Game Expectation
Let the elements in a payoff matrix be denoted a_(ij), where the is are player A's strategies and the js are player B's strategies. Player A can get at least
| min_(j<=n)a_(ij) |
(1)
|
for strategy i. Player B can force player A to get no more than max_(j<=m)a_(ij) for a strategy j. The best strategy for player A is therefore
| max_(i<=m)min_(j<=n)a_(ij), |
(2)
|
and the best strategy for player B is
| min_(j<=n)max_(i<=m)a_(ij). |
(3)
|
In general,
| max_(i<=m)min_(j<=n)a_(ij)<=min_(j<=n)max_(i<=m)a_(ij). |
(4)
|
Equality holds only if a game saddle point is present, in which case the quantity is called the value of the game.
See also
Game, Game Saddle Point, Payoff Matrix, Strategy, ValueExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Game Expectation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GameExpectation.html