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lecture22 - Lecture 22: Competitive Analysis for Finance...

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Lecture 22: Competitive Analysis for Finance Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794–4400 http://www.cs.sunysb.edu/ skiena

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Competitive Ratio We say an online algorithm ALG is c -competitive if there is a constant α such that for all finite input sequences I , ALG ( I ) c · OPT ( I ) + α Note that the additive constant α is a fixed cost that becomes unimportant as the size of the problem increases. We do not particularly care about the run-time efficiency of ALG (except maybe that it is polynomial), but we do care about its competitive ratio c .
One-Way Trading A generalization of the price searching problem is to sell my entire assets over the trading period, but to remove the constraint that I must sell it all at once. Suppose I am trying to liquidate my position in a stock. I may be able to better optimize my expected performance by using this freedom. This is particularly true in real markets, as my sales serve to depress prices by increasing supply. For this problem, it turns out that there is no difference between what competitive ratio is achievable with and without randomization. What is a reasonable strategy?

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Threat-Based Strategies Suppose we know that a competitive ratio of c can be obtained. If the current price is high and I don’t sell, my adversary can drop it to m and keep it there the rest of the period. But if I do buy, the adversary can jack the price to M at least momentarily, and I will be in trouble if I have already sold everything. The threat-based strategy sells only when the price hits a new maximum. It sells just enough to ensure that we achieve a competitive ratio of c if the price drops to m for the rest of the game.
Randomized Strategies Analysis is needed to determine the optimal c value and also how much to buy in response to price changes. Clearly, we can achieve c = O (lg φ ) , since we can use the randomized strategy and sell all at once. We can simulate the randomized strategy deterministically by

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lecture22 - Lecture 22: Competitive Analysis for Finance...

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