Lecture8

# Lecture8 - MS&E111 Introduction to Optimization Prof Amin...

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Lecture 8 Introduction to Optimization May 1-3, 2006 Prof. Amin Saberi 1 Two player Zero-Sum 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 zero-sum 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 two-person zero-sum games, of which our example is a

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Lecture8 - MS&E111 Introduction to Optimization Prof Amin...

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