Lecture 8
Introduction to Optimization
May 13, 2006
Prof. Amin Saberi
1
Two player ZeroSum games
In this section, we consider games in which each of two opponents selects a strategy and
receives a payoﬀ contingent on both his own and his opponents selection. We restrict atten
tion here to zerosum games  those in which a payoﬀ to one player is a loss to his opponent.
Let us deﬁne the basic concepts in the below problem setting.
Example:(drug running)
A South American drug lord is trying to get as many of his
shipments across the border as possible. He has a ﬂeet of boats available to him, and each
time he sends a boat, he can choose one of three ports at which to unload. He could choose
to unload in San Diego, Los Angeles, or San Francisco.
The US Coast Guard is trying to intercept as many of the drug shipments as possible but only
has suﬃcient resources to cover one port at a time. Moreover, the chance of intercepting a
drug shipment diﬀers from port to port. A boat arriving at a port closer to South America will
have more fuel with which to evade capture than one arriving farther away. The probabilities
of interception are given by the following table:
Port
Probability of interception
San Diego
1/3
Los Angeles
1/2
San Francisco
3/4
The drug lord considers sending each boat to San Diego, but the Coast Guard realizing this
would always choose to cover San Diego, and only 2/3 of his boats would get through. A
better strategy would be to pick a port at random (each one picked with 1/3 probability).
Then, the Coast Guard should cover port 3, since this would maximize the number of
shipments captured. In this scenario, 3/4 of the shipments would get through, which is
better than 2/3. But is this the best strategy?
Clearly, the drug lord should consider randomized strategies. But what should he optimize?
We consider as an objective maximizing the probability that a ship gets through, assuming
that the Coast Guard knows the drug lord’s choice of randomized strategy. We now formalize
this solution concept for general twoperson zerosum games, of which our example is a
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 Spring '06
 UNKNOWN
 Optimization, Game Theory, San Diego, linear program, drug lord

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