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 12345678lots StrategyRRRRRBBBB Cost50100150200250300300300300

6. Algorithm 1: Buy Times skiing 12345678lots Cost of A 300300300300300300300300300 Opt Cost 50100150200250300300300300 Ratio6321.51.21111 Competitive ratio = 6

7. Algorithm 2: Rent Times skiing 12345678lots Cost of A 50100150200250300350400lots Opt Cost 50100150200250300300300300 Ratio6321.51.211.21.33lots Competitive ratio = lots

8. Algorithm 3: Rent 6 times and then buy Times skiing 12345678lots Cost of A 50100150200250300600600600 Opt Cost 50100150200250300300300300 Ratio111111222 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.