}](g_81.gif) |
(1) |
for Strategy
. Player B can force player A to get no more than
for a Strategy
. The
best Strategy for player A is therefore
![\begin{displaymath}
\max_{i\leq m}\min_{j\leq n} a_{ij},
\end{displaymath}](g_83.gif) |
(2) |
and the best Strategy for player B is
![\begin{displaymath}
\min_{j\leq n}\max_{i\leq m} a_{ij}.
\end{displaymath}](g_84.gif) |
(3) |
In general,
![\begin{displaymath}
\max_{i\leq m} \min_{j\leq n} a_{ij} \leq \min_{j\leq n}\max_{i\leq m} a_{ij}.
\end{displaymath}](g_85.gif) |
(4) |
Equality holds only if a Saddle Point is present, in which case the quantity is called
the Value of the game.
See also Game, Payoff Matrix, Saddle Point (Game), Strategy, Value
© 1996-9 Eric W. Weisstein
1999-05-25