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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
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
algorithm 1 buy
Algorithm 1: Buy

Competitive ratio = 6

algorithm 2 rent
Algorithm 2: Rent

Competitive ratio = lots

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.