Quantum Game Theory

Let’s play a coordination game: You and I are each asked a single question, either “Do you like cats?” or “Do you like dogs?”. Our questions are determined by independent coin flips. We both win if our answers differ, unless we’re both asked about dogs, in which case we both win if our answers match.

Here’s a pretty good strategy we could agree on in advance: We’ll contrive to always differ. Whatever we’re asked, I’ll say yes and you say no. That way we win 3/4 of the time.

Can we do any better? No, if we live in a world governed by classical physics. Yes, if we live in the world we actually inhabit—the world of quantum mechanics.

All we need is a pair of entangled particles, easy enough to create in the laboratory. If I get the cat question, I’ll measure my particle’s spin (which is either up or down) and answer “yes” or “no” accordingly. If I get the dog question, I’ll do the same thing, but first I’ll rotate my measuring apparatus by 90 degrees. You do the same, but start with your measuring apparatus rotated 45 degrees from
mine.

The thing about entangled particles is that the outcomes of these measurements are correlated in a very particular way, and remain so forever, even if the particles are separated. In particular, our answers will differ about 85% of the time unless we both make “dog” measurements, in which case they’ll agree about 85% of the time. Overall, then, we’ll have about an 85% win rate. Those particular correlations would be impossible to achieve with any set of measurements if our electrons obeyed the laws of classical physics.

(More precisely, our win rate is cos²(pi/8).)

I took this example from a beautiful paper by Richard Cleve, Peter Hoyer, Benjamin Toner and John Watrous. (The paper has a lot of other cool examples too.) The moral is that game theory changes dramatically when players have access to quantum technology—which might sound very science fictiony at the moment but probably won’t in another couple of decades.