How do you model the future?

1
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.

2
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.

3
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

4
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

5
Optimal Strategy
Times skiing 1 2 3 4 5 6 7 8 lots
Strategy R R R R R B B B B
Cost 50 100 150 200 250 300 300 300 300
6
Algorithm 1 Buy
Times skiing 1 2 3 4 5 6 7 8 lots
Cost of A 300 300 300 300 300 300 300 300 300
Opt Cost 50 100 150 200 250 300 300 300 300
Ratio 6 3 2 1.5 1.2 1 1 1 1

• Competitive ratio 6

7
Algorithm 2 Rent
Times skiing 1 2 3 4 5 6 7 8 lots
Cost of A 50 100 150 200 250 300 350 400 lots
Opt Cost 50 100 150 200 250 300 300 300 300
Ratio 6 3 2 1.5 1.2 1 1.2 1.33 lots

• Competitive ratio lots

8
Algorithm 3 Rent 6 times and then buy
Times skiing 1 2 3 4 5 6 7 8 lots
Cost of A 50 100 150 200 250 300 600 600 600
Opt Cost 50 100 150 200 250 300 300 300 300
Ratio 1 1 1 1 1 1 2 2 2

• Competitive ratio 2

9
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.