CS261: A Second Course in Algorithms Lecture #10: The Minimax Theorem and Algorithms for Linear Programming∗

Recall rock-paper-scissors (or roshambo). Two players simultaneously choose one of rock, paper, or scissors, with rock beating scissors, scissors beating paper, and paper beating rock. Here’s an idea: what if I made you go first? That’s obviously unfair — whatever you do, I can respond with the winning move. But what if I only forced you to commit to a probability distribution over rock, paper, and scissors? (Then I respond, then nature flips coins on your behalf.) If you prefer, imagine that you submit your code for a (randomized) algorithm for choosing an action, then I have to choose my action, and then we run your algorithm and see what happens. In the second case, going first no longer seems to doom you. You can protect yourself by randomizing uniformly among the three options — then, no matter what I do, I’m equally likely to win, lose, or tie. The minimax theorem states that, in general games of “pure competition,” a player moving first can always protect herself by randomizing appropriately.