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How do you model the future?. Stochastic approach: The future can be modeled as a distribution over possible events. Very successful in many contexts. Alternative: Think of the future as an adversary, do well against all possible future outcomes. Toy Example: Ski Optimization.

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How do you model the future?

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How do you model the future

How do you model the future?

  • Stochastic approach: The future can be modeled as a distribution over possible events.

  • Very successful in many contexts.

  • Alternative: Think of the future as an adversary, do well against all possible future outcomes.


Toy example ski optimization

Toy Example: Ski Optimization

  • I decide to take up skiing.

  • Should I rent or buy skis?

  • Uncertainty:

    • Will I like skiing?

    • Will there be snow?

    • Will I break my leg?

    • Will the government outlaw skiing?

  • I want to have a good strategy against all possible outcomes In this case an outcome is the number of times I wind up going skiing.


  • Ski rental

    Ski Rental

    • A pair of skis (and boots) costs $300.

    • A ski rental costs $50.

    • What should you do?

    • How do you evaluate if you did the right thing?

    • You give a strategy (algorithm)

    • You compare against how well someone who knows that future could do.

    • You take the worst case and call that the competitive ratio


    Ski rental1

    Ski Rental

    • Let A be my algorithm.

    • Let OPT be the behavior of someone who knows the future

    • Consider any realization of the future I (number of times I actually ski)

    • Competetive ratio

    • We want a strategy with a small competitive ratio


    Optimal strategy

    Optimal Strategy


    Algorithm 1 buy

    Algorithm 1: Buy

    Competitive ratio = 6


    Algorithm 2 rent

    Algorithm 2: Rent

    Competitive ratio = lots


    Algorithm 3 rent 6 times and then buy

    Algorithm 3: Rent 6 times and then buy

    Competitive ratio = 2


    Lessons

    Lessons

    • Without knowing the future, you can guarantee that no matter what happens, you will never spend more than twice what anyone could have spent.

    • A good algorithm balances different bad outcomes

    • If you allow randomization, you can decrease the competetive ratio to e/(e-1), around 1.58.


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