maximize their return. Linear algebra can be used to describe this situation. In this using
exactly how linear algebra enters into formulating strategies.
Key Words: Two person zero-sum game, Payoff matrix, Strategy, Expected payoff, Optimal strategy, Value of a
game, Saddle point, Fundamental theorem of a 2-person zero-sum game.
References: Basic books on Operations Research (a branch of applied mathematics which studies these types
of problems) are a good place to look for information.
http://www.zweigmedia.com/pdfs/GameTheory.pdf
Problem 1: Suppose that a game pays off according to the following table:
B
i ii iii iv
A i -4 6 -4 1
ii 5 -7 3 8
iii -8 0 6 -2
(a) Suppose that player A uses strategy i half of the time, strategy iii half of the time, and strategy ii none of
the time. Suppose also that player B uses each of the four strategies one fourth of the time. Find the expected
payoff of the game.
(b) If player B keeps his strategy the same as in part (a), what strategy should player A choose to maximize her
expected payoff?
(c) If player A keeps her strategy the same as in part (a), what strategy should player B choose to maximize his
expected payoff?
Problem 2: Determine the optimal pure strategies under the minimax criterion for both players. Indicate
determined, and give its value if this is the case.
