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17_Backward_Induction - Econ 251 Fall 2006 Dynamic Choice...

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Econ 251 Fall 2006 Dynamic Choice: American Options and Backward Induction 1 Plans and Actions How can somebody know what to buy and sell (or how to act) at time n unless he has a clear view about the consequences of his actions for time n +1 ? But how can he know what the consequences are for time n + 1 unless he has thought out what he needs for time n + 2 ? To put the problem in other words, how can you know what choice to make today until you know what choices you will make tomorrow? And tomorrow, how can you know what choice to make without knowing what choice to make today? Choices in fi nance that occur over di ff erent time periods are often called American options. So how does a trader know how to use his American option? 1.1 Marriage Problem Consider a politically incorrect man who fi gures he will sequentially meet 1000 women in his life. He is sure that any woman he asks to marry him will say yes, but the trouble is, once he has passed over a girl, he can never go back and ask her later. He assumes that the suitability of the women are chosen randomly, independently and uniformly distributed on [0,1], but that he only sees the suitability of a woman by meeting her. Thus the suitability of girl 3, v 3 , is a number drawn at random somewhere between 0 and 1, with equal probability of being at any point. After seeing her, the man knows v 3 , but he does not yet know v 4 or any other value ν t with n > 3 before he must decide whether to propose to lady 3. What should he do? Let us assume that the man wants to pursue the strategy that maximizes the expected suitability of the woman he ends up with. He cannot guarantee anything for sure, but let us suppose he is indi ff erent between a girl with suitability .5 and a girl whose quality he does not know, because on average the latter girl will also have quality .5. After seeing the quality of a candidate, the man has an option: take the woman or not. On what basis should he decide? All he knows is her suitability, and how many women are left. So his strategy must be a threshold level changing over time. Let θ ( t ) represent the threshold level for woman t . If her suitability is above θ ( t ) , he should take her. If her suitability is below θ ( t ) , he should go on. optimal strategy if v t θ ( t ) stay with miss t if v t < θ ( t ) move on How high should θ ( t ) be? And how should θ ( t ) depend on t ? It is clear that if the man has fewer women left to see, he is in a worse situation. So he must lower θ ( t ) as t grows. But can we say anything more de fi nite? Clearly θ ( t ) should be the quality that the man can expect to end up with if he plays the game with 1000 t women left. If he can expect to get θ ( t ) (on average) from his choice among the remaining 1000 t women, there is no point in stopping with a girl before that with quality 1
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less than θ ( t ) . Conversely, if lady t has suitability better than or equal to the θ ( t ) he can expect by continuing, he should stop with her.
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17_Backward_Induction - Econ 251 Fall 2006 Dynamic Choice...

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