1. Introduction and the "listening to expert advice" problem

Where this lecture sits in the course

• The linear programming block is finished: simplex, ellipsoid and interior point have all been covered 0:09.
• Today's tool — multiplicative weights (MW) — can also solve linear programs, but it is much more than an LP solver; after this topic the course stops solving LPs altogether 0:27.
• Informal poll in class: only two students had seen multiplicative weights before 0:44.

The motivating scenario

• Stock trading: many self-proclaimed experts tell you what to buy; you need a strategy for deciding whom to listen to 1:31.
• The framing deliberately echoes competitive analysis from earlier lectures 1:50.

Formal setup

Element	Definition
Horizon	T days 1:50
Experts	n experts 1:50
Daily signal	Each expert says yes or no ("will it rain?", "will this stock go up?") 2:24
Your move	Commit to a decision using all expert opinions 2:57
Feedback	At the end of each day reality is revealed; you have unbounded memory of all past advice 5:27
Goal	Make few mistakes over the T days, with a provable guarantee 2:57

Why absolute guarantees are impossible

• The experts may know nothing and emit random junk; an adversarial reality can always contradict you, so "few mistakes in absolute terms" is hopeless 3:56 4:14.
• Fix (the whole point of the model): measure success competitively — be nearly as good as the best expert in hindsight, i.e. the expert with the fewest mistakes after the T days 4:30 5:15.