Two-Person Zero-Sum Games

  • Peter Morris
Part of the Universitext book series (UTX)


Let \( \vec \pi \) be a game in normal form with strategy sets X1,...,X N We say that this game is zero-sum if
$$ \sum\limits_{i = 1}^N {{\pi _i}\,({x_1},\,.\,.\,.\,,\,{x_N})\, = \,0,} $$
for every choice of x i X i , 1 ≤ iN. The corresponding definition for a game in extensive form states that the sum of the components of \( \vec p(v) \) is zero for each terminal vertex v. This condition is certainly true for ordinary recreational games. It says that one player cannot win an amount unless the other players jointly lose the same amount. Nonrecreational games, however, tend not to be zero-sum. Competitive situations in economics and international politics are often of the type where the players can jointly do better by playing appropriately, and jointly do worse by playing stupidly. The phrase “zero-sum game” has entered the language of politics and business.


Saddle Point Optimal Strategy Mixed Strategy Pure Strategy Expected Payoff 
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Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Peter Morris
    • 1
  1. 1.Mathematics DepartmentPenn State UniversityUniversity ParkUSA

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